On a degenerate singular elliptic problem

TL;DR

This paper establishes existence, uniqueness, and regularity of solutions for a class of degenerate singular elliptic boundary value problems involving weighted p-Laplacian operators.

Contribution

It provides new mathematical results on the well-posedness and regularity of solutions for a broad class of degenerate singular elliptic equations with weights in the Muckenhoupt class.

Findings

Existence of solutions under given conditions.

Uniqueness of solutions for the problem.

Regularity results for solutions.

Abstract

In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by \begin{gather*} \begin{cases} -\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u^\delta}\,\,\text{ in }\,\,\Omega, u>0\text{ in }\Omega,\\ u = 0 \text{ on } \partial\Omega, \end{cases} \end{gather*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 2$, $w$ belong to the Muckenhoupt class $A_p$ for some $1<p<\infty$, $f$ is a nonnegative function belong to some Lebesgue space and $\delta>0$.

Equations

$$\{\begin{array}{l}-\mathrm{div}(w(x)|\nabla u{|}^{p-2}\nabla u)=\frac{f(x)}{u^{\delta }}\text{ in }\Omega ,\\ u>0\text{ in }\Omega ,\\ u=0\text{ on }\partial \Omega ,\end{array}$$

$$\begin{gathered}\begin{cases}-\operatorname{div}\big{(}\mathcal{A}(x,\nabla u)\big{)}=\frac{f(x)}{u^{\delta}}\text{ in }\Omega,\\ u>0\text{ in }\Omega,\\ u=0\text{ on }\partial\Omega,\end{cases}\end{gathered}$$

$$\big{<}\mathcal{A}(x,\zeta_{1})-\mathcal{A}(x,\zeta_{2}),\zeta_{1}-\zeta_{2}\big{>}\geq c\,|\zeta_{1}-\zeta_{2}|^{\gamma}\big{\{}\overline{\mathcal{A}}(x,\zeta_{1},\zeta_{2})\big{\}}^{1-\frac{\gamma}{p}}w(x),$$

$$\overline{\mathcal{A}}(x,\zeta_{1},\zeta_{2}):=\frac{1}{w(x)}\big{(}\big{<}\mathcal{A}(x,\zeta_{1}),\zeta_{1}\big{>}+\big{<}\mathcal{A}(x,\zeta_{2}),\zeta_{2}\big{>}\big{)}.$$

$$\begin{gathered}\begin{cases}Lu:=-\operatorname{div}\big{(}M(x)|\nabla u|^{p-2}\nabla u\big{)}=\frac{f(x)}{u^{\delta}}\,\,\text{ in }\,\,\Omega,\\ u>0\text{ in }\Omega,\\ u=0\text{ on }\partial\Omega,\end{cases}\end{gathered}$$

$$|M(x)\zeta |\le w(x)|\zeta |,M(x)\zeta \cdot \zeta \ge w(x)|\zeta {|}^2,\text{ for a.e.}x\in \Omega ,\forall \zeta \in {\mathbb{R}}^N.$$

$$\Delta_{p,w}u:=\text{div}\big{(}w(x)|\nabla u|^{p-2}\nabla u\big{)}.$$

$$\Big{(}\frac{1}{|B|}\int_{B}w\,dx\Big{)}\Big{(}\frac{1}{|B|}\int_{B}w^{-\frac{1}{p-1}}\,dx\Big{)}^{p-1}\leq c_{p,w}.$$

$$W^{1,p}(\Omega ,w)=\{u:\Omega \to \mathbb{R}\text{ measurable }:||u|{|}_{1,p,w}<\infty \},$$

$$||u||_{1,p,w}=\Big{(}\int_{\Omega}|u(x)|^{p}w(x)\,dx\Big{)}^{\frac{1}{p}}+\Big{(}\int_{\Omega}|\nabla u|^{p}w(x)\,dx\Big{)}^{\frac{1}{p}}.$$

$$W^{1,p}_{0}(\Omega,w)=\overline{\big{(}C_{c}^{\infty}(\Omega),||.||_{1,p,w}\big{)}}.$$

$${\int }_{\Omega }|\varphi {|}^{p}w(x)dx\le c{\int }_{\Omega }|\nabla \varphi {|}^{p}w(x)dx\forall \varphi \in C_c^{\infty }(\Omega ),$$

$$||u||_{1,p,w}=\Big{(}\int_{\Omega}|\nabla u(x)|^{p}w(x)\,dx\Big{)}^{\frac{1}{p}}.$$

$$W^{1,p}(\Omega ,w)\hookrightarrow L^p(\Omega ,w)$$

$$A_{s}:=\Big{\{}w\in A_{p}:w^{-s}\in L^{1}(\Omega)\,\,\text{for some}\,\,s\in\big{[}\frac{1}{p-1},\infty\big{)}\cap\big{(}\frac{N}{p},\infty\big{)}\Big{\}}.$$

$$W^{1,p}(\Omega ,w)\hookrightarrow W^{1,p_s}(\Omega )\hookrightarrow \{\begin{array}{ll}{L}^q(\Omega ),& \text{for }{p}_s\le q\le p_s^{\ast },\text{in case of }1\le p_s<N,\\ L^q(\Omega ),& \text{for }1\le q<\infty ,\text{in case of }{p}_s=N,\\ C(\overline{\Omega }),& \text{in case of }{p}_s>N,\end{array}$$

$${\int }_{\Omega }|u(x){|}^{p_s}dx$$

$$\le ({\int }_{\Omega }|u(x){|}^{p}w(x)dx{)}^{\frac{p_s}{p}}({\int }_{\Omega }w(x{)}^{-s}dx{)}^{\frac{1}{s+1}},$$

$$||u|{|}_{L^{p_s}(\Omega )}\le ({\int }_{\Omega }w(x{)}^{-s}dx{)}^{\frac{1}{ps}}({\int }_{\Omega }|u(x){|}^{p}w(x)dx{)}^{\frac{1}{p}}.$$

$$||\nabla u|{|}_{L^{p_s}(\Omega )}\le ({\int }_{\Omega }w(x{)}^{-s}dx{)}^{\frac{1}{ps}}({\int }_{\Omega }|\nabla u{|}^{p}w(x)dx{)}^{\frac{1}{p}}.$$

$$||u|{|}_{W^{1,p_s}(\Omega )}\le ||w^{-s}|{|}_{L^1(\Omega )}^{\frac{1}{ps}}||u|{|}_{1,p,w}.$$

$$W^{1,p}(\Omega ,w)\hookrightarrow W^{1,p_s}(\Omega )$$

$$W^{1,p}(\Omega ,w)\hookrightarrow L^q(\Omega )$$

$$-\mathrm{div}\mathcal{A}(x,\nabla u)=0$$

$${\int }_{\Omega }\mathcal{A}(x,\nabla u)\cdot \nabla \varphi dx\ge 0$$

$$v_{n}:=\Big{(}\phi_{n}^{\frac{1}{\alpha}}-\epsilon\Big{)}^{+}.$$

$$||v_{n}||_{X}^{p}=\int_{\Omega}\,w(x)|\nabla v_{n}|^{p}\,dx\leq\int_{\big{\{\phi_{n}>\epsilon^{\alpha}\big{\}}}}\,w(x)\epsilon^{\alpha(\frac{1}{\alpha}-1)}|\nabla\phi_{n}|^{\alpha}\,dx\leq M,$$

$$\displaystyle-\operatorname{div}\big{(}\mathcal{A}(x,\nabla u_{n})\big{)}=G_{n}\text{ in }\Omega.$$

$$|<G_n,\varphi >|\le C_K||\varphi |{|}_{L^{\infty }(\Omega )},$$

$$L_{\mu }(s):=\{\begin{array}{ll}s,& \text{for }|s|\le \mu ,\\ \mu \frac{s}{|s|},& \text{for }|s|>\mu .\end{array}$$

$$\displaystyle\int_{\Omega}\phi_{K}\big{\{}\mathcal{A}(x,\nabla u_{n})-\mathcal{A}(x,\nabla u)\big{\}}\cdot\nabla L_{\mu}(u_{n}-u)\,dx$$

Full text

[TABLE]

where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ with $N\geq 2$, $w$ belong to the Muckenhoupt class $A_{p}$ for some $1<p<\infty$, $f$ is a nonnegative function belong to some Lebesgue space and $\delta>0$.

1. Introduction

In this article, we establish existence, uniqueness and regularity results to the following degenerate singular elliptic boundary value problem:

[TABLE]

where $\delta>0$, $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ with $N\geq 2$ and $f$ is a nonnegative function belong to some Lebesgue space but not identically zero. The function $\mathcal{A}:\Omega\times\mathbb{R}^{N}\to\mathbb{R}^{N}$ is Carathéodory by which we mean

• •

the function $\mathcal{A}(\cdot,s)$ is measurable on $\Omega$ for every $s\in\mathbb{R}^{N}$, and

• •

the function $\mathcal{A}(x,\cdot)$ is continuous on $\mathbb{R}^{N}$ for a.e. $x\in\Omega$.

Moreover, the following additional hypothesis on the function $\mathcal{A}$ will be imposed throughout the paper.

• (H1)

For any $w$ belong to the Muckenhoupt class $A_{p}$ (defined in section 2),

• (H2)

(Growth) $|\mathcal{A}(x,\zeta)|\leq|\zeta|^{p-1}w(x)$, for a.e. $x\in\Omega$,  $\forall$  $\zeta\in\mathbb{R}^{N}$.

• (H3)

(Degeneracy) $\mathcal{A}(x,\zeta)\cdot\zeta\geq|\zeta|^{p}w(x)$, for a.e. $x\in\Omega$,  $\forall$  $\zeta\in\mathbb{R}^{N}$.

• (H4)

(Homogeneity) $\mathcal{A}(x,t\zeta)=t\,|t|^{p-2}\,\mathcal{A}(x,\zeta)$, for $t\in\mathbb{R}$, $t\neq 0$.

• (H5)

(Strong Monotonicity) For \gamma=\text{max}\,\big{\{}p,2\big{\}},

[TABLE]

for some positive constant $c$ where $\overline{\mathcal{A}}$ is defined as

[TABLE]

A prototype of the equation (1.1) is given by the following boundary value problem

[TABLE]

where $M(\cdot)$ is a continuous function with values in the set of $N\times N$ symmetric matrix satisfying

[TABLE]

In case of $M(x)=w(x)I$, where $I$ is the $N\times N$ identity matrix, the operator $L$ reduces to the weighted $p$-Laplace operator $\Delta_{p,w}$ defined by

[TABLE]

We observe that for $w=1$, $\Delta_{p,w}u=\Delta_{p}u$, which is the standard $p$-Laplace operator. For the constant weight $w$, singular problems of type (1.2) has been widely studied in the last three decades, see [2, 3, 4, 5, 7, 8, 9, 10, 20, 21, 22, 23, 30] and the references therein. We would like to point out some historical developments made in this direction which are closely related to the problem (1.2).

The case of $M=I$ and $p=2$ with Dirichlet boundary condition is settled in the pioneering work of Crandall et al [13], where the existence of a unique classical solution $u\in C^{2}(\Omega)\cap C(\overline{\Omega})$ to the equation (1.2) is proved for any $\delta>0$. This solution $u\in W_{0}^{1,2}(\Omega)$ if and only if $\delta<3$, and for $\delta>1$, $u$ does not belong to $C^{1}(\overline{\Omega})$ is proved by Lazer-McKenna [28] for a positive H$\ddot{\text{o}}$lder continuous data $f$.

Boccardo-Orsina [7] studied the semilinear case $p=2$ for a constant weight function $w$ and nonnegative (not identically zero) data $f$ in some Lebesgue space to obtain existence and regularity results for any $\delta>0$ to the problem (1.2). De Cave [14] generalised these results in the quasilinear case $1<p<N$. Further Canino et al [9] proved existence in addition to uniqueness results to the problem (1.2) in the full range $1<p<\infty$ for $M=I$. In summary depending on $p$ and the nonlinearity $f$, authors in [7, 9, 14] proved existence of a solution $u\in W_{0}^{1,p}(\Omega)$ if $0<\delta<1$ and $u\in W^{1,p}_{loc}(\Omega)$ such that $u^{\frac{\delta+p-1}{p}}\in W_{0}^{1,p}(\Omega)$ (this was the meaning of $u=0$ on $\partial\Omega$) if $\delta\geq 1$ to the problem (1.2). Moreover, we emphasize that when $0<\delta<1$ under the assumption $f\in L^{1}(\Omega)$, authors in [7] (for $p=2$) proved existence result in a larger Sobolev space than $W_{0}^{1,2}(\Omega)$, whereas if $2-\delta+\frac{\delta-1}{N}\leq p<N$, author in [14] proved existence result in a larger Sobolev space than $W_{0}^{1,p}(\Omega)$ to the problem (1.2). When $f$ is a Radon measure, existence results to singular $p$-Laplace equations has also been investigated in the recent past and we refer the reader to De Cave et al [15], De Cave-Oliva [16] and the references therein.

In contrast to [7, 9, 14], a natural question can be posed to say what happens to the equation (1.2) in the presence of a nonconstant weight function $w$? Indeed, our main motive in this paper is to answer this question affirmatively by providing a certain class of weight function (which may vanish or blow up near the origin) to ensure existence, uniqueness and regularity results analogous to [7, 9, 14] for the more general weighted singular problem (1.1).

We have started with choosing the weight function $w$ in the class of Muckenhoupt weight $A_{p}$ whose theory is well developed, see [11, 17, 19, 24, 26, 31, 33]. Such class of weights was firstly introduced by Muckenhoupt [31], where the author proved these are the only class of weights such that the Hardy-Littlewood maximal operator is bounded from the weighted Lebesgue space into itself and thus plays a very significant role in harmonic analysis.

Due to the presence of the weight function solutions of (1.1) are investigated in a weighted Sobolev space (see section $2$ for definition). We mainly adapt the approximation approach introduced by the authors in [7] along [9, 14] although there are some difficulties we will face in our setting. To be more precise, by regularizing the right hand size of (1.1) we prove existence of a uniform positve and bounded solutions to the approximated problem (3.2). But in contrast to [7], weak convergence is not enough to pass the limit in the approximated problem (3.2). In this concern a gradient convergence theorem is proved by the author in [14] which allows to pass the limit (see also [9]). Here, we establish a counterpart of gradient convergence theorem in our setting (see Theorem 2.11) by following the idea from Boccardo-Murat [6] in order to pass the limit in the equation (3.2) and obtain our existence results. The availability of embedding results in the classical Sobolev space $W^{1,p}(\Omega)$ (see [1, 18]) is one of the main ingredient in [7, 9, 14]. Such embeddings are not readily available in our setting which we establish here for a subclass of $A_{p}$ (see Theorem 2.6). Then following the idea from [7, 14] choosing suitable test functions into the equation (3.2) along with an application of our embedding theorem we obtain regularity results depending on the summability of $f$. Finally, to obtain uniqueness results, we establish a variational inequality (see Lemma 2.13) further avoiding the use of boundary continuity of solutions to the regularized $p$-Laplace equations (see e.g., [29, 32, 34, 35]) as implemented by the authors in [9].

Notations: Throughout the paper, the following notations will be used:

• •

$X:=W_{0}^{1,p}(\Omega,w)$.

• •

$X^{*}:=\text{Dual space of }X$.

• •

$||u||_{X}:=||u||_{1,p,w}$.

• •

$c,c_{i}$, $i\in\mathbb{N}$ will denote constants whose values may vary depending on the situation from line to line or even in the same line.

• •

$|S|:=$ Lebesgue measure of a set $S$.

• •

$T_{\eta}(s):=\text{min}\{\eta,s\}$ for $\eta>0$, $s\geq 0$.

• •

$B(x,r):$ Ball of radius $r$ with center $x$.

This paper is organized as follows: In section 2, we present some preliminary results. In section 3, existence and regularity results and in section 4, uniqueness results are proved.

2. Preliminaries

In this section, we present some basic properties of $A_{p}$ weights and a brief literature of the corresponding weighted Sobolev space. For a more general theory we refer the reader to look at the nice surveys by Drábek et al [17], Fabes et al [19], Heinonen et al [24] and Kilpe$\ddot{\text{l}}$ainen [26].

2.1. Muckenhoupt Weight

Definition 2.1**.**

Let $w$ be a locally integrable function in $\mathbb{R}^{N}$ such that $0<w<\infty$ a.e. in $\mathbb{R}^{N}$. Then for $1<p<\infty$, we say that $w$ belong to the Muckenhoupt class $A_{p}$ or $w$ is an $A_{p}$-weight, if there exists a positive constant $c_{p,w}$ (called the $A_{p}$ constant of $w$) depending only on $p$ and $w$ such that for all balls $B$ in $\mathbb{R}^{N}$,

[TABLE]

Example:

• •

$w(x)=|x|^{\alpha}\in A_{p}$ if and only if $-N<\alpha<N(p-1)$, see [24, 26].

Definition 2.2**.**

(Weighted Sobolev Space:) For any $w\in A_{p}$, define the weighted Sobolev space $W^{1,p}(\Omega,w)$ by

[TABLE]

where

[TABLE]

• •

Observe that if $0<c\leq w\leq d$ for some constants $c$ and $d$, the weighted Sobolev space $W^{1,p}(\Omega,w)$ becomes the classical Sobolev space $W^{1,p}(\Omega)$.

• •

The fact $w\in A_{p}$ implies $w\in L^{1}_{loc}(\Omega)$ and hence $C_{c}^{\infty}(\Omega)\subset W^{1,p}(\Omega,w)$. Therefore we can introduce the space

[TABLE]

• •

Both the spaces $W^{1,p}(\Omega,w)$ and $W^{1,p}_{0}(\Omega,w)$ are uniformly convex Banach spaces with respect to the norm $||.||_{1,p,w}$, see [24].

Definition 2.3**.**

We say that $u\in W^{1,p}_{loc}(\Omega,w)$ if and only if $u\in W^{1,p}(\Omega^{\prime},w)$ for every $\Omega^{\prime}\Subset\Omega$.

Theorem 2.4**.**

(Poincaré inequality [24]) For any $w\in A_{p}$, we have

[TABLE]

for some positive constant $c$ independent of $\phi$.

Using Theorem 2.4, an equivalent norm to $(\ref{norm1})$ on the space $W^{1,p}_{0}(\Omega,w)$ can be defined by

[TABLE]

2.2. Embedding Theorems

The following compactness result follows from Chua et al [11].

Theorem 2.5**.**

(Theorem 2.2, [11]) Let $w\in A_{p}$ with $1<p<\infty$, then the inclusion map

[TABLE]

is compact.

For the rest of the paper, we assume the weight function $w\in A_{s}$ unless otherwise stated where $A_{s}$ is a subclass of $A_{p}$ given by

[TABLE]

For example, $w(x)=|x|^{\alpha}$ with $-\frac{N}{s}<\alpha<\frac{N}{s}$ belong to $A_{s}$ for any s\in\big{[}\frac{1}{p-1},\infty\big{)}\cap\big{(}\frac{N}{p},\infty\big{)}, provided $1<p<N$. This subclass allows one to shift from the weighted Sobolev space into the classical Sobolev space using the idea of [17]. Indeed, we prove the following embedding theorem.

Theorem 2.6**.**

(Embedding from weighted to classical Sobolev space)

• •

For any $w\in A_{s}$, we have the following continuous inclusion map

[TABLE]

where $p_{s}=\frac{ps}{s+1}\in[1,p)$.

• •

Moreover, the above embeddings are compact except for $q=p_{s}^{*}$ in case of $1\leq p_{s}<N$.

• •

The same result holds for the space $W_{0}^{1,p}(\Omega,w)$.

Proof.

Let $u\in W^{1,p}(\Omega,w)$. Since $\frac{p}{p_{s}}>1$, using H$\ddot{\mbox{o}}$lder inequality with exponents $\frac{p}{p_{s}}$ and $(\frac{p}{p_{s}})^{\prime}=s+1$, we obtain

[TABLE]

which implies

[TABLE]

Replacing $u$ by $\nabla u$, similarly we obtain

[TABLE]

Adding (2.3) and (2.4) we have

[TABLE]

Hence the embedding

[TABLE]

is continuous. The rest of the proof follows from the classical Sobolev embedding theorem (Theorem 1.1.3 of Ambrosetti-Arcoya [1]). ∎

Remark 2.7**.**

Observe that the fact s\in\big{[}\frac{1}{p-1},\infty\big{)}\cap\big{(}\frac{N}{p},\infty\big{)} implies that $p_{s}^{*}>p$. Therefore, by Theorem 2.6 there exists a constant $q>p$ such that the inclusion

[TABLE]

is continuous. The existence of such $q$ is an important tool to prove some a priori estimates later, see [17] for more applications.

Now we state two important theorems on $\mathcal{A}$ superharmonic functions, for the definition of such functions we refer the reader to [24].

Theorem 2.8**.**

(Theorem 7.12, [24]) A nonconstant $\mathcal{A}$ superharmonic function cannot attain its infimum in $\Omega$.

Theorem 2.9**.**

(Corollary 7.18, [24]) If $u\in W^{1,p}_{loc}(\Omega,w)$ is a weak supersolution of the equation

[TABLE]

in $\Omega$, i.e.

[TABLE]

whenever $\phi\in C^{\infty}_{c}(\Omega)$ is nonnegative, then there exists $\mathcal{A}$ superharmonic function $v$ such that $v=u$ a.e.

Theorem 2.10**.**

Let $u\in W^{1,p}_{loc}(\Omega,w)$ be positive a.e. in $\Omega$ and $\alpha\geq 1$ be such that $u^{\alpha}\in X$. Then for every $\epsilon>0$, we have $(u-\epsilon)^{+}\in X$.

Proof.

Since $u^{\alpha}\in X$, there exists a sequence of nonnegative functions $\{\phi_{n}\}\in C_{c}^{\infty}(\Omega)$ such that $\phi_{n}$ converges to $u^{\alpha}$ in the norm of $X$. Set

[TABLE]

Observe that, since $\alpha\geq 1$, one has

[TABLE]

where $M$ is a constant independent of $n$, since $||\phi_{n}||_{X}\leq c$ for some positive constant $c$ independent of $n$. Therefore, the sequence $v_{n}$ is uniformly bounded in $X$ and by the reflexivity of $X$, it follows that $(u-\epsilon)^{+}\in X$. ∎

Theorem 2.11**.**

(Gradient Convergence Theorem) Given $n\in\mathbb{N}$ and $w\in A_{s}$, consider the following equation

[TABLE]

Assume that $u_{n}\to u$ weakly in $W^{1,p}(\Omega,w)$. In addition, suppose $G_{n}$ satisfies

[TABLE]

for all $\phi\in C_{c}^{\infty}(\Omega)\,\,\mbox{with\,\,support}\,\phi\subset\,K$, where $C_{K}$ is a constant depending on $K$. Then, upto a subsequence $\nabla u_{n}\to\nabla u$ pointwise a.e. in $\Omega$.

Proof.

In the unweighted case this theorem is proved in Theorem 2.1 of [6] and following the same arguments we present the proof in the weighted case as follows:

Step 1. Fix a compact set $K\subset\Omega$ and a function $\phi_{K}\in C_{c}^{\infty}(\Omega)$ such that $0\leq\phi_{K}\leq 1$ and $\phi_{K}\equiv 1$ on $K$. Define the truncated function

[TABLE]

Then $v_{n}=\phi_{K}\,L_{\mu}(u_{n}-u)\in W_{0}^{1,p}(\Omega,w)$ with compact support.

[TABLE]

Now,

[TABLE]

Since $u_{n}\to u$ weakly in $W^{1,p}(\Omega,w)$, by Theorem 2.5 the sequence $I_{n}$ converges to [math] as $n\to\infty$. Moreover, since the sequence $L_{\mu}(u_{n}-u)\to 0$ weakly in $W^{1,p}(\Omega,w)$ as $n\to\infty$, one has the sequence

[TABLE]

converges to [math] as $n\to\infty$. Now, by the given condition we have $|<G_{n},v_{n}>|\leq c_{K}\mu$.

Step 2. Fix $\theta\in(0,1)$ and define the sequence of function

[TABLE]

We denote by

[TABLE]

Therefore

[TABLE]

By Theorem 2.6, we have $u_{n}\to u$ strongly in $L^{p_{s}}(\Omega)$. Therefore $|G_{n}^{\mu}|\to 0\text{ as }n\to\infty.$ Also, the sequence $\{e_{n}\}$ is bounded in $L^{1}(\Omega)$, since

[TABLE]

for some constant $M$ independent of $n$. By Step $1$ and the fact $\phi_{K}\equiv 1$ on $K$, we obtain

[TABLE]

Letting $\mu\to 0$, we have $e_{n}^{\theta}\to 0$ in $L^{1}(K)$. Therefore upto a subsequence $e_{n}(x)\to 0$ a.e. in $\Omega$ and using the hypothesis (H5), we obtain upto a subsequence $\nabla u_{n}\to\nabla u$ pointwise a.e. in $\Omega$. ∎

Moreover, we will use the following three important results, see Ciarlet [12] for Theorem 2.12-Theorem 2.13 and Kinderlehrer-Stampacchia [27] for Theorem 2.14 respectively.

Theorem 2.12**.**

(Theorem 9.14, [12]) Let $V$ be a real reflexive Banach space and let $A:V\to V^{*}$ be a coercive and demicontinuous monotone operator. Then $A$ is surjective, i.e., given any $f\in V^{*}$, there exists $u\in V$ such that $A(u)=f$. If $A$ is strictly monotone, then $A$ is also injective.

Theorem 2.13**.**

([12]) Let $U$ be a nonempty closed and convex subset of a real separable reflexive Banach space and let $A:V\to V^{*}$ be a coercive and demicontinuous monotone operator. Then for every $f\in V^{*}$, there exists $u\in U$ such that

[TABLE]

Moreover, if $A$ is strictly monotone, then $u$ is unique.

Theorem 2.14**.**

(Lemma B.1, [27]) Let $\phi(t),\,k_{0}\leq t<\infty,$ be nonnegative and nonincreasing such that

[TABLE]

where $c,l,m$ are positive constants with $m>1$. Then $\phi(k_{0}+d)=0,$ where

[TABLE]

3. Existence and regularity results

Definition 3.1**.**

A function $u\in W^{1,p}_{loc}(\Omega,w)$ is said to be a weak solution of the problem (1.1), if for every $K\Subset\Omega$, there exists a positive constant $c_{K}$ such that $u\geq c_{K}>0$ in $K$ and for all $\phi\in C_{c}^{1}(\Omega)$, one has

[TABLE]

where by $u=0$ on $\partial\Omega$, we mean that for some $\alpha\geq 1$, the function $u^{\alpha}\in X$.

Our main existence and regularity results in this paper reads as follows:

3.1. The case $0<\delta<1$

Theorem 3.2**.**

For any $0<\delta<1$, the problem (1.1) has at least one weak solution in $X$, if

1. (a)

f\in L^{m}(\Omega),\,m=\big{(}\frac{p_{s}^{*}}{1-\delta}\big{)}^{{}^{\prime}}, provided $1\leq p_{s}<N$, or 2. (b)

$f\in L^{m}(\Omega)$* for some $m>1$, provided $p_{s}=N$, or* 3. (c)

$f\in L^{1}(\Omega)$* for $p_{s}>N$.*

Theorem 3.3**.**

Let $0<\delta<1$, then the solution $u$ given by Theorem 3.2 satisfies the following properties:

1. (a)

For $1\leq p_{s}<N,$

1. (i)

if $f\in L^{m}(\Omega)$ for some m\in\big{[}(\frac{p_{s}^{*}}{1-\delta})^{{}^{\prime}},\frac{p_{s}^{*}}{p_{s}^{*}-p}\big{)}, then $u\in L^{t}(\Omega)$, $t=p_{s}^{*}\,\gamma$ where $\gamma=\frac{(\delta+p-1)m^{{}^{\prime}}}{(pm^{{}^{\prime}}-p_{s}^{*})}$. 2. (ii)

if $f\in L^{m}(\Omega)$ for some $m>\frac{p_{s}^{*}}{p_{s}^{*}-p}$, then $u\in L^{\infty}(\Omega)$. 2. (b)

Let $p_{s}=N$ and assume $q>p$. Then if $f\in L^{m}(\Omega)$ for some m\in\big{(}(\frac{q}{1-\delta})^{\prime},\frac{q}{q-p}\big{)}, we have $u\in L^{t}(\Omega)$, $t=p\,\gamma$ where $\gamma=\frac{pm^{\prime}}{pm^{\prime}-q}$. 3. (c)

For $p_{s}>N$ and $f\in L^{1}(\Omega)$, we have $u\in L^{\infty}(\Omega)$.

3.2. The case $\delta=1$

Theorem 3.4**.**

For $\delta=1$ with any $p_{s}$, the problem (1.1) has at least one weak solution in $X$, provided $f\in L^{1}(\Omega)$.

Theorem 3.5**.**

Let $\delta=1$, then the solution $u$ given by Theorem 3.4 satisfies the following properties:

1. (a)

For $1\leq p_{s}<N$,

1. (i)

if $f\in L^{m}(\Omega)$ for some m\in\big{(}1,\frac{p_{s}^{*}}{p_{s}^{*}-p}\big{)}, then $u\in L^{t}(\Omega)$, $t=p_{s}^{*}\gamma$, where $\gamma=\frac{pm^{{}^{\prime}}}{(pm^{{}^{\prime}}-p_{s}^{*})}$. 2. (ii)

if $f\in L^{m}(\Omega)$ for some $m>\frac{p_{s}^{*}}{p_{s}^{*}-p}$, then $u\in L^{\infty}(\Omega)$. 2. (b)

Let $p_{s}=N$ and $q>p$. Then if $f\in L^{m}(\Omega)$ for some m\in\big{(}1,\frac{q}{q-p}\big{)}, we have $u\in L^{t}(\Omega)$, $t=q\,\gamma$, where $\gamma=\frac{pm^{{}^{\prime}}}{pm^{{}^{\prime}}-q}$. 3. (c)

For $p_{s}>N$ and $f\in L^{1}(\Omega)$, we have $u\in L^{\infty}(\Omega)$.

3.3. The case $\delta>1$

Theorem 3.6**.**

For $\delta>1$ with any $p_{s}$, the problem (1.1) has at least one weak solution, say $u$ in $W^{1,p}_{loc}(\Omega,w)$ such that $u^{\frac{\delta+p-1}{p}}\in X$, provided $f\in L^{1}(\Omega)$.

Theorem 3.7**.**

Let $\delta>1,$ then the solution $u$ given by Theorem 3.6 satisfies the following properties:

1. (a)

For $1\leq p_{s}<N,$

1. (i)

if $f\in L^{m}(\Omega)$ for some m\in\big{(}1,\frac{p_{s}^{*}}{p_{s}^{*}-p}\big{)}, then $u\in L^{t}(\Omega)$ where $t=p_{s}^{*}\,\gamma$, where $\gamma=\frac{(\delta+p-1)m^{\prime}}{pm^{\prime}-p_{s}^{*}}$. 2. (ii)

if $f\in L^{m}(\Omega)$ some $m>\frac{p_{s}^{*}}{p_{s}^{*}-p}$, then $u\in L^{\infty}(\Omega)$. 2. (b)

Let $p_{s}=N$ and assume $q>p$. Then if $f\in L^{m}(\Omega)$ for some m\in\big{(}1,\frac{q}{q-p}\big{)}, we have $u\in L^{t}(\Omega)$, $t=q\,\gamma$, where $\gamma=\frac{(\delta+p-1)m^{{}^{\prime}}}{pm^{{}^{\prime}}-q}$. 3. (c)

For $p_{s}>N$ and $f\in L^{1}(\Omega)$, we have $u\in L^{\infty}(\Omega)$.

3.4. Preliminaries

For $n\in\mathbb{N}$, define f_{n}(x):=\text{min}\,\big{\{}f(x),n\big{\}} and consider for $\delta>0$, the approximated problem

[TABLE]

Definition 3.8**.**

A function $u\in X$ is said to be a weak solution of the problem (3.2) if $u>0$ in $\Omega$ and for all $\phi\in X$, one has

[TABLE]

Define the operator $J:X\to X^{*}$ by

[TABLE]

Lemma 3.9**.**

$J$* is a surjective and strictly monotone operator.*

Proof.

The proof follows applying Theorem 2.12, since

1. (1)

Boundedness: Using the H$\ddot{\text{o}}$lder’s inequality and hypothesis (H2) we obtain

[TABLE]

Hence $J$ is bounded. 2. (2)

Demicontinuity: Let $u_{n}\to u$ in the norm of $X$, then $w^{\frac{1}{p}}\nabla u_{n}\to w^{\frac{1}{p}}\nabla u$ in $L^{p}(\Omega)$. Therefore upto a subsequence $u_{n_{k}}$ of $u_{n}$, we have $\nabla u_{n_{k}}(x)\to\nabla u(x)$ pointwise for a.e. $x\in\Omega$. Since the function $\mathcal{A}(x,\cdot)$ is continuous in the second variable, we have

[TABLE]

pointwise for a.e. $x\in\Omega.$ Now using the growth condition (H2), we obtain

[TABLE]

where $||u_{n_{k}}||_{X}\leq c$. Therefore since the sequence w^{-\frac{1}{p}}\mathcal{A}\big{(}x,\nabla u_{n_{k}}(x)\big{)} is uniformly bounded in $L^{\frac{p}{p-1}}(\Omega),$ we have w^{-\frac{1}{p}}\mathcal{A}\big{(}x,\nabla u_{n_{k}}(x)\big{)}\to w^{-\frac{1}{p}}\mathcal{A}\big{(}x,\nabla u(x)\big{)} weakly in $L^{\frac{p}{p-1}}(\Omega)$, see Jakszto [25]. Since the weak limit is independent of the choice of the subsequence $u_{n_{k}},$ it follows that

[TABLE]

weakly. Now $\phi\in X$ implies the function $w^{\frac{1}{p}}\nabla\phi\in L^{p}(\Omega)$ and therefore by the weak convergence, we obtain

[TABLE]

$\mbox{as}\;n\to\infty$ and hence $J$ is demicontinuous. 3. (3)

Coercivity: Using (H3), we have the inequality

[TABLE]

Therefore $J$ is coercive. 4. (4)

Strict monotonicity: Using the strong monotonicity condition (H5), for all $u\neq v\in X$, we have

[TABLE]

∎

Lemma 3.10**.**

The operator $J^{-1}:X^{*}\to X$ is bounded and continuous.

Proof.

Using the H$\ddot{\text{o}}$lder’s inequality for all $u,v\in X$, we have the estimate

[TABLE]

which implies the operator $J^{-1}$ is bounded. Suppose by contradiction $J^{-1}$ is not continuous, then there exists $g_{k}\to g$ strongly in $X^{*}$ such that $||J^{-1}(g_{k})-J^{-1}(g)||_{X}\geq\gamma$ for some $\gamma>0.$ Denote by $u_{k}=J^{-1}(g_{k})$ and $u=J^{-1}(g).$ Therefore, using (H3) we have

[TABLE]

which implies

[TABLE]

Since $g_{k}\to g$ strongly in $X^{*}$, we have the sequence $\{u_{k}\}$ uniformly bounded in $X.$ Therefore upto subsequence there exists $u^{1}\in X$ such that $u_{k}\to u^{1}$ weakly in $X$. Now

[TABLE]

Since $J(u_{k})\to J(u)$ in $X^{*}$ and $u_{k}\to u^{1}$ weakly in $X$, both the terms

[TABLE]

converges to [math] as $k\to\infty$. Therefore,

[TABLE]

Putting $v=u_{k}$ and $u=u^{1}$ in the inequality (3.4) we obtain $||u_{k}||_{X}\to||u^{1}||_{X}.$ Therefore by the uniform convexity of $X,$ it follows that $u_{k}\to u^{1}$ in $X$ which together with the convergence $J(u_{k})\to J(u)$ in $X^{*}$ implies that $J(u^{1})=J(u)$. Now the injectivity of $J$ implies $u=u^{1}$, a contradiction to our assumption. Hence $J^{-1}$ is continuous. ∎

Lemma 3.11**.**

Let $\zeta_{k}$, $\zeta\in X$ satisfies,

[TABLE]

for all $\phi\in X$ where $<,>$ denotes the dual product between $X^{*}$ and $X$. If $h_{k}\to h$ in $X^{*}$, then we have $\zeta_{k}\to\zeta$ in $X$.

Proof.

By the strict monotonicity of $J$, we have $J(\zeta)=h$ and $J(\zeta_{k})=h_{k}$. Therefore applying Lemma 3.10, if $h_{k}\to h$ in $X^{*}$ then $J^{-1}(h_{k})\to J^{-1}(h)$ i.e. $\zeta_{k}\to\zeta$ as $k\to\infty$. Hence the proof. ∎

Using Lemma 3.9, we can define the operator $A:L^{p_{s}}(\Omega)\to X$ by $A(v)=u$ where $u\in X$ is the unique weak solution of the problem

[TABLE]

i.e., for all $\phi\in X$,

[TABLE]

Lemma 3.12**.**

The map $A:L^{p_{s}}(\Omega)\to X$ is continuous as defined above.

Proof.

Let $v_{k}\to v$ in $L^{p_{s}}(\Omega)$. Suppose $A(v_{k})=\zeta_{k}$ and $A(v)=\zeta$. Then for every fixed $n\in\mathbb{N}$ and for all $\phi\in X$, we have

[TABLE]

Denote by

[TABLE]

Now, by Theorem 2.6, one has

[TABLE]

Now since \big{|}{\big{(}|v_{k}|+\frac{1}{n}\big{)}^{-\delta}}-{\big{(}|v|+\frac{1}{n}\big{)}^{-\delta}}\big{|}\leq 2n^{\delta+1} and $v_{k}\to v$ in $L^{p_{s}}(\Omega)$, upto a subsequence $v_{k_{l}}\to v$ pointwise a.e. in $\Omega$. As a consequence of the Lebesgue dominated theorem, we obtain \Big{|}\Big{|}{\big{(}|v_{k_{l}}|+\frac{1}{n}\big{)}^{-\delta}}-{\big{(}|v|+\frac{1}{n}\big{)}^{-\delta}}\Big{|}\Big{|}_{L^{p_{s}^{{}^{\prime}}}(\Omega)}\to 0 as $k_{l}\to\infty$. Since the limit is independent of the choice of the subsequence, we have \Big{|}\Big{|}{\big{(}|v_{k}|+\frac{1}{n}\big{)}^{-\delta}}-{\big{(}|v|+\frac{1}{n}\big{)}^{-\delta}}\Big{|}\Big{|}_{L^{p_{s}^{{}^{\prime}}}(\Omega)}\to 0 as $k\to\infty$. Therefore by Lemma 3.11, we have $\zeta_{k}\to\zeta$ as $k\to\infty$. Hence $A:L^{p_{s}}(\Omega)\to X$ is a continuous map. ∎

Theorem 3.13**.**

For any $p_{s}\geq 1$ the following holds:

1. (1)

The problem (3.2) has a unique weak solution, say $u_{n}$ in $X\cap L^{\infty}(\Omega)$ for every fixed $n\in\mathbb{N}$, 2. (2)

$u_{n+1}\geq u_{n}$* for every $n\in\mathbb{N}$, and* 3. (3)

For every $K\Subset\Omega$ there exists a positive constant $C_{K}$ (independent of $n$) such that $u_{n}\geq C_{K}>0$ in $K$.

Proof.

1. (1)

Existence: Define

[TABLE]

Let $v_{i}\in S$ and $A(v_{i})=u_{i}$ for $i=1,2$. Using $u_{i}$ as test function in (3.5) we obtain

[TABLE]

where $c(n)$ is a constant depending on $n$ but not on $u_{i}$, $i=1,2$. Therefore, by Lemma 3.12 and the compactness of the inclusion

[TABLE]

together with the inequality $(\ref{uniform estimate for fixed n})$, it follows that the map

[TABLE]

Observe that,

[TABLE]

Hence the set $S$ is bounded in $L^{p_{s}}(\Omega).$ By Schaefer’s Fixed Point Theorem, there exists a fixed point of the map $A$, say $u_{n}$ i.e. $A(u_{n})=u_{n}$ and hence $u_{n}\in X$ is a solution of (3.2).

$L^{\infty}$-estimate: For any $k>1$, define the set

[TABLE]

Choosing

[TABLE]

as a test function in (3.3) together with the H$\ddot{\text{o}}$lder inequality and Remark 2.7, we obtain

[TABLE]

Therefore we get

[TABLE]

where $c$ depends on $n$. Now for $1<k<h$, by the Remark 2.7, we obtain

[TABLE]

Hence we obtain the inequality

[TABLE]

Now $q>p$ implies $\frac{p^{\prime}q}{pq^{\prime}}>1$, therefore by Theorem 2.14, we obtain

[TABLE]

where $c$ is a constant dependent on $n$. 2. (2)

Monotonicity: Let $u_{n}$ and $u_{n+1}$ satisfies the equations

[TABLE]

and

[TABLE]

respectively for all $\phi\in X$. Choosing $\phi=(u_{n}-u_{n+1})^{+}\in X$ and using the inequality $f_{n}(x)\leq f_{n+1}(x)$ we obtain after subtracting the equation (3.7) from (3.8)

[TABLE]

Again using the strong monotonicity condition (H5), we have

• •

for $p\geq 2$,

[TABLE]

• •

and for $1<p<2$,

[TABLE]

which gives $u_{n+1}\geq u_{n}$.

Uniqueness: The uniqueness of $u_{n}$ follows by arguing similarly as in monotonicity. 3. (3)

Choosing \phi=\text{min}\big{\{}u_{n},0\big{\}} as a test function in the equation (3.3) we get $u_{n}\geq 0$ in $\Omega$. Applying Theorem 2.8 we get $u_{1}>0$ in $\Omega$. Hence by the monotonicity and Theorem 2.9 there exists $C_{K}>0$ (Independent of $n$) such that $u_{n}\geq C_{K}>0$ for every $K\Subset\Omega$.

∎

3.5. Proof of the existence and regularity results

3.6. The case $0<\delta<1$

Proof of Theorem 3.2.

Let $0<\delta<1$.

1. (a)

Let $1\leq p_{s}<N$. Choosing $\phi=u_{n}\in X$ as a test function in the equation (3.3) and using H$\ddot{\text{o}}$lder’s inequality together with the continuous embedding $X\hookrightarrow L^{p_{s}^{*}}(\Omega)$ we obtain

[TABLE]

Since $\delta+p-1>0$, we have $||u_{n}||_{X}\leq c,$ where $c$ is a constant independent of $n$. Therefore one can apply Theorem 2.11 to conclude that upto a subsequence $\nabla u_{n_{k}}\to\nabla u$ pointwise a.e. in $\Omega$. Since the function $\mathcal{A}(x,\cdot)$ is continuous, we have w^{{-}\frac{1}{p}}(x)\,\mathcal{A}\big{(}x,\nabla u_{n_{k}}(x)\big{)}\to w^{-\frac{1}{p}}(x)\,\mathcal{A}\big{(}x,\nabla u(x)\big{)} pointwise for a.e. $x\in\Omega$. Now we observe that

[TABLE]

Since the sequence $w^{-\frac{1}{p}}\mathcal{A}(x,\nabla u_{n_{k}})$ is uniformly bounded in $L^{\frac{p}{p-1}}(\Omega)$, the sequence w^{-\frac{1}{p}}\mathcal{A}\big{(}x,\nabla u_{n_{k}}(x)\big{)}\to w^{-\frac{1}{p}}\mathcal{A}\big{(}x,\nabla u(x)\big{)} weakly in $L^{\frac{p}{p-1}}(\Omega)$. As the weak limit is independent of the choice of the subsequence $u_{n_{k}},$ it follows that w^{-\frac{1}{p}}\mathcal{A}\big{(}x,\nabla u_{n}(x)\big{)}\to w^{-\frac{1}{p}}\mathcal{A}\big{(}x,\nabla u(x)\big{)} weakly. Now $\phi\in X$ implies the function $w^{\frac{1}{p}}\nabla\phi\in L^{p}(\Omega)$ and hence by the weak convergence, we obtain

[TABLE]

Moreover, by Theorem 3.13 we have $u\geq u_{n}\geq c_{K}>0$ for every $K\Subset\Omega$. Since for $\phi\in C_{c}^{1}(\Omega)$, one has

[TABLE]

and \frac{f_{n}}{\big{(}u_{n}+\frac{1}{n}\big{)}^{\delta}}\,\phi\to\frac{f}{u^{\delta}}\,\phi pointwise a.e. in $\Omega$ as $n\to\infty$, by the Lebesgue dominated convergence theorem we obtain

[TABLE]

Therefore we have for all $\phi\in C_{c}^{1}(\Omega)$,

[TABLE]

and hence $u\in X$ is a weak solution of (1.1). 2. (b)

Let $p_{s}=N$. Choosing $\phi=u_{n}\in X$ as a test function in (3.3) and using H$\ddot{\text{o}}$lder inequality together with the continuous embedding $X\hookrightarrow L^{q}(\Omega),\,\,q\in[1,\infty),$ we obtain

[TABLE]

where $c$ is a constant independent of $n$. Since $\delta+p-1>0$ we have the sequence $\{u_{n}\}$ is uniformly bounded in $X$. Now arguing similarly as in case (a) we obtain the required result. 3. (c)

Let $p_{s}>N$. Choosing $\phi=u_{n}\in X$ as a test function in (3.3) and using H$\ddot{\text{o}}$lder inequality together with the continuous embedding $X\hookrightarrow L^{\infty}(\Omega)$ we obtain

[TABLE]

Since $\delta+p-1>0$, we have $||u_{n}||_{X}\leq c,$ where $c$ is a constant independent of $n$. Therefore the sequence $\{u_{n}\}$ is uniformly bounded in $X$. Arguing similarly as in (a) we obtain the required result.

∎

Proof of Theorem 3.3.

1. (a)

Let $1\leq p_{s}<N$, then $p_{s}^{*}>p$.

1. (i)

We observe that

• •

for m=\big{(}\frac{p_{s}^{*}}{1-\delta}\big{)}^{\prime} i.e., $(1-\delta)m^{\prime}=p_{s}^{*}$, we have $\gamma=\frac{(\delta+p-1)m^{{}^{\prime}}}{(pm^{{}^{\prime}}-p_{s}^{*})}=1$ and

• •

m\in\big{(}(\frac{p_{s}^{*}}{1-\delta})^{{}^{\prime}},\frac{p_{s}^{*}}{p_{s}^{*}-p}\big{)} implies $\gamma=\frac{(\delta+p-1)m^{{}^{\prime}}}{(pm^{{}^{\prime}}-p_{s}^{*})}>1$.

Note that $(p\gamma-p+1-\delta)m^{\prime}=p_{s}^{*}\gamma$ and choosing $\phi=u_{n}^{p\gamma-p+1}\in X$ as a test function in $(\ref{weak solution of the approximated problem})$ we obtain

[TABLE]

Now using the continuous embedding $X\hookrightarrow L^{p_{s}^{*}}(\Omega)$ and the fact $\frac{p}{p_{s}^{*}}-\frac{1}{m^{\prime}}>0$ we obtain $||u_{n}^{\gamma}||_{L^{p_{s}^{*}}(\Omega)}\leq c,$ where $c$ is independent of $n$ implies the sequence $\{u_{n}^{\gamma}\}$ is uniformly bounded in $L^{t}(\Omega)$ where $t=p_{s}^{*}\gamma$. Therefore the pointwise limit $u$ belong to $L^{t}(\Omega)$ e.g., see [25]. 2. (ii)

Let $m>\frac{p_{s}^{*}}{p_{s}^{*}-p}$ and for $k>1$, choosing $\phi_{k}=(u_{n}-k)^{+}\in X$ as a test function in (3.3) we obtain after using H$\ddot{\text{o}}$lder’s and Young’s inequality with $\epsilon\in(0,1)$

[TABLE]

where A(k)=\big{\{}x\in\Omega:u_{n}\geq k\text{ a.e. in }\Omega\big{\}}. Since $m>\frac{p_{s}^{*}}{p_{s}^{*}-p}$, we have $m>p_{s}^{*}{{}^{\prime}}$. Using H$\ddot{\text{o}}$lder’s inequality in the above estimate we obtain

[TABLE]

where $c$ is a constant independent of $n$. Now using the continuous embedding $X\hookrightarrow L^{p_{s}^{*}}(\Omega)$ we obtain for $1<k<h$,

[TABLE]

Therefore

[TABLE]

Since ${\frac{p^{\prime}p_{s}^{*}}{pp_{s}^{*}{{}^{\prime}}}\frac{1}{(\frac{m}{p_{s}^{*}{{}^{\prime}}})^{{}^{\prime}}}}>1$, by Theorem 2.14, we have $||u_{n}||_{L^{\infty}(\Omega)}\leq c,$ where $c$ is a constant independent of $n$. Therefore we have $u\in L^{\infty}(\Omega)$. 2. (b)

Let $p_{s}=N$ and $q>p$. Observe that

• •

for m=\big{(}\frac{q}{1-\delta}\big{)}^{\prime} i.e., $(1-\delta)m^{\prime}=q$, we have $\gamma=\frac{(\delta+p-1)m^{{}^{\prime}}}{(pm^{{}^{\prime}}-q)}=1$ and

• •

m\in\big{(}(\frac{q}{1-\delta})^{{}^{\prime}},\frac{pm^{\prime}}{pm^{\prime}-q}\big{)} implies $\gamma=\frac{(\delta+p-1)m^{{}^{\prime}}}{(pm^{{}^{\prime}}-q)}>1$.

Note that $(p\gamma-p+1-\delta)m^{\prime}=q\,\gamma$ and choosing $\phi=u_{n}^{p\gamma-p+1}\in X$ as a test function in $(\ref{weak solution of the approximated problem})$ we obtain

[TABLE]

Now using the continuous embedding $X\hookrightarrow L^{q}(\Omega)$ and the fact $\frac{p}{q}-\frac{1}{m^{\prime}}>0$ we obtain $||u_{n}^{\gamma}||_{L^{q}(\Omega)}\leq c,$ where $c$ is independent of $n$ implies the sequence $\{u_{n}^{\gamma}\}$ is uniformly bounded in $L^{t}(\Omega)$ where $t=q\,\gamma$. Therefore $u$ belong to $L^{t}(\Omega)$. 3. (c)

Follows from Theorem 3.2 using the continuous embedding $X\hookrightarrow L^{\infty}(\Omega)$.

∎

3.7. The case $\delta=1$

Proof of Theorem 3.4.

Let $\delta=1$ and $f\in L^{1}(\Omega)$. Then choosing $\phi=u_{n}\in X$ as a test function in $(\ref{weak solution of the approximated problem})$ for any $p_{s}\geq 1$, we obtain $||u_{n}||^{p}_{X}\leq||f||_{L^{1}(\Omega)}.$ Now arguing similarly as in Theorem 3.2 we obtain the existence of weak solution $u\in X$ of (1.1). ∎

Proof of Theorem 3.5.

1. (a)

Let $1\leq p_{s}<N$, then $p_{s}^{*}>p$.

1. (i)

Observe that m\in\big{(}1,\frac{p_{s}^{*}}{p_{s}^{*}-p}\big{)} implies $\gamma=\frac{pm^{{}^{\prime}}}{(pm^{{}^{\prime}}-p_{s}^{*})}>1$. Now choosing $\phi=u_{n}^{p\gamma-p+1}\in X$ as a test function in $(\ref{weak solution of the approximated problem})$ together with the continuous embedding $X\hookrightarrow L^{p_{s}^{*}}(\Omega)$ and arguing similarly as in part (i) of Theorem 3.3 we obtain the required result. 2. (ii)

Follows arguing similarly as in part (ii) of Theorem 3.3. 2. (b)

Let $p_{s}=N$ and $q>p$. Observe that m\in\big{(}1,\frac{q}{q-p}\big{)} implies $\gamma=\frac{pm^{\prime}}{pm^{\prime}-q}>1$. Choosing $\phi=u_{n}^{p\gamma-p+1}\in X$ as a test function in $(\ref{weak solution of the approximated problem})$ together with the continuous embedding $X\hookrightarrow L^{q}(\Omega)$ and proceeding similarly as in part (b) of Theorem 3.3 we obtain the required result. 3. (c)

Follows from Theorem 3.4 using the continuous embedding $X\hookrightarrow L^{\infty}(\Omega)$.

∎

3.8. The case $\delta>1$

Proof of Theorem 3.6.

Let $\delta>1$ and $f\in L^{1}(\Omega)$ with $p_{s}\geq 1$. By Theorem 3.13 for every fixed $n\in\mathbb{N}$ we have $u_{n}\in L^{\infty}(\Omega)$ (the bound may depend on $n$). Choosing $\phi=u_{n}^{\delta}\in X$ as a test function in $(\ref{weak solution of the approximated problem})$ (which is admissible since $\delta>1$ and $u_{n}\in L^{\infty}(\Omega)$ by Theorem 3.13) we obtain

[TABLE]

which implies

[TABLE]

where $c$ is independent of $n$. Therefore the sequence \Big{\{}u_{n}^{\frac{\delta+p-1}{p}}\Big{\}} is uniformly bounded in $X$. Let $\phi\in C_{c}^{\infty}(\Omega)$ and consider $v_{n}=\phi^{p}\,u_{n}\in X$. We observe that

[TABLE]

and using Young’s inequality for $\epsilon\in(0,1)$, we obtain for some positive constant $c_{\epsilon}$,

[TABLE]

Now choosing $\phi=v_{n}\in X$ as a test function in (3.3) and using the estimates (3.9), (3.10), we obtain

[TABLE]

where $K$ is the support of $\phi$ and $c_{\phi}$ is a constant depending on $\phi$. Therefore we have

[TABLE]

Now since the sequence \Big{\{}u_{n}^{\frac{\delta+p-1}{p}}\Big{\}} is uniformly bounded in $X$ we have the sequence $\{u_{n}\}$ is uniformly bounded in $W^{1,p}_{loc}(\Omega,w)$. Now arguing similarly as in Theorem 3.2, we obtain $u\in W^{1,p}_{loc}(\Omega,w)$ is a weak solution of (1.1). The fact that $u^{\frac{\delta+p-1}{p}}\in X$ follows from the uniform boundedness of the sequence \Big{\{}u_{n}^{\frac{\delta+p-1}{p}}\Big{\}} in $X$. ∎

Proof of Theorem 3.7.

1. (a)

Let $1\leq p_{s}<N$, then $p_{s}^{*}>p$.

1. (i)

Observe that m\in\big{(}1,\frac{p_{s}^{*}}{p_{s}^{*}-p}\big{)} implies $\gamma=\frac{(\delta+p-1)m^{\prime}}{pm^{\prime}-p_{s}^{*}}>\frac{\delta+p-1}{p}>1$, since $\delta>1$. Now choosing $\phi=u_{n}^{p\gamma-p+1}\in X$ as a test function in $(\ref{weak solution of the approximated problem})$ together with the continuous embedding $X\hookrightarrow L^{p_{s}^{*}}(\Omega)$ and arguing similarly as in part (i) of Theorem 3.3 the result follows. 2. (ii)

Follows by arguing similarly as in part (ii) of Theorem 3.3. 2. (b)

Let $p_{s}=N$ and $q>p$. Observe that $\delta>1$, m\in\big{(}1,\frac{q}{q-p}\big{)} implies $\gamma=\frac{(\delta+p-1)m^{\prime}}{pm^{\prime}-q}>1$. Choosing $\phi=u_{n}^{p\gamma-p+1}\in X$ as a test function in $(\ref{weak solution of the approximated problem})$ together with the continuous embedding $X\hookrightarrow L^{q}(\Omega)$ and proceeding similarly as in part (b) of Theorem 3.3 we obtain the required result. 3. (c)

Follows from Theorem 3.6 using the continuous embedding $X\hookrightarrow L^{\infty}(\Omega)$.

∎

4. Uniqueness results

In this section we state and prove our main uniqueness results.

4.1. The case $0<\delta\leq 1$

Theorem 4.1**.**

For any $0<\delta\leq 1$ and $w\in A_{p}$, the problem (1.1) admits at most one weak solution in $W_{0}^{1,p}(\Omega,w)$ for any non-negative $f\in L^{1}(\Omega)$.

Proof.

Let $0<\delta\leq 1$, $w\in A_{p}$ be arbitrary and $u_{1}$, $u_{2}\in X$ are two solutions of the equation $(\ref{main})$. The fact $(u_{1}-u_{2})^{+}\in X$ allows us to choose $\{\varphi_{n}\}\in C_{c}^{\infty}(\Omega)$ converging to $(u_{1}-u_{2})^{+}$ in $||\cdot||_{X}$. Now setting, \psi_{n}:=\text{min}\,\big{\{}(u_{1}-u_{2})^{+},\varphi^{+}_{n}\big{\}}\in X\cap L^{\infty}_{c}(\Omega) as a test function in (1.1) we get

[TABLE]

Passing to the limit and using the strong monotonicity condition (H5), $(u_{1}-u_{2})^{+}=0$ a.e. in $\Omega$ which implies $u_{1}\leq u_{2}$. Similarly changing the role of $u_{1}$ and $u_{2}$, we get $u_{2}\leq u_{1}$. Therefore, $u_{1}\equiv u_{2}$. ∎

4.2. The case $\delta>1$

Theorem 4.2**.**

Let $\delta>1$ and $w\in A_{s}$. Then the problem (1.1) has at most one weak solution in $W^{1,p}_{loc}(\Omega,w)$ if

• (a)

$f\in L^{m}(\Omega)$* for some $m=(p_{s}^{*})^{\prime}$, provided $1\leq p_{s}<N$, or*

• (b)

$f\in L^{m}(\Omega)$* for some $m>1$, provided $p_{s}=N$, or*

• (c)

$f\in L^{1}(\Omega)$* for $p_{s}>N$.*

Remark 4.3**.**

In case $w\equiv 1$ our main results in this paper will hold by replacing $p_{s}$ by $p$. Moreover, since $(p^{*})^{\prime}<\frac{N}{p}$, Theorem 4.2 improves the range of $f$ in Theorem 1.5 of [9] to get the uniqueness provided $1<p<N$.

Preliminaries: Define for $k>0$ and $\delta>1$, the truncated function

[TABLE]

Definition 4.4**.**

We say that $v(>0)\in W_{loc}^{1,p}(\Omega,w)$ is a super-solution of the problem (1.1), if for every $K\Subset\Omega$, there exists a positive constant $c_{K}$ such that $v\geq c_{K}>0$ in $K$ and for every non-negative $\phi\in C_{c}^{1}(\Omega)$, one has

[TABLE]

Definition 4.5**.**

We say that $v(>0)\in W_{loc}^{1,p}(\Omega,w)$ is a sub-solution of the problem (1.1), if for every $K\Subset\Omega$, there exists a positive constant $c_{K}$ such that $v\geq c_{K}>0$ in $K$ and for every non-negative $\phi\in C_{c}^{1}(\Omega)$, one has

[TABLE]

For a fixed super-solution $v$ of (1.1), consider the following non-empty closed and convex set

[TABLE]

Lemma 4.6**.**

There exists $z\in\mathbb{K}$ such that for every non-negative $\phi\in C_{c}^{1}(\Omega)$, one has

[TABLE]

Proof.

Under the assumptions on $f$ and applying Theorem 2.6 one can define the operator $J_{k}:X\to X^{*}$ for every $u,\psi\in X$ by

[TABLE]

Following the same arguments as in Lemma 3.9, it follows that $J_{k}$ is demicontinuous, coercive and strictly monotone. As a consequence of Theorem 2.13, there exists a unique $z\in\mathbb{K}$ such that for every $\psi\in\mathbb{K}$, one has

[TABLE]

Let us consider a real valued function $g\in C_{c}^{\infty}(\mathbb{R})$ such that $0\leq g\leq 1$, $g\equiv 1$ in $[-1,1]$ and $g\equiv 0$ in $(-\infty,-2]\cup[2,\infty)$. Define the function $\phi_{h}:=g(\frac{z}{h})\phi$ and \phi_{h,t}:=\text{min}\,\big{\{}z+t\phi_{h},v\big{\}} with $h\geq 1$ and $t>0$ for a given non-negative $\phi\in C_{c}^{1}(\Omega)$. Then by the inequality (4.4), we have

[TABLE]

By (H5), we have

[TABLE]

Therefore,

[TABLE]

where

[TABLE]

Let us denote by

[TABLE]

Set $\Omega=S_{v}\cup S_{v}^{c},$ where $S_{v}:=\{x\in\Omega:\phi_{h,t}(x)=v(x)\}$ and $S_{v}^{c}:=\Omega\setminus S_{v}.$ Observe that $g(x)=g_{v}(x)=0$ on $S_{v}^{c}$ and $g(x)=g_{v}(x)$ on $S_{v}$. This gives from (4.6),

[TABLE]

Since $v$ is a super-solution of (1.1), choosing $(z+t\phi_{h}-\phi_{h,t})$ as a test function in (4.1) and using the fact that $\phi_{h,t}=v$ on $S_{v}$, we obtain

[TABLE]

Since $I\geq 0$ and $\phi_{h,t}-z\leq t\phi_{h}$, using the inequality (4.7), we get

[TABLE]

Therefore letting $t\to 0$, we obtain

[TABLE]

As $h\to\infty$, we obtain

[TABLE]

Hence the proof. ∎

Proof of Theorem 4.2.

Suppose $u,v\in W^{1,p}_{loc}(\Omega,w)$ both are solutions of the problem (1.1). Then, we can assume that $u$ is a sub-solution and $v$ is a super-solution of (1.1). By the given condition on $f$, one can use Lemma 4.6 to get the existence of $z\in\mathbb{K}$ satisfying the inequality (4.3). Let $\epsilon=2k^{-\frac{1}{\delta}}$ for $k>0$. Since $u=0$ on $\partial\Omega$, one can use Theorem 2.10 to obtain $(u-z-\epsilon)^{+}\in X$. Applying Lemma 4.6, for any $\eta>0$, by standard density arguments one has

[TABLE]

Since $(u-z-\epsilon)^{+}\in X$, there exists a sequence $\phi_{n}\in C_{c}^{\infty}(\Omega)$ such that $\phi_{n}\to(u-z-\epsilon)^{+}$ in $||\cdot||_{X}$. Denote by

[TABLE]

and since $u$ is a sub-solution of (1.1), we obtain

[TABLE]

Since $w|\nabla u|^{p}$ is integrable in the support of $(u-z-\epsilon)^{+}$, one can pass to the limit as $n\to\infty$ and obtain

[TABLE]

By using (4.8), (4.9), the fact $\epsilon>k^{-\frac{1}{\delta}}$ together with (H5), we obtain for \gamma:=\text{max}\big{\{}p,2\big{\}},

[TABLE]

Since $\eta>0$ is arbitrary, we have $u\leq z+2k^{-\frac{1}{\delta}}\leq v+2k^{-\frac{1}{\delta}}.$ Letting $k\to\infty$, we get $u\leq v$ a.e. in $\Omega$. Arguing similarly we obtain $v\leq u$ a.e. in $\Omega$. Hence $u\equiv v$. ∎

Acknowledgement

The author would like to show his sincere gratitude to Dr. Kaushik Balfor somefruitful discussion on the topic. The author was supported by NBHM FellowshipNo: 2-39(2)-2014 (NBHM-RD-II-8020-June 26, 2014).