Existence of positive solutions for a class of singular elliptic problems with convection term and critical exponential growth

Abstract

This paper uses the Galerkin method to investigate the existence of positive solution to a class of singular elliptic problems given by

$$\begin{aligned} \textstyle\begin{cases} -\Delta u= \displaystyle \frac {\lambda _{0}}{u^{\beta _{0}}} + \Lambda _{0} |\nabla u|^{\gamma _{0}}+ \frac{f_{0}(u)}{|x|^{\alpha _{0}}}+ h_{0}(x), \ \ u>0 \ \ \text{in} \ \Omega , \\ u=0 \ \text{on} \ \ \partial \Omega , \end{cases}\displaystyle \end{aligned}$$

where $\Omega \subset \mathbb{R}^{2}$ is a bounded smooth domain, $0<\beta _{0}$, $\gamma _{0} \leq 1$, $\alpha _{0} \in [0,2)$, $h_{0}(x)\geq 0$, $h_{0}\neq 0$, $h_{0}\in L^{\infty}(\Omega )$, $0<\|h_{0}\|_{\infty} < \lambda _{0} < \Lambda _{0}$, and $f_{0}$ are continuous functions. More precisely, $f_{0}$ has a critical exponential growth, that is, the nonlinearity behaves like $\exp (\overline{\Upsilon}s^{2})$ as $|s| \to \infty$, for some $\overline{\Upsilon}>0$.

1 Introduction

The purpose in the first part of this article is to prove the existence of solution to the following class of singular problems

$$ \textstyle\begin{cases} -\Delta u= \displaystyle \frac {\lambda _{0}}{u^{\beta _{0}}} + \Lambda _{0}|\nabla u|^{\gamma _{0}}+ \frac{f_{0}(u)}{|x|^{\alpha _{0}}}+h_{0}(x)\text{ in }\Omega \text{,} \\ u>0\text{ in }\Omega \text{,} \\ u=0\text{ on } \partial \Omega \text{,} \end{cases} $$

(1.1)

where is a bounded smooth domain $0<\beta _{0}$, $\gamma _{0}\leq 1$, $0< \|h_{0}\|_{\infty}< \lambda _{0}< \Lambda _{0}$, and $\alpha _{0} \in [0,2)$ are real parameters.

Elliptic problems with singularities are a class of partial differential equations (PDEs) that arise in various fields of science and engineering. These problems are characterized by the presence of singularities in the coefficients or boundary conditions of the elliptic PDE, which can have a significant impact on the behavior of the solutions. Singularities can occur at specific points or along curves, and they often represent physical phenomena or geometrical features that require special treatment in the mathematical modeling of the problem.

The presence of singularities in elliptic problems can complicate the analysis and numerical solution of these equations. Understanding and handling these singularities is crucial for obtaining accurate and physically meaningful solutions in practical applications. Researchers in various fields, including mathematics, physics, engineering, and computational science, have explored numerous approaches and techniques for tackling elliptic problems with singularities.

To delve deeper into this topic and explore the methods and theories used in the analysis of elliptic problems with singularities, one may refer to the key references [6, 9, 12], and [13].

On the other hand, nonlinear elliptic problems with gradient-dependent growth are a class of partial differential equations involving a nonlinear term in the equation, which depends on the gradient (or derivative) of the solution. These problems are typically expressed as follows:

$$ -\Delta u= f(x,u,\nabla u) \ \ in \ \ \Omega , \ \ u=0 \ \ on \ \ \partial \Omega . $$

Understanding and analyzing such problems is crucial in many areas of science and engineering, including nonlinear elasticity, porous media flow, phase transition, and optimal control problems with pointwise constraints. For more information on physical motivation for this class of problems, one can refer to [4, 19], and [21].

We are going back to our problem to enunciate the hypotheses on the functions $h_{0}$ and $f_{0}$. More precisely, the functions are continuous satisfying the following properties:

$(f_{1})$:

There exists $\overline{\Upsilon}> 0$ such that the exponential growth conditions at infinity are given by:

$$ \lim _{t\rightarrow \infty} \frac{f_{0}(t)}{\exp \left (\Upsilon _{0}\vert t\vert ^{2}\right )}=0 \ \text{ for } \Upsilon _{0}>\overline{\Upsilon}\ \text{ and }\ \lim _{t \rightarrow \infty} \frac{f_{0}(t)}{\exp \left (\Upsilon _{0}\vert t\vert ^{2}\right )}= \infty \ \text{ for }\ \ 0< \Upsilon _{0}< \overline{\Upsilon}. $$

$(f_{2})$:

The growth condition at the origin:

$$ \lim _{t\rightarrow 0^{+}}\frac{f_{0}(t)}{t}=0. $$

$(h_{1})$:

$h_{0}\in L^{\infty}(\Omega )$, $h_{0}(x) \geq 0$ and $h_{0}(x)\neq 0$.

Since we are looking for positive solution, in this paper, we consider $f_{0}(t)=0$ for all $t<0$.

The main result is:

Theorem 1.1

Assume that conditions $(f_{1})-(f_{2})$ and $(h_{1})$ hold. Then, there exists $\lambda ^{*}>0$ such that the problem (1.1) has a positive weak solution for every $\Lambda _{0} \in (0,\lambda ^{*})$.

Problems involving singularities, exponential growth, or dependence on the solution’s gradient have been extensively studied in recent years. For example, after the excellent article by Adimurthi and Sandeep [2], which introduced the version of the Trudinger-Moser inequality with a singular term, several articles have emerged discussing this topic. In [1], the authors demonstrated the existence of two solutions to the problem

$$ -\Delta u= \lambda \left (\frac{p(x)}{u^{\delta}} + h(x,u) \exp (4 \pi u^{2})\right )\ \ in \ \ \Omega \subset \mathbb{R}^{2}, \ \ u=0 \ \ on \ \ \partial \Omega , $$

where $0< p \in L^{\infty}(\Omega )$, $0< \lambda$ and $0<\delta <3$.

In [10], the authors investigated the following problem

$$ -\Delta u = h(u)\frac{\exp (\alpha u^{2})}{|x|^{\gamma}} \ \ in \ \ \Omega \subset \mathbb{R}^{2}, \ \ u=0 \ \ on \ \ \partial \Omega . $$

The authors established some existence results for a class of critical elliptic problems with singular exponential nonlinearities, but they do not assume any global sign conditions on the nonlinearity, which makes our results new even in the nonsingular case.

In [8], the authors apply minimax methods to obtain existence and multiplicity of weak solutions to singular and nonhomogeneous elliptic equation of the following form

$$ -\Delta _{N} u= \frac{f(x,u)}{|x|^{\alpha}} + h(x) \ \ in \ \ \Omega \subset \mathbb{R}^{N}, \ \ u=0 \ \ on \ \ \partial \Omega , $$

where f has the maximal growth on s.

In [3], the authors use the Galerkin method to demonstrate the existence of solutions to the problem

$$ -\Delta u= h(x,u) + \lambda g(x,\nabla u)\ \ in \ \ \Omega \subset \mathbb{R}^{2}, \ \ u=0 \ \ on \ \ \partial \Omega , $$

where h has sublinear and singular terms, and g is a continuous function with $0 \leq \lambda$.

Using the nonlinear domain decomposition method, in [5], the authors provide a sufficient condition for the problem

$$ -\Delta u -\lambda \rho (x)|\nabla u|^{2}= \epsilon V(x) \exp (u) \ \ in \ \ \Omega \subset \mathbb{R}^{2}, \ \ u=0 \ \ on \ \ \partial \Omega . $$

The version for $\mathbb{R}^{N}$ of this class of problems was studied in [11]. More precisely, the authors investigated the problem

$$ -\Delta u = p(x) \left (g(u) + f(u) + |\nabla u|^{a}\right ) \ \ in \ \ \mathbb{R}^{N}, \ \ N\geq 3, $$

where $0< a<1$, and p is a positive weight. Under the hypothesis that f is a nondecreasing function with sublinear growth and g is decreasing and unbounded around the origin, they established the existence of a ground state solution vanishing at infinity. The arguments used by the authors rely essentially on the maximum principle.

Before finishing the introduction part, we would like to mention the study with other techniques of some problems related to the ones we are studying, such as [17, 18], and [22].

Below, we list what we believe are the main contributions of our paper:

• 1) We complete the results in [1] because, in our article, we consider the convection term and the critical exponential growth with singularity.

• 2) Unlike what was studied in [8] and [10], in our article, we are considering the convection term and singularity in u.

• 3) We also complete the results in [3, 5], and [11] because we are considering exponential growth with singularity, which makes the estimates more delicate.

The plan of the paper is the following: In Sect. 2, we recall some preliminary results. In Sect. 3, we study an auxiliary problem. We show the existence of solution to the auxiliary problem in Sect. 4, where we prove Theorem 1.1.

2 Preliminary results

Let us consider the Sobolev space $W_{0}^{1,2}(\Omega )$ endowed with the norm

$$ \Vert u\Vert =\left (\int\limits_{\Omega}\vert \nabla u\vert ^{2} dx\right )^{\frac{1}{2}}\text{.} $$

We say that $u\in W_{0}^{1,2}(\Omega )$ is a weak solution to the problem (1.1) if $u>0$ in Ω and it verifies

$$ \int \limits_{\Omega}\nabla u\ \nabla \phi \ dx - \lambda _{0} \int\limits_{\Omega} \frac{\phi}{u^{\beta _{0}}} dx - \Lambda _{0}\int\limits_{\Omega} |\nabla u|^{\gamma _{0}}\phi dx-\int\limits_{\Omega} \frac{f_{0}(u)}{|x|^{\alpha _{0}}}\phi \ dx-\int \limits_{\Omega} h_{0}(x)\phi \ dx=0\text{,} $$

for all $\phi \in W_{0}^{1,2}(\Omega )$.

First, we recall some important results by Adimurthi-Sandeepr [2] and Hardy-Sobolev [14].

Theorem 2.1

(A singular Trudinger-Moser inequality) Let Ω be a bounded domain in $\mathbb{R}^{2}$, $u\in W_{0}^{1,2}(\Omega )$ and $\Upsilon >0$, then

$$ \frac{\exp \left (\Upsilon u^{2}\right )}{|x|^{\alpha}}\in L^{1}( \Omega ), $$

and there exists a constant $M>0$ such that

$$ \sup \limits_{ \Vert u\Vert \leq 1}\int \limits_{\Omega}\frac{\exp \left (\Upsilon u^{2}\right )}{|x|^{\alpha}} dx \leq M $$

if, only if, $\Upsilon \leq 2\pi (2-\alpha )$.

Theorem 2.2

(Hardy-Sobolev inequality) If $u\in C^{1}(\overline{\Omega})\cap W_{0}^{1,2}(\Omega )$, then $\dfrac{u}{Cd^{\tau}}\in L^{r}(\Omega )$, for $\frac{1}{r}=\frac{1}{2}-\frac{1-\tau}{2}$, $0< \tau \leq 1$ and

$$ \left \vert \dfrac{u}{Cd^{\tau}} \right \vert _{L^{r}(\Omega )}\leq \vert \nabla u \vert _{L^{2}(\Omega )}, $$

where $d(x)=dist(x,\partial \Omega )$, and C is a positive constant independent of x.

We observe that, from $(f_{1})$–$(f_{2})$, for all $\delta >0$ and for all $\alpha >\alpha _{0}$, there exists $C_{\delta}>0$ such that

$$ \vert f_{0}(t)t\vert \leq {\delta}\vert t\vert ^{2}+C_{\delta}\vert t \vert ^{q_{0}}\exp \left (\Upsilon \vert t \vert ^{2}\right ), $$

(2.1)

for all $q_{0}\geq 0$. In this paper, we will use $q_{0}>2$.

3 An auxiliary problem

For each $\varepsilon >0$, we consider the following auxiliary problem

$$ \textstyle\begin{cases} -\Delta u=\dfrac{\lambda _{0}}{( u+\varepsilon )^{\beta _{0}}}+ \Lambda _{0} |\nabla u|^{\gamma _{0}}+ \frac{f_{0}(u)}{|x|^{\alpha _{0}}}+ h_{0}(x),\text{ in }\Omega \text{,} \\ u>0\text{ in }\Omega \text{,} \\ u=0\text{ on } \partial \Omega \text{,} \end{cases} $$

(3.1)

where the function $h_{0}$, $f_{0}$ satisfies the hypotheses of the Theorem 1.1.

To prove Theorem 1.1, we first show the existence of a solution to problem (3.1). For this, we will use the Galerkin method together with the following fixed point theorem, see [20] and [16, Theorem 5.2.5].

Lemma 3.1

Let be a continuous function such that $\langle G(\xi ),\xi \rangle \geq 0$ for every with $\vert \xi \vert = r$ for some $r>0$. Then, there exists $z_{0}\in \overline{B}_{r}(0)$ such that $G(z_{0})=0$.

The main result in this section is the following:

Lemma 3.2

For each $0<\varepsilon <1$, there exists $\lambda ^{*}>0$ such that the problem (3.1) has a positive weak solution for every $\lambda _{0}, \Lambda _{0} \in (0,\lambda ^{*})$.

Proof

Let $B=\{e_{1},e_{2},\ldots ,e_{m},\ldots \}$ be a Schauder basis of $W_{0}^{1,2}(\Omega )$. For each , let

$$ W_{m}=[e_{1},e_{2},\ldots ,e_{m}] $$

be the finite-dimensional space generated by $\{e_{1},e_{2},\ldots ,e_{m}\}$. Note that the spaces $(W_{m},\Vert \cdot \Vert _{m})$ and are isometrically isomorphic by natural mapping

given by

$$ u=\sum _{j=1}^{m}\xi _{j}e_{j}\mapsto S(u)=\xi =(\xi _{1},\xi _{2}, \ldots ,\xi _{m})\text{,} $$

where

$$ \vert \xi \vert _{s}=\sum _{j=1}^{m}\vert \xi _{j}\vert . $$

Moreover,

$$ \Vert u\Vert _{m}=\vert \xi \vert _{s}=\vert S(u)\vert _{s}\text{.} $$

(3.2)

In the rest of the text, we will identify $u \in W_{m}$ with $\xi \in \mathbb{R}^{m}$ via T isometry. For each , define the function such that

$$ G(\xi )=G(\xi _{1},\xi _{2},\ldots ,\xi _{m})=(G_{1}(\xi ),G_{2}(\xi ), \ldots ,G_{m}(\xi )), $$

where ,

$$\begin{aligned} G_{j}(\xi )&=\int \limits_{\Omega}\nabla u\nabla e_{j} dx - \lambda _{0}\int \limits_{\Omega}\frac{e_{j}}{( u +\varepsilon )^{\beta _{0}}}dx\\ &\quad - \Lambda _{0} \int \limits_{\Omega} |\nabla u|^{\gamma _{0}} e_{j} dx-\int \limits_{\Omega} \frac{f_{0}(u)}{|x|^{\alpha _{0}}}e_{j} dx -\int \limits_{\Omega} h_{0}(x)e_{j} dx \text{,} \end{aligned}$$

$j=1,2,\ldots ,m$ and $u=\displaystyle \sum _{j=1}^{m}\xi _{j}e_{j}\in W_{m}$. Therefore,

$$\begin{aligned} \langle G(\xi ),\xi \rangle &=\sum _{j=1}^{m} G_{j}(\xi )\xi _{j}= \int \limits_{\Omega}\vert \nabla u\vert ^{2}dx - \lambda _{0}\int \limits_{\Omega}\frac{u}{( u +\varepsilon )^{\beta _{0}}} dx\\ &\quad - \Lambda _{0} \int\limits_{\Omega} |\nabla u|^{\gamma _{0}}u dx- \int\limits_{\Omega}\frac{f_{0}(u)}{|x|^{\alpha _{0}}}u dx - \int \limits_{\Omega}h_{0}(x)u dx. \end{aligned}$$

Note that

$$ \int \limits_{\Omega}\frac{u}{( u +\varepsilon )^{\beta _{0}}} dx\leq \int\limits_{\Omega}|u|^{1-\beta _{0}} dx\leq \widehat{C} \|u\|. $$

(3.3)

Moreover,

$$ \int\limits_{\Omega} |\nabla u|^{\gamma _{0}}u dx\leq \left (\int \limits_{\Omega} |\nabla u|^{2} dx\right )^{\gamma _{0}/2} \left (\int \limits_{\Omega} | u|^{2/(1-\gamma _{0})} dx\right )^{(1-\gamma _{0})/2}. $$

(3.4)

Using the Hölder inequality for $\frac{3}{\gamma _{0}}$ and $\frac{3-\gamma _{0}}{3}$, we have

$$\begin{aligned} \displaystyle \int _{\Omega}\frac{|u|^{2}}{|x|^{\gamma _{0}}}dx\leq \left (\displaystyle \int _{\Omega}|u|^{\frac{6}{3-\gamma _{0}}}dx \right )^{\frac{3-\gamma _{0}}{3}} \left (\displaystyle \int _{\Omega} \frac{1}{|x|^{3}}dx\right )^{\frac{\gamma _{0}}{3}}. \end{aligned}$$

(3.5)

Using (2.1), (3.5), and the Sobolev embedding, there exists positive constant $C_{1}$ such that

$$ \int \limits_{\Omega}\frac{f_{0}(u)}{|x|^{\alpha _{0}}}u dx\leq \delta C_{1} \Vert u\Vert ^{2}+C_{\delta} \int\limits_{\Omega}\vert u\vert ^{q_{0}} \frac{[\exp \left (\Upsilon \vert u\vert ^{2}\right )]}{|x|^{\alpha _{0}}}dx. $$

(3.6)

It follows from (3.3), (3.4), and (3.6) that

$$ \begin{aligned} \langle G(\xi ),\xi \rangle &\geq (1-\delta C_{1})\Vert u\Vert ^{2} - ( \lambda _{0} \widehat{C}+ \widetilde{C}\|h_{0}\|_{\infty}) \|u\| \\ &\quad -C \Lambda _{0} \|u\|^{\gamma _{0} +1} - C_{\delta}\int \limits_{\Omega}\vert u\vert ^{q_{0}} \frac{[\exp \left (\Upsilon \vert u\vert ^{2}\right )]}{|x|^{\alpha _{0}}}dx, \end{aligned} $$

(3.7)

for some $\widetilde{C}>0$. Since $0<\|h_{0}\|_{\infty}< \lambda _{0} < \Lambda _{0}$, we get

$$ \begin{aligned} \langle G(\xi ),\xi \rangle& \geq (1-\delta C_{1})\Vert u\Vert ^{2} - \Lambda _{0}( \widehat{C}+ \widetilde{C}) \|u\|\\ &\quad -C\Lambda _{0} \|u\|^{ \gamma _{0} +1} - C_{\delta}\int \limits_{\Omega}\vert u\vert ^{q_{0}} \frac{[\exp \left (\Upsilon \vert u\vert ^{2}\right )]}{|x|^{\alpha _{0}}}dx. \end{aligned} $$

(3.8)

Using Hölder’s inequality with $s,s'>1$ with s sufficiently close to 1, such that $\dfrac{1}{s}+\dfrac{1}{s'}=1$, we get

$$ C_{\delta}\int \limits_{\Omega}\vert u\vert ^{q_{0}} \frac{[\exp \left (\Upsilon \vert u\vert ^{2}\right )]}{|x|^{\alpha _{0}}}dx \leq C_{\delta}\left (\int \limits_{\Omega}\vert u\vert ^{q_{0}s'}dx\right )^{\frac{1}{s'}}\left ( \int \limits_{\Omega} \frac{[\exp \left (\Upsilon s\vert u\vert ^{2}\right )]}{|x|^{\alpha _{0}}}dx \right )^{\frac{1}{s}}. $$

Since $q_{0}>2$ and $s'>1$, by the Sobolev embedding, there exists $\widetilde{C_{1}}>0$ such that

$$ C_{\delta}\int \limits_{\Omega}\vert u\vert ^{q_{0}} \frac{[\exp \left (\Upsilon \vert u\vert ^{2}\right )]}{|x|^{\alpha _{0}}}dx \leq C_{\delta}\widetilde{C_{1}}\Vert u\Vert ^{q_{0}}\left (\int \limits_{\Omega} \frac{[\exp \left (\Upsilon s\vert u\vert ^{2}\right )]}{|x|^{\alpha _{0}}}dx \right )^{\frac{1}{s}}. $$

(3.9)

Then, it follows from (3.8) and (3.9) that

$$\begin{aligned} \langle G(\xi ),\xi \rangle &\geq (1-\delta C_{1})\Vert u\Vert ^{2} - \Lambda _{0}( \widehat{C}+\widetilde{C}) \|u\|-C\Lambda _{0} \|u\|^{ \gamma _{0} +1} \\ &\quad -C_{\delta}\widetilde{C_{1}}\Vert u\Vert ^{q_{0}} \left (\int \limits_{\Omega} \frac{[\exp \left (\Upsilon s\vert u\vert ^{2}\right )]}{|x|^{s\alpha _{0}}}dx \right )^{\frac{1}{s}}. \end{aligned}$$

Assume now that $\Vert u\Vert =r$ for some $0< r<1$ to be chosen later. We have

$$\begin{aligned} \int \limits_{\Omega} \frac{[\exp \left (\Upsilon s\vert u\vert ^{2}\right )]}{|x|^{s\alpha _{0}}}dx =&\int \limits_{\Omega} \frac{\exp \left (\Upsilon s\Vert u\Vert ^{2}\left (\frac{\vert u\vert}{\Vert u\Vert}\right )^{2} \right )}{|x|^{s\alpha _{0}}}dx \end{aligned}$$

(3.10)

$$\begin{aligned} =&\int \limits_{\Omega} \frac{\exp \left (\Upsilon sr^{2}\left (\frac{\vert u\vert}{\Vert u\Vert}\right )^{2} \right )}{|x|^{s\alpha _{0}}}dx, \end{aligned}$$

(3.11)

and applying Theorem 2.1, we impose that

$$ r\leq \left [\frac{2\pi (2-s\alpha _{0})}{\Upsilon s}\right ]^{1/2}. $$

Therefore, there exists $M>0$ such that

$$\begin{aligned} \sup \limits_{ \Vert u\Vert \leq 1}\int \limits_{\Omega} \frac{\exp \left (\Upsilon sr^{2}\left (\frac{\vert u\vert}{\Vert u\Vert}\right )^{2} \right )}{|x|^{s\alpha _{0}}}dx \leq M, \end{aligned}$$

(3.12)

and hence,

$$\begin{aligned} \langle G(\xi ),\xi \rangle \geq & (1-\delta C_{1})r^{2} - \Lambda _{0}( \widehat{C}+ \widetilde{C}) r -C\Lambda _{0} r^{\gamma _{0} +1} - C_{ \delta}\widetilde{C_{1}} M^{{1}/{s}}r^{q_{0}} \\ \geq & (1-\delta C_{1})r^{2} - \Lambda _{0}( \widehat{C}+C+ \widetilde{C}) r - C_{\delta}\widetilde{C_{1}} M^{{1}/{s}}r^{q_{0}}. \end{aligned}$$

Now, it is necessary to choose r such that

$$ (1-\delta C_{1})r^{2} -C_{\delta}\widetilde{C_{1}} M^{{1}/{s}}r^{q_{0}} \geq \dfrac{r^{2}}{2}, $$

in others words,

$$ r\leq \left ( \frac{(1/2-\delta C_{1})}{C_{\delta}\widetilde{C_{1}} M^{\frac{1}{s}}} \right )^{\frac{1}{q_{0}-2}}. $$

Thus, considering $r\leq \min \left \lbrace 1, \left [ \frac{2\pi (2-s\alpha _{0})}{\Upsilon s}\right ]^{1/2},\left ( \frac{(1/2-\delta C_{1})}{C_{\delta}\widetilde{C_{1}} M^{\frac{1}{s}}} \right )^{\frac{1}{q_{0}-2}}\right \rbrace$, we get

$$ \langle G(\xi ),\xi \rangle \geq \dfrac{r^{2}}{2}- \Lambda _{0}( \widehat{C}+C+\widetilde{C}). $$

Furthermore, choosing

$$ \lambda ^{*}=\frac{r^{2}}{2(\widehat{C}+C+\widetilde{C})}, $$

we obtain

By virtue of Lemma 3.1, for every , there exists with $\vert y\vert _{s}\leq r<1$ such that $G(y)=0$. Thus, from (3.2), there exists $u_{m}\in W_{m}$ satisfying

(3.13)

such that

$$ \begin{aligned} \int \limits_{\Omega}\nabla u_{m}\nabla e_{j} dx &= \lambda _{0}\int \limits_{\Omega}\frac{e_{j}}{( u_{m} +\varepsilon )^{\beta _{0}}}dx+ \Lambda _{0}\int \limits_{\Omega} |\nabla u_{m}|^{\gamma _{0}} e_{j} dx\\ &\quad +\int \limits_{\Omega} \frac{f_{0}(u_{m})}{|x|^{\alpha _{0}}}e_{j} dx + \int \limits_{\Omega} h_{0}(x)e_{j} dx \text{, }j=1,2,\ldots ,m. \end{aligned} $$

(3.14)

Multiplying equality (3.14) by any constant $\sigma _{j}$, for each $j=1,2,\ldots ,m$, and adding them, we conclude

$$ \begin{aligned} \int \limits_{\Omega}\nabla u_{m}\nabla \phi dx &= \lambda _{0}\int \limits_{\Omega}\frac{\phi}{( u_{m} +\varepsilon )^{\beta _{0}}}dx+ \Lambda _{0}\int \limits_{\Omega} |\nabla u_{m}|^{\gamma _{0}} \phi dx \\ &\quad + \int \limits_{\Omega} \frac{f_{0}(u_{m})}{|x|^{\alpha _{0}}}\phi dx + \int \limits_{\Omega} h_{0}(x)\phi dx \text{, for all } \phi \in W_{m}, \end{aligned} $$

(3.15)

which shows that $u_{m}$ is an approximate weak solution to problem (3.1).

Since r is independent of m and $W_{m}\subset W_{0}^{1,2}(\Omega )$, for all , then $(u_{m})$ is a bounded sequence in $W_{0}^{1,2}(\Omega )$. Thus, for some subsequence, there exists $u\in W_{0}^{1,2}(\Omega )$ such that

$$ \textstyle\begin{cases} u_{m}\rightharpoonup u \text{ in } W_{0}^{1,2}(\Omega )\text{,} \\ u_{m}\rightarrow u \text{ in } L^{\theta}(\Omega ),\ \theta \geq 1 \text{,} \\ u_{m}(x)\rightarrow u(x)\text{ a.e in }\Omega \text{,} \\ \vert u_{m}(x)\vert \leq g(x)\in L^{\theta}(\Omega ) \text{ a.e in } \Omega \text{,}\ \theta \geq 1. \end{cases} $$

(3.16)

Fix and consider $m\geq k$, then $W_{k}\subset W_{m}$ and

$$ \begin{aligned} \int \limits_{\Omega}\nabla u_{m}\nabla \phi _{k} dx &=\lambda _{0}\int \limits_{\Omega}\frac{\phi _{k}}{( u_{m} +\varepsilon )^{\beta _{0}}}dx+ \Lambda _{0}\int \limits_{\Omega} |\nabla u_{m}|^{\gamma _{0}} \phi _{k} dx\\ &\quad +\int \limits_{\Omega} \frac{f_{0}(u_{m})}{|x|^{\alpha _{0}}}\phi _{k} dx + \int \limits_{\Omega} h_{0}(x)\phi _{k} dx \text{, } \forall \phi _{k}\in W_{k}. \end{aligned} $$

(3.17)

Since $\phi _{k}\in W_{k}$, note that

$$ \left \vert \frac{\phi _{k}}{( u_{m} +\varepsilon )^{\beta _{0}}} \right \vert \leq \frac{\vert \phi _{k}\vert}{\varepsilon ^{\beta _{0}}}\in L^{1}( \Omega ). $$

Using (3.16), we have

$$ \frac{\phi _{k}}{( u_{m}(x)+\varepsilon )^{\beta _{0}}}\rightarrow \frac{\phi _{k}}{( u(x)+\varepsilon )^{\beta _{0}}}\ \text{ a.e in } \Omega . $$

Therefore, we use [7, Theorem 4.2] to obtain that

$$ \int \limits_{\Omega}\frac{\phi _{k}}{( u_{m}+\varepsilon )^{\beta _{0}}}dx \rightarrow \int \limits_{\Omega} \frac{\phi _{k}}{( u+\varepsilon )^{\beta _{0}}}dx. $$

(3.18)

Note that $|\nabla u_{m}(x)|^{\gamma _{0}}\to |\nabla u(x)|^{\gamma _{0}}$ a.e in Ω. Moreover, $|\nabla u_{m}|^{\gamma _{0}} \in L^{2/\gamma _{0}}(\Omega )$. Hence, from the Brezis-Lieb lemma [15, Lemma 4.8], we have that

$$\begin{aligned} \int \limits_{\Omega} |\nabla u_{m}|^{\gamma _{0}} \phi _{k} dx \to \int \limits_{\Omega} |\nabla u|^{\gamma _{0}} \phi _{k} dx, \end{aligned}$$

(3.19)

for all $\phi \in L^{2/(2-\gamma _{0})}(\Omega )$. In particular, for all $\phi \in W_{k}$.

Now, since $f_{0}$ is a continuous function, using (3.16), we have

$$ \frac{f_{0}(u_{m}(x))\phi _{k}}{|x|^{\alpha _{0}}}\rightarrow \frac{f_{0}(u(x))\phi _{k}}{|x|^{\alpha _{0}}}\ \text{ a.e in }\Omega . $$

(3.20)

Using (2.1), we get

$$\begin{aligned} \frac{\vert f_{0}(u_{m}(x))\phi _{k}\vert}{|x|^{\alpha _{0}}} \leq \delta \frac{\vert u_{m}(x)\vert \vert \phi _{k}\vert}{|x|^{\alpha _{0}}}+C_{ \delta} \frac{\vert u_{m}(x)\vert ^{q_{0}-1}\exp \left (\Upsilon \vert u_{m}(x)\vert ^{2}\right ) \vert \phi _{k}\vert}{|x|^{\alpha _{0}}}. \end{aligned}$$

We will need to prove that the function defined by

$$ \widehat{g}(u_{m}(x)):=\delta \frac{\vert u_{m}(x)\vert \vert \phi _{k}\vert}{|x|^{\alpha _{0}}}+C_{ \delta} \frac{\vert u_{m}(x)\vert ^{q_{0}-1}\exp \left (\Upsilon \vert u_{m}(x)\vert ^{2}\right ) \vert \phi _{k}\vert}{|x|^{\alpha _{0}}} $$

is convergent in $L^{1}(\Omega )$. Indeed, we invoke (3.16) to obtain

$$ \frac{\vert u_{m}(x)\vert \vert \phi _{k}\vert}{|x|^{\alpha _{0}}} \rightarrow \frac{\vert u(x)\vert \phi _{k}\vert}{|x|^{\alpha _{0}}} \text{ a.e in }\Omega $$

(3.21)

and

$$ \frac{\vert u_{m}(x)\vert \vert \phi _{k}(x)\vert}{|x|^{\alpha _{0}}} \leq \frac{g(x)\vert \phi _{k}(x)\vert }{|x|^{\alpha _{0}}} \leq \frac{1}{2}g^{2}(x) + \frac{1}{2} \frac{|\phi _{k}(x)|^{2}}{|x|^{2\alpha _{0}}}. $$

(3.22)

Arguing as (3.5) and using (3.16), we conclude that

$$\begin{aligned} \displaystyle \int _{\Omega} \frac{\vert u_{m}\vert \vert \phi _{k}\vert}{|x|^{\alpha _{0}}}dx \to \displaystyle \int _{\Omega} \frac{\vert u\vert \vert \phi _{k}\vert}{|x|^{\alpha _{0}}}dx. \end{aligned}$$

(3.23)

Furthermore, from (3.16), we get

$$ \frac{\vert u_{m}(x)\vert ^{q_{0}-1}\exp \left (\Upsilon \vert u_{m}(x)\vert ^{2}\right )}{|x|^{\alpha _{0}}} \rightarrow \frac{\vert u(x)\vert ^{q_{0}-1}\exp \left (\Upsilon \vert u(x)\vert ^{2}\right )}{|x|^{\alpha _{0}}} \ \text{ a.e in }\Omega . $$

(3.24)

Now, considering $s,s'>1$ such that $\dfrac{1}{s}+\dfrac{1}{s'}=1$, with s sufficiently close to 1, we use (3.16) and the fact that $q_{0}>2$ to obtain

$$ \vert u_{m}\vert ^{q_{0}-1}\rightarrow \vert u\vert ^{q_{0}-1}\ \text{ in } L^{s'}(\Omega ). $$

(3.25)

Moreover, by (3.12), we have

$$\begin{aligned} \int \limits_{\Omega} \frac{\exp \left (\Upsilon s\vert u_{m}\vert ^{2}\right ) }{|x|^{\alpha _{0}}}dx \leq M. \end{aligned}$$

(3.26)

Hence, by (3.25), (3.26), and Hölder’s inequality, we get

$$\begin{aligned} \int \limits_{\Omega} \frac{\vert u_{m}\vert ^{q_{0}-1}\exp \left (\Upsilon \vert u_{m}\vert ^{2}\right )}{|x|^{\alpha _{0}}}dx \leq & \left (\int \limits_{\Omega}\vert u_{m}\vert ^{(q_{0}-1)s'} dx\right )^{\frac{1}{s'}} \left (\int \limits_{\Omega} \frac{\exp \left (\Upsilon s\vert u_{m}\vert ^{2}\right )}{|x|^{\alpha _{0}}}dx \right )^{\frac{1}{s}} \\ \leq & \vert u_{m}\vert ^{q_{0}-1}_{L^{s'}(\Omega )}M^{\frac{1}{s}}= \overline{M}. \end{aligned}$$

(3.27)

We use (3.24), (3.27), and [15, Theorem 4.8] to conclude that

$$ \frac{\vert u_{m}\vert ^{q_{0}-1}\exp \left (\Upsilon \vert u_{m}\vert ^{2}\right )}{|x|^{\alpha _{0}}} \rightharpoonup \frac{\vert u\vert ^{q_{0}-1}\exp \left (\Upsilon \vert u\vert ^{2}\right )}{|x|^{\alpha _{0}}}. $$

(3.28)

It follows from (3.28) that, for all $\phi _{k} \in W_{k}$, we get

$$ \int \limits_{\Omega} \frac{\vert u_{m}\vert ^{q_{0}-1}\exp \left (\Upsilon \vert u_{m}\vert ^{2}\right )}{|x|^{\alpha _{0}}} \vert \phi _{k} \vert dx\rightarrow \int \limits_{\Omega} \frac{\vert u\vert ^{q_{0}-1}\exp \left (\Upsilon \vert u\vert ^{2}\right )}{|x|^{\alpha _{0}}} \vert \phi _{k}\vert dx. $$

(3.29)

Therefore, by (3.23) and (3.29), we prove that

$$\begin{aligned} \int \limits_{\Omega}\widehat{g}(u_{m}(x))dx\rightarrow \delta \int \limits_{\Omega}\frac{\vert u(x)\vert ^{p_{0}-1}}{|x|^{\alpha _{0}}} \vert \phi _{k}\vert dx+C_{\delta}\int \limits_{\Omega} \frac{\vert u\vert ^{q_{0}-1}\exp \left (\Upsilon \vert u\vert ^{2}\right )}{|x|^{\alpha _{0}}} \vert \phi _{k}\vert dx, \end{aligned}$$

which shows the identity

$$ \begin{aligned} \int \limits_{\Omega}\nabla u\nabla \phi _{k} dx &=\lambda _{0}\int \limits_{\Omega}\frac{\phi _{k}}{( u +\varepsilon )^{\beta _{0}}}dx+ \Lambda _{0}\int \limits_{\Omega} |\nabla u|^{\gamma _{0}} \phi _{k} dx+\int \limits_{\Omega} \frac{f_{0}(u)}{|x|^{\alpha _{0}}}\phi _{k} dx \\ &\quad +\int \limits_{\Omega} h_{0}(x)\phi _{k} dx \text{, } \forall \phi _{k}\in W_{k}. \end{aligned} $$

(3.30)

Since is dense in $W_{0}^{1,2}(\Omega )$, we have

$$ \phi _{k}\rightarrow \phi \ \text{ as }\ k\rightarrow \infty . $$

Therefore, since $\phi \in W_{0}^{1,2}(\Omega )$ is arbitrary, it follows from (3.30) that

$$ \begin{aligned} \int \limits_{\Omega}\nabla u\nabla \phi dx &=\lambda _{0}\int \limits_{\Omega}\frac{\phi}{( u +\varepsilon )^{\beta _{0}}}dx+ \Lambda _{0} \int \limits_{\Omega} |\nabla u |^{\gamma _{0}} \phi dx+\int \limits_{\Omega} \frac{f_{0}(u)}{|x|^{\alpha _{0}}}\phi dx \\ &\quad +\int \limits_{\Omega} h_{0}(x)\phi dx, \end{aligned} $$

(3.31)

for all $\phi \in W_{0}^{1,2}(\Omega )$, which shows that u is a weak solution to problem (3.1).

Furthermore, $u> 0$ in Ω. In fact, since $f_{0}(t)=0$, $\forall t<0$, we use $\phi =u^{-}$ in (3.31) to obtain

$$ \int \limits_{\Omega}\vert \nabla u^{-}\vert ^{2} dx \leq 0, $$

which implies that $u^{-}=0$ and then $u=u^{+}\geq 0$. However, due to [3, Lemma 3.1], $u\in C^{2}(\Omega )\cap C^{1}(\overline{\Omega})$ and the maximum principle, we get $u>0$ in Ω. □

4 Proof of Theorem 1.1

For each , let $\varepsilon =\dfrac{1}{n}$ and $u_{\frac {1}{n}}=u_{n}$, where $u_{n}$ is a solution of auxiliary problem (3.1)

$$ \textstyle\begin{cases} -\Delta u_{n}=\dfrac{\lambda _{0}}{( u_{n}+\frac{1}{n})^{\beta _{0}}}+ \Lambda _{0} |\nabla u_{n}|^{\gamma _{0}}+ \frac{f_{0}(u_{n})}{|x|^{\alpha _{0}}}+ h_{0}(x),\text{ in }\Omega \text{,} \\ u>0\text{ in }\Omega \text{,} \\ u=0\text{ on } \partial \Omega \text{,} \end{cases} $$

obtained by Lemma 3.2. Note that, from $(f_{3})$, we get

$$\begin{aligned} \dfrac{\lambda _{0}}{( u_{n}+\frac{1}{n})^{\beta _{0}}}+\Lambda _{0} | \nabla u_{n}|^{\gamma _{0}}+\frac{f_{0}(u_{n})}{|x|^{\alpha _{0}}}+ h_{0}(x) \geq h_{0}(x). \end{aligned}$$

Then,

$$ -\Delta u_{n} \geq h_{0}(x)>0\ \text{ in }\Omega . $$

Considering $v\in W^{1,2}_{0}(\Omega )$ the unique positive solution to the problem

$$ \textstyle\begin{cases} -\Delta v= h_{0}(x)\ \text{ in }\Omega , \\ v>0\text{ in }\Omega \text{,} \\ v=0\text{ on } \partial \Omega \end{cases} $$

(4.1)

and using comparison principle, we conclude that

(4.2)

which implies that $u_{n}(x)\nrightarrow 0$, for each $x\in \Omega$.

Now, from (3.16), we get

$$ u_{m}\rightharpoonup u_{n}\ \text{ in } W_{0}^{1,2}(\Omega ) \ \text{ as }\ m\rightarrow +\infty , $$

and it follows from (3.13) that

Therefore, r does not depend on n, which shows that $(u_{n})$ is a bounded sequence in $W^{1,2}_{0}(\Omega )$. Thus, since $W_{0}^{1,2}(\Omega )$ is a reflective Banach space, for some subsequence, there exists $u\in W_{0}^{1,2}(\Omega )$ such that

$$ \textstyle\begin{cases} u_{n}\rightharpoonup u \text{ in } W_{0}^{1,2}(\Omega )\text{,} \\ u_{n}\rightarrow u \text{ in } L^{\theta}(\Omega ),\ \theta \geq 1 \text{,} \\ u_{n}(x)\rightarrow u(x)\text{ a.e in }\Omega \text{,} \\ \vert u_{n}(x)\vert \leq g(x)\in L^{\theta}(\Omega ) \text{ a.e in } \Omega \text{,}\ \theta \geq 1. \end{cases} $$

(4.3)

Recall from (3.31) that

$$ \begin{aligned} \int \limits_{\Omega}\nabla u_{n}\nabla \phi dx&=\lambda _{0}\int \limits_{\Omega}\frac{\phi}{( u_{n} +\frac{1}{n})^{\beta _{0}}}dx+ \Lambda _{0}\int \limits_{\Omega} |\nabla u_{n} |^{\gamma _{0}} \phi dx+\int \limits_{\Omega} \frac{f_{0}(u_{n})}{|x|^{\alpha _{0}}}\phi dx \\ &\quad +\int \limits_{\Omega} h_{0}(x)\phi dx\text{, }\forall \phi \in W_{0}^{1,2}( \Omega ). \end{aligned} $$

(4.4)

By same computation in (3.19) and (3.29), we obtain

$$\begin{aligned} \int \limits_{\Omega} |\nabla u_{n}|^{\gamma _{0}} \phi dx \to \int \limits_{\Omega} |\nabla u|^{\gamma _{0}} \phi dx, \end{aligned}$$

(4.5)

and

$$ \int \limits_{\Omega} \frac{f_{0}(u_{n})}{|x|^{\alpha _{0}}}\phi dx \rightarrow \int \limits_{\Omega}\frac{ f_{0}(u)\phi}{|x|^{\alpha _{0}}} dx\text{, } \forall \phi \in W_{0}^{1,2}(\Omega ). $$

(4.6)

Note that, from (4.3), we get

$$ \frac{\phi}{\left (u_{n}(x)+\frac{1}{n}\right )^{\beta _{0}}} \rightarrow \frac{\phi}{u(x)^{\beta _{0}}}\ \text{ a.e in }\Omega . $$

(4.7)

Since $v\in C^{1}(\overline{\Omega})$ and Ω is a smooth bounded domain in $\mathbb{R}^{N}$, by the maximum principle that $u_{n}(x)\geq v(x)>Cd(x)>0$, where $d(x)=dist(x,\partial \Omega )$, and C is a positive constant that does not depend on x. Thus,

$$ |\frac{\phi}{(u_{n}(x)+\frac{1}{n})^{\beta _{0}}} | \leq | \frac{\phi}{u_{n}(x)^{\beta _{0}}}|\leq | \frac{\phi}{v(x)^{\beta _{0}}}|\leq |\frac{\phi}{Cd(x)^{\beta _{0}}}| \ \text{ a.e in }\Omega . $$

Hence, we invoke Theorem 2.2 to obtain $\left \vert \frac{\phi}{Cd(x)^{\beta _{0}}} \right \vert \in L^{r}( \Omega )$. Therefore, by (4.7) and [7, Theorem 4.2], we get

$$ \int \limits_{\Omega} \frac{\phi}{\left (u_{n}+\frac{1}{n}\right )^{\beta _{0}}} dx \rightarrow \int \limits_{\Omega} \frac{\phi}{u^{\beta _{0}}}dx\text{, }\forall \phi \in W_{0}^{1,2}( \Omega ). $$

(4.8)

Letting $n\rightarrow +\infty$ in (4.4), we use (4.5), (4.6), and (4.8) to conclude that

$$\begin{aligned} \int \limits_{\Omega}\nabla u\nabla \phi dx&=\lambda _{0}\int \limits_{\Omega}\frac{\phi}{ u^{\beta _{0}}}dx+ \Lambda _{0}\int \limits_{\Omega} |\nabla u |^{\gamma _{0}} \phi dx+\int \limits_{\Omega} \frac{f_{0}(u)}{|x|^{\alpha _{0}}}\phi dx \\ &\quad +\int \limits_{\Omega} h_{0}(x)\phi dx\text{, }\forall \phi \in W_{0}^{1,2}( \Omega )\text{,} \end{aligned}$$

which proves that $u\in W_{0}^{1,2}(\Omega )$ is a weak solution to problem (1.1).