Sign-changing bubble tower solutions for the supercritical Hénon-type equations

Abstract

This paper deals with the following supercritical Hénon-type equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=|x|^\alpha |u|^{p_\alpha -1-\varepsilon }u~~&{}\text {in}~\Omega ,\\ u=0~~&{}\text {on}~\partial \Omega , \end{array}\right. } \end{aligned}$$

where $\alpha >-2$, $\varepsilon >0$, $p_\alpha =\frac{N+2+2\alpha }{N-2}$, $N\ge 3$, $\Omega$ is a smooth bounded domain in $\mathbb {R}^N$ containing the origin. For $\varepsilon >0$ small enough, it is shown that if $\alpha$ is not an even integer, the above problem has sign-changing bubble tower solutions, which blow up at the origin. It seems that this is the first existence result of sign-changing bubble tower solutions for the supercritical Hénon-type equation.

1 Introduction and main results

In this paper, we consider the existence of sign-changing bubble tower solutions for the following supercritical Hénon-type equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=|x|^\alpha |u|^{p_\alpha -1-\varepsilon }u~~&{}\text {in}~\Omega ,\\ u=0~~&{}\text {on}~\partial \Omega , \end{array}\right. } \end{aligned}$$

(1.1)

where $\alpha >-2$, $\varepsilon >0$, $p_\alpha =\frac{N+2+2\alpha }{N-2}$, $N\ge 3$, $\Omega$ is a smooth bounded domain in $\mathbb {R}^N$ containing the origin.

When $\Omega$ is the unit ball $B_1(0)$ of $\mathbb {R}^N$, problem (1.1) becomes the well-known Hénon equation, i.e.,

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=|x|^\alpha u^{p}, \; u>0, &{} \text {in}\; B_1(0),\\ u=0, &{}\text {on}\; \partial B_1(0). \end{array}\right. } \end{aligned}$$

(1.2)

Problem (1.2) was proposed by Hénon in [15] when he studied rotating stellar structures, which has attracted a lot of interest in recent years. Ni [21] first considered (1.2) and proved that it possesses a positive radial solution when $p\in (1,p_\alpha )$. Due to the appearance of the weighted term $|x|^\alpha$, the classical moving plane method in [13] cannot be applied to problem (1.2) when $\alpha >0$. Therefore it is quite natural to ask whether problem (1.2) with $\alpha >0$ has non-radial solutions. Based on numerical results in [4], Smets, Su and Willem [27] obtained the existence of non-radial solutions for $1<p<\frac{N+2}{N-2}$, when $\alpha$ is large enough. For $p=\frac{N+2}{N-2}-\varepsilon$, Cao and Peng [5] showed that the ground state solution is non-radial and blows up near the boundary of $B_1(0)$ as $\varepsilon \rightarrow 0$. Later on, Peng [22] constructed multiple boundary peak solutions for problem (1.2). When $p=\frac{N+2}{N-2}$, Serra [26] proved that problem (1.2) has a non-radial solution provided $\alpha$ is large enough. More recently, Wei and Yan [28] showed that there are infinitely many non-radial positive solutions for problem (1.2) with $\alpha >0$. For other results related to the Hénon-type problems, see [1, 2, 6, 16, 24] and the references therein.

On the other hand, using the Pohozaev-type identity [25], we know that for $p\ge p_\alpha$ there are no nontrivial solutions to problem (1.2). So it seems more interesting whether there are solutions for $p\in (\frac{N+2}{N-2},p_\alpha )$. When $p=p_\alpha -\varepsilon$ with small $\varepsilon >0$, Gladiali and Grossi in [11] showed that there exists a solution concentrating at origin provided $0<\alpha \le 1$. By the results in [12], the same results still hold when $\alpha$ is not an even integer. In [17], the asymptotic behavior of the radial solutions obtained by Ni in [21] was analyzed as $\varepsilon \rightarrow 0^+$. More recently, Liu and Peng [19] constructed large number of peak solutions for (1.2) with $p=\frac{N+2}{N-2}+\varepsilon$. However, as far as we know, it seems that there are no results on existence of sign-changing solutions for (1.2) when $p\in (\frac{N+2}{N-2},p_\alpha )$

Our purpose in the present paper is to construct sign-changing bubble tower solutions for (1.1) which blow up at the origin. It seems that this is the first existence result of sign-changing bubble tower solutions for (1.1).

The main result of this paper is as follows.

Theorem 1.1

Assume that $N\ge 3$, $\alpha >-2$ is not an even integer, then for any $k\in \mathbb {N}^+$, there exists $\varepsilon _k>0$ such that for $\varepsilon \in (0,\varepsilon _k)$, problem (1.1) has a sign-changing bubble tower solution $u_\varepsilon$ with exactly k nodal sets in $\Omega$.

Remark 1.2

When $\alpha =0$ and $\Omega$ has some symmetry property, problem (1.1) has been studied in [3] and [23]. Our results do not need any symmetry property of $\Omega$. Further more, compared with the classical Hénon equation where $\alpha >0$, our result covers the more general case $\alpha >-2$.

Remark 1.3

If $\Omega$ is the unit ball of $\mathbb {R}^N$ , then Theorem 1.1 holds for all $\alpha >-2$. Actually, we can construct a sign-changing radial bubble tower solution $u_\varepsilon$. Considering the transformation $\texttt {w}(s)=u(r),~~r=s^{\frac{2}{\alpha +2}}$, problem (1.1) can be changed into the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\texttt {w}^{\prime \prime }-\frac{M-1}{s}{} \texttt {w}^\prime =\frac{4}{(2+\alpha )^2}|\texttt {w}|^{\frac{M+2}{M-2}-\varepsilon }{} \texttt {w} , &{} \text {in}\; (0,1),\\ \texttt {w}^\prime (0)=\texttt {w}(1)=0, &{} \end{array}\right. } \end{aligned}$$

(1.3)

where $M=\frac{2(N+\alpha )}{2+\alpha }$. When M is an integer, problem (1.3) was studied in [3]. However, problem (1.3) can be dealt with in a similar way if M is not an integer.

Let us outline the main idea to prove Theorem 1.1. To do this, we introduce a few notations first. For $x\in \mathbb {R}^N$ and $\mu >0$, set

$$\begin{aligned} U_{\mu }(x)=C_{\alpha ,N}\left( \frac{\mu ^{\frac{2+\alpha }{2}}}{\mu ^{2+\alpha }+|x|^{2+\alpha }}\right) ^{\frac{N-2}{2+\alpha }},\,\, C_{\alpha ,N}=((N+\alpha )(N-2))^{\frac{N-2}{4+2\alpha }}. \end{aligned}$$

It is well known from [12, 14] that $U_{\mu }(x)$ are the only radial solutions of

$$\begin{aligned} -\Delta u=|x|^\alpha u^{p_\alpha }, u>0~\text {in}~\mathbb {R}^N. \end{aligned}$$

(1.4)

We define the following Emden–Fowler-type transformation

$$\begin{aligned} v(y,\Theta )=\mathcal {T}(u)(y,\Theta )=\left( \frac{p_\alpha -1}{2}\right) ^{\frac{2}{p_\alpha -1}}r^{\frac{N-2}{2}}u(r,\Theta ), \end{aligned}$$

(1.5)

where

$$\begin{aligned} r=e^{-\frac{p_\alpha -1}{2}y}, \Theta \in \mathbb {S}^{N-1}. \end{aligned}$$

Define

$$\begin{aligned} D=\{(y,\Theta )\in \mathbb {R}\times \mathbb {S}^{N-1}:(e^{-\frac{p_\alpha -1}{2}y},\Theta )\in \Omega \}. \end{aligned}$$

After these changes of variables, problem (1.1) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} L(v)=\sigma _\varepsilon e^{-\frac{2+\alpha }{2}\varepsilon y}|v|^{p_\alpha -1-\varepsilon }v~~&{}\text {in}~D,\\ v=0~~&{}\text {on}~\partial D, \end{array}\right. } \end{aligned}$$

(1.6)

where

$$\begin{aligned} \sigma _\varepsilon =\left( \frac{p_\alpha -1}{2}\right) ^{\frac{2\varepsilon }{p_\alpha -1}} \end{aligned}$$

and

$$\begin{aligned} L(v)=-v^{\prime \prime }+\frac{(2+\alpha )^2}{4}v-\left( \frac{p_\alpha -1}{2}\right) ^2 \Delta _{\mathbb {S}^{N-1}}v. \end{aligned}$$

The energy functional corresponding to problem (1.6) is

$$\begin{aligned} \begin{aligned} I_\varepsilon (v)&=\frac{1}{2}\int _D\left( |v^\prime |^2+\frac{(2+\alpha )^2}{4} |v|^2\right) \mathrm{d}y\mathrm{d}\Theta +\frac{1}{2}\left( \frac{p_\alpha -1}{2}\right) ^2\int _D|\nabla _{\mathbb {S}^{N-1}}v|^2\mathrm{d}y\mathrm{d}\Theta \\&\quad -\,\frac{\sigma _\varepsilon }{p_\alpha +1-\varepsilon }\int _D e^{-\frac{2+\alpha }{2}\varepsilon y}|v|^{p_\alpha +1-\varepsilon }\mathrm{d}y\mathrm{d}\Theta . \end{aligned} \end{aligned}$$

It is easy to see that

$$\begin{aligned} I_\varepsilon (v)=\left( \frac{p_\alpha -1}{2}\right) ^{\frac{p_\alpha +3}{p_\alpha -1}}J_\varepsilon (u), \end{aligned}$$

(1.7)

where

$$\begin{aligned} J_\varepsilon (u)=\frac{1}{2}\int _\Omega |\nabla u|^2dx-\frac{1}{p_\alpha +1-\varepsilon }\int _\Omega |x|^\alpha |u|^{p_\alpha +1-\varepsilon }dx. \end{aligned}$$

We observe that W(y) is the unique solution of the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} W^{\prime \prime }-\frac{(2+\alpha )^2}{4}W+W^{p_\alpha }=0~~\text {in}~\mathbb {R},\\ W^\prime (0)=0,\quad W(y)>0, \\ W(y)\rightarrow 0~~~\text {as}~~y\rightarrow \pm \infty , \end{array}\right. } \end{aligned}$$

(1.8)

where

$$\begin{aligned} W(y)=\gamma _{\alpha ,N} \frac{e^{-\frac{2+\alpha }{2}y}}{\left( 1+e^{-\frac{(2+\alpha )^2}{N-2}y}\right) ^{\frac{N-2}{2+\alpha }}}, ~~\gamma _{\alpha ,N}=\left( \frac{(2+\alpha )^2(N+\alpha )}{N-2}\right) ^{\frac{N-2}{4+2\alpha }}. \end{aligned}$$

We denote the function $\mathrm{PU}_\mu :=U_\mu +R_\mu$, which is the projection onto $H_0^1(\Omega )$ of the function $U_\mu$, that is,

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \mathrm{PU}_\mu =|x|^\alpha U_\mu ^{p_\alpha }~~&{}\text {in}~\Omega ,\\ \mathrm{PU}_\mu =0~~&{}\text {on}~\partial \Omega . \end{array}\right. } \end{aligned}$$

Then, we have

$$\begin{aligned} R_\mu =-C_{\alpha ,N}\mu ^{\frac{N-2}{2}}H(x,0)+O\left( \mu ^{\frac{N+2+2\alpha }{2}}\right) , \end{aligned}$$

where H(x, 0) is the Robin function.

For given $\Lambda _i>0,~i=1,2,\ldots ,k,$ set

$$\begin{aligned} \begin{aligned} \xi _1&= -\frac{1}{2+\alpha }\log \varepsilon +\frac{2}{2+\alpha }\log \Lambda _1, \\ \xi _{i+1}-\xi _i&=-\frac{2}{2+\alpha } \log \varepsilon -\frac{2}{2+\alpha }\log \Lambda _{i+1},~~i=1,2,\ldots ,k-1. \end{aligned} \end{aligned}$$

(1.9)

Let us write

$$\begin{aligned} W_i(y)=W(y-\xi _i),~~~V_i(y)=W_i(y)+\Pi _i(y),~~~V(y)=\sum _{i=1}^k(-1)^iV_i(y), \end{aligned}$$

(1.10)

where

$$\begin{aligned} \Pi _i(y)=\mathcal {T}(R_{\mu _i}),~\mu _i=e^{-\frac{p_\alpha -1}{2}\xi _i}. \end{aligned}$$

We will prove Theorem 1.1 by verifying the following result.

Theorem 1.4

Suppose that $\alpha >-2$ is not an even integer. Then for any integer $k\ge 1$, there exists $\varepsilon _k>0$ such that for $\varepsilon \in (0,\varepsilon _k)$, problem (1.6) has a pair of solutions $v_\varepsilon$ and $-v_\varepsilon$ of the form

$$\begin{aligned} v_\varepsilon =V+\phi _\varepsilon , \end{aligned}$$

where $\Vert \phi _\varepsilon \Vert _{L^\infty }\rightarrow 0$, as $\varepsilon \rightarrow 0$.

Remark 1.5

Using the Emden–Fowler-type transformation (1.5), we can give the explicit expression of solution to problem (1.1), that is,

$$\begin{aligned} u_\varepsilon (x)=C_{\alpha ,N}\sum _{i=1}^k(-1)^i \left( \frac{M_i^{\frac{2+\alpha }{2}}\varepsilon ^{\frac{(2+\alpha )(2i-1)}{2(N-2)}}}{M_i^{2+\alpha }\varepsilon ^{\frac{(2+\alpha )(2i-1)}{N-2}}+|x|^{2+\alpha }}\right) ^{\frac{N-2}{2+\alpha }}(1+o(1)), \end{aligned}$$

where $M_i, i=1,2,\ldots ,k$ are some certain positive constants (see (4.3)) and $o(1)\rightarrow 0$ uniformly on compact subsets of $\Omega$ as $\varepsilon \rightarrow 0$ .

The proof of Theorem 1.4 is motivated by [7, 23]. More precisely, we will use the Lyapunov–Schmidt reduction argument to prove Theorem 1.4, which reduces the construction of the solutions to a finite-dimensional variational problem. As a final remark, we point out that bubble tower concentration phenomena have been observed in [3, 7, 8, 10, 18, 20, 23] near the critical Sobolev exponent, i.e., $\alpha =0$. However, as far as we know, there are no such results for $\alpha \ne 0$.

This paper is organized as follows. In Sect. 2, we give some basic estimates and asymptotic expansion. In Sect. 3, we will carry out the finite-dimensional reduction argument and the main results will be proved in Sect. 4.

2 Energy expansion

In this section, we give some estimates and asymptotic expansion used in the later sections.

Lemma 2.1

For fixed $\delta >0$ and $\delta<\Lambda _i<\delta ^{-1}, i=1,2,\ldots ,k$, we have the following estimates:

$$\begin{aligned}&\displaystyle \int _D |V|^{p_\alpha +1}\mathrm{d}y\mathrm{d}\Theta = k\omega _{N-1}\int _{\mathbb {R}}W^{p_\alpha +1}+o(1), \end{aligned}$$

(2.1)

$$\begin{aligned}&\displaystyle \int _D\left( |V|^{p_\alpha +1}-|V|^{p_\alpha +1-\varepsilon }\right) \mathrm{d}y\mathrm{d}\Theta = k\omega _{N-1}\varepsilon \int _{\mathbb {R}}W^{p_\alpha +1}\log W+o(\varepsilon ), \end{aligned}$$

(2.2)

$$\begin{aligned}&\displaystyle \int _D y|V|^{p_\alpha +1}\mathrm{d}y\mathrm{d}\Theta =\left( \sum _{\ell =1}^k\xi _\ell \right) \omega _{N-1}\int _{\mathbb {R}}W^{p_\alpha +1}+o(1), \end{aligned}$$

(2.3)

$$\begin{aligned}&\displaystyle \int _{D_\ell }W_i^{p_\alpha }W_j\mathrm{d}y\mathrm{d}\Theta =o\left( \varepsilon \right) ,~~i\ne \ell , \end{aligned}$$

(2.4)

$$\begin{aligned}&\displaystyle \int _{D_\ell }W_\ell ^{p_\alpha }W_j\mathrm{d}y\mathrm{d}\Theta =a_3 e^{-\frac{2+\alpha }{2}|\xi _\ell -\xi _j|}+o\left( \varepsilon \right) ,~~j\ne \ell , \end{aligned}$$

(2.5)

$$\begin{aligned}&\displaystyle \int _{D_\ell }\left( |V_\ell |^{p_\alpha +1}-|V|^{p_\alpha +1}+(p_\alpha +1)V_\ell ^{p_\alpha }\sum _{j\ne \ell }(-1)^{\ell +j}V_j\right) \mathrm{d}y\mathrm{d}\Theta =o\left( \varepsilon \right) , \end{aligned}$$

(2.6)

$$\begin{aligned}&\displaystyle \int _{D_\ell }\left( W_\ell ^{p_\alpha }-V_\ell ^{p_\alpha }\right) V_j\mathrm{d}y\mathrm{d}\Theta = o\left( \varepsilon \right) ,~~~j\ne \ell , \end{aligned}$$

(2.7)

where $a_3=\gamma _{\alpha ,N}\omega _{N-1}\int _{\mathbb {R}}e^{-\frac{2+\alpha }{2}y}W^{p_\alpha }$, $D_\ell =\{(y,\Theta )\in D: \eta _\ell \le y<\eta _{\ell +1}\}$, $\eta _1=0, \eta _\ell =\frac{\xi _{\ell -1}+\xi _\ell }{2},~\ell =2,\ldots ,k, \eta _{k+1}=+\infty$.

Proof

The results are similar to Lemma 4.4 in [23], we omit the details. $\square$

Next, we will calculate the asymptotic expansions of the energy functional $I_\varepsilon (V)$.

Proposition 2.2

For any $\delta >0$, there exists $\varepsilon _0>0$ such that for $\varepsilon \in (0,\varepsilon _0)$, we have the following asymptotic expansion

$$\begin{aligned} \begin{aligned} I_\varepsilon (V)=k a_0+ka_1\varepsilon -\frac{k^2}{2}a_4\varepsilon \log \varepsilon +\varepsilon \Psi _k(\Lambda )+\varepsilon R_\varepsilon (\Lambda ), \end{aligned} \end{aligned}$$

(2.8)

where

$$\begin{aligned} \Psi _k(\Lambda )=ka_4\log \Lambda _1+\frac{a_2H(0,0)}{\Lambda _1^2}+\sum _{\ell =2}^k\left( a_3\Lambda _\ell -(k-\ell +1)a_4\log \Lambda _\ell \right) \end{aligned}$$

and $R_\varepsilon \rightarrow 0$, as $\varepsilon \rightarrow 0$ uniformly in $C^1$-norm on the set of $\Lambda _i$’s with $\delta<\Lambda _i<\delta ^{-1}$, $i=1,2,\ldots ,k$. Here $a_i, i=0,1,\ldots ,4$, are given by

$$\begin{aligned} {\left\{ \begin{array}{ll}\displaystyle a_0=\frac{2+\alpha }{2(N+\alpha )}\left( \frac{p_\alpha -1}{2}\right) ^{\frac{p_\alpha +3}{p_\alpha -1}}\int _{\mathbb {R}^N}|x|^\alpha U_1^{p_\alpha +1},\\ \displaystyle a_1=\frac{\omega _{N-1}}{p_\alpha +1}\left( \int _{\mathbb {R}}W^{p_\alpha +1}\log W-\frac{1}{p_\alpha +1}\int _{\mathbb {R}}W^{p_\alpha +1} -\frac{2}{p_\alpha -1}\log \frac{p_\alpha -1}{2}\int _{\mathbb {R}}W^{p_\alpha +1}\right) ,\\ \displaystyle a_2=\frac{2+\alpha }{2(N+\alpha )}\left( \frac{p_\alpha -1}{2}\right) ^{\frac{p_\alpha +3}{p_\alpha -1}}C_{\alpha ,N}\int _{\mathbb {R}^N}|x|^\alpha U_1^{p_\alpha },\\ \displaystyle a_3=\gamma _{\alpha ,N}\omega _{N-1}\int _{\mathbb {R}}e^{-\frac{2+\alpha }{2}y}W^{p_\alpha },\\ \displaystyle a_4=\frac{\omega _{N-1}}{p_\alpha +1}\int _{\mathbb {R}}W^{p_\alpha +1}. \end{array}\right. } \end{aligned}$$

Proof

The proof is standard, and we only give a sketch here.

Note that

$$\begin{aligned} \begin{aligned} I_{\varepsilon }(V)=&I_0(V)+\frac{1}{p_\alpha +1}\int _D|V|^{p_\alpha +1}-\frac{\sigma _\varepsilon }{p_\alpha +1-\varepsilon }\int _De^{-\frac{2+\alpha }{2}\varepsilon y}|V|^{p_\alpha +1-\varepsilon }\\ =&I_0(V)-\frac{1}{p_\alpha +1}\int _D\left( e^{-\frac{2+\alpha }{2}\varepsilon y}-1\right) |V|^{p_\alpha +1}+k a_1\varepsilon +o(\varepsilon ). \end{aligned} \end{aligned}$$

It follows from Lemma 2.1 that

$$\begin{aligned} \begin{aligned} \frac{1}{p_\alpha +1}\int _D\left( e^{-\frac{2+\alpha }{2}\varepsilon y}-1\right) |V|^{p_\alpha +1}&=-\frac{(2+\alpha )\varepsilon }{2(p_\alpha +1)}\int _Dy|V|^{p_\alpha +1}+o(\varepsilon )\\&=-\varepsilon \frac{a_4(2+\alpha )}{2}\sum _{j=1}^k\xi _j+o(\varepsilon ). \end{aligned} \end{aligned}$$

It is easy to check that

$$\begin{aligned} I_0(V)-\sum _{i=1}^kI_0(V_i)= \frac{1}{p_\alpha +1}\int _D\left( \sum _{i=1}^kV_i^{p_\alpha +1}-|V|^{p_\alpha +1}\right) +\sum _{i,j=1,i>j}^k(-1)^{i+j}\int _D W_i^{p_\alpha }V_j. \end{aligned}$$

Since $\Pi _j=O(e^{-\frac{2+\alpha }{2}\xi _j})=O(\varepsilon ^{\frac{3}{2}})$, $j\ge 2$, from Lemma 2.1, we have

$$\begin{aligned} \begin{aligned}&I_0(V)-\sum _{i=1}^kI_0(V_i)\\&\quad =\frac{1}{p_\alpha +1} \sum _{\ell =1}^k\int _{D_\ell }\left( V_\ell ^{p_\alpha +1}-|V|^{p_\alpha +1}+(p_\alpha +1)\sum _{j<\ell }(-1)^{\ell +j}W_\ell ^{p_\alpha }V_j\right) +o(\varepsilon )\\&\quad =-\sum _{\ell =1}^k\sum _{j>\ell }(-1)^{\ell +j}\int _{D_\ell }W_\ell ^{p_\alpha }W_j+o(\varepsilon )\\&\quad =a_3\sum _{\ell =1}^{k-1}e^{-\frac{2+\alpha }{2}|\xi _{\ell +1}-\xi _\ell |}+o(\varepsilon ). \end{aligned} \end{aligned}$$

Next, we estimate $I_0(V_i)$, $i=1,2,\ldots ,k$.

Recall that

$$\begin{aligned} V_i=W_i+\Pi _i, ~~~\Pi _i(y)=\mathcal {T}(R_{\mu _i}),~\mu _i=e^{-\frac{p_\alpha -1}{2}\xi _i} \end{aligned}$$

and

$$\begin{aligned} I_0(V_i)=\left( \frac{p_\alpha -1}{2}\right) ^{\frac{p_\alpha +3}{p_\alpha -1}}J_0(PU_{\mu _i}). \end{aligned}$$

Thus, we find

$$\begin{aligned} J_0(PU_{\mu _i})= & {} \frac{2+\alpha }{2(N+\alpha )}\int _{\mathbb {R}^N}|x|^\alpha U_1^{p_\alpha +1}\\&+\frac{2+\alpha }{2(N+\alpha )}C_{\alpha ,N}H(0,0)\mu _i^{N-2}\int _{\mathbb {R}^N}|x|^\alpha U_1^{p_\alpha }+O(\mu _i^N). \end{aligned}$$

Since $\mu _i=e^{-\frac{p_\alpha -1}{2}\xi _i}$, we find

$$\begin{aligned} \sum _{i=1}^kI_0(V_i)=ka_0+a_2H(0,0)e^{-(2+\alpha )\xi _1}+o(\varepsilon ). \end{aligned}$$

Hence, we can deduce

$$\begin{aligned} \begin{aligned} I_\varepsilon (V)&=ka_0+ka_1\varepsilon +a_2H(0,0)e^{-(2+\alpha )\xi _1} +a_3\sum _{\ell =1}^{k-1}e^{-\frac{2+\alpha }{2}|\xi _{\ell +1}-\xi _\ell |}\\&\quad +\,a_4\varepsilon \frac{2+\alpha }{2}\sum _{\ell =1}^k\xi _\ell +o(\varepsilon ). \end{aligned} \end{aligned}$$

By the definition of $\xi _i, i=1,2,\ldots ,k$, we can obtain (2.8) immediately and the proof of Proposition 2.2 is concluded. $\square$

3 The finite-dimensional reduction

In this section, we perform the finite-dimensional procedure, which reduces problem (1.6) to a finite-dimensional problem on $\mathbb {R}_+$.

For given $\xi _i, ~i=1,2,\ldots ,k$, let

$$\begin{aligned} \Vert \phi \Vert _*=\sup \limits _{(y,\Theta )\in D}\left( \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}\right) ^{-1}|\phi (y,\Theta )|, \end{aligned}$$

where $\sigma >0$ is a small constant. We denote $\mathcal {C}_*$ by the continuous function space defined on D with finite norm defined as above.

Define

$$\begin{aligned} \tilde{Z}_i(x)=\mu _i\frac{\partial U_{\mu _i}}{\partial \mu _i}, \quad \quad \mu _i=e^{-\frac{p_\alpha -1}{2}\xi _i},~i=1,2,\ldots ,k. \end{aligned}$$

Then, $\tilde{Z}_i(x)$ solves

$$\begin{aligned} -\Delta \tilde{Z}_i(x)=p_\alpha U_{\mu _i}^{p_\alpha -1}\tilde{Z}_i(x) ~~\text {in}~\mathbb {R}^N. \end{aligned}$$

Let $P\tilde{Z}_i$ be the projection onto $H_0^1(\Omega )$ of the function $\tilde{Z}_i(x)$, that is,

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta P\tilde{Z}_i=p_\alpha U_{\mu _i}^{p_\alpha -1}\tilde{Z}_i(x) ~~&{}\text {in}~\Omega ,\\ P\tilde{Z}_i=0&{}\text {on} ~\partial \Omega . \end{array}\right. } \end{aligned}$$

Set

$$\begin{aligned} Z_i(y,\Theta )=\mathcal {T}(P\tilde{Z}_i)(y,\Theta ). \end{aligned}$$

Then, $Z_i$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} L(Z_i)=p_\alpha W_i^{p_\alpha -1}W^\prime _i ~~&{}\text {in}~D,\\ Z_i=0&{}\text {on}~\partial D. \end{array}\right. } \end{aligned}$$

First, we consider the following linear problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbb {L}_\varepsilon (\phi )=h+\sum _{j=1}^kc_jZ_j~~~&{}\text {in}~~D,\\ \phi =0&{}\text {on}~~\partial D,\\ \int _D Z_i\phi \mathrm{d}y\mathrm{d}\Theta =0,~~i=1,2,\ldots ,k, \end{array}\right. } \end{aligned}$$

(3.1)

where $c_i$, $i=1,2,\ldots ,k$, are some constants and

$$\begin{aligned} \mathbb {L}_\varepsilon (\phi )=L(\phi ) -(p_\alpha -\varepsilon )\sigma _\varepsilon e^{-\frac{2+\alpha }{2}\varepsilon y}|V|^{p_\alpha -1-\varepsilon }\phi . \end{aligned}$$

Lemma 3.1

Assume that there are sequences $\varepsilon _n\rightarrow 0$ and points $0<\xi _1^n<\xi ^n_2<\cdots <\xi _k^n$ with

$$\begin{aligned} \xi _1^n\rightarrow \infty ,~~\min \limits _{1\le i\le k-1}(\xi _{i+1}^n-\xi _{i}^n)\rightarrow +\infty ,~~ \xi _k^n=o(\varepsilon _n^{-1}), \end{aligned}$$

such that $\phi _n$ solves (3.1) for scalars $c_i^n$ and $h_n$ with $\Vert h_n\Vert _*\rightarrow 0$, then $\lim \limits _{n\rightarrow \infty }\Vert \phi _n\Vert _*=0$.

Proof

We will first show that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\Vert \phi _n\Vert _{L^\infty }=0. \end{aligned}$$

Arguing by contradiction, we may assume that $\Vert \phi _n\Vert _{L^\infty }=1$. Multiplying (3.1) by $Z_\ell ^n$ and integrating by parts, we find

$$\begin{aligned} \sum _{i=1}^kc_i^n\int _D Z_i^nZ_\ell ^n\mathrm{d}y\mathrm{d}\Theta =\int _D\mathbb {L}_{\varepsilon _n}(Z_\ell ^n)\phi _n\mathrm{d}y\mathrm{d}\Theta -\int _D h_nZ_\ell ^n\mathrm{d}y\mathrm{d}\Theta . \end{aligned}$$

Note that

$$\begin{aligned} \int _D Z_i^nZ_\ell ^n\mathrm{d}y\mathrm{d}\Theta =C\delta _{i\ell }+o(1) \end{aligned}$$

where $\delta _{i\ell }$ is the Kronecker’s delta function. This defines an almost diagonal system in the $c_i^n$’s as $n\rightarrow \infty$.

Thus, we have

$$\begin{aligned} \begin{aligned} \sum _{i=1}^kc_i^n\int _D Z_i^nZ_\ell ^n=&\int _D\left[ L(Z_\ell ^n) -(p_\alpha -\varepsilon _n)\sigma _{\varepsilon _n}e^{-\frac{2+\alpha }{2}\varepsilon _n y}|V|^{p_\alpha -1-\varepsilon _n}Z_\ell ^n\right] \phi _n -\int _D h_nZ_\ell ^n. \end{aligned} \end{aligned}$$

(3.2)

But

$$\begin{aligned} L(Z_\ell ^n)=p_\alpha W^{p_\alpha -1}(y-\xi _\ell ^n)W^\prime (y-\xi _\ell ^n), \end{aligned}$$

by the dominated convergence theorem, we know that $\lim \limits _{n\rightarrow \infty }c_i^n=0$. Assume that $(y_n,\Theta _n)\in D$ is such that $|\phi _n(y_n,\Theta _n)|=1$, we claim that there is an $\ell \in \{1,\ldots ,k\}$ and a fixed $R>0$, such that $|\xi ^n_\ell -y_n|\le R$ for n large enough. Otherwise, we can suppose that $|\xi ^n_\ell -y_n|\rightarrow +\infty$ as $n\rightarrow +\infty$ for any $\ell =1,2,\ldots ,k$. Then either $|y_n|\rightarrow +\infty$ or $|y_n|$ is bounded. Assume first that $|y_n|\rightarrow +\infty$.

Define

$$\begin{aligned} \tilde{\phi }_n(y,\Theta )=\phi _n(y+y_n,\Theta ). \end{aligned}$$

By the standard elliptic regularity theory, we may assume that $\tilde{\phi }_n$ converges uniformly over compact sets to a function $\tilde{\phi }$. Set $\tilde{\psi }=\mathcal {T}^{-1}(\tilde{\phi })$, then we have

$$\begin{aligned} \Delta \tilde{\psi }=0~~~\text {in}~~\mathbb {R}^N{\setminus }\{0\}. \end{aligned}$$

Due to $\Vert \tilde{\phi }_n\Vert _{L^\infty }=1$, we see that $|\tilde{\psi }(x)|\le |x|^{-\frac{N-2}{2}}$. Hence, $\tilde{\psi }$ can extend smoothly to 0 to be a harmonic function in $\mathbb {R}^N$ with this decay condition. So, $\tilde{\phi }=0$ gives a contradiction. The fact that $|y_n|$ cannot be bounded can be handled in similar way. Thus, there exists an integer $\ell \in \{1,\ldots ,k\}$ and a positive number $R>0$ such that for n large enough, $|y_n-\xi _\ell ^n|\le R.$

Define again

$$\begin{aligned} \tilde{\phi }_n(y,\Theta )=\phi _n(y+\xi _\ell ^n,\Theta ). \end{aligned}$$

Thus, $\tilde{\phi }_n$ converges uniformly over compact sets to a function $\tilde{\phi }$. Set again that $\tilde{\psi }=\mathcal {T}^{-1}(\tilde{\phi })$. Hence, $\tilde{\psi }$ is a nontrivial solution of

$$\begin{aligned} \Delta \tilde{\psi }+p_\alpha |x|^\alpha U_1^{p_\alpha -1}\tilde{\psi }=0~~\text {in}~~\mathbb {R}^N{\setminus }\{0\}. \end{aligned}$$

Moreover, $|\tilde{\psi }(x)|\le C|x|^{-\frac{N-2}{2}}$. Therefore, we obtain a classical solution in $\mathbb {R}^N{\setminus }\{0\}$ decaying at infinity. It follows from [12] that it equals a linear combination of the $\{\tilde{Z}_i\}$ provided that $\alpha$ is not an even integer. However, the orthogonality conditions imply $\tilde{\phi }=0$. This is again a contradiction. Thus, we can deduce that $\lim \limits _{n\rightarrow \infty }\Vert \phi _n\Vert _{L^\infty }=0.$

Next we shall establish that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\Vert \phi _n\Vert _*\rightarrow 0. \end{aligned}$$

Now we see that (3.1) possesses the following form

$$\begin{aligned} -\phi _n^{\prime \prime }+\frac{(2+\alpha )^2}{4}\phi _n-\left( \frac{p_\alpha -1}{2}\right) ^2 \Delta _{\mathbb {S}^{N-1}}\phi _n=g_n, \end{aligned}$$

(3.3)

where

$$\begin{aligned} g_n=h_n+(p_\alpha -\varepsilon _n)\sigma _\varepsilon e^{-\frac{2+\alpha }{2}\varepsilon _n y}|V|^{p_\alpha -1-\varepsilon _n}\phi _n+\sum _{i=1}^n c_i^nZ_i^n. \end{aligned}$$

If $0<\sigma <\min \{p_\alpha -1,1\}$, we find

$$\begin{aligned} |g_n(y)|\le \theta _n\sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i^n|}~~\text {with}~~\theta _n\rightarrow 0. \end{aligned}$$

Choosing $C>0$ large enough, we see that

$$\begin{aligned} \varphi _n(y)=C\theta _n\sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i^n|} \end{aligned}$$

is a supersolution of (3.3), and $-\varphi _n(y)$ will be a subsolution of (3.3). Thus,

$$\begin{aligned} |\phi _n|\le C\theta _n\sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i^n|}. \end{aligned}$$

The following proposition is a direct consequence of Proposition 1 in [9] combining with Lemma 3.1.

Proposition 3.2

There exist positive numbers $\varepsilon _0, \delta _0, R_0$, such that if

$$\begin{aligned} R_0<\xi _1,~~R_0<\min \limits _{i=1,\ldots ,k-1}(\xi _{i+1}-\xi _{i}),~~\xi _k<\frac{\delta _0}{\varepsilon }, \end{aligned}$$

(3.4)

then for all $0<\varepsilon <\varepsilon _0$ and $h\in \mathcal {C}_*$, problem (3.1) has a unique solution $\phi =T_\varepsilon (h)$. Moreover, there exists $C>0$ such that

$$\begin{aligned} \Vert T_\varepsilon (h)\Vert _*\le C\Vert h\Vert _*,~~~|c_i|\le C\Vert h\Vert _*. \end{aligned}$$

For later purposes, we need to understand the differentiability of the operator $T_\varepsilon$ on the variables $\xi _i$. We will use the notation $\xi =(\xi _1,\xi _2,\ldots ,\xi _k)$. We also consider the space $L(\mathcal {C}_*)$ of the linear operator of $\mathcal {C}_*$. We have the following result.

Proposition 3.3

Under the same assumptions of Proposition 3.2, the map $\xi \rightarrow T_\varepsilon$ with values in $L(\mathcal {C}_*)$ is of class $C^1$. Besides, there is a constant $C>0$ such that

$$\begin{aligned} \Vert D_\xi T_\varepsilon \Vert _{L({\mathcal {C}_*})}\le C \end{aligned}$$

uniformly on the vectors $\xi$ satisfying (3.4).

Proof

Fix $h\in \mathcal {C}_*$, and let $\phi =T_\varepsilon (h)$. We are interested in studying the differentiability of $\phi$ with respect to $\xi _\ell$ for $\ell =1,2,\ldots ,k$. Recall that $\phi$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbb {L}_\varepsilon (\phi )=h+\sum _{j=1}^kc_jZ_j~~~&{}\text {in}~~D,\\ \phi =0&{}\text {on}~~\partial D,\\ \int _D Z_i\phi \mathrm{d}y\mathrm{d}\Theta =0,~~i=1,2,\ldots ,k, \end{array}\right. } \end{aligned}$$

for certain constants $c_i$. Differentiating the above equation with respect to $\xi _\ell ,~\ell =1,\ldots ,k.$ Define $Y=\partial _{\xi _\ell }\phi$ and $d_i=\partial _{\xi _\ell }c_i$, we find

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbb {L}_\varepsilon (Y)=(p_\alpha -\varepsilon )\sigma _\varepsilon e^{-\frac{2+\alpha }{2}\varepsilon y}(\partial _{\xi _\ell }|V|^{p_\alpha -1-\varepsilon })\phi +c_\ell \partial _{\xi _\ell }Z_\ell +\sum _{j=1}^kd_jZ_j~~~&{}\text {in}~~D,\\ Y=0&{}\text {on}~~\partial D,\\ \int _D (YZ_i+\phi \partial _{\xi _\ell }Z_i)\mathrm{d}y\mathrm{d}\Theta =0,~~i=1,2,\ldots ,k. \end{array}\right. } \end{aligned}$$

Set $\chi =Y-\sum _{i=1}^kb_iZ_i$, where the constants $b_i$ satisfy

$$\begin{aligned} \begin{aligned}&\sum _{i=1}^k b_i\int _D Z_iZ_j\mathrm{d}y\mathrm{d}\Theta =0,~~~j\ne \ell ,\\&\sum _{i=1}^k b_i\int _D Z_iZ_\ell \mathrm{d}y\mathrm{d}\Theta =-\int _D\phi \partial _{\xi _\ell }Z_\ell \mathrm{d}y\mathrm{d}\Theta . \end{aligned} \end{aligned}$$

This is also an almost diagonal system and $Y=\chi +\sum _{j=1}^kb_jZ_j,$ where$\int _D\chi Z_j \mathrm{d}y\mathrm{d}\Theta =0,~j=1,2,\ldots ,k.$ Moreover, it is easy to see that $\chi$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbb {L}_\varepsilon (\chi )=g+\sum _{j=1}^kd_jZ_j~~~&{}\text {in}~~D,\\ \chi =0&{}\text {on}~~\partial D,\\ \int _D\chi Z_j \mathrm{d}y\mathrm{d}\Theta =0,~j=1,2,\ldots ,k, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} g=(p_\alpha -\varepsilon )\sigma _\varepsilon e^{-\frac{2+\alpha }{2}\varepsilon y}(\partial _{\xi _\ell }|V|^{p_\alpha -1-\varepsilon })\phi +c_\ell \partial _{\xi _\ell }Z_\ell -\sum _{j=1}^k b_j\mathbb {L}_\varepsilon (Z_j). \end{aligned}$$

Then, we find

$$\begin{aligned} \chi =T_\varepsilon (g) \end{aligned}$$

and

$$\begin{aligned} \partial _{\xi _\ell }\phi =T_\varepsilon (g)+\sum _{j=1}^k b_jZ_j. \end{aligned}$$

By Proposition 3.2, we find

$$\begin{aligned} \Vert T_\varepsilon (g)\Vert _*\le C\Vert g\Vert _*. \end{aligned}$$

Since

$$\begin{aligned} \Vert g\Vert _*\le C\left( \Vert \phi \Vert _*+|c_\ell |+\sum _{j=1}^k|b_j|\right) \end{aligned}$$

and

$$\begin{aligned} |b_i|\le C\Vert \phi \Vert _*,~~|c_i|\le C\Vert h\Vert _*,~~\Vert \phi \Vert _*\le C\Vert h\Vert _*. \end{aligned}$$

Thus, we can obtain that $\Vert \partial _{\xi _\ell }\phi \Vert _*\le C\Vert h\Vert _*$, and $\partial _{\xi _\ell }\phi$ depends continuously on $\xi$ for this norm. $\square$

Now we consider

$$\begin{aligned} {\left\{ \begin{array}{ll} L(V+\phi )-\sigma _\varepsilon e^{-\frac{2+\alpha }{2}\varepsilon y}|V+\phi |^{p_\alpha -1-\varepsilon }(V+\phi ) =\sum \limits _{j=1}^kc_jZ_j~~~\text {in}~D,\\ \phi =0~~~\text {on}~\partial D,\\ \int _D Z_i\phi \mathrm{d}y\mathrm{d}\Theta =0,\quad i=1,2,\ldots ,k. \end{array}\right. } \end{aligned}$$

(3.5)

In order to solve problem (3.5), we rewrite it as

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbb {L}_\varepsilon (\phi )=N_\varepsilon (\phi )+R_\varepsilon +\sum \limits _{j=1}^kc_jZ_j~~~\text {in}~~D,\\ \phi =0\quad \text {on}~~\partial D,\\ \int _D Z_i\phi \mathrm{d}y\mathrm{d}\Theta =0,\quad i=1,2,\ldots ,k, \end{array}\right. } \end{aligned}$$

(3.6)

where

$$\begin{aligned} N_\varepsilon (\phi )=\sigma _\varepsilon e^{-\frac{2+\alpha }{2}\varepsilon y} \left( |V+\phi |^{p_\alpha -1-\varepsilon }(V+\phi )-|V|^{p_\alpha -1-\varepsilon }V-(p_\alpha -\varepsilon )|V|^{p_\alpha -1-\varepsilon }\phi \right) \end{aligned}$$

and

$$\begin{aligned} R_\varepsilon= & {} \sigma _\varepsilon e^{-\frac{2+\alpha }{2}\varepsilon y}|V|^{p_\alpha -1-\varepsilon }V-\sum _{i=1}^k(-1)^iW_i^{p_\alpha }. \end{aligned}$$

Let us fix a large number $M>0$, $\xi$ satisfies the following conditions

$$\begin{aligned} \xi _1>\frac{1}{2}\log \frac{1}{M\varepsilon },~~ \min \limits _{1\le i\le k-1}(\xi _{i+1}-\xi _i)>\log \frac{1}{M\varepsilon },~~ \xi _k<k\log \frac{1}{M\varepsilon }. \end{aligned}$$

(3.7)

In order to prove that (3.6) is uniquely solvable in the set that $\Vert \phi \Vert _*$ is small, we need to estimate $R_\varepsilon$ and $N_\varepsilon (\phi )$.

Lemma 3.4

If $N\ge 3$, then

$$\begin{aligned} \begin{aligned}&\Vert N_\varepsilon (\phi )\Vert _*\le C\Vert \phi \Vert _*^{\min \{p_\alpha -\varepsilon ,2\}},\\&\left\| \frac{\partial N_\varepsilon (\phi )}{\partial \phi }\right\| _*\le C\Vert \phi \Vert _*^{\min \{p_\alpha -1-\varepsilon ,1\}}. \end{aligned} \end{aligned}$$

(3.8)

Proof

Since

$$\begin{aligned} |N_\varepsilon (\phi )|\le {\left\{ \begin{array}{ll} C|\phi |^{p_\alpha -\varepsilon },~~&{}p_\alpha -1\le 1,\\ C|V|^{p_\alpha -2-\varepsilon }\phi ^2+C|\phi |^{p_\alpha -\varepsilon },~&{}p_\alpha -1>1. \end{array}\right. } \end{aligned}$$

First, we consider the case $p_\alpha -1\le 1$.

$$\begin{aligned} \begin{aligned} |N_\varepsilon (\phi )|&\le C\Vert \phi \Vert _*^{p_\alpha -\varepsilon }\left( \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}\right) ^{p_\alpha -\varepsilon }\\&\le C\Vert \phi \Vert _*^{p_\alpha -\varepsilon }\left( \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}\right) . \end{aligned} \end{aligned}$$

where we have used the fact that

$$\begin{aligned} \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}\le C. \end{aligned}$$

Thus, the result follows.

Now we show the result holds for $p_\alpha -1>1$.

$$\begin{aligned} \begin{aligned} |N_\varepsilon (\phi )|\le&C\Vert \phi \Vert _*^2|V|^{p_\alpha -2-\varepsilon }\left( \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}\right) ^2 +C\Vert \phi \Vert _*^{p_\alpha -\varepsilon }\left( \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}\right) ^{p_\alpha -\varepsilon }\\ \le&C\left( \Vert \phi \Vert _*^{p_\alpha -\varepsilon }+\Vert \phi \Vert _*^2\right) \left( \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}\right) . \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} \Vert N_\varepsilon (\phi )\Vert _*\le C\Vert \phi \Vert _*^{\min \{{p_\alpha -\varepsilon ,2}\}}. \end{aligned}$$

The other terms can be estimated similarly, and the proof of the lemma is completed. $\square$

Lemma 3.5

If $N\ge 3$, then

$$\begin{aligned} \Vert R_\varepsilon \Vert _*\le C\varepsilon ^{\frac{1+\tau }{2}},~~\Vert \partial _\xi R_\varepsilon \Vert _*\le C\varepsilon ^{\frac{1+\tau }{2}}, \end{aligned}$$

(3.9)

where $\tau >0$ is a small constant.

Proof

We give here the proof of the first one only. The second one can be obtained similarly.

Note that

$$\begin{aligned} R_\varepsilon= & {} (\sigma _\varepsilon -1)e^{-\frac{2+\alpha }{2}\varepsilon y}|V|^{p_\alpha -1-\varepsilon }V+e^{-\frac{2+\alpha }{2}\varepsilon y}\left( |V|^{p_\alpha -1-\varepsilon }V-|V|^{p_\alpha -1}V\right) \\&+\,|V|^{p_\alpha -1}V\left( e^{-\frac{2+\alpha }{2}\varepsilon y}-1\right) +|V|^{p_\alpha -1}V-\sum _{i=1}^k(-1)^iV_i^{p_\alpha }\\&+\sum _{i=1}^k(-1)^iV_i^{p_\alpha }-\sum _{i=1}^k(-1)^iW_i^{p_\alpha }\\= & {} :J_1+J_2+J_3+J_4+J_5. \end{aligned}$$

Recalling that

$$\begin{aligned} V=\sum _{i=1}^k(-1)^iV_i,~~0\le V_i\le W_i. \end{aligned}$$

Thus, we find

$$\begin{aligned} \begin{aligned}&|J_1|\le C\varepsilon e^{-\frac{2+\alpha }{2}\varepsilon y}|V|^{p_\alpha -\varepsilon }\le C\varepsilon \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}, \\&|J_2|\le C\varepsilon |\log V||V|^{p_\alpha -1}\le C\varepsilon \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|},\\&|J_3|=\left| \left( e^{-\frac{2+\alpha }{2}\varepsilon y}-1\right) |V|^{p_\alpha -1}V\right| \le C\varepsilon y|V|^{p_\alpha }\le C\varepsilon \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}. \end{aligned} \end{aligned}$$

Next we estimate $J_4$ and $J_5$.

Define

$$\begin{aligned} \chi _\ell =\frac{\xi _{\ell -1}+\xi _\ell }{2},\,\,\,\,\ell =1,2,\ldots ,k+1,~\text {where}~~\xi _0=\inf \limits _{(y,\Theta )\in D}|y|,~~\xi _{k+1}=+\infty . \end{aligned}$$

Thus, for $\chi _\ell \le y<\chi _{\ell +1}$, we have

$$\begin{aligned} \begin{aligned} |J_4|&=\Big ||V|^{p_\alpha -1}V-\sum _{i=1}^k(-1)^iV_i^{p_\alpha }\Big |\le CV_\ell ^{p_\alpha -1}\left( \sum _{j\ne \ell }V_j\right) \\&\le \,C\sum _{j\ne \ell }e^{-\frac{2+\alpha }{2}(p_\alpha -1)|y-\xi _\ell |}e^{-\frac{2+\alpha }{2}|y-\xi _j|}\\&\le \,Ce^{-\frac{2+\alpha }{2}\sigma |y-\xi _\ell |}\sum _{j\ne \ell }e^{-\frac{2+\alpha }{2}(p_\alpha -\sigma -1)|y-\xi _\ell |}e^{-\frac{2+\alpha }{2}|y-\xi _j|}\\&\le \,Ce^{-\frac{2+\alpha }{2}\sigma |y-\xi _\ell |}\sum _{j\ne \ell }e^{-\frac{(2+\alpha )(1+\tau )}{4}|\xi _\ell -\xi _{\ell -1}|}\\&\le \,C\varepsilon ^{\frac{1+\tau }{2}}\sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} |J_5|&=\left| \sum _{i=1}^k\left( V_i^{p_\alpha }-W_i^{p_\alpha }\right) \right| \le C\sum _{i=1}^kW_i^{p_\alpha -1}|\Pi _i|\\&\le \,CR_{\mu _1}\left( e^{-\frac{p_\alpha -1}{2}y},\Theta \right) \sum _{i=1}^ke^{-\frac{2+\alpha }{2}(p_\alpha -1)|y-\xi _i|}e^{-\frac{2+\alpha }{2}y}, ~\mu _1=e^{-\frac{p_\alpha -1}{2}\xi _1}\\&\le \,C\varepsilon ^{\frac{1+\tau }{2}}\sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}. \end{aligned} \end{aligned}$$

Therefore, $\Vert R_\varepsilon \Vert _*\le C\varepsilon ^{\frac{1+\tau }{2}}$ and the results follow. $\square$

The next proposition enables us to reduce the problem of finding a solution for (1.6) to a finite-dimensional problem.

Proposition 3.6

Suppose that condition (3.7) holds. Then there exists a positive constant C such that, for $\varepsilon >0$ small enough, problem (3.6) admits a unique solution $\phi =\phi (\xi )$, which satisfies

$$\begin{aligned} \Vert \phi \Vert _*\le C\varepsilon ^{\frac{1+\tau }{2}}. \end{aligned}$$

Moreover, $\phi (\xi )$ is of class $C^1$ on $\xi$ with the $\Vert \cdot \Vert _*$-norm, and

$$\begin{aligned} \Vert D_\xi \phi \Vert _*\le C\varepsilon ^{\frac{1+\tau }{2}}, \end{aligned}$$

where $\tau >0$ is a small constant.

Proof

Define

$$\begin{aligned} A_\varepsilon (\phi ):=T_\varepsilon (N_\varepsilon (\phi )+R_\varepsilon ), \end{aligned}$$

then we know that problem (3.6) is equivalent to the fixed point problem $\phi =A_\varepsilon (\phi )$. We will use the contraction mapping theorem to solve it.

Set

$$\begin{aligned} E_\rho =\{\phi \in \mathcal {C}_*:\Vert \phi \Vert _*\le \rho \varepsilon ^{\frac{1+\tau }{2}}\}, \end{aligned}$$

where $\rho >0$ will be fixed later.

We will show that $A_\varepsilon$ is a contraction map from $E_\rho$ to $E_\rho$.

In fact, for $\varepsilon >0$ small enough, we find

$$\begin{aligned} \begin{aligned} \Vert A_\varepsilon (\phi )\Vert _*\le C\Vert N_\varepsilon (\phi )+R_\varepsilon \Vert _* \le C\left( (\rho \varepsilon )^{\min \{p_\alpha -\varepsilon ,2\}}+\varepsilon ^{\frac{1+\tau }{2}}\right) \le \rho \varepsilon ^{\frac{1+\tau }{2}}, \end{aligned} \end{aligned}$$

provided $\rho$ is chosen large enough, but independent of $\varepsilon$.

Thus, $A_\varepsilon$ maps $E_\rho$ into itself. Moreover,

$$\begin{aligned} \begin{aligned} |N_\varepsilon (\phi _1)-N_\varepsilon (\phi _2)|\le&|\partial _\phi N_\varepsilon (t\phi _1+(1-t)\phi _2)||\phi _1-\phi _2|\\ \le&C\left( \rho \varepsilon ^{\frac{1+\tau }{2}}\right) ^{\min \{p_\alpha -1-\varepsilon ,1\}}|\phi _1-\phi _2|. \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} \begin{aligned} \Vert A_\varepsilon (\phi _1)-A_\varepsilon (\phi _2)\Vert _*\le&C\left( \rho \varepsilon ^{\frac{1+\tau }{2}}\right) ^{\min \{p_\alpha -1-\varepsilon ,1\}}\Vert \phi _1-\phi _2\Vert _*\\ \le&\frac{1}{2}\Vert \phi _1-\phi _2\Vert _*. \end{aligned} \end{aligned}$$

Thus, there is a unique $\phi \in E_\rho$, such that $\phi =A_\varepsilon (\phi )$.

Now we consider the differentiability of $\xi \rightarrow \phi (\xi )$.

Let

$$\begin{aligned} B(\xi ,\phi )=\phi -T_\varepsilon (N_\varepsilon (\phi )+R_\varepsilon ). \end{aligned}$$

First, we have $B(\xi ,\phi (\xi ))=0$. Let us write

$$\begin{aligned} D_\phi B(\xi ,\phi )[\psi ]=\psi -T_\varepsilon (\psi D_\phi N_\varepsilon (\phi ))=\psi +M(\psi ), \end{aligned}$$

where

$$\begin{aligned} M(\psi )=-T_\varepsilon (\psi D_\phi N_\varepsilon (\phi )). \end{aligned}$$

From (3.8), we find

$$\begin{aligned} \Vert M(\psi )\Vert _*\le C\varepsilon ^{\frac{1+\tau }{2}\min \{p_\alpha -1-\varepsilon ,1\}}\Vert \psi \Vert _*. \end{aligned}$$

Thus, the linear operator $D_\phi B(\varepsilon ,\phi )$ is invertible in $\mathcal {C}_*$ with uniformly bounded inverse depending continuously on its parameters. Differentiating with respect to $\xi$, we deduce

$$\begin{aligned} D_\xi B(\xi ,\phi )=-D_\xi T_\varepsilon [N_\varepsilon (\phi )+R_\varepsilon ]-T_\varepsilon [D_\xi N_\varepsilon (\xi ,\phi )+D_\xi R_\varepsilon ], \end{aligned}$$

where all these expressions depend continuously on their parameters.

By the implicit function theorem, we see that $\phi (\xi )$ is of class $C^1$ and

$$\begin{aligned} D_\xi \phi =-\left( D_\phi B(\xi ,\phi )\right) ^{-1}[D_\xi B(\xi ,\phi )]. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert D_\xi (\phi )\Vert _*\le C\left( \Vert N_\varepsilon (\phi )+R_\varepsilon \Vert _*+\Vert D_\xi N_\varepsilon (\xi ,\phi )\Vert _*+\Vert D_\xi R_\varepsilon \Vert _*\right) \le C\varepsilon ^{\frac{1+\tau }{2}}. \end{aligned}$$

The proof of Proposition 3.6 is concluded. $\square$

4 Proof of the main result

In this section, we will prove Theorem 1.1. As deduced in the introduction, we need to verify Theorem 1.4. To do this, we will choose $\xi$ such that $V+\phi$ is a solution of (1.6), where $\phi$ is the map obtained in Proposition 3.6.

Recall that

$$\begin{aligned} \begin{aligned} I_\varepsilon (v)=\,&\frac{1}{2}\int _D\left( |v^\prime |^2+\frac{(2+\alpha )^2}{4} |v|^2\right) \mathrm{d}y\mathrm{d}\Theta +\frac{1}{2}\left( \frac{p_\alpha -1}{2}\right) ^2\int _D|\nabla _{\mathbb {S}^{N-1}}v|^2\mathrm{d}y\mathrm{d}\Theta \\&-\frac{\sigma _\varepsilon }{p_\alpha +1-\varepsilon }\int _D e^{-\frac{2+\alpha }{2}\varepsilon y}|v|^{p_\alpha +1-\varepsilon }\mathrm{d}y\mathrm{d}\Theta . \end{aligned} \end{aligned}$$

(4.1)

Define

$$\begin{aligned} K_\varepsilon (\xi )=I_\varepsilon (V+\phi ). \end{aligned}$$

It is now well known that if $\xi$ is a critical point of $K_\varepsilon (\xi )$, then $V+\phi$ is a solution of (1.6). Next, we will prove that $K_\varepsilon (\xi )$ has a critical point. To this end, we need the next lemma, which is important in finding the critical point of $K_\varepsilon$.

Lemma 4.1

The following expansion holds

$$\begin{aligned} K_\varepsilon (\xi )=I_\varepsilon (V)+O(\varepsilon ^{1+\tau }), \end{aligned}$$

(4.2)

where $O(\varepsilon ^{1+\tau })$ is uniformly in the $C^1$-sense on the vectors $\xi$ satisfying (3.4).

Proof

Using the Taylor expansion

$$\begin{aligned} F(u+v)=F(u)+dF(u)[v]+\int _0^1(1-t)d^2F(u+tv)[v,v]dt \end{aligned}$$

and the fact that $\nabla I_\varepsilon (V+\phi )[\phi ]=0$, we have

$$\begin{aligned} \begin{aligned}&I_\varepsilon (V+\phi )-I_\varepsilon (V)=\int _0^1\nabla ^2I_\varepsilon (V+t\phi )[\phi ,\phi ]tdt\\&\quad =\int _0^1\left( \int _D(N_\varepsilon (\phi )+R_\varepsilon )\phi +(p_\alpha -\varepsilon )\sigma _\varepsilon \int _D e^{-\frac{2+\alpha }{2}\varepsilon y}\right. \nonumber \\&\qquad \left. \times \left( |V|^{p_\alpha -1-\varepsilon }-|V+t\phi |^{p_\alpha -1-\varepsilon }\right) \phi ^2\right) tdt. \end{aligned} \end{aligned}$$

Since $\Vert \phi \Vert _*\le C\varepsilon ^{\frac{1+\tau }{2}}$, we find that

$$\begin{aligned} \begin{aligned} \int _D|(N_\varepsilon (\phi )+R_\varepsilon )\phi |\le&C(\Vert N_\varepsilon (\phi )\Vert _*+\Vert R_\varepsilon \Vert _*)\Vert \phi \Vert _* \int _D\left( \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}\right) ^2\\ \le&C(\Vert N_\varepsilon (\phi )\Vert _*+\Vert R_\varepsilon \Vert _*)\Vert \phi \Vert _*={O(\varepsilon ^{{1+\tau }})}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\int _D\left| |V|^{p_\alpha -1-\varepsilon }-|V+t\phi |^{p_\alpha -1-\varepsilon }\right| \phi ^2\\&\quad \le C\Vert \phi \Vert _*^2\int _D \left( \sum _{i=1}^ke^{-\frac{2+\alpha }{2}\sigma |y-\xi _i|}\right) ^2\\&\quad \le C\Vert \phi \Vert _*^2. \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} I_\varepsilon (V+\phi )=I_\varepsilon (V)+O(\varepsilon ^{1+\tau }). \end{aligned}$$

Differentiating with respect to $\xi _\ell$, we see that

$$\begin{aligned} \begin{aligned}&\partial _{\xi _\ell }\left( I_\varepsilon (V+\phi )-I_\varepsilon (V)\right) \\&\quad =\int _0^1\int _D\partial _{\xi _\ell }\left[ (N_\varepsilon (\phi )+R_\varepsilon )\phi \right] tdt\\&\qquad +(p_\alpha -\varepsilon )\sigma _\varepsilon \int _0^1\int _D e^{-\frac{2+\alpha }{2}\varepsilon y}\partial _{\xi _\ell } \left[ \left( |V|^{p_\alpha -1-\varepsilon }-|V+t\phi |^{p_\alpha -1-\varepsilon }\right) \phi ^2\right] tdt. \end{aligned} \end{aligned}$$

In a similar way, we have that

$$\begin{aligned} \partial _{\xi _\ell }I_\varepsilon (V+\phi )=\partial _{\xi _\ell }I_\varepsilon (V)+O(\varepsilon ^{1+\tau }). \end{aligned}$$

Thus, the result follows. $\square$

Proof of Theorem 1.4

Recalling that

$$\begin{aligned} \begin{aligned} \xi _1 =&-\frac{1}{2+\alpha }\log \varepsilon +\frac{2}{2+\alpha }\log \Lambda _1, \\ \xi _{i+1}-\xi _i=&-\frac{2}{2+\alpha } \log \varepsilon -\frac{2}{2+\alpha }\log \Lambda _{i+1},~~i=1,2,\ldots ,k-1, \end{aligned} \end{aligned}$$

where $\delta<\Lambda _i<\frac{1}{\delta }$, $\delta >0$ is a fixed constant. To simplify the notation, we denote $\Lambda =(\Lambda _1,\Lambda _2,\ldots ,\Lambda _k)$. Thus, it is sufficient to find a critical point of the function

$$\begin{aligned} \widetilde{K}_\varepsilon (\Lambda )=\varepsilon ^{-1}\left( K_\varepsilon (\xi (\Lambda ))-ka_0\right) . \end{aligned}$$

From Lemma 4.1 and Proposition 2.2, we have

$$\begin{aligned} \widetilde{K}_\varepsilon (\Lambda )=\Psi _k(\Lambda )+ka_1-\frac{k^2}{2}a_4\log \varepsilon +o(1), \end{aligned}$$

where the term o(1) goes to 0 uniformly as $\varepsilon \rightarrow 0$.

It is easy to see that the function

$$\begin{aligned} \Lambda _1\rightarrow ka_4\log \Lambda _1+\frac{a_2H(0,0)}{\Lambda _1^2} \end{aligned}$$

has a stable minimum point $\Lambda _1^*=\left( \frac{2a_2H(0,0)}{ka_4}\right) ^{\frac{1}{2}}$ on $(0,+\infty )$, and for $i=2,\ldots ,k$, the function

$$\begin{aligned} \Lambda _i\rightarrow a_3\Lambda _i-(k-i+1)a_4\log \Lambda _i \end{aligned}$$

also has a stable minimum point $\Lambda _i^*=\frac{(k-i+1)a_4}{a_3}$ on $(0,+\infty )$. Thus, the function $\Psi _k(\Lambda )$ has a stable minimum point $\Lambda ^*=(\Lambda _1^*,\ldots ,\Lambda _k^*)$. Therefore, for $\varepsilon$ small enough, there exists a critical point $\Lambda ^\varepsilon =(\Lambda _1^\varepsilon ,\ldots ,\Lambda _k^\varepsilon )$ of the function $\widetilde{K}_\varepsilon (\Lambda )$, such that $\Lambda _i^\varepsilon \rightarrow \Lambda _i^*$ as $\varepsilon \rightarrow 0$ for $i=1,2,\ldots ,k$.

For the $\Lambda _i^\varepsilon ~~(i=1,\ldots ,k)$ obtained above, let

$$\begin{aligned} \xi _1^\varepsilon =\frac{2}{2+\alpha }\log \frac{\Lambda _1^\varepsilon }{\varepsilon ^{\frac{1}{2}}}, ~~\xi _i^\varepsilon =\frac{2}{2+\alpha }\log \frac{\Lambda _1^\varepsilon }{\Lambda _2^\varepsilon \ldots \Lambda _i^\varepsilon \varepsilon ^{\frac{2i-1}{2}}}, ~~i=2,3,\ldots ,k. \end{aligned}$$

Hence, $\xi ^\varepsilon =(\xi _1^\varepsilon ,\ldots ,\xi _k^\varepsilon )$ is a critical point of $K_\varepsilon (\xi )$ and $V+\phi (\xi ^\varepsilon )$ is a solution of (1.6). $\square$

Proof of Theorem 1.1

Note that $\Lambda _i^\varepsilon =\Lambda _i^*+o(1), i=1,2,\ldots ,k$ as $\varepsilon \rightarrow 0$. Then

$$\begin{aligned} \begin{aligned} \xi _1^\varepsilon&=\frac{2}{2+\alpha }\log \frac{\Lambda _1^*}{\varepsilon ^{\frac{1}{2}}}+o(1),\\ \xi _i^\varepsilon&=\frac{2}{2+\alpha }\log \frac{\Lambda _1^*}{\Lambda _2^*\ldots \Lambda _i^*\varepsilon ^{\frac{2i-1}{2}}}+o(1), ~~i=2,3,\ldots ,k. \end{aligned} \end{aligned}$$

Using the fact that $e^{-\frac{p_\alpha -1}{2}\xi ^\varepsilon _i}=M_i\varepsilon ^{\frac{2i-1}{N-2}}(1+o(1))$, $i=1,\ldots ,k$, where

$$\begin{aligned} \begin{aligned} M_1=\left( \frac{1}{\Lambda _1^*}\right) ^{\frac{2}{N-2}},\,\,\, M_i=\left( \frac{\Lambda _2^*\ldots \Lambda _i^*}{\Lambda _1^*}\right) ^{\frac{2}{N-2}},~i=2,\ldots ,k. \end{aligned} \end{aligned}$$

(4.3)

Thus, by the transformation (1.5), we find

$$\begin{aligned} u_\varepsilon (x)= C_{\alpha ,N}\sum _{i=1}^k(-1)^i \left( \frac{M_i^{\frac{2+\alpha }{2}}\varepsilon ^{\frac{(2+\alpha )(2i-1)}{2(N-2)}}}{M_i^{2+\alpha }\varepsilon ^{\frac{(2+\alpha )(2i-1)}{N-2}}+|x|^{2+\alpha }}\right) ^{\frac{N-2}{2+\alpha }}(1+o(1)), \end{aligned}$$

where $o(1)\rightarrow 0$ uniformly on compact subsets of $\Omega$ as $\varepsilon \rightarrow 0$.

Let

$$\begin{aligned} \begin{aligned} \hat{u}_\varepsilon (x)&=\sum _{i=1}^k(-1)^i \left( \frac{M_i^{\frac{2+\alpha }{2}}\varepsilon ^{\frac{(2+\alpha )(2i-1)}{2(N-2)}}}{M_i^{2+\alpha }\varepsilon ^{\frac{(2+\alpha )(2i-1)}{N-2}}+|x|^{2+\alpha }}\right) ^{\frac{N-2}{2+\alpha }}\\&=\sum _{i=1}^k(-1)^i\left( \frac{1}{M_i^{\frac{2+\alpha }{2}}\varepsilon ^{\frac{(2+\alpha )(2i-1)}{2(N-2)}} +M_i^{-\frac{2+\alpha }{2}}\varepsilon ^{-\frac{(2+\alpha )(2i-1)}{2(N-2)}}|x|^{2+\alpha }}\right) ^{\frac{N-2}{2+\alpha }}. \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} u_\varepsilon (x)=C_{\alpha ,N}\hat{u}_\varepsilon (x)(1+o(1)). \end{aligned}$$

(4.4)

Set $S_\varepsilon ^j=\{x\in \mathbb {R}^N: |x|=\varepsilon ^{\frac{2j-1}{N-2}}\}, j=1,2,\ldots ,k$, and choose a compact subset $K\subset \Omega$ such that, for $\varepsilon$ small enough, $S_\varepsilon ^j\subset K$ for $j=1,2,\ldots ,k$.

Then, for $x\in S_\varepsilon ^j$, we have

$$\begin{aligned} \begin{aligned} \hat{u}_\varepsilon (x)&=\sum _{i=1}^k(-1)^i\left( \frac{1}{M_i^{\frac{2+\alpha }{2}}\varepsilon ^{\frac{(2+\alpha )(2i-1)}{2(N-2)}} +M_i^{-\frac{2+\alpha }{2}}\varepsilon ^{\frac{(2+\alpha )(4j-2i-1)}{2(N-2)}}}\right) ^{\frac{N-2}{2+\alpha }}\\&=\varepsilon ^{-\frac{2j-1}{2}}\sum _{i=1}^k(-1)^i\left( \frac{1}{M_i^{\frac{2+\alpha }{2}}\varepsilon ^{\frac{(2+\alpha )(i-j)}{(N-2)}} +M_i^{-\frac{2+\alpha }{2}}\varepsilon ^{\frac{(2+\alpha )(j-i)}{(N-2)}}}\right) ^{\frac{N-2}{2+\alpha }}\\&=(-1)^j\varepsilon ^{-\frac{2j-1}{2}}\left( \frac{1}{\big (M_j^{\frac{2+\alpha }{2}}+M_j^{-\frac{2+\alpha }{2}}\big )^{\frac{N-2}{2+\alpha }}}+o(1)\right) . \end{aligned} \end{aligned}$$

Thus, for $\varepsilon >0$ small enough, $(-1)^j\hat{u}_\varepsilon >0$ on $S_\varepsilon ^j$, $j=1,2,\ldots ,k$, which implies that $(-1)^ju_\varepsilon >0$ on $S_\varepsilon ^j$. Therefore, $u_\varepsilon$ has at least k nodal domains $\Omega _1,\ldots ,\Omega _k$ such that $\Omega _i$ contains the sphere $S_\varepsilon ^i$.

Next we show that, for $\varepsilon$ small enough, $u_\varepsilon$ has at most k nodal sets. Thanks to Proposition 2.2, Lemma 4.1, (1.7) and (1.10), we have

$$\begin{aligned} J_\varepsilon (PU_{\mu _i})\rightarrow \frac{(2+\alpha )}{2(N+\alpha )}\int _{\mathbb {R}^N}|x|^\alpha U_1^{p_\alpha +1},\quad i=1,2,\ldots ,k,~~\text {as}~\varepsilon \rightarrow 0 \end{aligned}$$

(4.5)

and

$$\begin{aligned} J_\varepsilon (u_\varepsilon )\rightarrow \frac{(2+\alpha )k}{2(N+\alpha )}\int _{\mathbb {R}^N}|x|^\alpha U_1^{p_\alpha +1},~~\text {as}~\varepsilon \rightarrow 0. \end{aligned}$$

(4.6)

Argue by contradiction, we can assume that there exists another nodal domain denoted by $\Omega _{k+1}$. If $\alpha >0$, we find that

$$\begin{aligned} \left( \int _{\Omega _{k+1}}|u_\varepsilon |^{\frac{2N}{N-2}}\right) ^{\frac{N-2}{N}}\le C\int _{\Omega _{k+1}}|x|^\alpha |u_\varepsilon |^{p_\alpha +1-\varepsilon }. \end{aligned}$$

(4.7)

Hence,

$$\begin{aligned} \left( \int _{\Omega _{k+1}}|u_\varepsilon |^{\frac{2N}{N-2}}\right) ^{\frac{N-2}{N}}\le C\Vert u_\varepsilon \Vert ^{\frac{2\alpha }{N-2}-\varepsilon }_{L^\infty (\Omega _{k+1})}\int _{\Omega _{k+1}} |u_\varepsilon |^{\frac{2N}{N-2}}. \end{aligned}$$

By (4.4), we see that $\Vert u_\varepsilon \Vert _{L^\infty (\Omega _{k+1})}\le C$. Thus, $\int _{\Omega _{k+1}}|u_\varepsilon |^{\frac{2N}{N-2}}\ge C>0,$ which implies $J_\varepsilon (u_\varepsilon )>\frac{(2+\alpha )k}{2(N+\alpha )}\int _{\mathbb {R}^N}|x|^\alpha U_1^{p_\alpha +1}$. This is a contradiction with (4.6). If $-2<\alpha <0$, by Hardy inequality, we obtain that $\int _{\Omega }|x|^\alpha |u|^{p_\alpha +1}\le C\big (\int _{\Omega }|\nabla u|^2\big )^{\frac{p_\alpha +1}{2}}$. Similar to the case $\alpha =0$ in [23], we still have that $J_\varepsilon (u_\varepsilon )>\frac{(2+\alpha )k}{2(N+\alpha )}\int _{\mathbb {R}^N}|x|^\alpha U_1^{p_\alpha +1}$ and the proof of Theorem 1.1 is finished. $\square$