Regularizing Effects for a Singular Elliptic Problem

Abstract

In this paper, we prove existence and regularity results for a nonlinear elliptic problem of p-Laplacian type with a singular potential like ${\displaystyle \frac{f}{u^{\gamma }}}$ and a lower order term $bu$, where u is the solution and b and f are only assumed to be summable functions. We show that, despite the lack of regularity of the data, for suitable choices of the function b in the lower order term, a strong regularizing effect appears. In particular we exhibit the existence of bounded solutions. Worth notice is that this result fails if $b\equiv 0$, i.e., in absence of the lower order term. Moreover, we show that, if the singularity is “not too large” (i.e., $\gamma \le 1$), such a regular solution is also unique.

1. Introduction

In this paper, we study the existence and regularity of the solutions to a class of nonlinear elliptic problems whose prototype is

$$\left\{\begin{array}{cc}-{\mathsf{\Delta }}_{p}u+b\left(x\right)u={\displaystyle \frac{f}{u^{\gamma }}},\hfill & \mathrm{in} \mathsf{\Omega },\hfill \\ u=0,\hfill & \mathrm{on} \partial \mathsf{\Omega },\hfill \end{array}\right.$$

(1)

where $\gamma >0$, $\mathsf{\Omega }$ is a bounded open set of ${\mathbb{R}}^N$, and the data $b\left(x\right)$ and $f\left(x\right)$ are non-negative functions belonging to $L^1\left(\mathsf{\Omega }\right)$.

There are a large number of papers investigating this kind of problems starting from the 1970s to today (see [1,2,3,4,5,6,7,8,9,10,11,12] and the references therein), since it has a particular interest from a physical point of view. For example (see [13]), for $p=2$ and $N=3$, system (1) describes (in the stationary case) the distribution of the temperature $u\left(x\right)$ at the point $x\in \mathsf{\Omega }$ of a region $\mathsf{\Omega }$ occupied by an electrical conductor where the local voltage drop in $\mathsf{\Omega }$ is described by f, the resistivity by $u^{\gamma }$, and $b\left(x\right)$ acts as a coolant in order to limit overheating. Moreover, equations with singular nonlinearity appear also in the contest of chemical heterogeneous catalysts (see [14,15,16] and the references therein) and in the study of non-Newtonian fluids (see [17] and the references therein).

The aim of this paper is to study the effect of the lower order term $b\left(x\right)u$ on the solutions of the problem.

We recall that, in absence of this term, i.e., when $b\equiv 0$, the model case (1) with $p=2$ (together with linear smooth perturbation of the Laplacian operator) has been investigated in [3,7,8,9,12], and in [1]. In more detail, in [3,7,8,9,12], the authors studied classical solutions in presence of regular data f, while weak solutions for the less regular case of datum f in Lebesgue spaces was treated in [1]. In this paper, the authors show that the exponent $\gamma$ influences the regularity of the solutions. As a matter of fact, if $\gamma >1$ (this case is denoted “strong singularity”), then there exists a solution $u\in H_{loc}^1\left(\mathsf{\Omega }\right)$ satisfying $u^{{\displaystyle \frac{\gamma +1}{2}}}\in H_0^1\left(\mathsf{\Omega }\right)$, even if f belongs only to $L^1\left(\mathsf{\Omega }\right)$, while when the singularity is not too large, i.e., if $\gamma <1$ (this case is referred “mild singularity”), the solution u belongs to the energy space $H_0^1\left(\mathsf{\Omega }\right)$ only if f is sufficiently regular, i.e., belonging to $L^m\left(\mathsf{\Omega }\right)$ with $m={\left({\displaystyle \frac{2^{*}}{1-\gamma }}\right)}^{\prime }$. In addition, for every $\gamma >0$ the constructed solution is bounded if $f\in L^m\left(\mathsf{\Omega }\right)$ with $m>{\displaystyle \frac{N}{2}}$, as it happens in the well known non singular case $\gamma =0$. Further results when $p=2$ and $b\equiv 0$ can be found in [5,6]. The extension of the results of [1] to values $p\ne 2$ was obtained in [4], while generalization to the case of measure data when $p=2$ can be found in [10] and in [11]. We recall that in the previous papers [4,10,11] (as in [1]) the lower order term does not appear (i.e., $b\left(x\right)\equiv 0$) and the regularity of the solutions depend on the regularity of the datum f and, for example, are not bounded if f is not regular enough.

The case $b\equiv 0$, $p=2$ and $\gamma =0$ was studied in [18]. The authors show that a regularizing effect can be obtained thanks to the presence of the term $b\left(x\right)u$ supposing that

$$\exists D>0\mathrm{s}.\mathrm{t}.f\left(x\right)\le Db\left(x\right).$$

(2)

Finally, the case $b\equiv 0$, $p=2$ and $\gamma \ne 0$ was studied in [2] when $\gamma \le 1$. In particular, in this paper, it is proved that the condition (2) also has a regularizing effect in this singular case, and produces the existence of a unique solution u belonging to $H_0^1\left(\mathsf{\Omega }\right)\cap L^{\infty }\left(\mathsf{\Omega }\right)$, even if f is only a summable function.

The aim of this paper is to complete these results, investigating the influence of the lower order term $bu$ in the general nonlinear case $p>1$ and $\gamma >0$.

In particular, we want to understand here if a regularizing effect appears also if $\gamma >1$. Moreover, another aim of our investigation is to know whether the smoothing phenomena together with the uniqueness results known in the case of mild singularity are peculiar to the fact that, in the equation, the linear Laplacian operator appears, or whether the same behavior of the solutions can be expected in the case of nonlinear operators of p-Laplacian type.

Here, we answer to the previous questions showing that, for every $1<p<N$, Assumption (2) always produces a regularizing effects on the solutions (therefore, both in case $\gamma >1$, and in the case $\gamma \le 1$). Thus, these phenomena remain true in the nonlinear case. Finally, we show that uniqueness of regular solutions holds also in the nonlinear setting. It is worth pointing out that our results hold for $\gamma >0$ and $p>1$, and thus generalize the results obtained in [2] when $p=2$ and $0<\gamma \le 1$, showing that the Assumption (2) produce a complete change in the behavior of the solution also in this more general setting.

We recall that when f is only assumed to be a summable function, it is well known that if $b\equiv 0$, it is not true that regularizing effects hold, and there is also no uniqueness of solutions; in other words, our results fail in absence of the lower order term $b\left(x\right)u$.

Hence, it is really amazing that the presence of the function $b\left(x\right)u\left(x\right)$, which could be very irregular, and may even not be a summable term, since here b is only assumed to be a summable function, can completely change the behavior of the solutions of these singular problems.

Notice that it remains an open problem to understand if, in the case of strong singularity, i.e., when $\gamma >1$, we have the uniqueness of the bounded solution. Despite what happens in the case $0<\gamma \le 1$, in which we succeed in exhibiting the existence of a unique weak solution u in $L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$ (see Theorem 2), in the case where $\gamma >1$, we do not know if there exists a solution belonging to $W_0^{1,p}\left(\mathsf{\Omega }\right)$. These open problems will be the subject of our future research. We point out that regularizing effects on the solutions of singular elliptic problems can appear also in presence of different type of lower order terms (see [19,20,21,22] and the references therein).

The plan of the paper is the following. In the next section, we state, in detail, our results whose proofs can be found in Section 3.

2. Main Results

Let us consider the following nonlinear singular elliptic problem:

$$\left\{\begin{array}{cc}-\mathrm{div}\left(a(x,\nabla u)\right)+b\left(x\right)u={\displaystyle \frac{f}{u^{\gamma }}},\hfill & \mathrm{in} \mathsf{\Omega },\hfill \\ u=0,\hfill & \mathrm{on} \partial \mathsf{\Omega }\hfill \end{array}\right.$$

(3)

where $\mathsf{\Omega }$ is a bounded open set of ${\mathbb{R}}^N,N\ge 2$, $\gamma >0$ is real parameter, the function $a:\mathsf{\Omega }\times {\mathbb{R}}^N\to {\mathbb{R}}^N$ is a Carathéodory function (i.e., it is measurable with respect to x for every $\xi$ and continuous with respect to $\xi$ for almost every $x\in \mathsf{\Omega }$) which satisfies the following structural assumptions for a.e. $x\in \mathsf{\Omega }$ and for every $\xi$ and $\eta$ in ${\mathbb{R}}^N$ with $\xi \ne \eta$

$$a(x,\xi )·\xi \ge {\alpha \left|\xi \right|}^p,1<p<N,$$

(4)

$$\left|a(x,\xi )\right|\le {\mathsf{\Lambda }\left|\xi \right|}^{p-1},$$

(5)

$$\left[a\right(x,\xi )-a(x,\eta \left)\right][\xi -\eta ]>0,$$

(6)

where $0<\alpha \le \mathsf{\Lambda }$. The functions $b\left(x\right)$ and $f\left(x\right)$ are such that

$$b\left(x\right)\ge 0,b\left(x\right)\in L^1\left(\mathsf{\Omega }\right),$$

(7)

$$f\left(x\right)\ge 0,f\left(x\right)\in L^1\left(\mathsf{\Omega }\right),$$

(8)

and there exists a nonnegative constant D such that

$$f\left(x\right)\le Db\left(x\right).$$

(9)

Before stating the main results, we recall our notion of solution to problem (3).

Definition 1. 

A weak solution to Problem (3) is a function u satisfying

$$u\ge 0a.e. in\mathsf{\Omega },$$

(10)

$$u^{{\displaystyle \frac{p+\max \{0,\gamma -1\}}{p}}}\in W_0^{1,p}\left(\mathsf{\Omega }\right),$$

(11)

$${\displaystyle \frac{f}{u^{\gamma }}}\in L_{loc}^1\left(\mathsf{\Omega }\right),$$

(12)

$$b\left(x\right)u\in L_{loc}^1\left(\mathsf{\Omega }\right),$$

(13)

$$a(x,\nabla u)\in L_{loc}^1\left(\mathsf{\Omega }\right),$$

(14)

and

$${\int }_{\mathsf{\Omega }}a(x,\nabla u)\nabla \phi +{\int }_{\mathsf{\Omega }}bu\phi ={\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}\phi ,$$

(15)

for every $\phi \in C_c^1\left(\mathsf{\Omega }\right)$.

Remark 1. 

We observe that the boundary condition $u=0$ on $\partial \mathsf{\Omega }$ in (3) is satisfied in the weak sense. In detail, for $0<\gamma \le 1$, the solution u belongs to $W_0^{1,p}\left(\mathsf{\Omega }\right)$, while if $\gamma >1$, the function $u^{{\displaystyle \frac{p+\gamma -1}{p}}}$ belongs to $W_0^{1,p}\left(\mathsf{\Omega }\right)$.

The main result of the paper is the following.

Theorem 1. 

Assume (4)–(9). Then, there exists a weak solution u to problem (3) in the sense of Definition (1). Moreover, it results $u\in L^{\infty }\left(\mathsf{\Omega }\right)$, and the following estimate holds true:

$${\parallel u\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)}\le D^{{\displaystyle \frac{1}{\gamma +1}}},$$

(16)

where D is as in (9).

Remark 2. 

We point out that the $L^{\infty }$-regularity result of Theorem 1 is rather surprising, since the datum f is only assumed to be a summable function. Thus, since this effect is not true when $b\equiv 0$, it reveals that the lower order term $bu$ has a very strong regularizing impact on the solutions of problem (3), both in the strong and mild singularity cases (i.e., for every value of γ).

Remark 3. 

We observe that we construct a solution u of problem (3) as a limit of a sequence $u_n$ of approximating problems where we remove the singularity and we introduce a sequence $f_n$ of regular functions which approximate f. In this way, we will obtain regular approximating solutions. In order to pass to the limit, we will prove suitable estimates, which will allow to use compactness results that will help us to pass to the limit in the approximating problems to derive the existence result stated in Theorem 1. We point out that this method (which is a classical method to prove existence results in absence of singular terms, i.e., when $\gamma =0$) has the advantage to show a concrete real way to construct and approximate the solution u of (3). In addition, it suggests that singular problems like (3) are “in some way” stable with respect to variations in the datum f.

In the particular case $0<\gamma \le 1$, the regular solution u given by Theorem 1 is also the unique regular solution of problem (3). In detail, we have the following result.

Theorem 2. 

Assume (4)–(9) and $0<\gamma \le 1$. Then, there exists a unique weak solution u to problem (3) belonging to $L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$ and satisfying (15) for every $\phi \in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$.

The paper is organized as follows: in Section 3 we prove Theorem 1, while the proof of Theorem 2 is given in Section 4.

3. Proof of Theorem 1

The proof proceeds by step: in Section 3.1, we introduce a sequence of approximating problems (see (18)), and we prove the existence of a solution $u_n\in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$ to these problems; in Section 3.2 we obtain some a priori estimates useful to the limit as $n\to +\infty$ in Section 3.3 and to obtain the results of Theorem 1.

3.1. Approximating Problems

Here, and throughout the paper, we denote by C a positive constant depending only on the variables in brackets, which can change from line to line.

For any measurable function v, we will denote $v^{+}$ and $v^{-}$, respectively, as follows:

$$v^{+}=\max (v,0),v^{-}=\max (-v,0).$$

(17)

In order to prove Theorem 1, we consider the following sequence of approximating problems:

$$\left\{\begin{array}{cc}-\mathrm{div}\left(a(x,\nabla u_n)\right)+b_n{u}_n={\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}},\hfill & \mathrm{in} \mathsf{\Omega },\hfill \\ u_n=0,\hfill & \mathrm{on} \partial \mathsf{\Omega },\hfill \end{array}\right.$$

(18)

where $b_n$ and $f_n$ are bounded functions defined as in [18] as follows:

$$f_n\left(x\right)={\displaystyle \frac{f\left(x\right)}{1+\frac{1}{n}f\left(x\right)}},b_n\left(x\right)={\displaystyle \frac{b\left(x\right)}{1+\frac{D}{n}b\left(x\right)}},$$

(19)

where D is the constant in Assumption (9).

We observe that, since the function $w\left(s\right)={\displaystyle \frac{s}{1+\frac{s}{n}}}$ is an increasing function, by (9), we deduce

$$0\le f_n\left(x\right)={\displaystyle \frac{f\left(x\right)}{1+\frac{1}{n}f\left(x\right)}}\le {\displaystyle \frac{Db\left(x\right)}{1+\frac{D}{n}b\left(x\right)}}=D{b}_n\left(x\right).$$

(20)

We prove that there exists $u_n\in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$, a weak solution of (18).

Proposition 1. 

Under Assumptions (4)–(9), there exists a nonnegative solution $u_n\in W_0^{1,p}\left(\mathsf{\Omega }\right)\cap L^{\infty }\left(\mathsf{\Omega }\right)$ of (18), i.e., satisfying

$${\int }_{\mathsf{\Omega }}a(x,\nabla u_n)\nabla \phi +{\int }_{\mathsf{\Omega }}{b}_n{u}_n\phi ={\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}\phi ,$$

for every $\phi \in W_0^{1,p}\left(\mathsf{\Omega }\right)$.

Proof. 

Let us fix $n\in \mathbb{N}$. Let v be a function in $L^p\left(\mathsf{\Omega }\right)$, and define $w=S\left(v\right)$ as the unique nonnegative solution in $W_0^{1,p}\left(\mathsf{\Omega }\right)$ to

$$\left\{\begin{array}{cc}-\mathrm{div}\left(a(x,\nabla w)\right)+b_{n}w={\displaystyle \frac{f_n}{\left(\right|v|+\frac{1}{n}{)}^{\gamma }}},\hfill & \mathrm{in} \mathsf{\Omega },\hfill \\ w=0,\hfill & \mathrm{on} \partial \mathsf{\Omega }.\hfill \end{array}\right.$$

(21)

Notice that w belongs to $L^{\infty }\left(\mathsf{\Omega }\right)$ being, by construction, $b_n$ and $f_n$ bounded functions.

Taking w as a test function in (21), by the ellipticity condition (4), we obtain

$$\alpha {\int }_{\mathsf{\Omega }}{|\nabla w|}^p+{\int }_{\mathsf{\Omega }}{b}_n{w}^2\le {\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{\left(\right|v|+\frac{1}{n}{)}^{\gamma }}}w,$$

which implies, given $b_n\ge 0$, that

$$\alpha {\int }_{\mathsf{\Omega }}{|\nabla w|}^p\le {\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{\left(\right|v|+\frac{1}{n}{)}^{\gamma }}}w\le n^{\gamma +1}{\int }_{\mathsf{\Omega }}\left|w\right|.$$

Applying the Poincare’s inequality and Holder’s inequality, we find

$${\int }_{\mathsf{\Omega }}{\left|w\right|}^p\le C{n}^{\gamma +1}{\left({\int }_{\mathsf{\Omega }}{\left|w\right|}^p\right)}^{{\displaystyle \frac{1}{p}}},$$

where $C=C(\alpha ,\mathsf{\Omega },p,N)$ is a constant independent of n.

By the previous inequality, it follows that

$${\parallel w\parallel }_{L^p\left(\mathsf{\Omega }\right)}\le {\left(C{n}^{\gamma +1}\right)}^{{\displaystyle \frac{1}{p-1}}},$$

hence, the ball of $L^p\left(\mathsf{\Omega }\right)$ of radius ${\left(C{n}^{\gamma +1}\right)}^{{\displaystyle \frac{1}{p-1}}}$ is invariant for S. Moreover, by Sobolev’s embedding Theorem, we have also that S is continuous and compact on $L^p\left(\mathsf{\Omega }\right)$, so that, by Schauder’s Fixed Point Theorem, we deduce that there exists $u_n$ in $W_0^{1,p}\left(\mathsf{\Omega }\right)\cap L^{\infty }\left(\mathsf{\Omega }\right)$, such that $u_n=S\left(u_n\right)$, which means that $u_n$ satisfies the following problem:

$$\left\{\begin{array}{cc}-\mathrm{div}\left(a(x,\nabla u_n)\right)+b_n{u}_n={\displaystyle \frac{f_n}{\left(\right|u_n{|+\frac{1}{n})}^{\gamma }}},\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ w=0,\hfill & \mathrm{on}\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

(22)

In order to prove that $u_n$ is nonnegative in $\mathsf{\Omega }$, we take $-u_n^{-}$ as test function in (22). We obtain

$${\int }_{\mathsf{\Omega }}a(x,\nabla u_n)\nabla u_n\nabla (-u_n^{-})+{\int }_{\mathsf{\Omega }}{b}_n{u}_n(-u_n^{-})={\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{\left(\right|u_n{|+\frac{1}{n})}^{\gamma }}}(-u_n^{-})\le 0,$$

which implies

$${\int }_{\mathsf{\Omega }}a(x,\nabla u_n)\nabla (-u_n^{-})\nabla (-u_n^{-})+{\int }_{\mathsf{\Omega }}{b}_n(-u_n^{-})(-u_n^{-})\le 0.$$

By (4), and dropping the second term in the left hand side, since it is nonnegative, we find

$$0\le \alpha {\int }_{\mathsf{\Omega }}{|\nabla u_n^{-}|}^p\le 0,$$

which implies that $u_n^{-}=0$ a.e. in $\mathsf{\Omega }$, i.e., $u_n\ge 0$ a.e. in $\mathsf{\Omega }$, and thus the proof of Proposition 1 is complete.

□

3.2. A Priori Estimates

We prove here some a priori estimates on the solution $u_n$ of the approximating problem (18).

Proposition 2. 

The following estimate holds true:

$$\left|\right|u_n{\left|\right|}_{L^{\infty }\left(\mathsf{\Omega }\right)}\le M\equiv D^{{\displaystyle \frac{1}{\gamma +1}}},$$

(23)

where D is as in (9).

Proof. 

Let $k\ge D$, where D, as recalled above, is the same constant in Assumption (9), and we choose $G_k\left(u_n\right)\in L^{\infty }\left(\mathsf{\Omega }\right)$ as test function in problem (18), where the function $G_k\left(s\right)$ is defined as follows:

$$G_k\left(s\right)={\left(\right|s|-k)}^{+}\mathrm{sign}\left(s\right).$$

By (4) and (20), we obtain

$$\alpha {\int }_{\mathsf{\Omega }}{|\nabla G_k\left(u_n\right)|}^p+{\int }_{\mathsf{\Omega }}{b}_n{u}_n{G}_k\left(u_n\right)\le D{\int }_{\mathsf{\Omega }}{b}_n{\displaystyle \frac{G_k\left(u_n\right)}{u_n^{\gamma }}}\le {\displaystyle \frac{D}{k^{\gamma }}}{\int }_{\mathsf{\Omega }}{b}_n{G}_k\left(u_n\right),$$

which implies

$$\alpha {\int }_{\mathsf{\Omega }}{|\nabla G_k\left(u_n\right)|}^p+k{\int }_{\mathsf{\Omega }}{b}_n{G}_k\left(u_n\right)-{\displaystyle \frac{D}{k^{\gamma }}}{\int }_{\mathsf{\Omega }}{b}_n{G}_k\left(u_n\right)\le 0.$$

(24)

We observe that, choosing $k\ge D^{{\displaystyle \frac{1}{\gamma +1}}}$, it follows

$$k{\int }_{\mathsf{\Omega }}{b}_n{G}_k\left(u_n\right)-{\displaystyle \frac{D}{k^{\gamma }}}{\int }_{\mathsf{\Omega }}{b}_n{G}_k\left(u_n\right)\ge 0.$$

Hence, by the previous inequality and (24), we deduce

$$\alpha {\int }_{\mathsf{\Omega }}{|\nabla G_k\left(u_n\right)|}^p\le 0,$$

and, consequently, it results (for every $k\ge D^{{\displaystyle \frac{1}{\gamma +1}}}$)

$$G_k\left(u_n\right)=0\mathrm{a}.\mathrm{e}.\mathrm{in}\mathsf{\Omega },$$

i.e., (remembering the definition of the function $G_k\left(s\right)$)

$$|u_n|\le k\mathrm{a}.\mathrm{e}.\mathrm{in}\mathsf{\Omega },$$

which implies the assertion (23), with M defined as follows:

$$M=D^{{\displaystyle \frac{1}{\gamma +1}}}.$$

(25)

□

Remark 4. 

We point out that the main ingredient in the previous proof has been Assumption (9). In detail, the presence of the lower-order term $b\left(x\right)u\left(x\right)$, with $b\left(x\right)$ also very irregular but related to f by Assumption (9) allows us to obtain the $L^{\infty }$-estimate (23), which will be crucial in order to construct a solution u of problem (3).

We now prove some energy estimates. We distinguish two cases according to the value of parameter $\gamma$.

We start with the case $0<\gamma \le 1$.

Proposition 3. 

Let $0<\gamma \le 1$. There exists a positive constant $C_1$, independent of n (see Formula (27) below), such that, for every $n\in \mathbb{N}$,

$${\int }_{\mathsf{\Omega }}{|\nabla u_n|}^p\le C_1.$$

(26)

Proof. 

In order to prove (26), taking as test function $u_n$ in problem (18), we obtain

$${\int }_{\mathsf{\Omega }}a(x,\nabla u_n)\nabla u_n+{\int }_{\mathsf{\Omega }}{b}_n{u}_n^2={\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{u}_n.$$

Noticing that

$${\int }_{\mathsf{\Omega }}{b}_n{u}_n^2\ge 0,$$

by (4) and (23), and observing that by (20), results in $f_n\le f$, we deduce

$${\int }_{\mathsf{\Omega }}{|\nabla u_n|}^p\le {\displaystyle \frac{1}{\alpha }}{\int }_{\mathsf{\Omega }}f{u}_n^{1-\gamma }\le C_1$$

where

$$C_1\equiv {\alpha }^{-1}{\parallel f\parallel }_{L^1\left(\mathsf{\Omega }\right)}{M}^{1-\gamma }.$$

(27)

□

Let $\gamma >1$. We want to show that, also in this case, we can give energy estimates on $u_n$. We have the following result.

Proposition 4. 

Let $\gamma >1$. For every $\phi \in C_c^1\left(\mathsf{\Omega }\right)$ with $\phi \ge 0$, there exists a positive constant $C_2$, independent of n (see Formula (32) below), such that, for every $n\in \mathbb{N}$,

$${\int }_{\mathsf{\Omega }}{|\nabla u_n|}^p{\phi }^p\le C_2.$$

(28)

Proof. 

Let $\phi \in C_c^1\left(\mathsf{\Omega }\right)$, satisfying $\phi \ge 0$. In order to prove (28), we choose as test function $(u_n-M){\phi }^p$, where M is as in (25). We obtain

$${\int }_{\mathsf{\Omega }}a(x,\nabla u_n)\nabla u_n{\phi }^p+{\int }_{\mathsf{\Omega }}a(x,\nabla u_n)(u_n-M)p{\phi }^{p-1}\nabla \phi +{\int }_{\mathsf{\Omega }}{b}_n{u}_n(u_n-M){\phi }^p=$$

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}(u_n-M){\phi }^p\le 0,$$

where, in the last step, we have used the fact that the test function is non positive. Using (4), by the previous inequality, we deduce

$$\alpha {\int }_{\mathsf{\Omega }}|\nabla u_n{|}^p{\phi }^p\le p{\int }_{\mathsf{\Omega }}|a(x,\nabla u_n)\left|\right|\nabla \phi |(M-u_n){\phi }^{p-1}+{\int }_{\mathsf{\Omega }}{b}_n{u}_n(M-u_n){\phi }^p.$$

(29)

Using (5), by (29), it follows that

$${\int }_{\mathsf{\Omega }}|\nabla u_n{|}^p{\phi }^p\le \mathsf{\Lambda }{\displaystyle \frac{p}{\alpha }}{\int }_{\mathsf{\Omega }}|\nabla u_n{|}^{p-1}|\nabla \phi |(M-u_n){\phi }^{p-1}+{\displaystyle \frac{M}{\alpha }}{\int }_{\mathsf{\Omega }}{b}_n{\phi }^p+$$

$$-{\displaystyle \frac{1}{\alpha }}{\int }_{\mathsf{\Omega }}{b}_n{u}_n^2{\phi }^p\le \mathsf{\Lambda }{\displaystyle \frac{Mp}{\alpha }}{\int }_{\mathsf{\Omega }}|\nabla u_n{|}^{p-1}|\nabla \phi |{\phi }^{p-1}+C_3,$$

(30)

where $C_3={\displaystyle \frac{M}{\alpha }}{\parallel b\parallel }_{L^1\left(\mathsf{\Omega }\right)}{\left|\right|\phi \left|\right|}_{L^{\infty }\left(\mathsf{\Omega }\right)}^p$.

Now, applying Young’s inequality, we can estimate the integral in the right hand side of (30) as follows:

$${\int }_{\mathsf{\Omega }}|\nabla u_n{|}^{p-1}|\nabla \phi |{\phi }^{p-1}\le {\displaystyle \frac{p-1}{p}}{\epsilon }^{p^{\prime }}{\int }_{\mathsf{\Omega }}|\nabla u_n{|}^p{\phi }^p+{\displaystyle \frac{1}{p{\epsilon }^p}}{\int }_{\mathsf{\Omega }}{|\nabla \phi |}^p.$$

(31)

Choosing $\epsilon ={\left({\displaystyle \frac{\alpha }{2(p-1)M\mathsf{\Lambda }}}\right)}^{{\displaystyle \frac{1}{p^{\prime }}}}$, by (30) and (31), we deduce that

$${\int }_{\mathsf{\Omega }}{|\nabla u_n|}^p{\phi }^p\le 2C_3$$

and thus estimate (28) follows with

$$C_2=2C_3=2{\displaystyle \frac{M}{\alpha }}{\parallel b\parallel }_{L^1\left(\mathsf{\Omega }\right)}{\parallel \phi \parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)}^p.$$

(32)

□

In order to give a sense to the boundary condition $u=0$ in the case $\gamma >1$, we prove the following result.

Proposition 5. 

Let $\gamma >1$. There exists a positive constant, $C_4$, independent of n (see Formula (35)), such that for every $n\in \mathbb{N}$,

$${\int }_{\mathsf{\Omega }}{|\nabla u_n^{{\displaystyle \frac{\gamma +p-1}{p}}}|}^p\le C_4.$$

(33)

Proof. 

Let us take $u_n^{\gamma }$ as test function in (18). Recalling that, (19) results in $0\le f_n\le f$, we obtain

$$\alpha \gamma {\int }_{\mathsf{\Omega }}|\nabla u_n{|}^p{u}_n^{\gamma -1}+{\int }_{\mathsf{\Omega }}{b}_n{u}_n^{\gamma +1}\le {\parallel f\parallel }_{L^1\left(\mathsf{\Omega }\right)}.$$

(34)

Observing that by (34)

$${\int }_{\mathsf{\Omega }}{b}_n{u}_n^{\gamma +1}\ge 0$$

it follows that

$${\int }_{\mathsf{\Omega }}{|\nabla u_n|}^p{u}_n^{\gamma -1}\le C_5,$$

where $C_5={\left(\alpha \gamma \right)}^{-1}{\parallel f\parallel }_{L^1\left(\mathsf{\Omega }\right)}$.

Since it results

$${\int }_{\mathsf{\Omega }}{|\nabla u_n|}^p{u}_n^{\gamma -1}={\displaystyle \frac{p^p}{{(\gamma +p-1)}^p}}{\int }_{\mathsf{\Omega }}{\left|\nabla u_n^{{\displaystyle \frac{\gamma +p-1}{p}}}\right|}^p,$$

we deduce that

$${\int }_{\mathsf{\Omega }}{\left|\nabla u_n^{{\displaystyle \frac{\gamma +p-1}{p}}}\right|}^p\le C_4,$$

where

$$C_4=C_5{\displaystyle \frac{{(\gamma +p-1)}^p}{p^p}}.$$

(35)

□

We prove now an estimate of the singular lower order term. In detail, we have the following result.

Proposition 6. 

For every nonnegative $\phi \in C_c^1\left(\mathsf{\Omega }\right)$, there exists a positive constant $C_6=C_6{(p,\phi ,M,|\left|b\right||}_{L^1\left(\mathsf{\Omega }\right)},\alpha ,\gamma )$, independent of n, such that, for every $n\in \mathbb{N}$,

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{\phi }^p\le C_6.$$

(36)

Proof. 

Let $\phi \in C_c^1\left(\mathsf{\Omega }\right)$ a nonnegative function arbitrarily fixed. Choosing ${\phi }^p\left(x\right)$ as test function in (18), we have

$$p{\int }_{\mathsf{\Omega }}a(x,\nabla u_n){\phi }^{p-1}\nabla \phi +{\int }_{\mathsf{\Omega }}{b}_n{u}_n{\phi }^p={\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{\phi }^p,$$

from which, using Assumptions (5) and (7), by Estimate (23), we obtain

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{\phi }^p\le p{\int }_{\mathsf{\Omega }}|a(x,\nabla u_n)|{\phi }^{p-1}|\nabla \phi |+{\parallel b\parallel }_{L^1\left(\mathsf{\Omega }\right)}{M\left|\right|\phi \left|\right|}_{L^{\infty }\left(\mathsf{\Omega }\right)}^p\le$$

$$\mathsf{\Lambda }p{\int }_{\mathsf{\Omega }}|\nabla u_n{|}^{p-1}{\phi }^{p-1}|\nabla \phi |+{\left|\right|b\left|\right|}_{L^1\left(\mathsf{\Omega }\right)}{M\left|\right|\phi \left|\right|}_{L^{\infty }\left(\mathsf{\Omega }\right)}^p.$$

Now, reasoning as in (31), by Estimate (26) if $\gamma \le 1$, and by Estimate (28) if $\gamma >1$, we deduce Estimate (36) with $C_6=C_6{(p,\phi ,M,|\left|b\right||}_{L^1\left(\mathsf{\Omega }\right)},\alpha ,\gamma )$.

□

We prove now the following uniform estimate near the singularity which will be essential in the proof of the convergence of the approximating term ${\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}$ in problem (18) to the singular term ${\displaystyle \frac{f}{u^{\gamma }}}$ in (3), and hence in proving that the sequence $u_n$ converges to the solution u of (3).

Proposition 7. 

For every $\mu >0$ and nonnegative $\phi \in C_c^1\left(\mathsf{\Omega }\right)$, there exist positive constants $C_7$ and $C_8$, independent of n and μ, such that, for every n,

$${\int }_{\mathsf{\Omega }\cap \{0\le u_n<\mu \}}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{\phi }^p\left(x\right)\le C_7\mu +C_8{\mu }^{{\displaystyle \frac{1}{p^{\prime }}}}.$$

(37)

Proof. 

Let $\mu >0$ and $\phi \in C_c^1\left(\mathsf{\Omega }\right)$ be nonnegative. Let $\delta >0$ be arbitrarily fixed. Choosing $v_{\delta }={\displaystyle \frac{T_{\delta }(-{(u_n-\mu )}^{-})}{\delta }}{\phi }^p\left(x\right)$ as test function in Problem (18), we obtain

$${\displaystyle \frac{1}{\delta }}{\int }_{\mathsf{\Omega }}a(x,\nabla u_n)\nabla (T_{\delta }(-{(u_n-\mu )}^{-}){\phi }^p\left(x\right)+p{\int }_{\mathsf{\Omega }}a(x,\nabla u_n){\displaystyle \frac{T_{\delta }(-{(u_n-\mu )}^{-})}{\delta }}{\phi }^{p-1}\nabla \phi$$

$$+{\int }_{\mathsf{\Omega }}{b}_n{u}_n{\displaystyle \frac{T_{\delta }(-{(u_n-\mu )}^{-})}{\delta }}{\phi }^p={\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}·{\displaystyle \frac{T_{\delta }(-{(u_n-\mu )}^{-})}{\delta }}{\phi }^p.$$

(38)

Observing that ${\displaystyle \frac{T_{\delta }(-{(u_n-\mu )}^{-})}{\delta }}=0$ on the set $\{x\in \mathsf{\Omega }:u_n\left(x\right)\ge \mu \}$, Equality (38) implies

$${\displaystyle \frac{1}{\delta }}{\int }_{\mathsf{\Omega }\cap \{\mu -\delta \le u_n\le \mu \}}a(x,\nabla u_n)\nabla u_n{\phi }^p+{\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}·{\displaystyle \frac{T_{\delta }\left({(u_n-\mu )}^{-}\right)}{\delta }}{\phi }^p$$

$$\le p{\int }_{\mathsf{\Omega }\cap \{0\le u_n\le \mu \}}|a(x,\nabla u_n)|{\phi }^{p-1}|\nabla \phi |+\mu {\int }_{\mathsf{\Omega }\cap \{0\le u_n\le \mu \}}{b}_n{\phi }^p.$$

Noting that, in view of (4), the first term in the left-hand side is nonnegative, by (5) and using Holder’s inequality, we obtain

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}·{\displaystyle \frac{T_{\delta }\left({(u_n-\mu )}^{-}\right)}{\delta }}{\phi }^p$$

$$\le p\mathsf{\Lambda }{\int }_{\mathsf{\Omega }\cap \{0\le u_n\le \mu \}}|\nabla u_n{|}^{p-1}{\phi }^{p-1}|\nabla \phi |+{\mu \parallel b\parallel }_{L^1\left(\mathsf{\Omega }\right)}{\left|\right|\phi \left|\right|}_{L^{\infty }\left(\mathsf{\Omega }\right)}^p$$

$$\le p\mathsf{\Lambda }{\left({\int }_{\mathsf{\Omega }\cap \{0\le u_n\le \mu \}}{|\nabla u_n|}^p{\phi }^p\right)}^{{\displaystyle \frac{p-1}{p}}}{\left({\int }_{\mathsf{\Omega }}{|\nabla \phi |}^p\right)}^{{\displaystyle \frac{1}{p}}}+\mu C_7,$$

where $C_7={\left|\right|b\left|\right|}_{L^1\left(\mathsf{\Omega }\right)}{\left|\right|\phi \left|\right|}_{L^{\infty }\left(\mathsf{\Omega }\right)}^p$. Moreover, using the test function $-{(u_n-\mu )}^{-}{\phi }^p\left(x\right)$ in (18), we obtain

$${\int }_{\mathsf{\Omega }\cap \{0\le u_n\le \mu \}}{|\nabla u_n|}^p{\phi }^p\left(x\right)\le C_9\mu ,$$

where $C_9=C_9{(\mathsf{\Lambda },p,|\mathsf{\Omega }|,M,\phi ,|\left|b\right||}_{L^1\left(\mathsf{\Omega }\right)})$ is a constant independent of n, $\mu$ and $\delta$.

Thus, we can conclude that

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}·{\displaystyle \frac{T_{\delta }\left({(u_n-\mu )}^{-}\right)}{\delta }}{\phi }^p\left(x\right)\le p\mathsf{\Lambda }{\left(C_9\mu \right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{\left({\int }_{\mathsf{\Omega }}{|\nabla \phi |}^p\right)}^{{\displaystyle \frac{1}{p}}}+C_7\mu =C_7\mu +C_8{\mu }^{{\displaystyle \frac{1}{p^{\prime }}}},$$

where $C_8=p\mathsf{\Lambda }{\left(C_9\mu \right)}^{{\displaystyle \frac{1}{p^{\prime }}}}{\left({\int }_{\mathsf{\Omega }}{|\nabla \phi |}^p\right)}^{{\displaystyle \frac{1}{p}}}=C_8(p,\mathsf{\Lambda },C_9,\phi )$.

Since ${\displaystyle \frac{T_{\delta }\left({(u_n-\mu )}^{-}\right)}{\delta }}$ a.e. converges to 1 on the set $\{x\in \mathsf{\Omega }:u_n\left(x\right)<\mu \}$ as $\delta \to 0$, by Lebesgue’s Theorem, we can pass to the limit obtaining

$${\int }_{\mathsf{\Omega }\cap \{0\le u_n<\mu \}}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{\phi }^p\left(x\right)\le C_7\mu +C_8{\mu }^{{\displaystyle \frac{1}{p^{\prime }}}}$$

and (37) holds true.

□

We prove now $L_{loc}^1-$ estimate for the sequence ${\left\{b_n{u}_n\right\}}_{n\in \mathbb{N}}$. In details, we show the following bound.

Proposition 8. 

For every nonnegative $\phi \in C_c^1\left(\mathsf{\Omega }\right)$ there exists a positive constant $C_{10}$, independent of n, such that for every n we have

$${\int }_{\mathsf{\Omega }}{b}_n{u}_n{\phi }^p\le C_{10}.$$

(39)

Proof. 

Let $k>0$ and the nonnegative function $\phi$ in $C_c^1\left(\mathsf{\Omega }\right)$ be arbitrarily fixed. Let us choose $v_k={\displaystyle \frac{T_k\left(u_n\right)}{k}}{\phi }^p$ as a test function in (18). By (4), we obtain

$${\int }_{\mathsf{\Omega }}{b}_n{u}_n{\displaystyle \frac{T_k\left(u_n\right)}{k}}{\phi }^p+{\int }_{\mathsf{\Omega }}a(x,\nabla u_n){\displaystyle \frac{T_k\left(u_n\right)}{k}}p{\phi }^{p-1}\nabla \phi \le {\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{\phi }^p{\displaystyle \frac{T_k\left(u_n\right)}{k}}.$$

Observing that ${\displaystyle \frac{T_k\left(u_n\right)}{k}}\le 1$, by (5) and (36), we deduce that

$${\int }_{\mathsf{\Omega }}{b}_n{u}_n{\displaystyle \frac{T_k\left(u_n\right)}{k}}{\phi }^p\le p\mathsf{\Lambda }{\int }_{\mathsf{\Omega }}|\nabla u_n{|}^{p-1}{\phi }^{p-1}|\nabla \phi |+C_6.$$

By the previous estimate, (26) (if $\gamma \le 1$) and (28) (if $\gamma >1$), we obtain

$${\int }_{\mathsf{\Omega }}{b}_n{u}_n{\displaystyle \frac{T_k\left(u_n\right)}{k}}{\phi }^p\le C_{10},$$

where $C_{10}=C_{10}{(p,\phi ,\mathsf{\Lambda },M,\parallel b\parallel }_{L^1\left(\mathsf{\Omega }\right)},\alpha ,\gamma )$ is a constant independent of k. Thus, passing to the limit as $k\to 0$, we deduce that (39) holds true.

In order to pass to the limit in (18), we now prove the equi-integrability of the sequence ${\left\{b_n{u}_n\right\}}_{n\in \mathbb{N}}$. To this aim, let us introduce the function $G_{k,\epsilon }\left(s\right)$, defined as follows:

$$G_{k,\epsilon }\left(s\right)=\left\{\begin{array}{c}0\mathrm{if}0<s<k\hfill \\ {\displaystyle \frac{s-k}{\epsilon }} \mathrm{if}k\le s\le k+\epsilon \hfill \\ 1\mathrm{if}s>k+\epsilon .\hfill \end{array}\right.$$

Taking $G_{k,\epsilon }\left(u_n\right)$ as test function in (18) we obtain, by (4),

$$\alpha {\epsilon }^{p-1}{\int }_{\mathsf{\Omega }}{|\nabla G_{k,\epsilon }\left(u_n\right)|}^p+{\int }_{\mathsf{\Omega }}{b}_n{u}_n{G}_{k,\epsilon }\left(u_n\right)\le {\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{G}_{k,\epsilon }\left(u_n\right).$$

(40)

Dropping the first term in the left hand side since nonnegative, Inequality (40) becomes

$${\int }_{\mathsf{\Omega }}{b}_n{u}_n{G}_{k,\epsilon }\left(u_n\right)\le {\int }_{A_k^n}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}\le {\displaystyle \frac{1}{k^{\gamma }}}{\parallel f\parallel }_{L^1\left(A_k^n\right)},$$

(41)

where $A_k^n=\{x\in \mathsf{\Omega }:u_n\left(x\right)>k\}$. Letting $\epsilon \to 0$, we obtain, from (41),

$${\int }_{A_k^n}{b}_n{u}_n\le {\displaystyle \frac{1}{k^{\gamma }}}{\parallel f\parallel }_{L^1\left(A_k^n\right)}\le {\displaystyle \frac{1}{k^{\gamma }}}{\parallel f\parallel }_{L^1\left(\mathsf{\Omega }\right)}.$$

(42)

Hence, by (42), for every measurable set E of $\mathsf{\Omega }$,

$${\int }_E{b}_n{u}_n\le {\int }_{A_k^n}{b}_n{u}_n+k{\int }_{E}b\le {\displaystyle \frac{1}{k^{\gamma }}}{\parallel f\parallel }_{L^1\left(\mathsf{\Omega }\right)}+k{\int }_{E}b,$$

(43)

which implies the equi-integrability of the sequence ${\left\{b_n{u}_n\right\}}_{n\in \mathbb{N}}$, being by assumption $b\in L^1\left(\mathsf{\Omega }\right)$.

□

3.3. Passing to the Limit and Conclusion of the Proof of Theorem 1

By Estimates (26) and (28), it follows that

$$u_n\rightharpoonup u\mathrm{weakly} \mathrm{in}{W}^{1,p}\left(\omega \right),\mathrm{for}\mathrm{every}\omega \subset \subset \mathsf{\Omega },$$

(44)

which implies

$$u_n\to u\mathrm{a}.\mathrm{e}.\mathrm{in}\mathsf{\Omega }.$$

(45)

Hence, by (26), (28), (33), (36), (39), (44). and (45), by applying Fatou’s Lemma, we obtain (11)–(13). Moreover, by (45), we also have that (10) holds true.

We now prove that the approximating term ${\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}$ converges in $L_{loc}^1\left(\mathsf{\Omega }\right)$ to a function $\tilde{f}$. This convergence will be essential to conclude the proof of Theorem 1 by showing that $\tilde{f}$ is, in reality, the singular term ${\displaystyle \frac{f}{u^{\gamma }}}$.

Proposition 9. 

For every $\omega \subset \subset \mathsf{\Omega }$, it results that

$${\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}\rightharpoonup \tilde{f}in{L}^1\left(\omega \right).$$

(46)

Proof. 

Let $\omega \subset \subset \mathsf{\Omega }$ be arbitrarily fixed. We prove (46) by means of the Dunford–Pettis Theorem. We start proving the following estimate:

$${\int }_{\omega }{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}\le C_{11},$$

(47)

with $C_{11}$ being a positive constant independent of n.

We observe that by (37), for every $\mu >0$, it results that

$${\int }_{\omega \cap \{0\le u_n<\mu \}}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}\le C_{12}[\mu +{\mu }^{{\displaystyle \frac{1}{p^{\prime }}}}],$$

(48)

with $C_{12}$ independent of n and $\mu$. Hence, it follows that

$${\int }_{\omega }{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}={\int }_{\omega \cap \{0\le u_n<\mu \}}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}+{\int }_{\omega \cap \{u_n\ge \mu \}}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}\le$$

(49)

$$\le C_{10}[\mu +{\mu }^{{\displaystyle \frac{1}{p^{\prime }}}}]+{\displaystyle \frac{\parallel f_n{\parallel }_{L^1\left(\mathsf{\Omega }\right)}}{{\mu }^{\gamma }}},$$

and, consequently, (47) holds true.

To conclude the proof, we need to show that, for every arbitrarily fixed $\epsilon >0$, there exists $\delta >0$ such that, for all $A\subset \omega$ with $\left|A\right|<\delta$, we have

$${\int }_A{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}\le \epsilon .$$

(50)

Choosing $\mu$ such that $C_{12}[\mu +{\mu }^{{\displaystyle \frac{1}{p^{\prime }}}}]<{\displaystyle \frac{\epsilon }{2}}$, and reasoning as in (49), we deduce that, for all $A\subset \omega$, we obtain

$${\int }_A{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}\le {\displaystyle \frac{\epsilon }{2}}+{\displaystyle \frac{{\parallel f\parallel }_{L^1\left(A\right)}}{{\mu }^{\gamma }}}.$$

(51)

Observing that, since f is a summable function, there exists $\delta >0$ such that, for all $A\subset \omega$ with $\left|A\right|<\delta$, it results

$${\int }_A{\displaystyle \frac{f}{{\mu }^{\gamma }}}\le {\displaystyle \frac{\epsilon }{2}}.$$

(52)

By (51) and (52), we obtain

$${\int }_A{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}\le \epsilon ,$$

for all $A\subset \omega$ with $\left|A\right|<\delta$ and, hence, (50) is proven. Thus, by the Dunford–Pettis Theorem, Assertion (46) follows.

□

We observe that, using (39), (43), and (45), by the Dunford–Pettis Theorem, it follows that, for every $\omega \subset \subset \mathsf{\Omega }$, it results

$$b_n{u}_n\rightharpoonup bu\mathrm{weakly}\mathrm{in}{L}^1\left(\omega \right).$$

(53)

Thus, for every $\omega \subset \subset \mathsf{\Omega }$, by (46) and (53), it follows that the following sequence:

$$g_n=-b_n{u}_n+{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}$$

is weakly convergent in $L^1\left(\omega \right)$. Applying Theorem 3.1 of [23] in the set $\omega$, we obtain

$$\nabla u_n\to \nabla u\mathrm{a}.\mathrm{e}.\mathrm{in}\omega .$$

(54)

By (5), (44), and (54), it follows that

$$a(x,\nabla u_n)\rightharpoonup a(x,\nabla u)\mathrm{in}{L}_{loc}^{p^{\prime }}\left(\mathsf{\Omega }\right).$$

Moreover, we observe that, by (47), it results

$$\left|\right\{u=0\}\cap \{f\ne 0\left\}\right|=0,$$

or, equivalently, $\{u=0\}\subseteq \{f=0\}$, except for a set with zero measure and consequently for any $\phi \in C_c^1\left(\mathsf{\Omega }\right)$, we have that

$${\int }_{\mathsf{\Omega }\cap \{u\ne 0\}}{\displaystyle \frac{f}{u^{\gamma }}}\phi ={\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}\phi .$$

The last step is to prove that, for any $\phi \in C_c^1\left(\mathsf{\Omega }\right)$, it results that

$${\int }_{\mathsf{\Omega }}\tilde{f}\phi ={\int }_{\mathsf{\Omega }\cap \{u\ne 0\}}{\displaystyle \frac{f}{u^{\gamma }}}\phi ,$$

(55)

where $\tilde{f}$ is the weak limit which appears in Proposition 9.

We observe that, thanks to (45), applying the dominated convergence theorem, we obtain

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{\chi }_{\{u_n\ge \mu \}}\phi \to {\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}{\chi }_{\{u\ge \mu \}}\phi ,$$

(56)

for every $\phi \in C_c^1\left(\mathsf{\Omega }\right)$ and $\mu >0$. By the following equality

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}\phi -{\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{\chi }_{\{u_n\ge \mu \}}\phi ={\int }_{\mathsf{\Omega }}{\displaystyle \frac{f_n}{{(u_n+\frac{1}{n})}^{\gamma }}}{\chi }_{\{u_n<\mu \}}\phi ,$$

and thanks to (46), (48) and (56), we deduce that for every $\phi \in C_c^1\left(\mathsf{\Omega }\right)$ and $\mu >0$, it results that

$$\left|{\int }_{\mathsf{\Omega }}\tilde{f}\phi -{\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}{\chi }_{\{u\ge \mu \}}\phi \right|\le C_{12}[\mu +{\mu }^{{\displaystyle \frac{1}{p^{\prime }}}}]{\left|\right|\phi \left|\right|}_{L^{\infty }\left(\mathsf{\Omega }\right)},$$

where we recall that $C_{12}$ is a constant independent of $\mu$. By the previous inequality and the arbitrariness of $\mu$, we can conclude that (55) follows and, hence, passing to the limit as $n\to +\infty$, we can conclude that u satisfies (15).

4. Proof of Theorem 2

The proof proceeds in two steps. In the first step, we prove that there exists a solution u to problem (3) that belongs to $L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$, and which satisfies (15) for every $\phi \in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$. In the second step, we conclude the proof, showing that such a solution is unique.

Step1: Existence. By Theorem 1 we know that there exists a weak solution u to problem (3) (in the sense of Definition (1)) that belongs to $L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$.

We show now that u satisfies (15) for every $\phi \in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$. To this aim, we start showing that for every nonnegative $\phi \in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$, it results that

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}\phi <+\infty .$$

(57)

Thus, let ${\phi }_n$ be a sequence of nonnegative functions belonging to $C_c^1\left(\mathsf{\Omega }\right)$ and satisfying

$${\phi }_n⟶\phi \mathrm{in}{W}_0^{1,p}\left(\mathsf{\Omega }\right),$$

$${\phi }_n⟶\phi \mathrm{a}.\mathrm{e}.\mathrm{in}\mathsf{\Omega },$$

$$\parallel {\phi }_n{\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)}\le C,\mathrm{for}\mathrm{every}n.$$

(58)

By (15), it results that

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}{\phi }_n={\int }_{\mathsf{\Omega }}a(x,\nabla u)\nabla {\phi }_n+{\int }_{\mathsf{\Omega }}bu{\phi }_n.$$

Since we know that u belongs to $W_0^{1,p}\left(\mathsf{\Omega }\right)$, by Assumption (5), we deduce that $a(x,\nabla u)\in {\left(L^{p^{\prime }}\left(\mathsf{\Omega }\right)\right)}^N$, and we obtain

$${\int }_{\mathsf{\Omega }}a(x,\nabla u)\nabla {\phi }_n⟶{\int }_{\mathsf{\Omega }}a(x,\nabla u)\nabla \phi .$$

Moreover, applying the dominated convergence theorem, it results that

$${\int }_{\mathsf{\Omega }}bu{\phi }_n⟶{\int }_{\mathsf{\Omega }}bu\phi .$$

Thus, it follows that

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}{\phi }_n⟶l_{\phi }\equiv {\int }_{\mathsf{\Omega }}a(x,\nabla u)\nabla \phi +{\int }_{\mathsf{\Omega }}bu\phi ,$$

and, consequently, the sequence ${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}{\phi }_n$ is bounded. Thus, by the Fatou’s lemma, we deduce that (57) holds true. We observe that, since for every $\phi \in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$, it results $\phi ={\phi }^{+}-{\phi }^{-}$, with ${\phi }^{\pm }\in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$, then it follows that (57) holds true for every $\phi \in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$ (without any sign condition).

We prove now that for every $\phi \in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$, it results that

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}\phi =l_{\phi }.$$

(59)

Notice that (59) implies that u satisfies (15) for every $\phi \in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$. To prove (59), we observe that there exists a function $H\in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$ and a subsequence ${\phi }_{n_k}$ of ${\phi }_n$ satisfying (for every k)

$$|{\phi }_{n_k}|\le H\left(x\right).$$

(60)

As a matter of fact, proceeding as in the proof of Theorem 4.9 in [24], but replacing in the construction of the subsequence ${\phi }_{n_k}$ the requirement that

$$\parallel {\phi }_{n_{k+1}}-{\phi }_{n_k}{\parallel }_{L^p\left(\mathsf{\Omega }\right)}\le {\displaystyle \frac{1}{2^k}}$$

with the request that

$$\parallel {\phi }_{n_{k+1}}-{\phi }_{n_k}{\parallel }_{W_0^{1,p}\left(\mathsf{\Omega }\right)}\le {\displaystyle \frac{1}{2^k}}$$

we obtain that there exists a function $h\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ satisfying

$$|{\phi }_{n_k}|\le h\left(x\right),$$

and thus (60) follows with $H\left(x\right)\equiv T_C\left(h\right)$, where C is as in (58).

By (57) (being H in $L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$), we deduce that ${\displaystyle \frac{f}{u^{\gamma }}}H\left(x\right)$ belongs to $L^1\left(\mathsf{\Omega }\right)$. Thus, we can apply the dominated convergence theorem, obtaining that

$${\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}{\phi }_{n_k}⟶{\int }_{\mathsf{\Omega }}{\displaystyle \frac{f}{u^{\gamma }}}\phi ,$$

and thus (59) holds true.

Step2: Uniqueness. Let u and v be two solutions of Problem (3) belonging to $L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$ and satisfying (15) for every $\phi \in L^{\infty }\left(\mathsf{\Omega }\right)\cap W_0^{1,p}\left(\mathsf{\Omega }\right)$. Taking as a test function $u-v$ in the problems satisfied by these solutions and subtracting the equalities obtained in this way, we deduce

$$\begin{array}{ccc}& & {\int }_{\mathsf{\Omega }}[a(x,\nabla u)-a(x,\nabla v)]\nabla (u-v)+{\int }_{\mathsf{\Omega }}b{(u-v)}^2=\hfill \\ & & {\int }_{\mathsf{\Omega }}\left({\displaystyle \frac{f}{u^{\gamma }}}-{\displaystyle \frac{f}{v^{\gamma }}}\right)(u-v),\hfill \end{array}$$

which implies, with the last integral in the right-hand side being non-positive, that

$${\int }_{\mathsf{\Omega }}[a(x,\nabla u)-a(x,\nabla v)]\nabla (u-v)\le 0.$$

The previous inequality together with Assumption (6) implies that $u=v$.