Caccioppolitype estimates and Hardytype inequalities derived from weighted pharmonic problems
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Abstract
We obtain Caccioppolitype estimates for nontrivial and nonnegative solutions to anticoercive partial differential inequalities of elliptic type involving weighted pLaplacian: \(\Delta _{p,a} u:= {\mathrm {div}}(a(x){\nabla }u^{p2}{\nabla }u)\ge b(x)\Phi (u)\chi _{\{u>0\}}\), where u is defined in a domain \(\Omega \). Using Caccioppolitype estimates, we obtain several variants of Hardytype inequalities in weighted Sobolev spaces.
Keywords
pHarmonic PDEs pLaplacian Nonlinear eigenvalue problems Degenerate PDEs Quasilinear PDEsMathematics Subject Classification
Primary 26D10 Secondary 35D30 35J60 35R451 Introduction
Our purpose is to investigate the two following issues: the qualitative theory of solutions to nonlinear problems and derivation of precise Hardytype inequalities. We contribute to the first of them by obtaining Caccioppolitype estimate for a priori not known solution, which in general is an important tool in the regularity theory. For the second issue, in most cases we assume that the solution to (1.1) is known and we use it in construction of Hardytype inequalities. Substituting \(a\equiv 1\) in our considerations, we retrieve several results obtained by the third author in [55], where she dealt with the partial differential inequality of the form \(\Delta _{p}u\ge \Phi \), admitting the function \(\Phi \) depending on u and x. Some of the inequalities derived in [55] and retrieved here are precise, i.e. they hold with the best constants. These cases are recalled in Remark 4.1. Furthermore, we provide here also new inequalities with the optimal constants, see Theorems 4.3 and 4.2. More new Hardy inequalites constructed using a solution to (1.1) are provided in Theorem 4.5.
The approach presented here and in the papers [55, 56] is a modification of techniques originating in [43]. In all of these papers, the investigations start with derivation of Caccioppolitype estimates for the solutions to nonlinear problems. The method was inspired by the wellknown nonexistence results by Pohozhaev and Mitidieri [51].
In contrast to the results from [43, 55], in this paper we admit the weighted pLaplacian: \(\Delta _{p,a}\) instead of the classical one in (1.1). Our main results are the Caccioppolitype estimate (Theorem 3.1) and the Hardytype inequality (Theorem 4.1). Some of the results obtained here are new even in the nonweighted case when \(a\equiv 1\), see Sect. 4.4 for discussion.
The discussion linking eigenvalue problems with Hardytype inequalities can be found in the paper by Gurka [39], which generalized earlier results by Beesack [8], Kufner and Triebel [46], Muckenhoupt [52], and Tomaselli [60]. See also related more recent paper by Ghoussoub and Moradifam [37]. Derivation of the Hardy inequalities on the basis of supersolutions to pharmonic differential problems can be found in papers by D’Ambrosio [22, 23, 24] and Barbatis et al. [5, 6]. Other interesting results linking the existence of solutions in elliptic and parabolic PDEs with Hardy type inequalities are presented in [2, 4, 36, 61, 62], see also references therein. We refer also to the contribution by the third author [56], where instead of the nonweighted pLaplacian in (1.1) one deals with the ALaplacian: \(\Delta _Au=\mathrm{div}\left( \frac{A(\nabla u)}{\nabla u^2}\nabla u\right) \), involving a function A from an Orlicz class. Similar estimates in the framework of nonlocal operators can be found e.g. in [12].
Let us present several reasons to investigate the partial differential inequality of the form \(\Delta _{p,a}u\ge b(x)\Phi (u){\chi _{\{u>0\}}}\) rather than a simple one \(\Delta _{p}u\ge \Phi (u)\).
We hope that by the investigation of the qualitative properties of supersolutions to degenerated PDEs, i.e. the Caccioppolitype estimate, and by constructions of Hardytype inequalities, we can get deeper insight into the theory of degenerated elliptic PDEs.
2 Preliminaries
Weighted Beppo Levi and Sobolev spaces. \(B_p\) weights: we deal a the special class of measures belonging to the class \(B_p(\Omega )\) from [45].
Definition 2.1
(Classes \(W(\Omega )\) and \(B_p(\Omega )\)) Let \(\Omega \subseteq \mathbf {R}^{n}\) be an open set and let \(W(\Omega )\) be the set of all Borel measurable real functions \(\varrho \) defined on \(\Omega \), such that \(0<\varrho (x)<\infty ,\) for a.e. \(x\in \Omega \). We say that a weight \(\varrho \in W(\Omega )\) satisfies \(B_p(\Omega )\)condition (\(\varrho \in B_p(\Omega )\) for short) if \( \varrho ^{1/(p1)}\in L^1_{{ loc}}(\Omega ).\)
The Hölder inequality leads to the following simple observation based on [45, Theorem 1.5]. For readers’ convenience we enclose the proof.
Proposition 2.1
Let \(\Omega \subset \mathbf {R}^n\) be an open set, \(p>1\) and \(\varrho \in B_p(\Omega )\). Then \( L^{p}_{\varrho ,loc}(\Omega )\subseteq L^1_{\mathrm{loc}}(\Omega ) \) and when \(u_k\rightarrow u\) locally in \(L^{p}_{\varrho }(\Omega )\) then also \(u_k\rightarrow u\) in \(L^1_{{ loc}}(\Omega )\).
Proof
Proposition 2.2
[45] \(p>1\), \(\Omega \subseteq {\mathbf {R}}^{n}\) be an open set and \(\varrho _1(\cdot )\in W(\Omega )\), \(\varrho _2(\cdot )\in B_p({\Omega })\). Then \(W^{1,p}_{(\varrho _1,\varrho _2)}(\Omega )\) defined by (2.3) equipped with the norm \(\Vert \cdot \Vert _{W_{(\varrho _1,\varrho _2)}^{1,p}(\Omega )}\) is a Banach space.
Remark 2.1
 (i)
As \(\nabla u^{p2}\nabla u\in L^{\frac{p}{p1}}_{a,loc}(\Omega ,{{\mathbf {R}}^{n}})\), then the righthand side in (2.4) is well defined for every \(w\in \mathcal{L}^{1,p}_a (\Omega )\) which is compactly supported in \(\Omega \).
 (ii)When \(u\in \mathcal{L}^{1,p}_a(\Omega )\), formula (2.4) extends for \(w\in W^{1,p}_{(b,a),0}(\Omega )\), whenever \(b\in W(\Omega )\). This follows from the estimatesTherefore in this case \(\Delta _{p,a}u\) can be also treated as an element of the dual to the Banach space \(W^{1,p}_{(b,a),0}(\Omega )\) denoted \((W^{1,p}_{(b,a),0}(\Omega ))^*\). We preserve the same notation \(\Delta _{p,a}u\) for this functional extension of formula (2.4).$$\begin{aligned} \langle \Delta _{p,a}u,w \rangle \le & {} \int _{\Omega }a\nabla u^{p1}\nabla w\,dx= \int _{\Omega }\left( a^{\frac{1}{p {'}}}\nabla u^{p1}\right) \left( a^{\frac{1}{p}}\nabla w\right) \,dx\\\le & {} \left( \int _{\Omega }\nabla u^p a\,dx\right) ^{1\frac{1}{p}} \left( \int _{\Omega }\nabla w^p a\,dx\right) ^{\frac{1}{p}}<\infty . \end{aligned}$$
Differential inequality. Our analysis is based on the following differential inequality.
Definition 2.2
Remark 2.2
 (i)
Inequality (2.5) can be interpreted as a variant of psuperharmonicity condition for the weighted pLaplacian defined by (2.1).
 (ii)
In the case of equation in (2.5): \(\Delta _{p,a} u= {b(x)\Phi (u)\chi _{\{u>0\}}},\) we deal with the solution of the nonlinear eigenvalue problem. It corresponds to the eigenvalue problem \(\Delta _{p} u= \lambda u^{p2}u,\) but in our case the role of an eigenvalue \(\lambda \) plays the function \(b(\cdot )\) and instead of the \(u^{p2}u\) we have now the possibly another expression \(\Phi (u)\chi _{\{u>0\}}\).
Assumption A

(a, b) \(a\in L^1_{loc}(\Omega )\cap B_p(\Omega )\), \(b(\cdot )\) is measurable;
 (\(\Psi ,g\)) The couple of continuous functions (\(\Psi ,g): (0,\infty ) \times (0,\infty ) \rightarrow (0,\infty ) \times (0,\infty )\), where \(\Psi \) is Lipschitz on every closed interval in \((0,\infty )\), satisfy the following compatibility conditions:
 (i) the inequalityholds with some constant \(C\in {\mathbf {R}}\) independent of t and \(\Psi \) is monotone (not necessarily strictly);$$\begin{aligned} g(t){\Psi }{'}(t)\le C {\Psi } (t)\quad {\mathrm { a.e.}}\quad {\mathrm {in}}\quad (0,\infty ) \end{aligned}$$(2.8)
 (ii) each of the functionsis nonincreasing or bounded in some neighbourhood of 0.$$\begin{aligned} {(0,\infty ) \ni }\ t\mapsto \Theta (t):=\Psi (t){g^{p1}(t)},\ \ \mathrm{and}\ \ t\mapsto \Psi (t)/g(t) \end{aligned}$$(2.9)

 (u) We assume that \(u\in \mathcal{L}^{1,p}_{a,loc}(\Omega )\) is nonnegative, (a, b) holds, \(\Phi : { (} 0,\infty ) \rightarrow { (} 0,\infty )\) is a continuous function, such that for every nonnegative compactly supported function \(w\in \mathcal{L}^{1,p}_a(\Omega )\) one has \(\int _{\Omega { \cap \{ u>0\} } } b(x) \Phi (u) w\,dx >\infty \) and \(b\Phi (u){ \chi _{ \{ u>0\} } }\in L^1_{loc}(\Omega )\). Moreover, let us consider the set \(\mathcal{A}\) of those \({\sigma }\in {\mathbf {R}}\) for whichWe suppose that$$\begin{aligned} b(x)\Phi (u)+{\sigma }\,\frac{ a(x)}{g(u)}\nabla u^p\ge 0 \quad {\mathrm { a.e.}}\ {\mathrm {in}}\ \Omega \cap \{ u>0\}. \end{aligned}$$(2.10)Since \(\mathrm{inf}\, \emptyset =+\infty \), \(\mathcal{A}\) can be neither an empty set nor unbounded from below.$$\begin{aligned} {\sigma }_0:= \inf \mathcal{A}= \inf \left\{ {\sigma }\in {\mathbf {R}}:\ {\sigma }\ \mathrm{satisfies}\ (2.10) \right\} \in {\mathbf {R}}. \end{aligned}$$(2.11)
 (a)
We suppose that (\(\Psi ,g\)) and (u) hold. Parameter \(\sigma \) satisfies \({\sigma }_0\le {\sigma }<C\), where C is given by (2.8) and \({\sigma }_0\) by (2.11).
 (b)
We suppose that (u) and (\(\Psi ,g\)) hold and we assume that for every \(R>0\) we have \(b^{+}(x)(\Phi \Psi ) (u)\chi _{0<u\le R}\in L^1_{loc}(\Omega )\).
 (c)
We suppose that (u) and (\(\Psi ,g\)) hold. We assume that for any compact subset \(K\subseteq \Omega \) we have
$$\begin{aligned}&\Psi (R)\int _{K\cap \{ u\ge R/2\} }\nabla u(x)^{p1}a(x)\, dx {\mathop {\rightarrow }\limits ^{R\rightarrow \infty }} 0,\\&\Psi (R)\int _{K\cap \{ u\ge R/2\} } \Phi (u)b(x)\, dx {\mathop {\rightarrow }\limits ^{R\rightarrow \infty }} 0. \end{aligned}$$  (a)
Comments on assumptions
Remark 2.3
 (i)
Assume that Condition (\(\Psi ,g\)), i) holds and, moreover, \(g'(t)\ge C\). Then \(\left( {\Psi }/{g} \right) '\le 0\) and \(\Psi (t)/g(t)\) is nonincreasing.
 (ii)
This condition is satisfied by examples of pairs from Table 1.
Example couples \((\Psi ,g)\) which satisfy condition (\(\Psi ,g\)) from Assumption A
\(\Psi (t)\)  g(t)  C  Remarks 

\(t^{\alpha }\)  t  \(\alpha \)  \(\alpha \in {\mathbf {R}}\) 
\(\left( t\log (a+t)\right) ^{1}\)  \(t\log (a+t)\)  \(\log a\)  \(a>1\) 
\(e^{t}\)  Bounded by C, \(g'\ge C\)  C  \(C>0\) 
\({e^{t}}/{t}\)  \(t/(1+t)\)  1  – 
The statement below shows that under Assumption A,(u) the function u cannot be constant almost everywhere in \(\Omega \). Moreover, in many cases \(\mathcal{A}\) is not empty and \(\mathrm{inf}\mathcal{A}\) is a real number.
Lemma 2.1
Suppose \(u\in \mathcal{L}^{1,p}_{a,loc}(\Omega )\) is a nonnegative solution to the PDI
\(\Delta _{p,a} u\ge {b(x)\Phi (u)\chi _{ \{ u>0\}}}\) in the sense of Definition 2.2, under all assumptions therein. Moreover, let \(b\ge 0\) a.e. in \(\Omega \). Then \(\sigma _0\) given by (2.11) exists and is finite if and only if u is not a constant function a.e. in \(\Omega \).
Proof
(\(\Longrightarrow \)) If \(\sigma _0\) is a finite number, then u cannot be constant. Indeed, for \(u\equiv Const\ge 0 \), condition (2.10) implies \(\mathcal{A}= (\infty ,\infty )\), which violates (2.11). \(\square \)
Remark 2.4
 (i)
When u is locally bounded.
 (ii)When \(b\ge 0\), \(u\in \mathcal{L}^p_{a,loc}(\Omega )\) and \(\Psi (R)/R\) is bounded at infinity. Indeed, we have from Hölder’s inequalityand \(Z_2(R):= \left( \int _{K\cap \{ u\ge R/2\} }\nabla u(x)^{p}a(x)\, dx \right) ^{1\frac{1}{p}}\rightarrow 0\) as \(R\rightarrow \infty \). On the other hand, by Czebyshev’s inequality applied to \(\mu (x)=a(x)dx\) on K, we get$$\begin{aligned} Z_1(R):=&\Psi (R)\int _{K\cap \{ u\ge R/2\} }\nabla u(x)^{p1}a(x)\, dx\\ \le&\Psi (R)\left( \int _{K\cap \{ u\ge R/2\} }\nabla u(x)^{p}a(x)\, dx \right) ^{1\frac{1}{p}} \left( \int _{K\cap \{ u\ge R/2\} }a(x)\, dx \right) ^{\frac{1}{p}} \end{aligned}$$Therefore, \(Z_1(R)\le \frac{\Psi (R)}{R} Z_2(R) Z_3(R)^{\frac{1}{p}}\rightarrow 0\) as \(R\rightarrow \infty \).$$\begin{aligned} \int _{K\cap \{ u\ge R/2\} }a(x)\, dx&=\mu (\{ x\in K : u(x)\ge R/2\}) \\&{\le } \frac{2^p}{R^p}\int _K u^p a(x)dx =: \frac{1}{R^p} Z_3(R). \end{aligned}$$
3 Caccioppolitype estimates
This section provides estimates which we call Caccioppolitype inequalities, see Lemma 3.1 and Theorem 3.1. In the classical setting the Caccioppoli inequality should involve u on the righthand side only and \(\nabla u\) exclusively on the lefthand side, see [38]. In our case on the righthand sides of inequalities (3.2) and (3.3) there is indeed no dependence on \(\nabla u\), when we estimate \(\chi _{\{ \nabla u\ne 0\}}\) by 1, while the lefthand side does involve \(\nabla u\). Nonetheless, this estimate is sufficient to play the role of Caccioppoli inequality for some purposes, e.g. in the studies on nonexistence [33, 43]
3.1 Formulation of results
Our first goal is to obtain the following estimate, which is the key tool in our considerations. We call it a ‘local estimate’, because it is stated on a part of the domain where u is not bigger than a given R.
Lemma 3.1
The above local estimate implies the following global estimate of Caccioppolitype for solutions to (3.1). It may be used to analyze various qualitative properties of them and is the main result of this section.
Theorem 3.1
3.2 Proof of Lemma 3.1
We use the following simple observations (see [55]).
Lemma 3.2
Lemma 3.3
Let \(u,\phi \) be as in the assumptions of Proposition 3.1 and let \(0<\delta <R\) be arbitrary. Then \(u_{\delta ,R}\in \mathcal{L}^{1,p}_{a,loc}(\Omega )\), \( {G}\in \mathcal{L}^{1,p}_{a}(\Omega )\) and G is compactly supported in \(\Omega \).
Remark 3.1
 (i)
We know that \(\mathcal{L}^{1,p}_{a,loc}(\Omega )\subseteq W^{1,1}_{loc}(\Omega )\). This inclusion, together with Nikodym ACL Characterization Theorem [50, Section 1.1.3], implies that we can verify if the function belongs to Sobolev space \(\mathcal{L}^{1,p}_{a,loc}(\Omega )\) by checking that it belongs to \(W^{1,1}_{loc}(\Omega )\) and that its derivatives computed almost everywhere belong to \(L^p_{a,loc}(\Omega )\). The fact that \(\Psi \) is locally Lipschitz is used to apply Lemma 3.3 in order to ensure that \(\Psi (u_{\delta ,R}(x))\) belongs to \(W^{1,1}_{loc}(\Omega )\).
 (ii)
The nonnegativity of function u allows to deduce that \( {G}\in \mathcal{L}^{1,p}_{a}(\Omega )\). This fact plays the crucial role in the proof of Lemma 3.1.
We will test the PDI against \(u_{\delta ,R}\) and pass to the limit with \(\delta \searrow 0\). To justify the convergence of some terms we need the following fact.
Lemma 3.4
(e.g. [43], Lemma 3.1) Let \(u\in W^{1,1}_{loc}(\Omega )\) be defined everywhere by (2.2) and let \(t\in {\mathbf {R}}\) be given. Then \( \{ x\in {\Omega } : u(x)=t \}\subseteq \{ x\in {\Omega } :\nabla u(x)=0\}\cup N,\) where N is a set of Lebesgue’s measure zero.
3.2.1 Proof of Lemma 3.1
At first we explain the strategy of the proof.
Let the quantities \(\Phi ,\Psi ,g,a,b,u\) be as in (3.1) and Assumption A, while \(\phi \) be as in the statement of the lemma.
The proof is performed in four steps:
3.2.2 Proof of Step 1
Introduction of parameters \(\delta \) and R is necessary as we need to move the quantities \(J_2,J_4\) in the estimates to the opposite sides of inequalities. For doing this we have to know that they are finite.
3.2.3 Proof of Step 2
For the proof of (3.8) we recall the nonnegative function \(\Theta (t):= \Psi (t) g^{p1}(t)\) given by (2.9), which is nonincreasing or bounded in the neighbourhood of zero.
To complete the proof of Step 2 it suffices to observe that for \(\delta \le \frac{R}{2} \) we have \(\tilde{C}(\delta ,R)\le \tilde{C}(R).\)
3.2.4 Proof of Step 3

(3a) \(\Psi \) is nonincreasing and \(\Psi /g\) is nonincreasing;

(3b) \(\Psi \) is increasing and \(\Psi /g\) is nonincreasing;

(3c) \(\Psi \) is nonincreasing and \(\Psi /g\) is bounded in some neighbourhood of 0;

(3d) \(\Psi \) is increasing and \(\Psi /g\) is bounded in some neighbourhood of 0.
3.2.5 Proof of Step 4 and thus of Lemma 3.1
4 Hardytype inequality
As a direct consequence of Caccioppolitype estimates for solutions to PDI, we obtain Hardytype inequality for a rather general class of test functions, i.e. Lipschitz and compactly supported functions. In this section we present our second main result, as well as we give comments on special instances holding with the optimal constants.
4.1 Main result
The following theorem is our main result on Hardytype inequalities.
Theorem 4.1
(Hardytype inequality) Suppose \(a\in L^1_{loc}(\Omega )\cap B_p(\Omega )\), \(b\in L^1_{loc}(\Omega )\). Assume that \(1<p<\infty \) and \(u \in \mathcal{L}^{1,p}_{a, loc}(\Omega )\) is a nonnegative solution to the PDI \(\Delta _{p,a} u\ge b(x)\Phi (u)\chi _{\{u>0\}}\) in the sense of Definition 2.2. Moreover, let Assumption A hold.
Proof
4.2 Special cases
The above Theorem 4.1 generalizes [55, Theorem 4.1], which implies several examples of inequalities with the best constants. Indeed, in the nonweighted case, i.e. when \(a(\cdot )= b(\cdot )\equiv 1\), Theorem 4.1, as well as Theorem 3.1, retrieves the results of [55]. In constrast with [55] our function \(\Psi \) need not be increasing here. Hence, broader class of measures \(\mu _1\) and \(\mu _2\) may appear in (4.1). Therefore our result generalizes that of [55] even in the nonweighted case. Below we mention special cases of [55, Theorem 4.1] that in particular results from Theorem 4.1.
Remark 4.1
The instance of Theorem 4.1 is the classical Hardy inequality on the halfline with powertype weights and optimal constant, see [55, Section 5.1]. Moreover, for a range of parameters the Hardy–Poincaré inequality with weights of a form \((1+x^\frac{p}{p1})^\alpha \) obtained in [57] (by application of [55]) is also sharp, while for another range it confirms some constants from [37] and [10]. This is commented below in detail.
In addition, the classical (unweighted) Poincaré’s inequality on an arbitrary bounded domain can be concluded from [55] and it is confirmed to hold with best constant in [26, Remark 7.6]. The inequality with weights of the form \(x^\alpha \exp \big (\beta x^\gamma \big )\) provided in [55, Theorem 5.5] can also be retrieved by the methods from [40] with the same constant, while the inequality with weights of the form \(x^{\alpha }\log ^{p+\alpha }(\mathrm{e}+x)\) from [42, Proposition 5.2] are comparable with [55, Theorem 5.8].
Results dealing with Hardy–Poincarétype inequalities and their sharpness
Inequality  Optimality  Comment 

Classical Hardy  [45]  Proven in [55] 
Hardy–Poincaré  Proven with explicitly computed spectrum of the associated linear operator  
In the case of Caffarelli–Kohn–Nirenberg weights \(x^\alpha \)  
[57]  
[44]  
Poincaré  [26]  Concluded from [55] 
Exponentialweighted Hardy  Expected  
Expected 
Hardy inequalities may be used to construct other inequalities.
Remark 4.2
4.3 Hardy inequalities resulted from existence theorems
This is the special case of inequality (2.5) for \(\Phi (u)= u^{p1}\). Our result reads as follows.
Theorem 4.2
Proof
We apply Theorem 4.1 with \(\Psi (t)=\frac{1}{t^{p1}}\), \(g(t)=t\), \(\Phi (t)=t^{p1}\), \(\sigma =0\) and verify that under our conditions Assumption A is satisfied. This gives (4.3).
Remark 4.3
Theorem 4.2 is known in the case \(a\equiv 1, b\equiv 1\), see [1] or Remark 1 on page 163 in [48].
Remark 4.4
We substitute the special value of \(\sigma =0\), in the proof of the above statement. Therefore, we do not expect that the inequality (4.3) holds with the best constant in general.
4.4 Sharp Hardy–Poincaré inequalities
Using the Talenti extremal profile given by (1.5) where \(\beta =0\) in our approach, one obtains the following theorem. Adopting the same nomenclature as in e.g. [10, 13], we call the following inequalities of Hardy–Poincaré type.
Theorem 4.3
Information about the proof
Remark 4.5
Remark 4.6
Such inequalities in the case \(p=2\) are very much of interest in the theory of nonlinear diffusions, where one investigates the asymptotic behavior of solutions of the equation \(u_t=\Delta u^m\), see [10]. The best constants in the case \(p=2\) have been obtained in [13], where also the whole spectrum of the associated operator has been explicitly calculated, by means of linear spectral methods, different from the one we use here. In the case of Caffarelli–Kohn–Nirenberg weights we refer to [14, 15, 29] for analogous results.
Remark 4.7
 (i)To our best knowledge our inequalities are new if \(r\ne 1\) in general. However, as an example dealing with \(r\ne 1\) and \(p=2\) we refer to the fourth line on page 434 in [10, Proposition 2], which is precisely our inequality from Theorem 4.3 with \(r=\gamma /n\), \(p=2\) (with the same constant)The proof of this inequality in [10] requires knowledge about the best constants in Sobolev inequality, which we do not need.$$\begin{aligned}&n(n2+\gamma )\int _{\mathbf {R}^n}\xi ^2 \left( 1+\frac{\gamma }{n}x^2\right) (1+x^2)^{\gamma 2}\, dx \\&\quad \le \, \int _{\mathbf {R}^n} \nabla \xi ^2 (1+x^2)^{\gamma }\, dx. \end{aligned}$$
 (ii)
We note that Theorem 4.3 provides a version of [10, Proposition 3] dealing with \(p=2\), for an arbitrary p. Thus, we can interpret it as therein that our method allows to find a sharp, in some cases, first spectral gap (or Poincaré inequality) with methods different from the already known ones, and that are suitable for generalization to more sophisticated operators.
 (iii)
We retrieve [57, Theorem 3.1] as a special case of Theorem 4.2 when one substitutes \(r=1\) and deals with \(\gamma >1\). In Theorem 4.3 we admit some range of negative \(\gamma \)s as well. The need to consider the negative \(\gamma \)s in case of \(p=2\) and to look for the best constants in such inequalities is visible in the studies on asymptotics of fast diffusion equations, see [13, Theorem 1] and its application in the proof of [13, Theorem 2].
We also present the following consequence of Theorem 4.1, which is in a sense complementary to Theorem 4.3, as it holds for with negative r. It essentially requres to deal with the weighted pLaplacian in the inequality (1.1).
Theorem 4.4
Proof
4.5 Other possible variants of Hardy inequality

(\(u_2\)) We assume that \(u\in \mathcal{L}^{1,p}_{a,loc}(\Omega )\) is nonnegative, (a, b) holds, \(\Phi : [0,\infty ) \rightarrow [0,\infty )\) is a continuous function, such that for every nonnegative compactly supported function \(w\in \mathcal{L}^{1,p}_a(\Omega )\) one has \(\int _{\Omega } b(x) \Phi (u) w\,dx >\infty \) and \(b(\cdot )\Phi (u)\in L^1_{loc}(\Omega )\).
Moreover, let us consider the set \(\mathcal{A}\) of those \({\sigma }\in {\mathbf {R}}\) for whichWe suppose that$$\begin{aligned} b(x)\Phi (u)+{\sigma }\,\frac{ a(x)}{g(u)}\nabla u^p\ge 0 \quad {\mathrm { a.e.}}\ {\mathrm {in}}\ \Omega \cap \{ u>0\}. \end{aligned}$$$$\begin{aligned} {\sigma }_0:= \inf \mathcal{A}= \inf \left\{ {\sigma }\in {\mathbf {R}}:\ {\sigma }\ \mathrm{satisfies}\ (2.10) \right\} \in {\mathbf {R}}. \end{aligned}$$
Theorem 4.5
Suppose \(a\in L^1_{loc}(\Omega )\cap B_p(\Omega )\), \(b\in L^1_{loc}(\Omega )\). Assume that \(1<p<\infty \) and \(u \in \mathcal{L}^{1,p}_{a, loc}(\Omega )\) is a nonnegative solution to the PDI (4.8). Moreover, let Assumption A holds where (u) substituted by (\(u_2\)) and where additionally \(\lim _{\delta \rightarrow 0}\Psi (\delta )=0\).
Sketch of the proof
From there we obtain the Caccioppoli estimates and Hardy inequalities, by precisely the same arguments as in the proofs in Theorems 3.1 and 4.1. \(\square \)
Remark 4.8
Notes
Acknowledgements
I.C. was supported by NCN Grant 2011/03/N/ST1/00111. The work of A.K. was supported by NCN Grant 2014/14/M/ST1/00600. P.D. was supported by the Grant Agency of Czech Republic, Project No. 18–03253S. This work originated when I.C. visited University of West Bohemia in Pilsen in February 2013. She want to thank Pilsen University for hospitality. This research was provided when A.K. was employed at Institute of Mathematics, Polish Academy of Sciences at Warsaw. She wants to thank IM PAN at Warsaw for hospitality. We also would like to thank the anonymous referee for many valuable comments which significantly improved the presentation of this paper.
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