Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

We derive Hardy inequalities in weighted Sobolev spaces via anticoercive partial differential inequalities of elliptic type involving A-Laplacian −ΔAu = −divA(∇u) ≥ Φ, where Φ is a given locally integrable function and u is defined on an open subset Ω⊆Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega \subseteq \mathbb{R}^n}$$\end{document}. Knowing solutions we derive Caccioppoli inequalities for u. As a consequence we obtain Hardy inequalities for compactly supported Lipschitz functions involving certain measures, having the form ∫ΩFA¯(|ξ|)μ1(dx)≤∫ΩA¯(|∇ξ|)μ2(dx),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_\Omega F_{\bar{A}}(|\xi|) \mu_1(dx) \leq \int_\Omega \bar{A}(|\nabla \xi|)\mu_2(dx),$$\end{document}where A¯(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{A}(t)}$$\end{document} is a Young function related to A and satisfying Δ′-condition, while FA¯(t)=1/(A¯(1/t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_{\bar{A}}(t) = 1/(\bar{A}(1/t))}$$\end{document}. Examples involving A¯(t)=tplogα(2+t),p≥1,α≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{A}(t) = t^p{\rm log}^\alpha(2+t), p \geq 1, \alpha \geq 0}$$\end{document} are given. The work extends our previous work (Skrzypczaki, in Nonlinear Anal TMA 93:30–50, 2013), where we dealt with inequality −Δpu ≥ Φ, leading to Hardy and Hardy–Poincaré inequalities with the best constants.

Hardy-type inequalities in Orlicz framework are considered in particular in [19]. We find there balance conditions for A, B-a pair of (not necessarily equal) Young functions for the validity of inequalities of the form where d is distance from the boundary of open and bounded domain Ω with Lipschitz-continuous boundary and ξ is sufficiently smooth function with compact support in Ω. We note that in (1.4), unlike in our inequality (1.1), B is the function of ξ/d 1+θ (x) and A is the function of (∇ξ)/d θ (x). Similar results dealing with conditions for validity of inequality

5)
Vol. 21 (2014) Hardy inequalities resulted from nonlinear problems 843 where d(x) is distance from the boundary of open and bounded Ω (sufficiently regular) and ξ is sufficiently smooth function with compact support in Ω, we find in the recent paper [14]. The conditions are expressed in the terms of capacities. Note that the left-hand side of (1.1) is of similar, but different type than left hand side in (1.5).
Our purpose is to give the constructive method of derivation of Hardy-Sobolev inequalities on the basis of nonlinear problems. We find such an approach in papers by Barbatis,Filippas,and Tertikas [7,8], where Hardy-Sobolev inequalities are derived on a domain where certain power of distance function is p-superharmonic. Futhermore, in papers of D'Ambrosio [23][24][25] the author derives inequality related to (1.1) involvingĀ(λ) = FĀ(λ) = λ p as the consequence of inequality −Δ p (u α ) ≥ 0 with certain constant α.
Our cosiderations are based on the methods from [47] developed further in [68]. The idea is as follows. In [47] the authors investigate nonexistence of nontrivial nonnegative weak solutions to the A-harmonic problem where Φ is a nonnegative function. Among other results, the authors derive Caccioppoli-type estimate for nonnegative weak solutions to (1.6).
In [68] we considered the case when with a locally integrable function Φ satisfying less restrictive conditions than in [47]. As it is shown in [68], the certain substitution in the derived Caccioppolitype inequality for solutions, implies the family of Hardy-type inequalities having the form where 1 < p < ∞, ξ : Ω → R is compactly supported Lipschitz function, and Ω is an open subset of R n . The involved measures μ 1,β (dx), μ 2,β (dx) depend on a certain parameter β and on u-a nonnegative weak solution to (1.7). It leads among other results to classical Hardy and Hardy-Poincaré inequalities with optimal constants (see [68,69], respectively). We retrieve this results as the special case here and therefore we confirm all the examples from [68,69].
Our goal now is to extend techniques from [68] to the more general situation when we deal with (1.2) instead of (1.7). We supply new weighted power-logarithmic Hardy-Sobolev inequalities of the form for compactly supported Lipschitz functions ξ, which result from the methods introduced in this paper.
The method may be used not only to construct new inequalities. We indicate also estimates for constants in the inequalities, which can be useful in investigating existence, as well as regularity in theory of partial differential equations in weighted Sobolev and Orlicz-Sobolev spaces.

Preliminaries Notation
In the sequel we assume that Ω ⊆ R n is an open subset not necessarily bounded.
By A-harmonic problems we understand those, which involve A-Laplace operator Δ A u = div(A(∇u)), understood in the weak sense, where A : R n → R n is a C 1 -function. Choosing A(λ) = |λ| p−2 λ we deal with the usual p-Laplacian.
As usual, C k (Ω) (respectively C k 0 (Ω)) denotes functions of class C k defined on an open set Ω ⊂ R n (respectively C k -functions on Ω with compact support). If f is defined on Ω, by fχ Ω we understand function f extended by 0 outside Ω. When V ⊆ R n , by |V | we denote its Lebesgue's measure. Having an arbitrary u ∈ L 1 loc (Ω) it is possible to define its value at every point by the formula We write f ∼ g if function f is comparable with function g, i.e. if there exist positive constants c 1 , c 2 such that for every x We deal with Δ 2 and Δ conditions defined below. Typical examples of functions satisfying the Δ -condition can be found among Zygmund-type logarithmic functions. Their construction is based on the following easy observation. Example 2.1. [42] The following functions satisfy Δ -condition: Let us state some useful facts and lemmas.
The following lemma comes from [44] (Lemma 4.2 therein), where the authors require F to be an N -function, whereas its proof holds for every Young function as well.
Moreover for every r, s > 0 the following estimate holds true

Remark 2.2.
We have the following observations. 1. When F (r) = r p , 1 p + 1 p = 1, we get r p−1 s ≤ 1 p r p + 1 p s p , equivalent to Young inequality qs ≤ q p p + s p p . 2. For general convex function F the latter inequality in (2.5) with finite constant D F is equivalent to F ∈ Δ 2 , while the condition d F > 1 is equivalent to F * ∈ Δ 2 (see [55], Theorem 4.3, or [43], Proposition 4.1). If d F and D F are the best possible in (2.5), they are called Simonenko lower and upper index of F , respectively (see e.g. [11,30,37,67]) for definition and discussion of their properties.

Orlicz-Sobolev spaces
By W 1,Ā (Ω) we mean the completion of the set 0 (Ω) (analogous notation is used for local Orlicz spaces LĀ loc (Ω)). Observe that we always have The following fact holds true.
where B andĀ are the same as in (2.1).

Remark 2.3.
Despite the formulation given in [47] involves N -functions instead of Young functions, the proof therein works for Young functions as well.
According to Fact 2.4 the right-hand side in (2.7) is well defined.

Differential inequality
The differential inequality we want to analyze is given by the following definition.

Definition 2.3.
Let Ω be any open subset of R n and Φ be the locally integrable function defined in Ω, such that for every nonnegative compactly supported if for every nonnegative compactly supported w ∈ W 1,Ā (Ω) we have

Remark 2.5. Examples when those conditions are satisfied in the caseĀ
To ensure that additionally Ψ(t)/g(t) is nonincreasing we have to assume that g (t) ≥ − C with C as in (2.11). Indeed, Ψ/g is nonincreasing because ) ≤ 0, This condition is also satisfied by pairs from Table 1. Table 1. Good pairs of Ψ and g

Caccioppoli estimates for solutions to −Δ A u ≥ Φ
Our main goal in this section is to obtain the following result.
Then the inequality holds for every nonnegative Lipschitz function φ with compact support in Ω, We call (3.1) Caccioppoli inequality because it involves ∇u on the lefthand side and only u on the right-hand side (see e.g. [18,41]).
We note that we do not assume that the right-hand side in (3.1) is finite. The proof is based on the idea of the proof of Theorem 3.1 from [68] inspired by the proof of Proposition 3.1 from [47].
Proof of Theorem 3.1 The proof follows by three steps.
Step 1. Derivation of local inequality. We obtain the following lemma. Then for every 0 < δ < R and every nonnegative Lipschitz function φ with compact support in Ω the inequality holds with Θ(u) given by (2.12) and Before we prove the theorem let us formulate the following facts.
Proof of Lemma 3.1.

Let us introduce some notatioñ
where Θ(u) is given by (2.12). Let us consider u δ,R and G defined by (3.4).
We have to introduce parameters δ and R to make sure that some quantities in the estimates, which we move to opposite sides of inequalities, are finite.
Step 2. Passing to the limit with δ 0.
We assumed in (Θ) that Θ is nonincreasing or bounded in the neighbourhood of zero. We start with the case when there exists κ > 0 such that for λ < κ the function Θ(λ) is nonincreasing. Without loss of generality we may consider κ ≤ R.
We divide the domain of integration Let us begin with integral over E κ . We consider δ → 0, so we may assume that δ < κ/2. Then for x ∈ E κ we have u + δ < κ. As function λ → Θ(λ) is nonincreasing when λ < κ, thus for fixed u and δ 0 the function δ → Θ(u+δ) is nondecreasing and so convergent monotonically almost everywhere to Θ(u). Therefore, due to the Lebesgue's Monotone Convergence Theorem lim δ→0 Eκ In the case of F κ we have κ/2 ≤ u + δ ≤ R. Over this domain Θ is a bounded function, so in particular on F κ We apply the Lebesgue's Dominated Convergence Theorem to deduce that lim δ→0 Fκ This completes the case of Θ nonincreasing in the neighbourhood of 0.
In the case when Θ is bounded in the neighbourhood of 0, we note that Θ is bounded also on every interval [0, R], where R > 0. Hence, we can use previous computations dealing with F κ in case κ = 0. To finish the proof of this step we note that (3.14) says that when δ 0 the first integral on the right-hand side of (3.2) is convergent to the first integral on the right-hand side of (3.12). To deal with the second expression we note that for δ ≤ R whereC(δ, R),C 2 (δ, R),C 1 (δ, R),C(R) are given by (3.3), (3.6), (3.7), (3.13), respectively.
We can pass to the limit with δ → 0 on the left-hand side of (3.2) due to the Lebesgue's Monotone Convergence Theorem as an expression in brackets is nonnegative by (2.13) and the whole integrand therein is nonincreasing by assumption (Ψ).
Step 3. We let R → ∞ and finish the proof.
We are going to let R → ∞ in (3.12). Without loss of generality we can assume that the integral in the right-hand side of (3.1) is finite, as otherwise the inequality follows trivially. Note that as B(|∇u|) ∇u, ∇φ and Φφ are integrable, we have lim R→∞ C(R) = 0. Therefore, (3.1) follows from (3.12) by the Lebesgue's Monotone Convergence Theorem.

Hardy type inequalities
Our most general conclusion resulting from Theorem 3.1 reads as follows. Then for every Lipschitz function ξ with compact support in Ω we have where which ensures that supp φ = supp ξ. This holds because as a Young function A is superlinear and we have lim s→∞Ā (s) = ∞. Furthermore, FĀ(t) is a locally Lipschitz function. We obtain it from Lemma 2.1 which implies The last term above is bounded for bounded t. Indeed, when t ≤ C 0 , we Therefore, FĀ(t) is locally Lipshitz. The composition of locally Lipshitz function FĀ(t) with Lipschitz and bounded ξ, i.e. FĀ(ξ) = φ, is Lipschitz. We verify that Ω∩supp φĀ |∇φ| φ φ dx < ∞. Note that for every compactly supported Lipschitz function ξ we have ΩĀ (|∇ξ|) dx < ∞. Therefore, it suffices to prove that LetĀ −1 be the inverse function ofĀ. AsĀ ∈ Δ we note that for each pair of x, y ≥ 0 we havē .
Hence, taking x = |∇φ| φ and y = φ, we obtain from (4.7) on every x where φ(x) > 0. Now we show that on every x where φ(x) > 0 we have (4.9) Vol. 21 (2014) Hardy inequalities resulted from nonlinear problems 855 Hence, we have |∇φ| φ ξ ≤ DĀ|∇ξ|, which is exactly (4.9). Summing up the estimates (4.8) and (4.9) we obtain (4.6) Thus the assumptions of Theorem 3.1 are satisfied and we obtain (3.1). The substitution φ = FĀ(ξ), equivalently taking transforms the left-hand side of (3.1) into the left-hand side of (4.2). What remains to show is that the right-hand side in (3.1) is estimated as follows This is a direct consequence of (4.6). The proof is complete. Examples dealing with variousĀ, Ψ and g are given in the following sections.

Links with existing results
In this section we present how our result is related to several other ones.

Results of Cianchi [19]
In paper by Cianchi [19] one finds necessary and sufficient conditions for the Hardy inequality

Results of Buckley and Hurri-Syrjänen [14] and Buckley and Koskela [15]
In [14] the authors consider inequality where d(x) is distance from the boundary of Lipschitz domain Ω and ξ is sufficiently smooth function with compact support in Ω ⊆ R n . It is assumed that complement of Ω satisfies the fatness condition, while Ψ belongs to the class of functions called G(p, q, C). The typical representants of such functions are power-logarithmic type functions like e.g. Ψ(t) = t p log α + t, Ψ a (t) = t p log α (a + t), p, a > 1, α ≥ 0.
The case of (5.2) with θ = 1 is considered in [15]. Let us mention that when one considers Theorem 4.1 with we get FĀ(t) = t p log −α (a + 1/t) ∼Ā(t). Thus our inequality (4.2) cannot be compared directly with (5.2). Furthermore, we deal also with the weight functions outside FĀ.
Then for every Lipschitz function ξ with compact support in Ω we have Proof. We apply Theorem 4.1 withĀ(t) = t log α (a+t). We note that according to (4.1) we have , when λ > 0 and FĀ(0) = 0. (5.6) For the constant estimate (4.5) we note that ifĀ(t) = t log α (a + t), α > 0, a > 1, then CĀ ≤ (2/ log a) α , dĀ = 1 and DĀ ≤ 1 + α/ log a. Vol. 21 (2014) Hardy inequalities resulted from nonlinear problems 857 The reverse Stein inequality has the form [50,70] and deals with measurable functions φ supported in some ball B ⊆ R n , where log + (λ) = log(λ)χ λ>1 , Mφ(x) = sup B x 1 |B| B |φ(y)|dy, where supremum is taken over all balls containing x, is the Hardy-Littlewood maximal function. In the one-dimensional case one can deduce from the above inequality the following Hardy inequality where R > 0 and φ has bounded support in R + , as we have 1 When in our inequality (5.3) we considerĀ(t) = t log(a + t) where a > 1, we obtain inequalities having the form: As for big arguments λ we have FĀ(λ) ∼ λ, (5.3) is similar to (5.7). That is why we call it Stein-type inequality. Below we derive inequalities which are similar to (5.7) and come as a consequence of Corollary 5.1. For the related results we refer e.g to [56,59,60] and their references.
To obtain (5.9) we observe that u(x) = x α solves the equation Due to Corollary 5.1 (5.3) holds with Thus, we have (5.9).
In [68] we show that the above theorem implies classical Hardy inequality with the optimal constant, as well as various other weighted Hardy inequalities e.g. with radial and exponential weights.
with the analysis of constants is obtained in [69]. See also the related papers [9,36]. Theorem 4.1 enables us to derive inequalities with various measures, which can have more general form than those from [68]. In the construction of measures we are not restricted to the pair Ψ(t) = t −β , g(t) = t as in [68], but we can consider also other pairs of admitted functions satisfying assumption (Ψ), e.g. pairs from Table 1.

New results. Inequalities with power-logarithmic functions
We collect here a few examples of the inequality (4.2) with several choices of A.

Preliminary preparations
In this part we derive two lemmas which will be used in the sequel.
Vol. 21 (2014) Hardy inequalities resulted from nonlinear problems 861 We reach the goal by computing the weights according to Lemma 6.1 and dividing both sides by the constant.
We notice that we can estimate the constant C as in (6.7). Indeed, due to the above method, we have Moreover, according to Facts 2.2 and 2.3, CĀ ≤ ( 2 log 2 ) α , dĀ = p ≤ DĀ ≤ p + α log 2 .

Inequalities on (0, ∞).
Applying Ψ(t) = t −C , g(t) = t in Lemma 6.2, we obtain the following result. Proof. We apply Lemma 6.2. It suffices now to check that the pair Ψ(t) = t −C , g(t) = t with C > 0 satisfies the assumption (Ψ) i) and ii) and finally we compute the weights.