1 Introduction

We consider in this paper, the following problem

$$\begin{aligned} \left\{ \begin{array}{ll} -Lu=\dfrac{f}{u^{\gamma (x) }} &{}\quad in~\Omega , \\ u=0 &{}\quad on~\Omega , \\ u\ge 0 &{}\quad in~\Omega , \end{array} \right. \end{aligned}$$
(1)

where

$$\begin{aligned} Lu=\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\right] , \end{aligned}$$

\(\gamma (x)>0\) is assumed to be a regular function, say for example \(\gamma (x)\in C(\overline{\Omega })\), and \(\Omega \) is a bounded regular domain in \({\mathbb {R}}^{N}\). We will assume without loss of generality that \(2\le p_{1}\le p_{2}\le \cdots \le p_{N}\) and that f is a non negative function belonging to a suitable Lebesgue space \(L^{m}\left( \Omega \right) .\)

When the differential operator is assumed to be semilinear, and \(\gamma (x)=\gamma \), Boccardo and Orsina in their leading work [2], obtained existence and regularity of the solution, and this was generalized to the case of the p-laplacian in [7], and to the the case of the anisotropic operator L in [14].

In the very recent work [3] the authors consider a singular semilinear elliptic problem with variable exponent \(\gamma (x)\), they obtained existence and regularity of the solution, under some conditions on the behavior of the function \(\gamma (x)\) near the boundary of \(\Omega \).

There exists a huge literature, devoted to the study of the anisotropic operator L, as it has many applications in fluid dynamics, and physical phenomena with anisotropic diffusion, we cite for example [8–11], and the references therein.

When a singular nonlinearity is considered in interaction with different types of differential operators as the laplacian or the p-laplcian, we invite the reader to see the works [1, 4–6, 10, 12, 15, 16, 18].

Problem (1) is associated to the following anisotropic Sobolev spaces

$$\begin{aligned} W^{1,(p_{i})}\left( \Omega \right) =\left\{ v\in W^{1,1}\left( \Omega \right) ;\partial _{i}v\in L^{p_{i}}\left( \Omega \right) \right\} \end{aligned}$$

and

$$\begin{aligned} W_{0}^{1,(p_{i})}\left( \Omega \right) =W^{1,(p_{i})}\left( \Omega \right) \cap W_{0}^{1,1}\left( \Omega \right) \end{aligned}$$

endowed by the usual norm

$$\begin{aligned} \left\| v\right\| _{W_{0}^{1,(p_{i})}\left( \Omega \right) }=\sum \limits _{i=1}^{N}\left\| \partial _{i}v\right\| _{L^{p_{i}}\left( \Omega \right) }. \end{aligned}$$

Definition 1.1

We will say that \(u\in W_{0}^{1,(p_{i})}\left( \Omega \right) \) is an “energy” solution to (1) if and only if

$$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\partial _{i}\varphi =\int \limits _{\Omega }\frac{f\varphi }{u^{\gamma (x) }}\ \ \ \ \ \ \ \ \forall \varphi \in C_{0}^{1}\left( \Omega \right) , \end{aligned}$$

and we will say that u is a “weak” solution to (1) if \(\partial _{i}u^{p_{i}-1}\in L^1(\Omega )\), \(\frac{f}{u^{\gamma (x) }} \in L^1_{loc}(\Omega )\), and one has the identity

$$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\partial _{i}\varphi =\int \limits _{\Omega }\frac{f\varphi }{u^{\gamma (x) }}\ \ \ \ \ \ \ \ \forall \varphi \in C_{0}^{1}\left( \Omega \right) . \end{aligned}$$

We will also very often use the following indices

$$\begin{aligned} \frac{1}{\overline{p}}=\frac{1}{N}\sum \limits _{i=1}^{N}\frac{1}{p_{i}} \end{aligned}$$

and

$$\begin{aligned} \overline{p}^{*}=\frac{N\overline{p}}{N-\overline{p}},~p_{\infty }=\max \left\{ p_{N},\overline{p}^{*}\right\} \end{aligned}$$

The following Theorem states some anisotropic Sobolev type inequalities, for more details we refer to the early works [13, 17, 20].

Theorem 1.2

There exists a positive constant C,  depending only on \(\Omega \), such that for every \( v\in W_{0}^{1,(p_{i})}\left( \Omega \right) ,\) we have

$$\begin{aligned}&\left\| v\right\| _{L^{\overline{p}^{*}}\left( \Omega \right) }^{p_{N}}\le C\sum \limits _{i=1}^{N}\left\| \partial _{i}v\right\| _{L^{p_{i}}\left( \Omega \right) }^{p_{i}}, \end{aligned}$$
(2)
$$\begin{aligned}&\left\| v\right\| _{L^{r}\left( \Omega \right) }\le C\prod \limits _{i=1}^{N}\left\| \partial _{i}v\right\| _{L^{p_{i}}\left( \Omega \right) }^{\frac{1}{N}}~~~\forall r\in \left[ 1, \overline{p}^{*}\right] \end{aligned}$$
(3)

and \(\forall v\in W_{0}^{1,(p_{i})}\left( \Omega \right) \cap L^{\infty }\left( \Omega \right) ,\overline{p}<N\)

$$\begin{aligned} \left( \int \limits _{\Omega }\left| v\right| ^{r}\right) ^{\frac{N}{p} -1}\le C\prod \limits _{i=1}^{N}\left( \int \limits _{\Omega } \left| \partial _{i}v\right| ^{p_{i}}\left| v\right| ^{t_{i}p_{i}}\right) ^{\frac{1}{p_{i}}}, \end{aligned}$$
(4)

for every r and \(t_{j}\) chosen in such a way to have

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{1}{r}=\frac{\gamma (x) _{i}(N-1)-1+\frac{1}{p_{i}}}{t_{i}+1} \\ \sum \limits _{i=1}^{N}\gamma (x) _{i}=1. \end{array} \right. \end{aligned}$$

In the whole paper, C will denote a constant that may change from line to line.

2 Approximation problems

All the results obtained in this section, are direct consequences of the ones presented in [2, 14], but for the reader convenience we present them in details.

Let us first consider the following approximation problems

$$\begin{aligned} \left\{ \begin{array}{ll} -Lu_{n}=\frac{f_{n}}{\left( u_{n}+\frac{1}{n}\right) ^{\gamma (x) }} &{}\quad in~\Omega , \\ u_{n}=0 &{}\quad on~\Omega , \\ u_{n}\ge 0 &{}\quad in~\Omega , \end{array} \right. \end{aligned}$$
(5)

where \(f_{n}=T_{n}(f).\)

Recalling that

$$\begin{aligned} T_{n}(s)=\left\{ \begin{array}{ll} n\frac{s}{\left| s\right| } &{} \quad if~\left| s\right| >n \\ s &{} \quad if~\left| s\right| \le n \end{array} \right. \end{aligned}$$

Lemma 2.1

The problem (5) has a solution in \(W_{0}^{1,(p_{i})}\left( \Omega \right) .\)

Proof

We will follow the same reasoning as in [2].

Fix \(n\in \mathbb {N~}\), and let \(v\in L_{\overline{p}^{*}}\left( \Omega \right) .\) Consider the equation

$$\begin{aligned} -Lw=\frac{f_{n}}{\left( \left| v\right| +\frac{1}{n}\right) ^{\gamma (x) }} , \end{aligned}$$
(6)

it is clear that the previous problem has a unique solution whenever the right hand side belongs to \(L^{s}\left( \Omega \right) \) with \(s\ge p_{\infty }^{\prime }\) see for instance [8, 9]. Denoting \( w=S(v),\)

using w as test function in (6), we obtain

$$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}w\right| ^{p_{i}}=\int \limits _{\Omega }\frac{wf_{n}}{\left( \left| v\right| +\frac{1 }{n}\right) ^{\gamma (x) }}\le n^{\gamma (x) +1}\int \limits _{\Omega }\left| w\right| \end{aligned}$$

by Sobolev inequality (2),

$$\begin{aligned} \left\| w\right\| _{L^{\overline{p}^{*}}\left( \Omega \right) }^{p_{N}}\le C\sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}w\right| ^{p_{i}}, \end{aligned}$$

by Hölder inequality

$$\begin{aligned} \int \limits _{\Omega }\left| w\right| \le \left( \int \limits _{\Omega }\left| w\right| ^{\overline{p}^{*}}\right) ^{\frac{1}{\overline{ p}^{*}}}. \end{aligned}$$

Hence

$$\begin{aligned} \left\| w\right\| _{L^{\overline{p}^{*}}\left( \Omega \right) }^{p_{N}}\le Cn^{\gamma (x) +1}\left\| w\right\| _{L^{\overline{p}^{*}}\left( \Omega \right) }, \end{aligned}$$

and then

$$\begin{aligned} \left\| w\right\| _{L^{\overline{p}^{*}}\left( \Omega \right) }\le C^{\prime }\left( n^{\gamma (x) +1}\right) ^{\frac{1}{p_{N}-1}}=R_{N}, \end{aligned}$$

which means that the ball of radius \(R_{N}\) in \(L^{\overline{p}^{*}}\left( \Omega \right) \) is invariant by S, and so by Sobolev embedding and Schauder’s fixed point theorem we conclude that the approximation problem (5) has a solution in \(W_{0}^{1,(p_{i})}\left( \Omega \right) \), for every fixed n.

\(\square \)

Lemma 2.2

The sequence \(\left\{ u_{n}\right\} _{n}\) is increasing with respect to n.

Proof

We recall that \(f_{n}=T_{n}\left( f\right) \) and so \(0\le f_{n}\le f_{n+1}\)

$$\begin{aligned} -Lu_{n}=\frac{f_{n}}{\left( u_{n}+\frac{1}{n}\right) ^{\gamma (x) }}\le \frac{ f_{n+1}}{\left( u_{n}+\frac{1}{n+1}\right) ^{\gamma (x) }} \end{aligned}$$

as

$$\begin{aligned} -Lu_{n+1}=\frac{f_{n+1}}{\left( u_{n+1}+\frac{1}{n+1}\right) ^{\gamma (x) }} \end{aligned}$$

and so one has that

$$\begin{aligned} -Lu_{n}+Lu_{n+1}\le & {} f_{n+1}\left[ \frac{1}{\left( u_{n}+\frac{1}{n+1} \right) ^{\gamma (x) }}-\frac{1}{\left( u_{n+1}+\frac{1}{n+1}\right) ^{\gamma (x) }} \right] \\\le & {} f_{n+1}\left[ \frac{\left( u_{n+1}+\frac{1}{n+1}\right) ^{\gamma (x) }-\left( u_{n}+\frac{1}{n+1}\right) ^{\gamma (x) }}{\left( u_{n}+\frac{1}{n+1} \right) ^{\gamma (x) }\left( u_{n+1}+\frac{1}{n+1}\right) ^{\gamma (x) }}\right] \end{aligned}$$

using \(\left( u_{n}-u_{n+1}\right) ^{+}\) as test function in the last inequality, the right hand side gives

$$\begin{aligned} f_{n+1}\left[ \frac{\left( u_{n+1}+\frac{1}{n+1}\right) ^{\gamma (x) }-\left( u_{n}+\frac{1}{n+1}\right) ^{\gamma (x) }}{\left( u_{n}+\frac{1}{n+1}\right) ^{\gamma (x) }\left( u_{n+1}+\frac{1}{n+1}\right) ^{\gamma (x) }}\right] \left( u_{n}-u_{n+1}\right) ^{+}\le 0. \end{aligned}$$

Now, taking into account the problems associated to \(u_n\) and to \(u_{n+1}\), it follows that

$$\begin{aligned} \int \limits _{\Omega }\left( -Lu_{n}+Lu_{n+1}\right) \left( u_{n}-u_{n+1}\right) ^{+}\le 0. \end{aligned}$$

Thus

$$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left( \left| \partial _{i}u_{n}\right| ^{p_{i}-2}\partial _{i}u_{n}-\left| \partial _{i}u_{n+1}\right| ^{p_{i}-2}\partial _{i}u_{n+1}\right) \partial _{i}\left( u_{n}-u_{n+1}\right) ^{+}\le 0. \end{aligned}$$

Integrating over the subset of \(\Omega \) where \(u_{n}\ge u_{n+1}\) and using the following inequality for \(p_{i}\ge 2\)

$$\begin{aligned} C_{0}\left| \partial _{i}\left( u_{n}-u_{n+1}\right) \right| ^{p_{i}}\le \left( \left| \partial _{i}u_{n}\right| ^{p_{i}-2}\partial _{i}u_{n}-\left| \partial _{i}u_{n+1}\right| ^{p_{i}-2}\partial _{i}u_{n+1}\right) \partial _{i}\left( u_{n}-u_{n+1}\right) \end{aligned}$$

we reach that

$$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}\left( u_{n}-u_{n+1}\right) ^{+}\right| ^{p_{i}}\le 0. \end{aligned}$$

Hence

$$\begin{aligned} u_n\le u_{n+1}, \end{aligned}$$

which allows us to conclude that \(\left\{ u_{n}\right\} _{n}\) is increasing with respect to n.

\(\square \)

Remark 2.3

We limit ourselves to the case \(p_{i}\ge 2\) because (at our knowledge), the operator L verify a strong maximal principle only in the case \(p_{i}\ge 2\) see for instance [8], maximal principle that will be necessary in the sequel.

Lemma 2.4

For all \(n\in {\mathbb {N}}\), \(u_{n}\) the solution to the approximation problem (5), is such that \(u_{n}\in L^{\infty }\left( \Omega \right) \) and for all \(K\subset \subset \Omega \), \(u_{n}\ge C_{K}>0.\)

Proof

By some modifications in the theory of Leray-Lions operators theory one can show the existence of solution to

$$\begin{aligned} -Lu_{1}=\frac{f_{1}}{\left( u_{1}+1\right) ^{\gamma (x) }} \end{aligned}$$

and so

$$\begin{aligned} -Lu_{1}=\frac{f_{1}}{\left( \left\| u_{1}\right\| _{\infty }+1\right) ^{\gamma (x) }}\ge 0 \end{aligned}$$

the strong maximum principle, and the monotonicity of \(\left\{ u_{n}\right\} _{n}\) give that \(u_{n}\ge C_{K}>0.\) The \(L^{\infty }\left( \Omega \right) \) estimate of \(\left\{ u_{n}\right\} _{n},\) is a direct consequence of Stampachia result [19], as done in [2]. \(\square \)

3 Passage to the limit

For fixed \(\delta \), let \(\Omega _{\delta }=\left\{ x\in \Omega ,dist(x,\partial \Omega )<\delta \right\} \)

Theorem 3.1

Let \(s=\dfrac{N\overline{p}}{N\left( \overline{p}-1\right) +\overline{p}}\) and \(f\in L^{s}\left( \Omega \right) ,\) assume that there exists a \(\delta >0\) such that \(\gamma (x)\le 1\) in \(\Omega _{\delta }\), then the sequence \(\left\{ u_{n}\right\} _{n}\) of solutions to (5), is bounded in \(W_{0}^{1,\left( p_{i}\right) }\left( \Omega \right) .\)

Proof

Put \( \omega _{\delta } = \Omega \backslash \overline{{\Omega _{\delta } }} \), by the previous results we know that \(u_{n}\ge C_{\omega _{\delta }}>0.\) Now using \(u_{n}\) as test function in (5) we obtain

$$\begin{aligned} \sum \limits _{i=1}^{N}\int _{\Omega }\left| \partial _{i}u_{n}\right| ^{p_{i}}= & {} \int _{\Omega }\frac{f_{n}(x)}{\left( u_{n}+\frac{1}{n}\right) ^{\gamma (x)}}u_{n} \\= & {} \int _{\overline{\Omega _{\delta }}}\frac{f_{n}(x)}{\left( u_{n}+\frac{1}{n }\right) ^{\gamma (x)}}u_{n}+\int _{\omega _{\delta }}\frac{f_{n}(x)}{\left( u_{n}+\frac{1}{n}\right) ^{\gamma (x)}}u_{n} \\\le & {} \int _{\overline{\Omega _{\delta }}}f(x)u_{n}^{1-\gamma (x)}+\int _{\omega _{\delta }}\frac{f(x)}{C_{\omega _{\delta }}^{\gamma (x)}} u_{n} \\\le & {} \int _{\overline{\Omega _{\delta }}\cap \left\{ u_{n}\le 1\right\} }f(x)+\int _{\overline{\Omega _{\delta }}\cap \left\{ u_{n}\ge 1\right\} }f(x)u_{n}+\int _{\omega _{\delta }}\frac{f(x)}{C_{\omega _{\delta }}^{\gamma (x)}}u_{n} \\\le & {} \left\| f\right\| _{L^{1}\left( \Omega \right) }+\left( 1+\left\| C_{\omega _{\delta }}^{-\gamma (x)}\right\| _{L^{\infty }\left( \Omega \right) }\right) \int _{\Omega }f(x)u_{n} \end{aligned}$$

Using Hölder and Sobolev inequalities, we then obtain

$$\begin{aligned} \sum \limits _{i=1}^{N}\int _{\Omega }\left| \partial _{i}u_{n}\right| ^{p_{i}}\le \left\| f\right\| _{L^{1}\left( \Omega \right) }+C\left( 1+\left\| C_{\omega _{\delta }}^{-\gamma (x)}\right\| _{L^{\infty }\left( \Omega \right) }\right) \left\| f\right\| _{L^{s}\left( \Omega \right) }\left[ \sum \limits _{i=1}^{N}\int _{\Omega }\left| \partial _{i}u_{n}\right| ^{p_{i}}\right] ^{\frac{1}{p_{N}}} \end{aligned}$$

which implies that

$$\begin{aligned} \sum \limits _{i=1}^{N}\int _{\Omega }\left| \partial _{i}u_{n}\right| ^{p_{i}}\le C \end{aligned}$$

where C is a constant independant of n. \(\square \)

Theorem 3.2

Let \(s=\dfrac{N\overline{p}}{N\left( \overline{p}-1\right) +\overline{p}}\) and \(f\in L^{s}\left( \Omega \right) ,\) assume that there exists a \(\delta >0 \) such that \(\gamma (x)\le 1\) in \(\Omega _{\delta }\), then problem (1) posses a solution \(u\in W_{0}^{1,\left( p_{i}\right) }\left( \Omega \right) . \)

Proof

By the previous proposition \(\left\{ u_{n}\right\} _{n}\) is bounded in \( W_{0}^{1,\left( p_{i}\right) }\left( \Omega \right) \), thus (up to a subsequence) \(\left\{ u_{n}\right\} _{n}\) converges weakly to some u in \( W_{0}^{1,\left( p_{i}\right) }\left( \Omega \right) .\) On the other hand,  \( \left\{ u_{n}\right\} _{n}\) converges strongly in \(L^{\theta }\left( \Omega \right) \) for \(\theta <\overline{p}^{*}\), thus \(\left\{ u_{n}\right\} _{n}\) converges to u almost everywhere in \(\Omega .\) So one has that for every \(\varphi \in C_{0}^{1}\left( \Omega \right) \)

$$\begin{aligned} \underset{n\rightarrow +\infty }{\lim }\sum \limits _{i=1}^{N}\int _{\Omega }\left| \partial _{i}u_{n}\right| ^{p_{i}-2}\partial _{i}u_{n}\partial _{i}\varphi =\sum \limits _{i=1}^{N}\int _{\Omega }\left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\partial _{i}\varphi \end{aligned}$$

By the fact that

$$\begin{aligned} 0\le \left| \frac{f_{n}(x)\varphi }{\left( u_{n}+\frac{1}{n}\right) ^{\gamma (x)}}\right| \le \left\| \varphi C_{\omega }^{-\gamma (x)}\right\| _{L^{\infty }\left( \Omega \right) }f(x) \end{aligned}$$

for every \(\varphi \in C_{0}^{1}\left( \Omega \right) \), whenever \(\varphi \ne 0\) and on the set where \(u_{n}\ge C_{\omega },\) \(\omega \) being the support of \(\varphi ; \) the dominated Lebesgue’s theorem permits us to conclude that

$$\begin{aligned} \underset{n\rightarrow +\infty }{\lim }\int _{\Omega }\frac{f_{n}(x)\varphi }{ \left( u_{n}+\frac{1}{n}\right) ^{\gamma (x)}}=\int _{\Omega }\frac{ f(x)\varphi }{u^{\gamma (x)}} \end{aligned}$$

by the sequel, the limit u of the sequence \(\left\{ u_{n}\right\} _{n}\) verify

$$\begin{aligned} \sum \limits _{i=1}^{N}\int _{\Omega }\left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\partial _{i}\varphi =\int _{\Omega }\frac{ f(x)\varphi }{u^{\gamma (x)}}. \end{aligned}$$

\(\square \)

Theorem 3.3

Assume that for some \(\gamma ^{*}>1\) and some \( \delta >0\) we have \(\left\| \gamma \right\| _{L^{\infty }\left( \Omega \right) }\le \gamma ^{*}.\) Provided that \(f\in L^{s}\left( \Omega \right) \) with \(s=\dfrac{N\left( \gamma ^{*}-1+\overline{p} \right) }{N\left( \overline{p}-1\right) +\overline{p}\gamma ^{*}},\) problem (1) has a solution u in \(L^{\alpha }\left( \Omega \right) \) with \(\alpha =\dfrac{N\left( \gamma ^{*}-1+\overline{p}\right) }{\left( N-\overline{p}\right) },\) belonging to \(W_{loc}^{1,\left( p_{i}\right) }\left( \Omega \right) .\)

Proof

Let us use \(u_{n}^{\gamma ^{*}}\) as test function in (5), so we obtain for every \(i=1,2,...,N\)

$$\begin{aligned} \int _{\Omega }\left| \partial _{i}u_{n}\right| ^{p_{i}}u_{n}^{\gamma ^{*}-1}\le & {} \int _{\overline{\Omega _{\delta }}}f(x)u_{n}^{\gamma ^{*}-\gamma (x)}+\int _{\omega _{\delta }}\frac{f(x)}{C_{\omega _{\delta }}^{\gamma (x)}}u_{n}^{\gamma ^{*}} \\\le & {} \left\| f\right\| _{L^{1}\left( \Omega \right) }+\left( 1+\left\| C_{\omega _{\delta }}^{-\gamma (x)}\right\| _{L^{\infty }\left( \Omega \right) }\right) \int _{\Omega }f(x)u_{n}^{\gamma ^{*}} \\\le & {} \left\| f\right\| _{L^{1}\left( \Omega \right) }+\left( 1+\left\| C_{\omega _{\delta }}^{-\gamma (x)}\right\| _{L^{\infty }\left( \Omega \right) }\right) \left( \int _{\Omega }f^{s}(x)\right) ^{\frac{ 1}{s}}\left( \int _{\Omega }u_{n}^{\gamma ^{*}\beta }\right) ^{\frac{1}{ \beta }} \end{aligned}$$

with \(\beta =\dfrac{N\left( \gamma ^{*}-1+\overline{p}\right) }{\left( N- \overline{p}\right) \gamma ^{*}},\) and so

$$\begin{aligned} \int _{\Omega }\left| \partial _{i}u_{n}\right| ^{p_{i}}u_{n}^{\gamma ^{*}-1}\le C_{1}+C_{2}\left( \int _{\Omega }u_{n}^{\gamma ^{*}\beta }\right) ^{\frac{1}{\beta }}, \end{aligned}$$

thus

$$\begin{aligned} \left( \int _{\Omega }\left| \partial _{i}u_{n}\right| ^{p_{i}}u_{n}^{\gamma ^{*}-1}\right) ^{\frac{1}{p_{i}}}\le \left( C_{1}+C_{2}\left( \int _{\Omega }u_{n}^{\gamma ^{*}\beta }\right) ^{\frac{ 1}{\beta }}\right) ^{\frac{1}{p_{i}}} \end{aligned}$$

which implies that

$$\begin{aligned} \prod \limits _{i=1}^{N}\left( \int _{\Omega }\left| \partial _{i}u_{n}\right| ^{p_{i}}u_{n}^{\gamma ^{*}-1}\right) ^{\frac{1}{ p_{i}}}\le \left( C_{1}+C_{2}\left( \int _{\Omega }u_{n}^{\gamma ^{*}\beta }\right) ^{\frac{1}{\beta }}\right) ^{\sum \limits _{i=1}^{N}\frac{1}{ p_{i}}}=\left( C_{1}+C_{2}\left( \int _{\Omega }u_{n}^{\alpha }\right) ^{ \frac{1}{\beta }}\right) ^{\frac{N}{\overline{p}}} \end{aligned}$$

with the following choice of exponents

$$\begin{aligned} \left\{ \begin{array}{l} t_{i}p_{i}=\gamma ^{*}-1 \\ r=\alpha =\dfrac{N\left( \gamma ^{*}-1+\overline{p}\right) }{\left( N- \overline{p}\right) } \\ \frac{1}{r}=\frac{\gamma _{i}\left( N-1\right) -1+\frac{1}{p_{i}}}{t_{i}+1} \end{array} \right. \end{aligned}$$

Sobolev inequality (4) gives

$$\begin{aligned} \left( \int _{\Omega }u_{n}^{\alpha }\right) ^{\frac{N}{\overline{p}}-1}\le \left( C_{1}+C_{2}\left( \int _{\Omega }u_{n}^{\alpha }\right) ^{\frac{1}{ \beta }}\right) ^{\frac{N}{\overline{p}}} \end{aligned}$$

and so

$$\begin{aligned} \left( \int _{\Omega }u_{n}^{\alpha }\right) ^{1-\frac{\overline{p}}{N}}\le C_{1}+C_{2}\left( \int _{\Omega }u_{n}^{\alpha }\right) ^{\frac{1}{\beta }} \end{aligned}$$

by the fact that

$$\begin{aligned} \frac{1}{\beta }<1-\frac{\overline{p}}{N} \end{aligned}$$

we conclude that \(\{u_{n}\}_{n}\) is bounded in \(L^{\alpha }\left( \Omega \right) \) with \(\alpha =\dfrac{N\left( \gamma ^{*}-1+\overline{p}\right) }{\left( N-\overline{p} \right) }\) and by the monotone convergence theorem, \(\{u_{n}\}_n\) converges strongly to \(u \in L^{\alpha }\left( \Omega \right) . \)

On the other side using \(u_{n}^{\gamma ^{*}}\) as test function in (5) we get

$$\begin{aligned} \sum \limits _{i=1}^{N}\int _{\Omega }\left| \partial _{i}u_{n}\right| ^{p_{i}}u_{n}^{\gamma ^{*}-1}\le C \end{aligned}$$

by strong maximum principle we have for every compact \(K\subset \subset \Omega \)

$$\begin{aligned} C_{K}^{\gamma ^{*}-1}\sum \limits _{i=1}^{N}\int _{\Omega }\left| \partial _{i}u_{n}\right| \le C \end{aligned}$$

thus we obtain weak convergence of \(\left\{ u_{n}\right\} _{n}\) to u in \( W_{loc}^{1,\left( p_{i}\right) }\left( \Omega \right) .\)

To complete the proof, we follow the same steps as in the previous Proposition. \(\square \)