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An exact estimate result for p-biharmonic equations with Hardy potential and negative exponents

  • Yanbin SangEmail author
  • Siman Guo
Open Access
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Abstract

In this paper, p-biharmonic equations involving Hardy potential and negative exponents with a parameter λ are considered. By means of the structure and properties of Nehari manifold, we give uniform lower bounds for \(\varLambda >0\), which is the supremum of the set of λ. When \(\lambda \in (0, \varLambda )\), the above problems admit at least two positive solutions.

Keywords

p-biharmonic equation Nehari manifold Positive solution Negative exponents 

1 Introduction and preliminaries

In this paper, we consider a p-biharmonic equation with Hardy potential and negative exponents:
$$ \textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}= f(x)u^{-q}+\lambda g(x)u ^{\gamma } & \text{in } \varOmega \setminus \{0\}, \\ u(x)>0 &\text{in } \varOmega \setminus \{0\}, \\ u=\Delta u=0 &\text{on } \partial \varOmega , \end{cases} $$
(1.1)
where \(0\in \varOmega \subset \mathbb{R}^{N}\) is a bounded smooth domain with \(1< p<\frac{N}{2}\), \(\Delta ^{2}_{p}u=\Delta (\vert \Delta u\vert ^{p-2} \Delta u)\) is the p-biharmonic operator. \(\lambda >0\) is a parameter, \(0<\mu <\mu _{N,p}=(\frac{(p-1)N(N-2p)}{p^{2}})^{p}\), \(0< q<1\) and \(p-1<\gamma <p^{*}-1\), where \(p^{*}=\frac{Np}{N-2p}\) is called the critical Sobolev exponent. \(f(x)\geq 0\), \(f(x)\not \equiv 0\), \(g(x)\) satisfies the requirement that the set \(\{x\in \varOmega : g(x)>0 \}\) has positive measures, \(\operatorname{supp}f \cap \{x\in \varOmega : g(x)>0 \} \neq \emptyset \) and \(f, g\in C(\overline{\varOmega })\). Biharmonic equations describe the sport of a rigid body and the deformations of an elastic beam. For example, this type of equation provides a model for considering traveling wave in suspension bridges [5, 16, 27, 30, 36]. Various methods and tools have been adopted to deal with singular problems, such that fixed point theorems [14], topological methods [37], Fourier and Laurent transformation [18, 19], monotone iterative methods [21], global bifurcation theory [12], and degree theory [22, 31].
In recent years, there was much attention focused on the existence, multiplicity and qualitative properties of solutions for p-biharmonic equations under Dirichlet boundary conditions or Navier boundary conditions with Hardy terms [4, 15, 17, 32, 34]. Xie and Wang [32] studied the following p-biharmonic equation with Dirichlet boundary conditions:
$$ \textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}= f(x,u) & \text{in } \varOmega , \\ u=\frac{\partial u}{\partial n}=0 & \text{on } \partial \varOmega , \end{cases} $$
(1.2)
where \(\frac{\partial }{\partial n}\) is the outer normal derivative. By using the variational method, the existence of infinitely many solutions with positive energy levels for (1.2) was established. Huang and Liu [15] considered the following p-biharmonic equation with Navier boundary conditions:
$$ \textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}= f(x,u) & \text{in } \varOmega , \\ u=\Delta u=0 & \text{on } \partial \varOmega , \end{cases} $$
(1.3)
where \(1< p<\frac{N}{2}\). By using invariant sets of gradient flows, the authors proved that (1.3) possesses a sign-changing solution. Furthermore, Yang, Zhang and Liu [34] showed that (1.3) has a positive solution, a negative solution and a sequence of sign-changing solutions when f satisfies appropriate conditions. Bhakta [4] established the qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with Hardy terms.
On the other hand, nonlinear biharmonic equations with negative exponents have been studied expensively [1, 6, 8, 13, 20]. Guerra [13] gave a complete description of entire radially symmetric solutions for the following biharmonic equation:
$$ \Delta ^{2} u=-u^{-q}, \qquad u>0 \quad \text{in } \mathbb{R}^{3}, $$
where \(q>1\). Moreover, Cowan et al. [8] dealt with the regularity of the extremal solution of the following fourth order boundary value problems:
$$ \textstyle\begin{cases} \Delta ^{2}u=\frac{\lambda }{(1-u)^{2}} &\text{in } \varOmega , \\ 0< u< 1 &\text{in } \varOmega , \\ u=\frac{\partial u}{\partial n}=0 &\text{on } \partial \varOmega . \end{cases} $$
Very recently, Ansari, Vaezpour and Hesaaraki [1] considered fourth order elliptic problem with the combinations of Hardy term and negative exponents,
$$ \textstyle\begin{cases} \Delta ^{2}u-\lambda M( \Vert \nabla u \Vert ^{2})\Delta u -\frac{\mu }{ \vert x \vert ^{4}}u= \frac{h(x)}{u^{\gamma }}+k(x) u^{\alpha } & \text{in }\varOmega , \\ u=\Delta u=0 & \text{on } \partial \varOmega , \end{cases} $$
(1.4)
where \(\varOmega \subset \mathbb{R}^{N}\) (\(N\geq 1\)) is a bounded \(C^{4}\)-domain, λ and μ are positive parameters and \(0<\alpha <1\), \(0<\gamma <1\) are constants. Here M, h and k are given continuous functions satisfying suitable hypotheses. By using the Galerkin method and the sharp angle lemma, the authors proved that problem (1.4) has a positive solution for \(0<\mu < (\frac{N(N-4)}{4} ) ^{2}\).
We say that \(u\in W:=W^{2,p}(\varOmega )\cap W_{0}^{1,p}(\varOmega )\) is a weak solution of (1.1), if for every \(\varphi \in W\), there holds
$$ \int _{\varOmega } \vert \Delta u \vert ^{p-2}\Delta u \Delta \varphi \,dx- \int _{\varOmega }\frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p-2}u \varphi \,dx= \int _{\varOmega }f(x)u^{-q}\varphi \,dx+\lambda \int _{\varOmega } g(x)u^{\gamma }\varphi \,dx. $$
(1.5)
The following Rellich inequality will be used in this paper:
$$ \int _{\varOmega } \vert \Delta u \vert ^{p}\,dx\geq \mu _{N,p} \int _{\varOmega }\frac{ \vert u \vert ^{p}}{ \vert x \vert ^{2p}}\,dx, \quad \forall u\in W, $$
and it is not achieved [9, 24]. For any \(u\in W\), and \(0<\mu <\mu _{N,p}\). The energy functional corresponding to (1.1) is defined by
$$ \begin{aligned}[b] I_{\lambda ,\mu }(u)={}&\frac{1}{p} \int _{\varOmega } \biggl( \vert \Delta u \vert ^{p}- \frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p} \biggr)\,dx- \frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \\ &{}- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx. \end{aligned} $$
(1.6)
For \(\mu \in [0,\mu _{N,p})\), W is equipped with the following norm:
$$ \Vert u \Vert ^{p}_{\mu }= \int _{\varOmega } \biggl( \vert \Delta u \vert ^{p}- \frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p} \biggr)\,dx. $$
Negative exponent term \(u^{-q}\) implies that \(I_{\lambda ,\mu }\) is not differential on W, therefore, critical point theory cannot be applied to the problem (1.1) directly. We consider the following manifold:
$$ \mathcal{M}= \biggl\{ u\in W: \Vert u \Vert ^{p}_{\mu } = \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx+\lambda \int _{\varOmega } g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} , $$
and make the following splitting for \(\mathcal{M}\):
$$\begin{aligned}& \mathcal{M}^{+}= \biggl\{ u\in \mathcal{M}: (p+q-1 ) \Vert u \Vert ^{p}_{ \mu } >\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} , \end{aligned}$$
(1.7)
$$\begin{aligned}& \mathcal{M}^{0}= \biggl\{ u\in \mathcal{M}: (p+q-1 ) \Vert u \Vert ^{p}_{ \mu } =\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} , \end{aligned}$$
(1.8)
$$\begin{aligned}& \mathcal{M}^{-}= \biggl\{ u\in \mathcal{M}: (p+q-1 ) \Vert u \Vert ^{p}_{ \mu } < \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} . \end{aligned}$$
(1.9)

In this paper, we will study the dependence of problem (1.1) on q, γ, f, g and Ω and evaluate the extremal value of λ related to multiplicity of positive solutions for problem (1.1). Our idea comes from [7, 28, 29]. Our results improve and complement previous ones obtained in [23, 25]. Denote \(\Vert u\Vert _{t}^{t}= \int _{\varOmega }\vert u\vert ^{t}\,dx\) and \(D^{2, p}(\mathbb{R}^{N})\) be the closure of \(C_{0}^{\infty }(\mathbb{R}^{N})\) with respect to the norm \((\int _{\mathbb{R}^{N}}\vert \Delta u\vert ^{p} \,dx )^{\frac{1}{p}}\).

\(\lambda _{1}\) denotes the smallest eigenvalue for
$$ \Delta _{p}^{2}u-\frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p-2}u=\lambda _{1} \vert u \vert ^{p-2}u, \quad x\in \varOmega \setminus \{0\}, u\in W, $$
(1.10)
and \(\varphi _{1}\) denotes the corresponding eigenfunction with \(\varphi _{1}>0\) in Ω [3, 10, 26, 33, 35]. The following minimization problem will be useful in the following discussions:
$$ S_{\mu }=\inf \biggl\{ \int _{\mathbb{R}^{N}} \biggl( \vert \Delta u \vert ^{p}- \frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p} \biggr)\,dx, u\in D^{2,p} \bigl(\mathbb{R}^{N} \bigr), \int _{\mathbb{R}^{N}} \vert u \vert ^{p^{*}}\,dx=1 \biggr\} >0, $$
(1.11)
and \(S_{\mu }\) is achieved by a family of functions [4, 11]. Thus, for every \(u\in W\setminus \{0\}\), \(\Vert u\Vert _{p^{*}}\leq \frac{ \Vert u\Vert _{\mu }}{\sqrt[p]{S_{\mu }}}\). Therefore, combining with the Hölder inequality, we deduce that
$$\begin{aligned}& \begin{aligned}[b] \int _{\varOmega } \vert u \vert ^{\gamma +1}\,dx&\leq \biggl[ \int _{\varOmega } \vert u \vert ^{(\gamma +1)\frac{p^{*}}{\gamma +1}}\,dx \biggr]^{\frac{ \gamma +1}{p^{*}}} \biggl( \int _{\varOmega }1\,dx \biggr)^{\frac{p^{*}-\gamma -1}{p^{*}}} \\ &= \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \Vert u \Vert ^{\gamma +1} _{p^{*}} \\ &\leq \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \biggl( \frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{\gamma +1}, \end{aligned} \end{aligned}$$
(1.12)
$$\begin{aligned}& \begin{aligned}[b] \int _{\varOmega } \vert u \vert ^{1-q}\,dx&\leq \biggl[ \int _{\varOmega } \vert u \vert ^{(1-q)\frac{p^{*}}{1-q}}\,dx \biggr]^{\frac{1-q}{p ^{*}}} \biggl( \int _{\varOmega }1\,dx \biggr)^{\frac{p^{*}-1+q}{p^{*}}} \\ &= \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}} \Vert u \Vert ^{1-q} _{p^{*}} \\ &\leq \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}} \biggl( \frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{1-q}, \end{aligned} \end{aligned}$$
(1.13)
and
$$ \begin{aligned}[b] \int _{\varOmega } \vert u \vert ^{1-q}\,dx&\leq \biggl[ \int _{\varOmega } \vert u \vert ^{(1-q)\frac{\gamma +1}{1-q}}\,dx \biggr]^{\frac{1-q}{ \gamma +1}} \biggl( \int _{\varOmega }1\,dx \biggr)^{\frac{\gamma +q}{\gamma +1}} \\ &= \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}} \Vert u \Vert ^{1-q} _{\gamma +1}. \end{aligned} $$
(1.14)

Our main results are stated in the following theorems.

Theorem 1.1

Assume that\(\lambda \in (0,\varLambda )\), where
$$\begin{aligned} \begin{aligned}[b] \varLambda \geq{}& T_{\mu }= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl( \frac{S_{\mu }}{ \vert \varOmega \vert ^{ \frac{p}{N}}} \biggr)^{\frac{q+\gamma }{p+q-1}} \\ >{}&0. \end{aligned} \end{aligned}$$
(1.15)
Then problem (1.1) admits at least two solutions\(u_{0}\in \mathcal{M}^{+}\), \(U_{0}\in \mathcal{M}^{-}\), with\(\Vert U_{0}\Vert _{\mu }> \Vert u_{0}\Vert _{\mu }\).

Corollary 1.2

Let\(U_{\lambda , \mu ,\varepsilon } \in \mathcal{M}^{-}\)be the solution of problem (1.1) with\(\gamma = \varepsilon +p-1\), where\(\lambda \in (0,T_{\mu })\). Then
$$\begin{aligned}& \Vert U_{\lambda , \mu ,\varepsilon } \Vert _{\mu }>C_{\mu ,\varepsilon } \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\varepsilon }} \end{aligned}$$
with
$$ C_{\mu ,\varepsilon }= \vert \varOmega \vert ^{\frac{1}{p}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl(1+ \frac{p+q-1}{\varepsilon } \biggr)^{ \frac{1}{p+q-1}} \biggl(\frac{ \vert \varOmega \vert ^{\frac{2}{N}}}{\sqrt[p]{S _{\mu }}} \biggr)^{\frac{1-q}{p+q-1}} \rightarrow \infty , \quad \textit{as }\varepsilon \rightarrow 0. $$
(1.16)

Theorem 1.3

There exists\(\lambda ^{*} =\lambda ^{*} (N, \varOmega , \mu , q, \gamma )>0\)such that problem (1.1) with\(f=g=1\)admits at least a positive solution for every\(0<\lambda <\lambda ^{*}\)and has no solution for every\(\lambda >\lambda ^{*}\).

2 Some lemmas

Lemma 2.1

Assume that\(\lambda \in (0,T_{\mu })\), where\(T_{\mu }\)is defined in (1.15). Then\(\mathcal{M}^{\pm }\neq \emptyset \)and\(\mathcal{M}^{0}=\{0\}\).

Proof

(i) We can choose \(u^{*}\in \mathcal{M}\setminus \{0 \}\) such that \(\int _{\varOmega }f(x)\vert u^{*}\vert ^{1-q}\,dx>0\) and \(\int _{\varOmega }g(x) \vert u^{*}\vert ^{\gamma +1}\,dx>0\) from the conditions imposed on f and g. Denote
$$\begin{aligned} \varphi _{\mu }(t) :=&\frac{1}{t^{\gamma }} \frac{d}{dt}I_{\lambda , \mu } \bigl(tu^{*} \bigr) \\ =&t^{p-1-\gamma } \bigl\Vert u^{*} \bigr\Vert _{\mu }^{p}-t^{-q- \gamma } \int _{\varOmega }f(x) \bigl\vert u^{*} \bigr\vert ^{1-q}\,dx- \lambda \int _{\varOmega }g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx, \quad t>0. \end{aligned}$$
Note that \(\varphi '_{\mu }(t)=(p-1-\gamma )t^{p-2-\gamma }\Vert u^{*}\Vert _{\mu }^{p}+(q+\gamma ) t^{-1-q-\gamma } \int _{\varOmega }f(x)\vert u^{*}\vert ^{1-q}\,dx\). Let \(\varphi '_{\mu }(t)=0\), we have
$$ t:=t_{\max }= \biggl[\frac{(\gamma -p+1) \Vert u^{*} \Vert _{\mu }^{p}}{(q+ \gamma )\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx} \biggr]^{\frac{1}{1-q-p}}. $$
(2.1)
It is easy to check that \(\varphi _{\mu }(t)\rightarrow -\infty \) as \(t\rightarrow 0^{+}\) and \(\varphi _{\mu }(t)\rightarrow -\lambda \int _{\varOmega } g(x)\vert u^{*}\vert ^{\gamma +1}\,dx<0\) as \(t\rightarrow \infty \). Furthermore, \(\varphi _{\mu }(t)\) attains its maximum at \(t_{\max }\). By (1.12) and (1.13), we obtain
$$\begin{aligned}& \varphi _{\mu }(t_{\max }) \\& \quad = \biggl[\frac{(\gamma -p+1) \Vert u^{*} \Vert _{\mu }^{p}}{(q+\gamma ) \int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx} \biggr]^{\frac{p-\gamma -1}{1-q-p}} \bigl\Vert u^{*} \bigr\Vert _{\mu }^{p} \\& \quad\quad{} - \biggl[\frac{(\gamma -p+1) \Vert u^{*} \Vert _{\mu }^{p}}{(q+ \gamma )\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx} \biggr]^{ \frac{-q-\gamma }{1-q-p}} \int _{\varOmega }f(x) \bigl\vert u^{*} \bigr\vert ^{1-q}\,dx \\& \quad \quad {} -\lambda \int _{\varOmega } g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx \\& \quad = \biggl(\frac{\gamma -p+1}{q+\gamma } \biggr)^{ \frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u^{*} \Vert _{\mu }^{p})^{\frac{- \gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}} \\& \quad\quad{} - \biggl( \frac{\gamma -p+1}{q+\gamma } \biggr) ^{\frac{-q- \gamma }{1-q-p}}\frac{( \Vert u^{*} \Vert _{\mu }^{p})^{ \frac{-\gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}} \\& \quad \quad {} -\lambda \int _{\varOmega } g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx \\& \quad = \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u^{*} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}}-\lambda \int _{\varOmega }g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx \\& \quad \geq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u^{*} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{ [ \Vert f \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}(\frac{ \Vert u^{*} \Vert _{\mu }}{\sqrt[p]{S_{ \mu }}})^{1-q} ]^{\frac{p-\gamma -1}{1-q-p}}} \\& \quad \quad {} -\lambda \Vert g \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \biggl( \frac{ \Vert u^{*} \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{\gamma +1} \\& \quad = \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \frac{(\sqrt[p]{S_{ \mu }})^{\frac{(1-q)(p-\gamma -1)}{1-q-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p ^{*}}\frac{p-\gamma -1}{1-q-p}}} \bigl\Vert u^{*} \bigr\Vert _{\mu } ^{\gamma +1} \\& \quad \quad {} -\lambda \Vert g \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p ^{*}}} \biggl(\frac{ \Vert u^{*} \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr) ^{\gamma +1} \\& \quad = \biggl[ \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \frac{(\sqrt[p]{S_{ \mu }})^{\frac{(1-q)(p-\gamma -1)}{1-q-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p ^{*}}\frac{p-\gamma -1}{1-q-p}}} \\& \quad \quad {}-\lambda \Vert g \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}}}{({\sqrt[p]{S_{\mu }}})^{ \gamma +1}} \biggr] \bigl\Vert u^{*} \bigr\Vert _{\mu }^{\gamma +1} \\& \quad :=A(\mu ,\lambda ) \bigl\Vert u^{*} \bigr\Vert _{\mu }^{\gamma +1} \\& \quad >0. \end{aligned}$$
(2.2)
When \(A(\mu ,\lambda )=0\), we get
$$ \begin{aligned} \lambda &= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl( \frac{\gamma -p+1}{q+ \gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \frac{(\sqrt[p]{S_{\mu }})^{\frac{(1-q)(p-\gamma -1)}{1-q-p}+\gamma +1}}{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}\frac{p- \gamma -1}{1-q-p} +\frac{p^{*}-\gamma -1}{p^{*}}}} \\ &= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl( \frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl[\frac{S_{\mu }}{ \vert \varOmega \vert ^{\frac{2p}{N}}} \biggr] ^{\frac{q+\gamma }{p+q-1}} =T_{\mu }, \end{aligned} $$
where we use the following two equalities:
$$ \frac{(1-q)(p-\gamma -1)}{1-q-p}+\gamma +1= \frac{p(q+\gamma )}{q+p-1}, $$
and
$$ \frac{(p^{*}-1+q)(p-\gamma -1)}{p^{*}(1-q-p)}+\frac{p^{*}-\gamma -1}{p ^{*}} =\frac{2p(q+\gamma )}{N(q+p-1)}. $$
In turn, this is also true. Hence \(A(\mu ,\lambda )=0\) if and only if \(\lambda =T_{\mu }\). Thus for \(\lambda \in (0,T_{\mu })\), we have \(A(\mu ,\lambda )>0\). Moreover, by (2.2), we derive that \(\varphi _{ \mu }(t_{\max })>0\). Consequently, there exist two numbers \(t_{\mu } ^{-}\) and \(t_{\mu }^{+}\) such that \(0< t_{\mu }^{-}< t_{\max }< t_{ \mu }^{+}\), and
$$ \varphi _{\mu } \bigl(t^{-}_{\mu } \bigr)=0=\varphi _{\mu } \bigl(t^{+}_{\mu } \bigr), \quad\quad \varphi '_{\mu } \bigl(t^{-}_{\mu } \bigr)>0> \varphi '_{\mu } \bigl(t^{+}_{\mu } \bigr). $$
It follows that \(t_{\mu }^{-}u^{*}\in \mathcal{M}^{+}\), and \(t_{\mu }^{+}u^{*}\in \mathcal{M}^{-}\). In fact, if \(\varphi _{\mu }(t)=0\), then
$$ \varphi _{\mu }(t)=t^{p-1-\gamma } \Vert u \Vert _{\mu }^{p}-t^{-q-\gamma } \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx-\lambda \int _{\varOmega } g(x) \vert u \vert ^{\gamma +1}\,dx=0, $$
namely
$$ \Vert tu \Vert _{\mu }^{p}= \int _{\varOmega }f(x) \vert tu \vert ^{1-q}\,dx +\lambda \int _{\varOmega }g(x) \vert tu \vert ^{\gamma +1}\,dx. $$
Hence \(tu\in \mathcal{M}\). Furthermore, if \(\varphi '_{\mu }(t)>0\), then
$$ (p-1-\gamma )t^{p-2-\gamma } \Vert u \Vert _{\mu }^{p} +(q+ \gamma )t^{-1-q- \gamma } \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx>0. $$
That is
$$ (p-1-\gamma ) \Vert tu \Vert _{\mu }^{p}+(q+\gamma ) \int _{\varOmega }f(x) \vert tu \vert ^{1-q}\,dx>0, $$
i.e.,
$$ (p-1-\gamma ) \Vert tu \Vert _{\mu }^{p}+(q+\gamma ) \biggl[ \Vert tu \Vert _{\mu }^{p} - \lambda \int _{\varOmega }g(x) \vert tu \vert ^{\gamma +1}\,dx \biggr]>0. $$
Note that \(tu\in \mathcal{M}\), we have
$$ (p+q-1) \Vert tu \Vert _{\mu }^{p}-\lambda (q+\gamma ) \int _{\varOmega }g(x) \vert tu \vert ^{\gamma +1}\,dx>0. $$
Thus \(tu\in \mathcal{M}^{+}\). By a similar argument, if \(\varphi _{ \mu }(t)=0\) and \(\varphi '_{\mu }(t)<0\), then \(tu\in \mathcal{M}^{-}\). Therefore, both \(\mathcal{M}^{+}\) and \(\mathcal{M}^{-}\) are non-empty sets for every \(\lambda \in (0,T_{\mu })\).
(ii) We claim that \(\mathcal{M}^{0}=\{0\}\). Otherwise, we suppose that there exists \(u_{*}\in \mathcal{M}^{0}\) and \(u_{*}\neq 0\). Since \(u_{*}\in \mathcal{M}^{0}\), we have
$$\begin{aligned}& (p+q-1 ) \Vert u_{*} \Vert ^{p}_{\mu }=\lambda ( \gamma +q ) \int _{\varOmega }g(x) \vert u_{*} \vert ^{\gamma +1} \,dx, \end{aligned}$$
moreover
$$ \begin{aligned} 0&= \Vert u_{*} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{*}^{1-q}\,dx -\lambda \int _{\varOmega }g(x)u_{*}^{\gamma +1}\,dx \\ &= \Vert u_{*} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{*}^{1-q}\,dx - \frac{p+q-1}{\gamma +q} \Vert u_{*} \Vert _{\mu }^{p} \\ &=\frac{\gamma -p+1}{\gamma +q} \Vert u_{*} \Vert _{\mu } ^{p}- \int _{\varOmega } f(x)u_{*}^{1-q}\,dx. \end{aligned} $$
For \(\lambda \in (0,T_{\mu })\) and \(u_{*}\neq 0\), combining with (2.2), we deduce that
$$ \begin{aligned} 0&< A(\mu ,\lambda ) \Vert u_{*} \Vert _{\mu }^{\gamma +1} \\ &\leq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u_{*} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\frac{\gamma -p+1}{q+\gamma } \Vert u_{*} \Vert _{\mu }^{p})^{\frac{p-\gamma -1}{1-q-p}}} - \biggl(\frac{q+p-1}{q+ \gamma } \biggr) \Vert u_{*} \Vert _{\mu }^{p}=0, \end{aligned} $$
which is a contradiction, Thus \(u_{*}=0\). That is, \(\mathcal{M}^{0}= \{0\}\). □

The gap structure in \(\mathcal{M}\) is embodied in the following lemma.

Lemma 2.2

Assume that\(\lambda \in (0,T_{\mu })\), then
$$\begin{aligned}& \Vert U \Vert _{\mu }>M_{\mu }(\lambda )>M_{\mu ,0}> \Vert u \Vert _{\mu }, \\& \Vert U \Vert _{\gamma +1}>N_{\mu }(\lambda )>N_{\mu ,0}> \Vert u \Vert _{\gamma +1}, \quad \forall u\in \mathcal{M}^{+}, U\in \mathcal{M}^{-}, \end{aligned}$$
where
$$\begin{aligned}& M_{\mu ,0}= \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \biggr] ^{\frac{1}{p+q-1}}, \\& M_{\mu }(\lambda )= \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }})^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}}, \\& N_{\mu ,0}= \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}+\frac{(p^{*}-1-\gamma )p}{p^{*}( \gamma +1)}}}{S_{\mu }} \biggr]^{\frac{1}{p+q-1}}, \\& N_{\mu }(\lambda )= \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}}. \end{aligned}$$

Proof

If \(u\in \mathcal{M}^{+}\subset \mathcal{M}\), then
$$ \begin{aligned} 0&< (p+q-1 ) \Vert u \Vert ^{p}_{\mu }- \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &= (p+q-1 ) \Vert u \Vert ^{p}_{\mu }- (\gamma +q ) \biggl[ \Vert u \Vert ^{p}_{\mu } - \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \biggr] \\ &= (p-\gamma -1 ) \Vert u \Vert ^{p}_{\mu }+ (\gamma +q ) \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx. \end{aligned} $$
We obtain from (1.13) that
$$ \begin{aligned} (\gamma -p+1 ) \Vert u \Vert ^{p}_{\mu }&< (\gamma +q ) \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \\ &\leq (\gamma +q ) \Vert f \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}} \biggl( \frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S _{\mu }}} \biggr)^{1-q}, \end{aligned} $$
which leads to
$$\begin{aligned}& \Vert u \Vert _{\mu }< \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \biggr] ^{\frac{1}{p+q-1}}=M_{\mu ,0}. \end{aligned}$$
By (1.12) and (1.14), we have
$$\begin{aligned} & (\gamma -p+1 ) \Vert u \Vert ^{p}_{\gamma +1} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \\ &\quad \leq (\gamma -p+1 )\frac{S_{\mu }}{ \vert \varOmega \vert ^{p\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1)}}} \biggl[ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}} \biggl(\frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr) ^{\gamma +1} \biggr] ^{\frac{p}{\gamma +1}} \\ &\quad = (\gamma -p+1 ) \Vert u \Vert ^{p}_{\mu } \\ &\quad < (\gamma +q ) \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \\ &\quad \leq (\gamma +q ) \Vert f \Vert _{\infty } \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}} \Vert u \Vert ^{1-q} _{\gamma +1}, \end{aligned}$$
which implies that
$$\begin{aligned}& \Vert u \Vert _{\gamma +1}< \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}+\frac{(p^{*}-1-\gamma )p}{p^{*}(\gamma +1)}}}{S_{\mu }} \biggr]^{\frac{1}{p+q-1}}=N_{\mu ,0}. \end{aligned}$$
If \(U\in \mathcal{M}^{-}\subset \mathcal{M}\), combining with (1.12), we derive that
$$ \begin{aligned} (p+q-1 ) \Vert U \Vert ^{p}_{\mu }&< \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U \vert ^{\gamma +1}\,dx \\ &\leq \lambda (\gamma +q ) \Vert g \Vert _{ \infty } \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \biggl( \frac{ \Vert U \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{\gamma +1}, \end{aligned} $$
which leads to
$$\begin{aligned}& \Vert U \Vert _{\mu }> \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }})^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}}=M_{\mu }( \lambda ). \end{aligned}$$
Furthermore
$$\begin{aligned}& (p+q-1 ) \Vert U \Vert ^{p}_{\gamma +1} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \\& \quad \leq (p+q-1 )\frac{S_{\mu }}{ \vert \varOmega \vert ^{p\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1)}}} \biggl[ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}} \biggl(\frac{ \Vert U \Vert _{\mu }}{\sqrt[p]{S_{ \mu }}} \biggr) \biggr]^{\frac{p}{\gamma +1}} \\& \quad = (p+q-1 ) \Vert U \Vert ^{p}_{\mu } \\& \quad < \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U \vert ^{\gamma +1}\,dx \\& \quad \leq \lambda (\gamma +q ) \Vert g \Vert _{ \infty } \Vert U \Vert ^{\gamma +1} _{\gamma +1}, \end{aligned}$$
which means that
$$\begin{aligned}& \Vert U \Vert _{\gamma +1}> \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}}=N_{\mu }(\lambda ). \end{aligned}$$
Therefore
$$\begin{aligned}& \lambda =T_{\mu }= \biggl( \frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl(\frac{S_{\mu }}{ \vert \varOmega \vert ^{ \frac{2p}{N}}} \biggr)^{\frac{q+\gamma }{p+q-1}} \\& \begin{aligned} \Leftrightarrow \quad M_{\mu }(\lambda )&= \biggl[\frac{p+q-1}{\lambda ( \gamma +q)} \frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }})^{ \gamma +1}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{ \gamma +1-p}} \\ & =\lambda ^{-\frac{1}{\gamma +1-p}} \biggl[\frac{p+q-1}{\gamma +q}\frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }}) ^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}} \\ & = \biggl(\frac{q+\gamma }{q+p-1} \biggr) ^{ \frac{1}{\gamma +1-p}} \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr) ^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}} \bigl( \Vert g \Vert _{\infty } \bigr)^{\frac{1}{\gamma +1-p}} \\ & \quad{}\times \frac{ \vert \varOmega \vert ^{\frac{2p}{N}\frac{q+\gamma }{(q+p-1)(\gamma +1-p)}}}{(S_{\mu })^{\frac{q+ \gamma }{(p+q-1) (\gamma +1-p)}}} \biggl[\frac{p+q-1}{\gamma +q}\frac{1}{ \Vert g \Vert _{\infty }}\frac{(\sqrt[p]{S _{\mu }}) ^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p^{*}}}} \biggr] ^{\frac{1}{\gamma +1-p}} \\ & = \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr)^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}}\frac{ \vert \varOmega \vert ^{ \frac{2p}{N} \frac{q+\gamma }{{(\gamma -p+1)(p+q-1)}}-\frac{p^{*}-1- \gamma }{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{p \frac{q+\gamma }{(\gamma -p+1)(p+q-1)} -\frac{\gamma +1}{\gamma +1-p}}} \\ & = \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \biggr] ^{ \frac{1}{p+q-1}}=M_{\mu ,0}, \end{aligned} \end{aligned}$$
where we have used the following facts:
$$ \begin{aligned} &\frac{2p}{N}\frac{q+\gamma }{{(\gamma -p+1)(p+q-1)}}- \frac{p^{*}-1- \gamma }{p^{*} (\gamma -p+1)} \\ &\quad =\frac{2p(p^{*}-p)}{2pp^{*}}\frac{q+\gamma }{ {(\gamma -p+1)(p+q-1)}} -\frac{p^{*}-1-\gamma }{p^{*}(\gamma -p+1)} \\ &\quad = \frac{(\gamma -p+1)(p^{*}+q-1)}{p^{*}(\gamma -p+1)(p+q-1)}, \end{aligned} $$
and
$$ p\frac{q+\gamma }{(\gamma -p+1)(p+q-1)}- \frac{\gamma +1}{\gamma +1-p} =\frac{pq-q\gamma +\gamma -p-q+1}{( \gamma -p+1)(p+q-1)}= \frac{1-q}{p+q-1}. $$
Similarly
$$\begin{aligned}& \lambda =T_{\mu }= \biggl( \frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl[\frac{S_{\mu }}{ \vert \varOmega \vert ^{ \frac{2p}{N}}} \biggr]^{\frac{q+\gamma }{p+q-1}}. \\& \Leftrightarrow \quad N_{\mu }(\lambda )= \biggl[\frac{p+q-1}{\lambda ( \gamma +q)} \frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} =\lambda ^{-\frac{1}{\gamma +1-p}} \biggl[\frac{p+q-1}{\lambda ( \gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} = \biggl(\frac{q+\gamma }{q+p-1} \biggr)^{ \frac{1}{\gamma +1-p}} \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr) ^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}} \bigl( \Vert g \Vert _{\infty } \bigr)^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} \quad{}\times \frac{ \vert \varOmega \vert ^{\frac{2p}{N}\frac{q+\gamma }{(q+p-1)(\gamma +1-p)}}}{(S_{\mu })^{\frac{q+ \gamma }{(p+q-1)(\gamma +1-p)}}} \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert W \Vert _{ \infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p^{*}( \gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} = \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr)^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}}\frac{ \vert \varOmega \vert ^{ \frac{2p}{N} \frac{q+\gamma }{(\gamma -p+1)(p+q-1)}-p\frac{p^{*}-1- \gamma }{p^{*}(\gamma +1) (\gamma +1-p)}}}{(S_{\mu })^{\frac{q+\gamma }{(\gamma -p+1)(p+q-1)} -\frac{1}{\gamma +1-p}}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} = \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}+\frac{(p^{*}-1-\gamma )p}{p^{*}(\gamma +1)}}}{S_{\mu }} \biggr]^{\frac{1}{p+q-1}}=N_{\mu ,0}, \end{aligned}$$
where we have applied the following equalities:
$$ \begin{aligned} &\frac{2p}{N}\frac{q+\gamma }{(\gamma -p+1)(p+q-1)}-p \frac{p^{*}-1- \gamma }{p^{*}(\gamma +1)(\gamma +1-p)} \\ &\quad =\frac{2p(p^{*}-p)}{2pp^{*}}\frac{q+\gamma }{ {(\gamma -p+1)(p+q-1)}} -\frac{p^{*}-1-\gamma }{p^{*}(\gamma -p+1)} \\ &\quad =\frac{\gamma +q}{\gamma +1}+p\frac{p^{*}-1- \gamma }{p^{*}(\gamma +1)}, \end{aligned} $$
and
$$ \frac{q+\gamma }{(\gamma -p+1)(p+q-1)}-\frac{1}{\gamma +1-p} =\frac{q+ \gamma -(p+q-1)}{(\gamma -p+1)(p+q-1)}= \frac{1}{p+q-1}. $$
Consequently, \(M_{\mu }(\lambda )=M_{\mu ,0}\) if and only if \(\lambda =T_{\mu }\) and \(N_{\mu }(\lambda )=N_{\mu ,0}\) if and only if \(\lambda =T_{\mu }\) respectively. This completes the proof of Lemma 2.2. □

Lemma 2.3

Assume that\(\lambda \in (0,T_{\mu })\). Then\(\mathcal{M}^{-}\)is a closed set inW-topology.

Proof

We choose a sequence \(\{U_{n}\}\) such that \(\{U_{n}\} \subset \mathcal{M}^{-}\) and \(U_{n}\rightarrow U_{0}\) with \(U_{0} \in W\). Then
$$ \begin{aligned} \Vert U_{0} \Vert _{\mu }^{p}&= \lim_{n\rightarrow \infty } \Vert U_{n} \Vert _{ \mu }^{p} \\ &=\lim_{n\rightarrow \infty } \biggl[ \int _{\varOmega }f(x) \vert U_{n} \vert ^{1-q}\,dx+ \lambda \int _{\varOmega } g(x) \vert U_{n} \vert ^{\gamma +1}\,dx \biggr] \\ &= \int _{\varOmega }f(x) \vert U_{0} \vert ^{1-q}\,dx+ \lambda \int _{\varOmega } g(x) \vert U_{0} \vert ^{\gamma +1}\,dx, \end{aligned} $$
and
$$ \begin{aligned} & (p+q-1 ) \Vert U_{0} \Vert ^{p}_{\mu }-\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U_{0} \vert ^{\gamma +1}\,dx \\ &\quad =\lim_{n\rightarrow \infty } \biggl[ (p+q-1 ) \Vert U _{n} \Vert ^{p}_{\mu }-\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U_{n} \vert ^{\gamma +1}\,dx \biggr]\leq 0. \end{aligned} $$
Hence \(U_{0} \in \mathcal{M}^{-} \cup \mathcal{M}^{0}\). By Lemma 2.2, we have
$$ \Vert U_{0} \Vert _{\mu }=\lim_{n\rightarrow \infty } \Vert U_{n} \Vert _{\mu } \geq M_{\mu ,0}>0, $$
that is, \(U_{0}\neq 0\). Combining with Lemma 2.1, we obtain \(U_{0}\notin \mathcal{M}^{0}\). Thus \(U_{0}\in \mathcal{M}^{-}\). Therefore \(\mathcal{M}^{-}\) is a closed set in W-topology for every \(\lambda \in (0,T_{\mu })\). □

Lemma 2.4

For\(u\in \mathcal{M}^{\pm }\), there exist a number\(\varepsilon >0\)and a continuous function\(\widetilde{g}(h)>0\)with\(h\in W\)and\(\Vert h\Vert <\varepsilon \)such that
$$ \widetilde{g}(0)=1, \quad\quad \widetilde{g}(h) (u+h)\in \mathcal{M}^{\pm }, \quad \forall h\in W, \Vert h \Vert < \varepsilon . $$

Proof

We only prove the case that \(\mathcal{M}^{+}\). Define a function \(\widetilde{F}: W\times \mathbb{R}^{+}\rightarrow \mathbb{R}\) by:
$$\begin{aligned}& \widetilde{F}(h,s)=s^{p-1+q} \Vert u+h \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert u+h \vert ^{1-q}\,dx-\lambda s^{\gamma +q} \int _{\varOmega }g(x) \vert u+h \vert ^{\gamma +1}\,dx. \end{aligned}$$
Note that \(u\in \mathcal{M}^{+}\), we obtain
$$\begin{aligned}& \widetilde{F}(0,1)= \Vert u \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx -\lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx=0, \end{aligned}$$
and
$$ \widetilde{F}_{s}(0,1)=(p-1+q) \Vert u \Vert _{\mu }^{p}-(q+\gamma )\lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx>0. $$
(2.3)
At \((0,1)\), using the implicit function theorem, we know that there exists \(\overline{\varepsilon }>0\) such that for \(h\in W\) and \(\Vert h\Vert <\overline{\varepsilon }\), the equation \(\widetilde{F}(h,s)=0\) has a unique continuous solution \(s=\widetilde{g}(h)>0\). Hence \(\widetilde{g}(0)=1\) and
$$ \begin{aligned} 0&=\widetilde{g}(h)^{p-1+q} \Vert u+h \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert u+h \vert ^{1-q}\,dx-\lambda \widetilde{g}(h)^{\gamma +q} \int _{\varOmega } g(x) \vert u+h \vert ^{\gamma +1}\,dx \\ &=\frac{ \Vert \widetilde{g}(h)(u+h) \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert \widetilde{g}(h)(u+h) \vert ^{1-q}\,dx -\lambda \int _{\varOmega }g(x) \vert \widetilde{g}(h)(u+h) \vert ^{\gamma +1}\,dx}{ \widetilde{g}(h)^{1-q}}, \end{aligned} $$
i.e.,
$$\begin{aligned}& \widetilde{g}(h) (u+h)\in \mathcal{M}, \quad \forall h\in W, \Vert h \Vert < \overline{ \varepsilon }, \end{aligned}$$
and
$$ \begin{aligned} \widetilde{F}_{s} \bigl(h, \widetilde{g}(h) \bigr)&=(p-1+q)\widetilde{g}(h)^{p+q-2} \Vert u+h \Vert _{\mu }^{p}-(q+ \gamma )\lambda \widetilde{g}(h)^{\gamma +q-1} \int _{\varOmega } g(x) \vert u+h \vert ^{\gamma +1}\,dx \\ &=\frac{(p-1+q) \Vert \widetilde{g}(h)(u+h) \Vert _{\mu }^{p}-(q+\gamma ) \lambda \int _{\varOmega } g(x) \vert \widetilde{g}(h)(u+h) \vert ^{\gamma +1}\,dx}{ \widetilde{g}^{2-q}(h)}, \end{aligned} $$
together with (2.3), these imply that we can choose \(\varepsilon >0\) small enough (\(\varepsilon <\overline{\varepsilon }\)) such that for every \(h\in W\) and \(\Vert h\Vert <\varepsilon \)
$$ (p-1+q) \bigl\Vert \widetilde{g}(h) (u+h) \bigr\Vert _{\mu }^{p}-(q+ \gamma )\lambda \int _{\varOmega } g(x) \bigl\vert \widetilde{g}(h) (u+h) \bigr\vert ^{\gamma +1}\,dx>0, $$
that is,
$$ \widetilde{g}(h) (u+h)\in \mathcal{M}^{+}, \quad \forall h\in W, \Vert h \Vert < \varepsilon . $$
This completes the proof of Lemma 2.3. □

3 Proof of Theorem 1.1

For every \(u\in \mathcal{M}\), by (1.13), we have
$$\begin{aligned} I_{\lambda ,\mu }(u)&=\frac{1}{p} \Vert u \Vert _{\mu }^{p}-\frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx - \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &=\frac{1}{p} \Vert u \Vert _{\mu }^{p}- \frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx- \frac{1}{\gamma +1} \biggl[ \Vert u \Vert _{\mu } ^{p}- \int _{\varOmega }f(x)u^{1-q}\,dx \biggr] \\ &= \biggl(\frac{1}{p}- \frac{1}{\gamma +1} \biggr) \Vert u \Vert _{\mu }^{p}- \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) \int _{\varOmega }f(x)u^{1-q}\,dx \\ &\geq \biggl(\frac{1}{p}-\frac{1}{\gamma +1} \biggr) \Vert u \Vert _{\mu } ^{p}- \biggl(\frac{1}{1-q} -\frac{1}{\gamma +1} \biggr) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \Vert u \Vert _{\mu }^{1-q} \\ &:=\mathcal{K}\bigl( \Vert u \Vert _{\mu }\bigr). \end{aligned}$$
(3.1)
Let
$$ \mathcal{K}'\bigl( \Vert u \Vert _{\mu }\bigr)= \biggl(1- \frac{p}{\gamma +1} \biggr) \Vert u \Vert _{\mu }^{p-1}- \biggl(1- \frac{1-q}{\gamma +1} \biggr) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \Vert u \Vert _{\mu }^{-q}=0. $$
We have
$$\Vert u \Vert _{\mu }:=\bigl( \Vert u \Vert _{\mu } \bigr)_{\min }= \biggl[\frac{(1-\frac{1-q}{ \gamma +1}) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S _{\mu }}) ^{1-q}}}{1-\frac{p}{\gamma +1}} \biggr]^{\frac{1}{p+q-1}}. $$
Since \(\mathcal{K}''(\Vert u\Vert _{\mu })>0\) for all \(\Vert u\Vert _{\mu }>0\) with \(\mathcal{K}(\Vert u\Vert _{\mu })\rightarrow 0\) as \(\Vert u\Vert _{\mu }\rightarrow 0\) and \(\mathcal{K}(\Vert u\Vert _{\mu })\rightarrow \infty \) as \(\Vert u\Vert _{ \mu }\rightarrow \infty \). Therefore \(\mathcal{K}(u)\) attains its minimum at \((\Vert u\Vert _{\mu })_{\min }\), and
$$ \begin{aligned} \mathcal{K} \bigl(\bigl( \Vert u \Vert _{\mu }\bigr)_{\min } \bigr)&= \biggl(\frac{1}{p}- \frac{1}{ \gamma +1} \biggr) \biggl[ \frac{(1-\frac{1-q}{\gamma +1}) \Vert f \Vert _{ \infty }\frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{ \mu }})^{1-q}}}{1-\frac{p}{\gamma +1}} \biggr]^{\frac{p}{p+q-1}} \\ &\quad {} - \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S _{\mu }})^{1-q}} \biggl[\frac{(1-\frac{1-q}{\gamma +1}) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}}}{1-\frac{p}{ \gamma +1}} \biggr] ^{\frac{1-q}{p+q-1}}. \end{aligned} $$
By (3.1), we deduce that
$$\begin{aligned}& \lim_{ \Vert u \Vert _{\mu }\rightarrow \infty }I_{\lambda ,\mu }(u) \geq \lim_{ \Vert u \Vert _{\mu }\rightarrow \infty } \mathcal{K}\bigl( \Vert u \Vert _{\mu }\bigr)=\infty , \end{aligned}$$
namely, \(I_{\lambda ,\mu }(u)\) is coercive on \(\mathcal{M}\). Combining with (3.1), we have
$$ I_{\lambda ,\mu }(u)\geq \mathcal{K}(u)\geq \mathcal{K} \bigl( \bigl( \Vert u \Vert _{ \mu }\bigr)_{\min } \bigr). $$
(3.2)
Thus \(I_{\lambda ,\mu }(u)\) is bounded below on \(\mathcal{M}\). According to Lemma 2.3, if \(\lambda \in (0,T_{\mu })\), then \(\mathcal{M}^{+} \cup \mathcal{M}^{0}\) and \(\mathcal{M}^{-}\) are two closed sets in \(\mathcal{M}\). Therefore, we apply the Ekeland variational principle [2] to derive a minimizing sequence \(\{u_{n}\}\subset \mathcal{M}^{+}\cup \mathcal{M}^{0}\) satisfying:
$$\begin{aligned}& (\mathrm{i}) \quad I_{\lambda ,\mu }(u_{n})< \inf _{\mathcal{M}^{+}\cup \mathcal{M}^{0}}I_{\lambda ,\mu }(u)+ \frac{1}{n}; \\& (\mathrm{ii}) \quad I_{\lambda ,\mu }(u)\geq I_{\lambda ,\mu }(u_{n})- \frac{1}{n} \Vert u-u _{n} \Vert , \quad \forall u\in \mathcal{M}^{+}\cup \mathcal{M}^{0}. \end{aligned}$$
Assume that \(u_{n}\geq 0\) on \(\varOmega \setminus \{0\}\). Note that \(I_{\lambda ,\mu }(u)\) is bounded below on \(\mathcal{M}\). By (3.2), we get
$$ \mathcal{K} \bigl(\bigl( \Vert u_{n} \Vert _{\mu }\bigr)_{\min } \bigr)\leq I_{\lambda ,\mu }(u_{n})< \inf_{\mathcal{M}^{+}\cup \mathcal{M}^{0}}I_{\lambda ,\mu }(u)+ \frac{1}{n}\leq C_{1}, $$
(3.3)
for n large enough and a positive constant \(C_{1}\). Hence \(\{u_{n}\}\) is bounded in \(\mathcal{M}\). Let us, for a subsequence, suppose that
$$ \textstyle\begin{cases} u_{n}\rightharpoonup u_{0} &\text{in }W, \\ u_{n}(x)\rightarrow u_{0}(x) & \text{a.e. in } \varOmega , \\ u_{n}\rightarrow u_{0} &\text{in } L^{1-q}(\varOmega ) \text{ and } L^{\gamma +1}(\varOmega ). \end{cases} $$
For every \(u\in \mathcal{M}^{+}\), we deduce from \(p>1\) that
$$ \begin{aligned} I_{\lambda ,\mu }(u)&=\frac{1}{p} \Vert u \Vert _{\mu }^{p}-\frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx - \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &=\frac{1}{p} \Vert u \Vert _{\mu }^{p}- \frac{1}{1-q} \biggl[ \Vert u \Vert _{\mu }^{p} - \lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr]- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ & = \biggl(\frac{1}{p}-\frac{1}{1-q} \biggr) \Vert u \Vert _{\mu }^{p}+ \biggl(\frac{1}{1-q}- \frac{1}{ \gamma +1} \biggr)\lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &< \biggl(\frac{1}{p}-\frac{1}{1-q} \biggr) \Vert u \Vert _{\mu }^{p} + \biggl(\frac{1}{1-q}-\frac{1}{ \gamma +1} \biggr)\frac{p+q-1}{\gamma +q} \Vert u \Vert _{\mu }^{p} \\ &=\frac{p+q-1}{\gamma +q} \biggl(\frac{1}{\gamma +1}-\frac{1}{p} \biggr) \Vert u \Vert _{\mu }^{p}< 0, \end{aligned} $$
which implies that \(\inf_{\mathcal{M}^{+}}I_{\lambda ,\mu }(u)<0\). For \(\lambda \in (0,T_{\mu })\), it follows from Lemma 2.1 that \(\mathcal{M}^{0}=\{0 \}\). Thus \(u_{n}\in \mathcal{M}^{+}\) for n large enough and \(\inf_{\mathcal{M}^{+}\cup \mathcal{M}^{0}}I_{\lambda ,\mu }(u) =\inf_{\mathcal{M}^{+}}I_{\lambda ,\mu }(u)<0\). Therefore
$$\begin{aligned}& I_{\lambda ,\mu }(u_{0})\leq \liminf_{n\rightarrow \infty }I _{\lambda ,\mu }(u_{n}) =\inf_{\mathcal{M}^{+}\cup \mathcal{M} ^{0}}I_{\lambda ,\mu }< 0, \end{aligned}$$
i.e., \(u_{0}\geq 0\) and \(u_{0}\neq 0\).
In the following, we prove that, when \(\lambda \in (0,T_{\mu })\),
$$ (p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx >\lambda ( \gamma -q+1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx. $$
(3.4)
For \(\{u_{n}\}\subset \mathcal{M}^{+}\), we have
$$\begin{aligned}& (p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx -\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\& \quad =\lim_{n\rightarrow \infty } \biggl[ (p+q-1 ) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\& \quad =\lim_{n\rightarrow \infty } \biggl\{ (p+q-1 ) \biggl[ \Vert u_{n} \Vert _{\mu }^{p}-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr]- \lambda (\gamma -p+1 ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr\} \\& \quad =\lim_{n\rightarrow \infty } \biggl[ (p+q-1 ) \Vert u _{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega }g(x)u_{n} ^{\gamma +1}\,dx \biggr] \geq 0. \end{aligned}$$
We suppose that
$$ (p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx=0. $$
(3.5)
It follows from \(u_{n}\in \mathcal{M}\), the weak lower semi-continuity of the norm and (3.5) that
$$ \begin{aligned} 0&=\lim_{n\rightarrow \infty } \biggl[ \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega } f(x)u_{n}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &\geq \Vert u_{0} \Vert _{\mu }^{p}- \int _{\varOmega } f(x)u_{0}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\ &= \textstyle\begin{cases} \Vert u_{0} \Vert _{\mu }^{p}-\lambda \frac{\gamma +q}{p+q-1} \int _{\varOmega } g(x)u_{0}^{\gamma +1}\,dx, \\ \Vert u_{0} \Vert _{\mu }^{p}-\lambda \frac{\gamma +q}{ \gamma -p+1} \int _{\varOmega } f(x)u_{0}^{1-q}\,dx. \end{cases}\displaystyle \end{aligned} $$
Hence, for every \(\lambda \in (0,T_{\mu })\) and \(u_{0}\neq 0\), combining with (2.2), we obtain
$$ \begin{aligned} 0&< A(\mu ,\lambda ) \Vert u_{0} \Vert _{\mu }^{\gamma +1} \\ &\leq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u_{0} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u_{0} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}}-\lambda \int _{\varOmega }g(x) \vert u_{0} \vert ^{\gamma +1}\,dx \\ &\leq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u_{0} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\frac{\gamma -p+1}{q+\gamma } \Vert u_{0} \Vert _{\mu }^{p})^{\frac{p-\gamma -1}{1-q-p}}} -\frac{p+q-1}{ \gamma +q} \Vert u_{0} \Vert _{\mu }^{p}=0, \end{aligned} $$
which is a contradiction. In view of (3.4), we get
$$ (p+q-1 ) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx\geq C_{2} $$
(3.6)
for n large enough and some positive constant \(C_{2}\). Since \(u_{n}\in \mathcal{M}\), we have
$$ (p+q-1 ) \Vert u_{n} \Vert _{\mu }^{p}- \lambda (\gamma +q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx\geq C_{2}>0. $$
(3.7)
Set \(\phi \in \mathcal{M}\) with \(\phi \geq 0\). Using Lemma 2.4, there exists \(\widetilde{g}_{n}(t)\) such that \(\widetilde{g}_{n}(0)=1\) and \(\widetilde{g}_{n}(t)(u_{n}+t\phi )\in \mathcal{M}^{+}\). Thus
$$\begin{aligned}& \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx=0 \end{aligned}$$
and
$$\begin{aligned}& \widetilde{g}_{n}^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p}-\widetilde{g} _{n}^{1-q}(t) \int _{\varOmega }f(x) (u_{n}+t\phi )^{1-q}\,dx- \lambda \widetilde{g}_{n} ^{\gamma +1}(t) \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1} \,dx=0. \end{aligned}$$
Therefore
$$ \begin{aligned} 0&= \bigl[\widetilde{g}_{n}^{p}(t)-1 \bigr] \Vert u_{n}+t\phi \Vert _{\mu }^{p}+ \bigl( \Vert u_{n}+t\phi \Vert _{\mu }^{p}- \Vert u_{n} \Vert _{\mu }^{p} \bigr) \\ &\quad {}- \bigl[ \widetilde{g}_{n} ^{1-q}(t)-1 \bigr] \int _{\varOmega }f(x) (u_{n}+t\phi )^{1-q}\,dx \\ &\quad {} - \int _{\varOmega }f(x) \bigl[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q} \bigr]\,dx- \lambda \bigl[ \widetilde{g}_{n}^{\gamma +1}(t)-1 \bigr] \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1}\,dx \\ &\quad {} -\lambda \int _{\varOmega }g(x) \bigl[(u_{n}+t\phi )^{\gamma +1}-u_{n}^{\gamma +1} \bigr]\,dx \\ & \leq \bigl[\widetilde{g}_{n}^{p}(t)-1 \bigr] \Vert u_{n}+t\phi \Vert _{\mu }^{p}+ \bigl( \Vert u_{n}+t\phi \Vert _{\mu }^{p}- \Vert u_{n} \Vert _{\mu } ^{p} \bigr) \\ &\quad {}- \bigl[ \widetilde{g}_{n} ^{1-q}(t)-1 \bigr] \int _{\varOmega }f(x) (u_{n}+t\phi )^{1-q}\,dx \\ &\quad {} - \lambda \bigl[\widetilde{g}_{n}^{\gamma +1}(t)-1 \bigr] \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1}\,dx- \lambda \int _{\varOmega }g(x) \bigl[(u_{n}+t\phi )^{\gamma +1}-u_{n}^{\gamma +1} \bigr]\,dx. \end{aligned} $$
Dividing by \(t>0\) and letting \(t\rightarrow 0\), we have
$$ \begin{aligned}[b] 0& \leq p \widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p}+p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad {}- (1-q ) \widetilde{g}'_{n}(0) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx \\ &\quad {} -\lambda (\gamma +1 )\widetilde{g}'_{n}(0) \int _{\varOmega } g(x)u_{n}^{\gamma +1}\,dx-\lambda ( \gamma +1 ) \int _{\varOmega } g(x)u_{n}^{\gamma }\phi \,dx \\ & =\widetilde{g}'_{n}(0) \biggl[p \Vert u_{n} \Vert _{\mu }^{p}- (1-q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &\quad {}+p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad {} -\lambda (\gamma +1 ) \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ & =\widetilde{g}'_{n}(0) \biggl[ (p+q-1 ) \Vert u_{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &\quad {} +p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx-\lambda (\gamma +1 ) \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx, \end{aligned} $$
(3.8)
where \(\widetilde{g}'_{n}(0)\) denotes the right derivative of \(\widetilde{g}_{n}(t)\) at zero. If it does not exist, \(\widetilde{g}'_{n}(0)\) should be replaced by \(\lim_{k\rightarrow \infty }\frac{\widetilde{g}_{n}(t_{k})- \widetilde{g}_{n}(0)}{t_{k}}\) for some sequence \(\{t_{k}\}_{k=1}^{ \infty }\) with \(\lim_{k\rightarrow \infty }t_{k} =0\) and \(t_{k}>0\).
Combining with (3.7) and (3.8), we have \(\widetilde{g}'_{n}(0)\neq - \infty \). Now we prove that \(\widetilde{g}'_{n}(0)\neq +\infty \). Otherwise, we suppose that \(\widetilde{g}'_{n}(0)=+\infty \). Note that \(\widetilde{g}_{n}(t)>\widetilde{g}_{n}(0)=1\) for n large enough, and
$$ \begin{aligned}[b] \bigl\vert \widetilde{g}_{n}(t)-1 \bigr\vert \cdot \Vert u_{n} \Vert +t\widetilde{g}_{n}(t) \Vert \phi \Vert &\geq \bigl\Vert \bigl[ \widetilde{g}_{n}(t)-1 \bigr]u_{n}+t\widetilde{g} _{n}(t)\phi \bigr\Vert \\ &= \bigl\Vert \widetilde{g}_{n}(t) (u_{n}+t\phi )-u_{n} \bigr\Vert . \end{aligned} $$
(3.9)
Using condition (ii) with \(u=\widetilde{g}_{n}(t)(u_{n}+t\phi ) \in \mathcal{M}^{+}\), we deduce that
$$\begin{aligned}& \bigl[\widetilde{g}_{n}(t)-1 \bigr] \cdot \frac{ \Vert u_{n} \Vert }{n}+t \widetilde{g}_{n}(t) \frac{ \Vert \phi \Vert }{n} \\ & \quad \geq \frac{1}{n} \bigl\Vert \widetilde{g}_{n}(t) ( u _{n}+t\phi )-u_{n} \bigr\Vert \\ & \quad \geq I_{\lambda ,\mu }(u_{n})- I_{\lambda ,\mu } \bigl( \widetilde{g}_{n} (t) (u_{n}+t\phi ) \bigr) \\ & \quad =\frac{1}{p} \Vert u_{n} \Vert _{\mu }^{p} -\frac{1}{1-q} \int _{\varOmega }f(x) \vert u_{n} \vert ^{1-q}\,dx- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx- \frac{1}{p}\widetilde{g}_{n} ^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p} \\ & \quad \quad {} +\frac{1}{1-q} \int _{\varOmega }f(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{1-q}\,dx+\frac{ \lambda }{\gamma +1} \int _{\varOmega }g(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{ \gamma +1}\,dx \\ & \quad =\frac{1}{p} \Vert u_{n} \Vert _{\mu }^{p}- \frac{1}{1-q} \biggl[ \Vert u_{n} \Vert _{\mu }^{p}- \lambda \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx \biggr]-\frac{\lambda }{ \gamma +1} \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx \\ & \quad \quad {} -\frac{1}{p}\widetilde{g}_{n}^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p}+ \frac{1}{1-q} \biggl[\widetilde{g}_{n}^{p}(t) \Vert u_{n}+t \phi \Vert _{ \mu }^{p}-\lambda \int _{\varOmega }g(x) \vert u_{n}+t\phi \vert ^{\gamma +1}\,dx \biggr] \\ & \quad \quad {} +\frac{\lambda }{\gamma +1} \widetilde{g}_{n}^{\gamma +1}(t) \int _{\varOmega }g(x) \vert u_{n}+t\phi \vert ^{\gamma +1}\,dx \\ & \quad = \biggl(\frac{1}{p}-\frac{1}{1-q} \biggr) \Vert u_{n} \Vert _{\mu } ^{p}+ \biggl( \frac{1}{1-q}- \frac{1}{\gamma +1} \biggr)\lambda \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx \\ & \quad \quad {} + \biggl(\frac{1}{1-q}-\frac{1}{p} \biggr) \widetilde{g}_{n} ^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p} \\ & \quad\quad{} - \biggl(\frac{1}{1-q}- \frac{1}{ \gamma +1} \biggr)\lambda \widetilde{g}_{n} ^{\gamma +1}(t) \int _{\varOmega }g(x) \vert u_{n}+t\phi \vert ^{\gamma +1}\,dx \\ & \quad = \biggl(\frac{1}{1-q}-\frac{1}{p} \biggr) \bigl( \Vert u_{n}+t \phi \Vert _{\mu }^{p}- \Vert u_{n} \Vert _{\mu }^{p} \bigr)+ \biggl( \frac{1}{1-q} -\frac{1}{p} \biggr) \bigl[\widetilde{g}_{n}^{p}(t)-1 \bigr] \Vert u_{n}+t \phi \Vert _{\mu }^{p} \\ & \quad \quad {} - \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) \lambda \widetilde{g}_{n}^{\gamma +1}(t) \int _{\varOmega }g(x) \bigl[ (u_{n}+t\phi )^{\gamma +1}-u _{n}^{\gamma +1} \bigr]\,dx \\ & \quad \quad {} - \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr)\lambda \bigl[ \widetilde{g}_{n}^{\gamma +1}(t)-1 \bigr] \int _{\varOmega }g(x) u_{n}^{\gamma +1}\,dx. \end{aligned}$$
Dividing by \(t>0\) and letting \(t\rightarrow 0\), we obtain
$$\begin{aligned}& \widetilde{g}'_{n}(0)\cdot \frac{ \Vert u_{n} \Vert }{n}+ \frac{ \Vert \phi \Vert }{n} \\ & \quad \geq \biggl(\frac{1}{1-q}-\frac{1}{p} \biggr)\cdot p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ & \quad \quad {}+ \biggl(\frac{1}{1-q}- \frac{1}{p} \biggr)\cdot p \widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p} \\ & \quad \quad {}-\lambda \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) (\gamma +1 ) \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ & \quad \quad {}-\lambda \biggl(\frac{1}{1-q} -\frac{1}{\gamma +1} \biggr) (\gamma +1 ) \widetilde{g}'_{n}(0) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \\ & \quad = \frac{p-1+q}{1-q} \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx + \frac{p-1+q}{1-q}\widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p} \\ & \quad \quad {}-\lambda \frac{\gamma +q}{1-q} \int _{\varOmega } g(x)u_{n}^{\gamma }\phi \,dx- \lambda \frac{\gamma +q}{1-q}\widetilde{g}'_{n}(0) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \\ & \quad = \frac{\widetilde{g}'_{n}(0)}{1-q} \biggl[ (p-1+q ) \Vert u_{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega } g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ & \quad \quad {}+\frac{p-1+q}{1-q} \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx-\lambda \frac{ \gamma +q}{1-q} \int _{\varOmega } g(x)u_{n}^{\gamma }\phi \,dx, \end{aligned}$$
that is,
$$ \begin{aligned}[b] \frac{ \Vert \phi \Vert }{n}&\geq \frac{\widetilde{g}'_{n}(0)}{1-q} \biggl[(p-1+q) \Vert u_{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega } g(x)u_{n}^{\gamma +1}\,dx- \frac{(1-q) \Vert u_{n} \Vert }{n} \biggr] \\ &\quad {} + \frac{p-1+q}{1-q} \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad {}-\lambda \frac{ \gamma +q}{1-q} \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx, \end{aligned} $$
(3.10)
which is not true since \(\widetilde{g}'_{n}(0)=+\infty \) and
$$\begin{aligned}& (p-1+q ) \Vert u_{n} \Vert _{\mu }^{p}- \lambda ( \gamma +q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx- \frac{(1-q) \Vert u_{n} \Vert }{n}\geq C _{2}-\frac{(1-q)C_{3}}{n}>0. \end{aligned}$$
It follows from (3.7), (3.8) and (3.10) that
$$\begin{aligned}& \bigl\vert \widetilde{g}_{n}^{\prime }(0) \bigr\vert \leq C_{4} \end{aligned}$$
for n sufficiently large and a suitable positive constant \(C_{4}\).
In the following, we prove that \(u_{0}\in \mathcal{M}^{+}\) is a solution of problem (1.1). By (3.9) and condition (ii) again, we have
$$\begin{aligned}& \frac{1}{n} \bigl[ \bigl\vert \widetilde{g}_{n}(t)-1 \bigr\vert \cdot \Vert u_{n} \Vert +t \widetilde{g}_{n}(t) \Vert \phi \Vert \bigr] \\& \quad \geq \frac{1}{n} \bigl\Vert \widetilde{g}_{n}(t) (u_{n}+t\phi )-u_{n} \bigr\Vert \\& \quad \geq I_{\lambda ,\mu }(u_{n})-I_{\lambda ,\mu } \bigl( \widetilde{g}_{n}(t) (u_{n}+t\phi ) \bigr) \\& \quad = \frac{1}{p} \Vert u_{n} \Vert _{\mu }^{p} -\frac{1}{1-q} \int _{\varOmega }f(x) \vert u_{n} \vert ^{1-q}\,dx- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx- \frac{1}{p}\widetilde{g}_{n} ^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p} \\& \quad \quad {} +\frac{1}{1-q} \int _{\varOmega }f(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{1-q}\,dx +\frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{ \gamma +1}\,dx \\& \quad =-\frac{\widetilde{g}_{n}^{p}(t)-1}{p} \Vert u_{n} \Vert _{\mu }^{p}- \frac{ \widetilde{g}_{n}^{p}(t)}{p} \bigl( \Vert u_{n}+t\phi \Vert _{\mu }^{p}- \Vert u _{n} \Vert _{\mu }^{p} \bigr) \\& \quad \quad {}+\frac{\widetilde{g}_{n}^{1-q}(t)-1}{1-q} \int _{\varOmega } f(x) (u_{n}+t\phi )^{1-q}\,dx \\& \quad \quad {} +\frac{1}{1-q} \int _{\varOmega }f(x) \bigl[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q} \bigr]\,dx +\frac{ \lambda (\widetilde{g}_{n}^{\gamma +1}(t)-1)}{\gamma +1} \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1}\,dx \\& \quad \quad {} +\frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \bigl[(u_{n}+t\phi )^{\gamma +1} -u_{n}^{\gamma +1} \bigr]\,dx. \end{aligned}$$
Dividing by \(t>0\) and letting \(t\rightarrow 0^{+}\), we derive that
$$\begin{aligned}& \frac{1}{n} \bigl[ \bigl\vert \widetilde{g}'_{n}(0) \bigr\vert \cdot \Vert u_{n} \Vert + \Vert \phi \Vert \bigr] \\ & \quad \geq -\widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx+ \widetilde{g}_{n}^{\prime }(0) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx \\ & \quad \quad {} +\lambda \widetilde{g}'_{n}(0) \int _{\varOmega }g(x)u_{n} ^{\gamma +1}\,dx+\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ & \quad \quad {}+\liminf _{t\rightarrow 0^{+}} \frac{1}{1-q} \int _{\varOmega }\frac{f(x)[(u_{n}+t\phi )^{1-q} -u_{n}^{1-q}]}{t}\,dx \\ & \quad =-\widetilde{g}'_{n}(0) \biggl[ \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ & \quad \quad {} - \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx +\lambda \int _{\varOmega }g(x) u_{n}^{\gamma }\phi \,dx \\ & \quad \quad {} +\liminf_{t\rightarrow 0^{+}}\frac{1}{1-q} \int _{\varOmega }\frac{f(x)[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q}]}{t}\,dx. \end{aligned}$$
Noting \(f(x) [(u_{n}+t\phi )^{1-q}-u_{n}^{1-q} ]\geq 0\), for every \(x \in \varOmega \) and \(t>0\), together with the Fatou lemma, we find that
$$ \liminf_{t\rightarrow 0^{+}} \biggl[\frac{f(x)[(u_{n}+t\phi )^{1-q}-u _{n}^{1-q}]}{t} \biggr] $$
is integrable, and
$$ \begin{aligned} & \int _{\varOmega }f(x)u_{n}^{-q}\phi \,dx \\ &\quad \leq \liminf_{t\rightarrow 0^{+}}\frac{1}{1-q} \int _{\varOmega } \frac{f(x)[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q}]}{t}\,dx \\ &\quad \leq \frac{ \vert \widetilde{g}'_{n}(0) \vert \Vert u_{n} \Vert + \Vert \phi \Vert }{n}+ \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad\quad{} -\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ &\quad \leq \frac{C_{3}C_{4}+ \Vert \phi \Vert }{n}+ \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx -\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx. \end{aligned} $$
Applying the Fatou lemma again, we have
$$ \begin{aligned} & \int _{\varOmega }f(x)u_{0}^{-q}\phi \,dx \\ &\quad = \int _{\varOmega } \Bigl[\liminf_{n\rightarrow \infty }f(x)u_{n} ^{-q}\phi \Bigr]\,dx \leq \liminf_{n\rightarrow \infty } \int _{\varOmega } f(x)u_{n}^{-q}\phi \,dx \\ &\quad \leq \liminf_{n\rightarrow \infty } \biggl[\frac{C_{3}C_{4}+ \Vert \phi \Vert }{n}+ \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad\quad{} -\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \biggr] \\ &\quad = \int _{\varOmega } \biggl( \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx -\lambda \int _{\varOmega }g(x)u_{0}^{\gamma }\phi \,dx. \end{aligned} $$
Since \(\int _{\varOmega }u_{0}^{-q}\varphi _{1}\,dx<\infty \), we have \(u_{0}>0\) a.e. in Ω. For every \(\phi \in \mathcal{M}\) and \(\phi \geq 0\), we have
$$ \begin{aligned}[b] & \int _{\varOmega } \biggl( \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx- \int _{\varOmega }f(x) u_{0}^{-q}\phi \,dx \\ &\quad{} -\lambda \int _{\varOmega }g(x)u_{0}^{\gamma }\phi \,dx\geq 0. \end{aligned} $$
(3.11)
Set \(\phi =u_{0}\) in (3.11), we derive that
$$\begin{aligned}& \Vert u_{0} \Vert _{\mu }^{p}= \int _{\varOmega } \biggl( \vert \Delta u_{0} \vert ^{p}-\mu \frac{ \vert u_{0} \vert ^{p}}{ \vert x \vert ^{2p}} \biggr)\,dx\geq \int _{\varOmega }f(x) u_{0}^{1-q}\,dx+\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx. \end{aligned}$$
Furthermore
$$ \begin{aligned}[b] \Vert u_{0} \Vert _{\mu }^{p}&\leq \liminf_{n\rightarrow \infty } \Vert u _{n} \Vert _{\mu }^{p}\leq \limsup _{n\rightarrow \infty } \Vert u_{n} \Vert _{\mu }^{p} \\ & =\limsup_{n\rightarrow \infty } \biggl[ \int _{\varOmega }f(x)u_{n}^{1-q}\,dx +\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &= \int _{\varOmega }f(x)u_{0}^{1-q}\,dx +\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx. \end{aligned} $$
(3.12)
Hence
$$ \Vert u_{0} \Vert _{\mu }^{p}= \int _{\varOmega }f(x)u_{0}^{1-q}\,dx +\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx. $$
(3.13)
Therefore \(u_{n}\rightarrow u_{0}\) in \(\mathcal{M}\) and \(u_{0}\in \mathcal{M}\). By (3.4), we have
$$ \begin{aligned} & (p+q-1 ) \Vert u_{0} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\ &\quad = (p+q-1 ) \biggl[ \int _{\varOmega }f(x)u_{0}^{1-q}\,dx+\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \biggr]- \lambda (\gamma +q ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\ &\quad = (p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx-\lambda ( \gamma -1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx>0, \end{aligned} $$
i.e., \(u_{0}\in \mathcal{M}^{+}\).
Next, we only need to show that \(u_{0}\) is a positive weak solution of problem (1.1). Define
$$ \varPhi =(u_{0}+\varepsilon \phi )^{+}, \quad \phi \in W, \varepsilon >0. $$
Substituting Φ into (3.11), combining with (3.12), we deduce that
$$\begin{aligned} 0&\leq \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \varPhi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \varPhi }{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q}\varPhi - \lambda g(x)u_{0} ^{\gamma }\varPhi \biggr]\,dx \\ &= \int _{\varOmega _{1}} \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \varPhi -\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \varPhi }{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} \varPhi -\lambda g(x)u_{0} ^{\gamma }\varPhi \biggr]\,dx \\ &\quad {} + \int _{\varOmega _{2}} \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \varPhi -\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \varPhi }{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} \varPhi -\lambda g(x)u_{0} ^{\gamma }\varPhi \biggr]\,dx \\ &= \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta (u_{0}+ \varepsilon \phi )-\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} (u_{0}+\varepsilon \phi )}{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} (u_{0}+ \varepsilon \phi ) \\ &\quad {} -\lambda g(x)u_{0}^{\gamma }(u_{0}+ \varepsilon \phi ) \biggr]\,dx \\ &\quad {} - \int _{\varOmega _{2}} \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta (u _{0}+\varepsilon \phi )-\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} (u_{0}+\varepsilon \phi )}{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} (u_{0}+ \varepsilon \phi ) \\ &\quad {} -\lambda g(x)u_{0}^{\gamma }(u_{0}+ \varepsilon \phi ) \biggr]\,dx \\ &= \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p}-\mu \frac{ \vert u_{0} \vert ^{p}}{ \vert x \vert ^{2p}}-f(x)u_{0}^{1-q}- \lambda g(x)u_{0}^{ \gamma +1} \biggr]\,dx \\ &\quad {} +\varepsilon \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma }\phi \biggr]\,dx \\ &\quad {} - \int _{\varOmega _{2}} \biggl[ \vert \Delta u_{0} \vert ^{p}+\varepsilon \vert \Delta u_{0} \vert ^{p-2} \Delta u_{0} \Delta \phi -\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} (u_{0}+ \varepsilon \phi )}{ \vert x \vert ^{2p}} \biggr]\,dx \\ &\quad {} - \int _{\varOmega _{2}} \bigl[-f(x) u_{0}^{-q}(u_{0}+ \varepsilon \phi )- \lambda g(x)u_{0}^{\gamma +1}-\varepsilon \lambda g(x)u_{0}^{\gamma } \phi \bigr]\,dx \\ &\leq \varepsilon \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma } \phi \biggr]\,dx \\ &\quad {} -\varepsilon \int _{\varOmega _{2}} \vert \Delta u_{0} \vert ^{p-2} \Delta u_{0} \Delta \phi \,dx+ \lambda \Vert g \Vert _{\infty } \int _{\varOmega _{2}} \vert \varepsilon \phi \vert ^{\gamma +1} \,dx+ \varepsilon \lambda \int _{\varOmega _{2}}g(x)u_{0}^{\gamma }\phi \,dx \\ &=\varepsilon \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma }\phi \biggr]\,dx \\ &\quad {} -\varepsilon \int _{\varOmega _{2}} \vert \Delta u_{0} \vert ^{p-2} \Delta u_{0} \Delta \phi \,dx+ \varepsilon \lambda \varepsilon ^{\gamma } \Vert g \Vert _{\infty } \int _{\varOmega _{2}} \vert \phi \vert ^{\gamma +1}\,dx+ \varepsilon \lambda \int _{\varOmega _{2}}g(x)u_{0}^{\gamma } \phi \,dx, \end{aligned}$$
where \(\varOmega _{1}=\{x\vert u_{0}(x)+\varepsilon \phi (x)>0, x\in \varOmega \}\) and \(\varOmega _{2}=\{x\vert u_{0}(x)+\varepsilon \phi (x)\leq 0, x\in \varOmega \}\). Since the measure of \(\varOmega _{2}\) tends to zero as \(\varepsilon \rightarrow 0\), we have \(\int _{\varOmega _{2}} \vert \Delta u_{0}\vert ^{p-2}\Delta u_{0} \Delta \phi \,dx \rightarrow 0\) as \(\varepsilon \rightarrow 0\). By the same arguments, we have \(\lambda \varepsilon ^{\gamma }\Vert g\Vert _{\infty } \int _{\varOmega _{2}}\vert \phi \vert ^{\gamma +1}\,dx \longrightarrow 0\) and \(\lambda \int _{\varOmega _{2}}g(x)u_{0}^{\gamma }\phi \,dx\longrightarrow 0\) as \(\varepsilon \rightarrow 0\). Dividing by ε and taking the limit for \(\varepsilon \rightarrow 0\), we deduce that
$$\begin{aligned}& \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma }\phi \biggr]\,dx\geq 0. \end{aligned}$$
Therefore \(u_{0}\) is a positive weak solution of problem (1.1).
We adopt the Ekeland variational principle again to derive a minimizing sequence \(U_{n}\subset \mathcal{M}^{-}\) for the minimization problem \(\inf_{\mathcal{M}^{-}} I_{\lambda ,\mu }\) such that for \(U_{n}\in \mathcal{M}\), \(U_{n} \rightharpoonup U_{0}\) weakly in \(\mathcal{M}\) and pointwise a.e. in Ω. By similar arguments to those in (3.4) and (3.6), for \(\lambda \in (0,T_{\mu })\), we have
$$ (p+q-1 ) \int _{\varOmega }f(x) \vert U_{0} \vert ^{1-q}\,dx- \lambda (\gamma -p+1 ) \int _{\varOmega }g(x) \vert U_{0} \vert ^{\gamma +1}\,dx< 0, $$
(3.14)
which leads to
$$\begin{aligned}& (p+q-1 ) \int _{\varOmega }f(x) \vert U_{n} \vert ^{1-q}\,dx- \lambda (\gamma -p+1 ) \int _{\varOmega }g(x) \vert U_{n} \vert ^{\gamma +1}\,dx \leq -C_{5}, \end{aligned}$$
for n large enough and a positive constant \(C_{5}\). Therefore \(U_{0}>0\) is the positive weak solution of problem (1.1). Furthermore \(U_{0}\in \mathcal{M}\). By (3.14), we obtain
$$ \begin{aligned} & (p+q-1 ) \Vert U_{0} \Vert _{\mu }^{p}- (q+\gamma ) \lambda \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx \\ & \quad = (p+q-1 ) \biggl[ \int _{\varOmega }f(x)U_{0}^{1-q}\,dx+\lambda \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx \biggr]- \lambda (\gamma +q ) \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx \\ & \quad = (p+q-1 ) \int _{\varOmega }f(x)U_{0}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx< 0, \end{aligned} $$
i.e., \(U_{0}\in \mathcal{M}^{-}\). According to Lemma 2.2, we know that problem (1.1) has at least two positive weak solutions \(u_{0}\in \mathcal{M}^{+}\) and \(U_{0}\in \mathcal{M}^{-}\) with \(\Vert U_{0}\Vert _{ \mu }>\Vert u_{0}\Vert _{\mu }\) for every \(\lambda \in (0,T_{\mu })\). This completes the proof of Theorem 1.1.

4 Proof of Corollary 1.2

For every \(U\in \mathcal{M}^{-}\), by Lemma 2.2, we deduce that
$$ \begin{aligned} \Vert U \Vert _{\mu }&>M_{\mu }( \lambda ) \\ & = \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }}\frac{(\sqrt[p]{S_{\mu }})^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}} \\ &= \biggl(\frac{1}{\lambda } \biggr)^{\frac{1}{\gamma +1-p}} \biggl(\frac{p+q-1}{ \gamma +q} \biggr)^{\frac{1}{\gamma +1-p}} \biggl(\frac{1}{ \Vert g \Vert _{ \infty }} \biggr)^{\frac{1}{\gamma +1-p}} \frac{(\sqrt[p]{S_{\mu }}) ^{\frac{\gamma +1}{\gamma +1-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1-p)}}} \\ &= (T_{\mu } )^{-\frac{1}{\gamma +1-p}} \biggl(\frac{p+q-1}{ \gamma +q} \biggr)^{\frac{1}{\gamma +1-p}} \biggl(\frac{1}{ \Vert g \Vert _{ \infty }} \biggr)^{\frac{1}{\gamma +1-p}} \frac{(\sqrt[p]{S_{\mu }}) ^{\frac{\gamma +1}{\gamma +1-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1-p)}}} \biggl(\frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{ \gamma +1-p}}. \end{aligned} $$
Combining with the definition of \(T_{\mu }\), we have
$$\begin{aligned} \Vert U \Vert _{\mu }&> \biggl(\frac{q+\gamma }{q+p-1} \biggr)^{\frac{1}{ \gamma +1-p}} \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr)^{ \frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \bigl( \Vert g \Vert _{\infty } \bigr)^{\gamma -p+1}\frac{ \vert \varOmega \vert ^{ \frac{2p}{N}\frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}}}{S_{\mu } ^{\frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}}} \\ & \quad{}\times \biggl(\frac{p+q-1}{\gamma +q} \biggr)^{ \frac{1}{\gamma +1-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) ^{\frac{1}{ \gamma +1-p}}\frac{(\sqrt[p]{S_{\mu }})^{ \frac{\gamma +1}{\gamma +1-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1-p)}}} \biggl(\frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{ \gamma +1-p}} \\ & = \biggl( \frac{q+\gamma }{\gamma -p+1} \biggr)^{ \frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl(\frac{ \vert \varOmega \vert ^{\frac{2p}{N}\frac{q+\gamma }{p+q-1} \frac{1}{ \gamma +1-p}-\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1-p)}}}{(\sqrt[p]{S _{\mu }})^{p\cdot \frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}- \frac{ \gamma +1}{\gamma +1-p}}} \biggr) \biggl( \frac{T_{\mu }}{\lambda } \biggr) ^{\frac{1}{\gamma +1-p}} \\ &= \vert \varOmega \vert ^{\frac{1}{p}} \biggl( \frac{q+\gamma }{ \gamma -p+1} \biggr)^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl( \frac{ \vert \varOmega \vert ^{\frac{2}{N}}}{\sqrt[p]{S _{\mu }}} \biggr) ^{\frac{1-q}{p+q-1}} \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\gamma +1-p}} \\ &= \vert \varOmega \vert ^{\frac{1}{p}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl(1+\frac{p+q-1}{\gamma -p+1} \biggr)^{ \frac{1}{p+q-1}} \biggl( \frac{ \vert \varOmega \vert ^{\frac{2}{N}}}{\sqrt[p]{S _{\mu }}} \biggr) ^{\frac{1-q}{p+q-1}} \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\gamma +1-p}}, \end{aligned}$$
where we adopted the following facts:
$$\begin{aligned}& \begin{aligned} &\frac{2p}{N}\frac{q+\gamma }{p+q-1} \frac{1}{\gamma +1-p}-\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1-p)} \\ &\quad =\frac{p^{*}-1+q}{p^{*}(p+q-1)}=\frac{\frac{Np}{N-2p}+q-1}{ \frac{Np}{N-2p}(p+q-1)} \\ &\quad =\frac{N(p+q-1)+2p(1-q)}{Np(p+q-1)}=\frac{1}{p}+\frac{2}{N}\cdot \frac{1-q}{p+q-1}, \end{aligned} \\& p\cdot \frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}-\frac{\gamma +1}{\gamma +1-p}= \frac{(1-q)(\gamma +1-p)}{(p+q-1)(\gamma +1-p)}= \frac{1-q}{p+q-1}. \end{aligned}$$
Let \(U_{\lambda , \mu ,\varepsilon }\in \mathcal{M}^{-}\) be the solution of problem (1.1) with \(\gamma =\varepsilon +p-1\), where \(\lambda \in (0,T_{\mu })\). Then
$$\begin{aligned}& \Vert U_{\lambda , \mu ,\varepsilon } \Vert _{\mu }>C_{\mu ,\varepsilon } \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\varepsilon }}, \end{aligned}$$
where \(C_{\mu , \varepsilon }\) is given in (1.16). This completes the proof of Corollary 1.2.

5 Proof of Theorem 1.3

For simplicity, we consider problem (1.1) with \(f=g=1\),
$$ \textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}=u^{-q}+\lambda u^{ \gamma } &\text{in } \varOmega \setminus \{0\}, \\ u(x)>0 &\text{in }\varOmega \setminus \{0\}, \\ u=\Delta u=0 &\text{on } \partial \varOmega . \end{cases} $$
(5.1)
Let us define
$$\begin{aligned}& \lambda ^{*}=\lambda ^{*}(N,\varOmega ,\mu ,q,\gamma )=\sup \bigl\{ \lambda >0: \text{problem (5.1) has a positive solution} \bigr\} . \end{aligned}$$
Using Theorem 1.1, we provide uniform estimates for \(\lambda ^{*}(N, \varOmega ,\mu ,q,\gamma )\).

Lemma 5.1

For\(1< p<\frac{N}{2}\), \(0<\mu <\mu _{N,p}\), \(0< q<1<\gamma <p^{*}-1\)and\(\varOmega \in \mathbb{U}\), where\(\mathbb{U}=\{\varOmega \in \mathbb{R}^{N}: \varOmega \textit{ is an open and bounded domain}\}\), we have
$$\begin{aligned}& 0< \lambda ^{-}\leq \lambda ^{*}\leq \lambda ^{+}< \infty , \end{aligned}$$
where
$$ \lambda ^{-}= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl( \frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl[\frac{S _{\mu }}{ \vert \varOmega \vert ^{\frac{2p}{N}}} \biggr]^{\frac{q+\gamma }{p+q-1}} $$
and
$$ \lambda ^{+}=\lambda _{1}^{\frac{\gamma +q}{q-1+p}} \biggl( \frac{\gamma -p+1}{\gamma +q} \biggr)^{\frac{\gamma -p+1}{q+p-1}}\frac{-1+p+q}{ \gamma +q}+ \frac{1}{2}. $$

Proof

(1) Assume that \(\lambda \in (0,\lambda ^{-})\), then problem (5.1) has at least two solutions. By the definition of \(\lambda ^{*}\), we have \(\lambda ^{*}\geq \lambda ^{-}>0\).

(2) Assume that (5.1) has a positive solution u. Integrating over Ω by multiplying (5.1) by \(\varphi _{1}\), we obtain
$$ \lambda _{1} \int _{\varOmega } \vert u \vert ^{p-2}u \varphi _{1}\,dx= \int _{\varOmega } \biggl(\Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}} \biggr) \varphi _{1}\,dx= \int _{\varOmega }u^{-q}\varphi _{1}\,dx+\lambda \int _{\varOmega }u^{\gamma } \varphi _{1}\,dx. $$
(5.2)
We claim that there exists \(\lambda ^{+}>0\) such that
$$ t^{-q}+\lambda ^{+}t^{\gamma }>\lambda _{1}t^{p-1}, \quad \forall t>0. $$
(5.3)
In fact, letting
$$ F_{\lambda }(t)=t^{-q}+\lambda t^{\gamma }- \lambda _{1}t^{p-1}=t^{ \gamma } \bigl(t^{-q-\gamma }+ \lambda -\lambda _{1}t^{-\gamma +p-1} \bigr) :=t^{ \gamma } \cdot G_{\lambda }(t), \quad t>0. $$
(5.4)
We have
$$ G'_{\lambda }(t)=(-\gamma -q)t^{-\gamma -q-1}+\lambda _{1} (\gamma -p+1)t ^{-\gamma +p-2}=0, $$
i.e.,
$$ t:=t_{\min }= \biggl(\frac{\gamma +q}{\lambda _{1} (\gamma -p+1)} \biggr) ^{\frac{1}{q-1+p}}. $$
Then \(G_{\lambda }(t)\) attains minimum at \(t_{\min }\), and
$$ G_{\lambda }(t_{\min })=\lambda +\lambda _{1}^{\frac{\gamma +q}{q-1+p}} \biggl(\frac{\gamma -p+1}{\gamma +q} \biggr)^{ \frac{\gamma -p+1}{q+p-1}}\frac{1-p-q}{\gamma +q}. $$
We may choose \(\lambda =\lambda _{1}^{\frac{\gamma +q}{q-1+p}} (\frac{ \gamma -p+1}{\gamma +q} )^{\frac{\gamma -p+1}{q+p-1}}\frac{-1+p+q}{ \gamma +q}+\frac{1}{2}=\lambda ^{+} >0\) such that
$$ G_{\lambda ^{+}}(t)\geq G_{\lambda ^{+}}(t_{\min })= \frac{1}{2}>0, \quad \text{for }t>0. $$
Therefore
$$ F_{\lambda ^{+}}(t)=t^{\gamma }\cdot G_{\lambda ^{+}}(t)>0 \quad \text{for }t>0. $$
Using (5.3) with \(t=u\), we have
$$ \int _{\varOmega }u^{-q}\varphi _{1}\,dx+\lambda ^{+} \int _{\varOmega } u^{\gamma }\varphi _{1}\,dx\geq \lambda _{1} \int _{\varOmega } \vert u \vert ^{p-2}u \varphi _{1}\,dx. $$
(5.5)
Combining with (5.2) and (5.5), we obtain \(\lambda \leq \lambda ^{+}\). Since λ is arbitrary, we have \(\lambda ^{*}\leq \lambda ^{+}< \infty \). □

Proof of Theorem 1.3

We only prove the case that \(0<\lambda <\lambda ^{*}\). By the definition of \(\lambda ^{*}\), there exists \(\overline{\lambda }\in (\lambda , \lambda ^{*})\) such that the problem
$$\begin{aligned}& \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}=u^{-q}+ \overline{ \lambda } u^{\gamma } \end{aligned}$$
has a positive solution, denoted by \(u_{\overline{\lambda }}\). It follows that
$$\begin{aligned}& \Delta ^{2}_{p}u_{\overline{\lambda }}-\mu \frac{ \vert u_{\overline{ \lambda }} \vert ^{p-2}u_{\overline{\lambda }}}{ \vert x \vert ^{2p}} =u_{\overline{ \lambda }}^{-q}+\overline{\lambda } u_{\overline{\lambda }}^{\gamma } \geq u_{\overline{\lambda }}^{-q}+\lambda u_{\overline{\lambda }}^{ \gamma }. \end{aligned}$$
Hence \(u_{\overline{\lambda }}\) is an upper solution of (5.1). Note that \(\lim_{t\rightarrow 0^{+}}G_{\lambda }(t)=\infty \), we can take \(\varepsilon >0\) small enough with \(\varepsilon \varphi _{1}< u_{\overline{ \lambda }}\) and \(G_{\lambda }(\varepsilon \varphi _{1})\geq 0\). Thus
$$\begin{aligned}& F_{\lambda }(\varepsilon \varphi _{1})= (\varepsilon \varphi _{1})^{ \gamma }G_{\lambda }(\varepsilon \varphi _{1})\geq 0, \quad \text{for all } \lambda >0, \end{aligned}$$
i.e.,
$$ \lambda _{1}(\varepsilon \varphi _{1})^{p-1} \leq (\varepsilon \varphi _{1})^{-q}+\lambda (\varepsilon \varphi _{1})^{\gamma }, \quad \text{for all } \lambda >0. $$
(5.6)
Combining with (1.10) and (5.6), we obtain
$$ \begin{aligned} \Delta ^{2}_{p}(\varepsilon \varphi _{1})-\mu \frac{ \vert (\varepsilon \varphi _{1}) \vert ^{p-2} (\varepsilon \varphi _{1})}{ \vert x \vert ^{2p}}&=\varepsilon ^{p-1} \biggl(\Delta ^{2}_{p}\varphi _{1}- \mu \frac{ \vert \varphi _{1} \vert ^{p-2} \varphi _{1}}{ \vert x \vert ^{2p}} \biggr) \\ & =\varepsilon ^{p-1}\lambda _{1} \vert \varphi _{1} \vert ^{p-1}= \lambda _{1}(\varepsilon \varphi _{1})^{p-1}\leq (\varepsilon \varphi _{1})^{-q}+\lambda (\varepsilon \varphi _{1})^{\gamma }, \end{aligned} $$
namely, \(\varepsilon \varphi _{1}\) is a lower solution of (5.1). Note that \(\Delta _{p}^{2}-\frac{\mu }{\vert x\vert ^{2p}}\) is monotone, then problem (5.1) has a positive solution \(u_{\lambda }\) with \(\varepsilon \varphi _{1}\leq u_{\lambda }\leq u_{\overline{\lambda }}\). □

6 Conclusions

In this paper, we study a class of p-biharmonic equations with Hardy potential and negative exponents. We establish the dependence of the above problem on q, γ, f, g and Ω and evaluate the extremal value of λ related to the multiplicity of positive solutions for this problem.

Notes

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Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Funding

This project is supported by the Natural Science Foundation of Shanxi Province (201601D011003), and the Natural Science Foundation of Shandong Province of China (ZR2017MA036).

Competing interests

The authors declare that they have no competing interests.

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© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceNorth University of ChinaTaiyuanP.R. China

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