1 Introduction

Problems involving a partial differential equation interacting with singular nonlinear terms, due to their importance, have been widely studied, and several works are available in the literature. In this short introduction, we will present a non-exhaustive content of the different works related to singular problems. We start by indicating to the reader the essential work [24] which contains almost all basic and advanced tools to study general singular problems. We also mention various works that consider semilinear problems with singular terms as [1, 6,7,8], and when the principal operator is a p-Laplacian like operator we mention [13, 30] and the references therein.

Problems involving the anisotropic operator

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

encounter a great interest, we cite for example [15,16,17], and the references therein.

We also mention the leading works [4, 21, 22] and the very recent one [12]; where the anisotropic operator is associated to a nonlinearity, and existence, uniqueness, multiplicity and non existence results are obtained by various ways.

When a singular term is associated to the anisotropic operator L there are some few recent results, for example in [23, 28, 31], existence and regularity of solutions to equation

$$\begin{aligned} Lu=\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\right] =\frac{f(x)}{u^{\gamma (x)}} +\lambda g(x,u) \end{aligned}$$

where g is a regular nonlinearity in u,  are obtained for the case \(\lambda =0\) by approximation methods, while in [18], the authors obtained existence results for \(\lambda \ne 0\) by monotonicity methods.

Systems involving operator L,  are less studied we cite in this direction the works [9, 18]. To the best of our knowledge, there are only two works in the literature studying problems where the anisotropic operator L is associated to nonlocal terms that are [5, 19]. We also mention [15, 29] and the references therein for results about variational techniques applied to the study of problems involving L.

Parabolic problems, with anisotropic operator L,  have also been studied by different authors we indicate here some of them [32, 34, 36].

Inspired by the works [7, 14, 20, 25] we consider in this paper the system

$$\begin{aligned} \left\{ \begin{array}{ll} -Lu=-\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\right] =p\dfrac{v^{q}}{u^{1-p}} &{}\quad in~\Omega , \\ -Lv=-\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}v\right| ^{p_{i}-2}\partial _{i}v\right] =qv^{q-1}u^{p} &{}\quad in~\Omega , \\ u>0\text { and }v>0 &{}\quad in~\Omega , \\ u=0\text { and }v=0 &{}\quad on~\partial \Omega , \end{array} \right. \end{aligned}$$
(1)

where \(\Omega\) is a bounded, open subset of \(\mathbb {R}^{N}\) \((N\ge 2)\) and, without loss of generality, \(1\le p_{1}\le p_{2}\le \cdots \le p_{N}.\) We will assume that \(\frac{1}{\overline{p}}=\frac{1}{N}\sum \nolimits _{i=1}^{N} \frac{1}{p_{i}},\ \overline{p}<N,\) \(0<p<1,\) \(q>1\) and \(p_{N}<p+q<\frac{N \overline{p}}{N-\overline{p}}=\overline{p}^{*}.\) In the whole paper we will denote \(\frac{\partial u}{\partial x_{i}}=\partial _{i}u.\) As the considered system is singular, we will use nondifferentiable variational techniques introduced in [2, 3].

The paper is organised as follows, after this brief introduction, we present some preliminaries dealing with the functional setting associated to our problem. In the third section we study approximating regular problems, where the singular nonlinearity is replaced by a regular one. In the fourth section we prove that the sequence of solutions to the approximating problems converges to the solution of problem (1). The paper is ended with some concluding remarks.

2 Preliminaries

The natural function spaces associated to the operator L are the 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 with the usual norm

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

We will also use very often the following indices

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

and

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

It will be assumed throughout this paper that \(p_{\infty }=\) \(\overline{p}^{*}<N,\) so in this case we will have that

$$\begin{aligned} W_{0}^{1,(p_{i})}\left( \Omega \right) \subset L^{r}\left( \Omega \right) \text { \ }\forall r\in \left[ 1,\overline{p}^{*}\right] \end{aligned}$$

this imbedding being compact whenever \(r<\overline{p}^{*}.\) Let us recall the following Sobolev type inequalities, we refer to the early works [27, 33, 35].

Theorem 1

$$\begin{aligned}&\left\| v\right\| _{L^{\overline{p}^{*}}\left( \Omega \right) }^{p_{N}}\le C\sum \limits _{i=1}^{N}\left\| \partial _{i}v\right\| _{L^{p_{i}}\left( \Omega \right) }^{p_{i}}, \end{aligned}$$
(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 such a way to have

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

We also have the following algebraic inequalities:

  • There exists a \(C>0\) not depending on \(\rho \in (0,1)\) such that for given \(\sigma _{i}>0,\) \(i=1,2\ldots N\) we have

    $$\begin{aligned} \sum \limits _{i=1}^{N}\sigma _{i}=\rho \Longrightarrow \sum \limits _{i=1}^{N} \frac{\sigma _{i}^{p_{i}}}{p_{i}}\ge C\rho ^{p_{N}} \end{aligned}$$
    (5)

  • For \(p_{i}\ge 2\)

    $$\begin{aligned} C\left| a-b\right| ^{p_{i}}\le \left( \left| a\right| ^{p_{i}-2}a-\left| b\right| ^{p_{i}-2}b\right) (a-b) \end{aligned}$$
    (6)

  • For \(1<p_{i}\le 2\)

    $$\begin{aligned} C\frac{\left| a-b\right| ^{2}}{\left( \left| a\right| +\left| b\right| \right) ^{2-p_{i}}}\le \left( \left| a\right| ^{p_{i}-2}a-\left| b\right| ^{p_{i}-2}b\right) (a-b) \end{aligned}$$
    (7)

We need as well to recall the following truncating functions

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

and

$$\begin{aligned} G_{n}(s)=s-T_{n}(s). \end{aligned}$$

Definition 1

We will say that positive \(u,v\in W_{0}^{1,(p_{i})}\left( \Omega \right)\) are solutions to (1) if and only if

$$\begin{aligned} \left\{ \begin{array}{ll} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\partial _{i}\varphi \right] =p\int \limits _{\Omega }\dfrac{v^{q}}{u^{1-p}}\varphi , &{}\quad \varphi \in C_{0}^{1}\left( \Omega \right) , \\ \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left[ \left| \partial _{i}v\right| ^{p_{i}-2}\partial _{i}v\partial _{i}\psi \right] =q\int \limits _{\Omega }v^{q-1}u^{p}\psi , &{}\quad \psi \in W_{0}^{1,(p_{i})}\left( \Omega \right) , \end{array} \right. \end{aligned}$$
(8)

We have the following comparison principle

Proposition 2

[9] Comparison principle

If \(u,v\in W_{0}^{1,(p_{i})}\) are such that

$$\begin{aligned} \left\{ \begin{array}{l} -Lu=-\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\right] \le -Lv=-\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}v\right| ^{p_{i}-2}\partial _{i}v\right] , \\ u\le v\quad \text {on}\,\,\partial \Omega , \end{array} \right. \end{aligned}$$
(9)

then \(u\le v\) a.e. in \(\Omega\).

The considered system (1) has a variational structure, so its solution can be seen as a critical point of the following functional

$$\begin{aligned} J(w,z)=\sum \limits _{i=1}^{N}\frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}w\right| ^{p_{i}}+\sum \limits _{i=1}^{N}\frac{1}{p_{i}} \int \limits _{\Omega }\left| \partial _{i}z\right| ^{p_{i}}-\int \limits _{\Omega }w_{+}^{p}z_{+}^{q} \end{aligned}$$
(10)

which is singular when \(p<1\) or/and \(q<1.\) Where, as usual \(u_{+}=\max \left\{ u,0\right\}\) and \(u_{-}=\max \left\{ -u,0\right\} =-\min \left\{ u,0\right\}\)

We will need to use the following theorem that can be find in [2, 14].

Theorem 3

(Mountain-Pass theorem for nondifferentiable functionals)

Let X,  Y be two Banach spaces with \(Y\subset X,\) and

$$\begin{aligned} F:X\rightarrow \mathbb {R} \end{aligned}$$

be a functional such that

  1. (H1)

    F is continuous on Y.

  2. (H2)

    F possesses a Gateaux dérivative in X,  \(\left\langle F^{\prime }(u),v\right\rangle\) through any direction \(v\in Y.\)

  3. (H3)

    For every fixed \(v\in Y\), the function \(\left\langle F^{\prime }(.),v\right\rangle\) is continuous in X.

  4. (H4)

    There exists \(\overline{u}\in Y\) such that

    $$\begin{aligned} c=\underset{\gamma \in \Gamma }{\inf }\underset{t\in \left[ 0,1\right] }{ \max }F\left( \gamma (t)\right) >\max \left\{ F(0),F(\overline{u})\right\} \end{aligned}$$

    with

    $$\begin{aligned} \Gamma =\left\{ \text { Continuous }\gamma :\left[ 0,1\right] \rightarrow Y; \text { }\gamma (0)=0\text { and }\gamma (1)=\overline{u}\right\} \end{aligned}$$
  5. (H5)

    Any sequence \(\left\{ u_{n}\right\} _{n}\) in Y possesses a convergent subsequence in X if, for some real positive sequence \(\left\{ M_{n}\right\} _{n}\) and \(\epsilon _{n}\rightarrow 0\) fulfills

    1. (a)

      \(\left\{ F(u_{n})\right\} _{n}\) is bounded

    2. (b)

      \(\left\| u_{n}\right\| _{Y}\le 2M_{n}\) for all n

    3. (c)

      \(\left| \left\langle F^{\prime }(u_{n}),v\right\rangle \right| \le \epsilon _{n}\left[ \frac{\left\| v\right\| _{Y}}{M_{n}} +\left\| v\right\| _{X}\right]\) for all \(v\in Y.\)

Then c is a critical value of F,  which means there exists \(u\in Y\smallsetminus \left\{ 0\right\}\)such that \(F(u)=c\) and \(\left\langle F^{\prime }(u),v\right\rangle =0\) for all \(v\in Y.\)

We also introduce the following function

$$\begin{aligned} g_{n}(s)=\frac{s^{2}}{\left( s+\frac{1}{n}\right) ^{2-p}} \end{aligned}$$
(11)

which will play the role of a smooth approximation of \(s^{p}\) when \(p<1.\) This function has the properties

$$\begin{aligned} \frac{s^{2}}{\left( s+1\right) ^{2-p}}\le g_{n}(s)\le s^{p}, \end{aligned}$$
(12)

and

$$\begin{aligned} \frac{1}{p_{N}}g_{n}^{\prime }(s)s+\left( \frac{q}{p_{N}}-1\right) g_{n}(s)\ge \frac{p+q-p_{N}}{p_{N}}g_{n}(s). \end{aligned}$$
(13)

and also

$$\begin{aligned} g_{n}^{\prime }(s)s\le 2s^{p} \end{aligned}$$
(14)

3 The approximating problems

As \(p<1,\) the functional J introduced in (10) is not differentiable (in w), so we will use the approximating functional

$$\begin{aligned} J_{n}(w,z)=\sum \limits _{i=1}^{N}\frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}w\right| ^{p_{i}}+\sum \limits _{i=1}^{N}\frac{1}{ p_{i}}\int \limits _{\Omega }\left| \partial _{i}z\right| ^{p_{i}}-\int \limits _{\Omega }g_{n}(w_{+})z_{+}^{q} \end{aligned}$$

with \(g_{n}(s)=\frac{s^{2}}{\left( s+\frac{1}{n}\right) ^{2-p}}.\)

Proposition 4

The functional \(J_{n}\) fulfils all conditions of Theorem 3 for \(X=W_{0}^{1,(p_{i})}\left( \Omega \right) \times W_{0}^{1,(p_{i})}\left( \Omega \right)\) and \(Y=\left( W_{0}^{1,(p_{i})}\left( \Omega \right) \cap L^{\infty }\left( \Omega \right) \right) \times \left( W_{0}^{1,(p_{i})}\left( \Omega \right) \cap L^{\infty }\left( \Omega \right) \right) .\)

Proof

  1. (H1)

    \(J_{n}:Y\rightarrow \mathbb {R}\) is continuous, indeed let \(\left\{ \left( w_{k},z_{k}\right) \right\} _{k}\subset Y\) be a strongly convergent sequence in Y,  \(\left( w_{k},z_{k}\right) \rightarrow \left( w,z\right)\) so as direct consequence one have that

    $$\begin{aligned} \sum \limits _{i=1}^{N}\frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}w_{k}\right| ^{p_{i}}\rightarrow \sum \limits _{i=1}^{N}\frac{1}{p_{i}} \int \limits _{\Omega }\left| \partial _{i}w\right| ^{p_{i}}, \end{aligned}$$
    $$\begin{aligned} \sum \limits _{i=1}^{N}\frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}w_{k}\right| ^{p_{i}}\rightarrow \sum \limits _{i=1}^{N}\frac{1}{p_{i}} \int \limits _{\Omega }\left| \partial _{i}w\right| ^{p}, \end{aligned}$$

    and as both \(\left\{ w_{k}\right\} _{k}\) and \(\left\{ z_{k}\right\} _{k}\) are bounded in \(L^{\overline{p}^{*}}\) and \(\frac{\overline{p}^{*}}{ p+q}>1\), so by Vitali’s theorem we get

    $$\begin{aligned} g_{n}(w_{n+})z_{n+}^{q}\rightarrow g_{n}(w_{+})z_{+}^{q}\text { in } L^{1}\left( \Omega \right) \end{aligned}$$

    which means that \(J_{n}\) is continuous in X and consequently it is also continuous in Y.

  2. (H2)

    The functional \(J_{n}\) is constructed in such a way to make (H2) verified.

  3. (H3)

    Let \(\left( \varphi ,\psi \right) \in Y\) and let \(\left\{ \left( w_{k},z_{k}\right) \right\} _{k}\subset X\) be a strongly convergent sequence in X,  \(\left( w_{k},z_{k}\right) \rightarrow \left( w,z\right) ,\) we need to prove that

    $$\begin{aligned} \left\langle J_{n}^{\prime }\left( w_{k},z_{k}\right) ,\left( \varphi ,\psi \right) \right\rangle \rightarrow \left\langle J_{n}^{\prime }\left( w,z\right) ,\left( \varphi ,\psi \right) \right\rangle . \end{aligned}$$

    Observing that

    $$\begin{aligned} \left\langle J_{n}^{\prime }\left( w_{k},z_{k}\right) ,\left( \varphi ,\psi \right) \right\rangle= & {} \int \limits _{\Omega }\left( J_{n_{w}}\varphi +J_{n_{z}}\psi \right) \\= & {} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}w_{k}\right| ^{p_{i}-2}\partial _{i}w_{k}\partial _{i}\varphi +\sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}z_{k}\right| ^{p_{i}-2}\partial _{i}z_{k}\partial _{i}\psi \\&-\int \limits _{\Omega }g_{n}^{\prime }(w_{k+})z_{k+}^{q}\varphi -q\int \limits _{\Omega }g_{n}(w_{k+})z_{k+}^{q-1}\psi . \end{aligned}$$

    The convergence of the two terms

    $$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}w_{k}\right| ^{p_{i}-2}\partial _{i}w_{k}\partial _{i}\varphi , \end{aligned}$$

    and

    $$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}z_{k}\right| ^{p_{i}-2}\partial _{i}z_{k}\partial _{i}\psi \end{aligned}$$

    is a consequence of the convergence of \(\left\{ \left( w_{k},z_{k}\right) \right\} _{k}.\) Now we deal with the third term

    $$\begin{aligned} \int \limits _{\Omega }g_{n}^{\prime }(w_{+})z_{+}^{q}\varphi , \end{aligned}$$

    one have that

    $$\begin{aligned} g_{n}^{\prime }(w_{k+})z_{k+}^{q}\rightarrow g_{n}^{\prime }(w_{+})z_{+}^{q} \text { a.e.} \end{aligned}$$

    and as

    $$\begin{aligned} \left| g_{n}^{\prime }(w_{k+})z_{k+}^{q}\right| \le 2n^{1-p}z_{k+}^{q} \end{aligned}$$

    by the assumption \(\frac{\overline{p}^{*}}{q}>1\) and as \(z_{k+}^{q}\) is bounded in \(L^{\frac{\overline{p}^{*}}{q}}\left( \Omega \right)\) so

    $$\begin{aligned} z_{k+}^{q}\rightarrow z_{+}^{q}\text { in }L^{1}\left( \Omega \right) , \end{aligned}$$

    and

    $$\begin{aligned} z_{k+}^{q}\rightarrow z_{+}^{q}\text { a.e.} \end{aligned}$$

    By Vitali’s theorem associated to the generalized Lebesgue theorem we arrive at the conclusion that

    $$\begin{aligned} g_{n}^{\prime }(w_{k+})z_{k+}^{q}\rightarrow g_{n}^{\prime }(w_{+})z_{+}^{q}~ \text {in }L^{1}\left( \Omega \right) . \end{aligned}$$

    By duality, and as \(\varphi \in L^{\infty }\left( \Omega \right)\) we obtain the convergence of the third term. For the last term

    $$\begin{aligned} q\int \limits _{\Omega }g_{n}(w_{k+})z_{k+}^{q-1}\psi , \end{aligned}$$

    we have that

    $$\begin{aligned} g_{n}(w_{k+})z_{k+}^{q-1}\rightarrow g_{n}(w_{+})z_{+}^{q-1}\text { a.e,} \end{aligned}$$

    and

    $$\begin{aligned} \left( \left\| w_{k}^{p}\right\| _{L^{\frac{\overline{p}^{*}}{p} }}\le C_{1}\text { and }\left\| z_{k}^{q-1}\right\| _{L^{\frac{ \overline{p}^{*}}{q-1}}}\le C_{2}\right) \Rightarrow \left\| g_{n}(w_{k+})z_{k+}^{q-1}\right\| ^{^{\frac{\overline{p}^{*}}{p+q-1} }}\le C_{3}, \end{aligned}$$

    as by hypothesis \(\frac{\overline{p}^{*}}{p+q-1}>1,\) we obtain the equiintegraility of \(g_{n}(w_{k+})z_{k+}^{q-1}\), Vitali’s theorem allows us to conclude that

    $$\begin{aligned} g_{n}(w_{k+})z_{k+}^{q-1}\rightarrow g_{n}(w_{+})z_{+}^{q-1}\text { in } L^{1}\left( \Omega \right) . \end{aligned}$$

    By duality, and as \(\psi \in L^{\infty }\left( \Omega \right)\) we obtain the convergence of the last term. In conclusion (H3) is fulfilled by \(J_{n}.\)

  4. (H4)

    Consequently by the properties of \(g_{n}\) Hölder, Sobolev and Young inequalities we have

    $$\begin{aligned} \int \limits _{\Omega }g_{n}(w_{+})z_{+}^{q}\le & {} \int \limits _{\Omega }w_{+}^{p}z_{+}^{q} \\\le & {} \left( \int \limits _{\Omega }w_{+}^{p}\right) ^{\frac{p}{\overline{p} ^{*}}}\left( \int \limits _{\Omega }z_{+}^{\frac{\overline{p}^{*}q}{ \overline{p}^{*}-p}}\right) ^{\frac{\overline{p}^{*}-p}{\overline{p} ^{*}}} \\\le & {} C\left( \int \limits _{\Omega }w_{+}^{p}\right) ^{\frac{p}{\overline{p} ^{*}}}\left( \int \limits _{\Omega }z_{+}^{\overline{p}^{*}}\right) ^{ \frac{q}{\overline{p}^{*}}} \\\le & {} C\left\| w\right\| _{W_{0}^{1,(p_{i})}}^{p}\left\| z\right\| _{W_{0}^{1,(p_{i})}}^{q} \\\le & {} \varepsilon \left\| w\right\| _{W_{0}^{1,(p_{i})}\left( \Omega \right) }^{p_{N}}+C(\varepsilon )\left\| z\right\| _{W_{0}^{1,(p_{i})}\left( \Omega \right) }^{\frac{qp_{N}}{qp_{N}-p}}. \end{aligned}$$

    Now using (5) consequently for \(\sigma _{i}=\left\| \partial _{i}w\right\| _{L^{p_{i}}\left( \Omega \right) }\) and \(\sigma _{i}=\left\| \partial _{i}z\right\| _{L^{p_{i}}\left( \Omega \right) }\) , associated to the latter inequation we obtain

    $$\begin{aligned} J_{n}(w,z)\ge & {} C_{1}\left\| w\right\| ^{p_{N}}+C_{2}\left\| z\right\| ^{p_{N}}-\varepsilon \left\| w\right\| ^{p_{N}}-C(\varepsilon )\left\| z\right\| ^{\frac{qp_{N}}{qp_{N}-p}} \\\ge & {} C\left( \left\| w\right\| ^{p_{N}}+\left\| z\right\| ^{p_{N}}\right) -\varepsilon \left\| w\right\| ^{p_{N}}-\varepsilon \left\| z\right\| ^{p_{N}}+\varepsilon \left\| z\right\| ^{p_{N}}-C(\varepsilon )\left\| z\right\| ^{\frac{qp_{N}}{qp_{N}-p}} \\\ge & {} \left( C-\varepsilon \right) \left( \left\| w\right\| ^{p_{N}}+\left\| z\right\| ^{p_{N}}\right) +\varepsilon \left\| z\right\| ^{p_{N}}-C(\varepsilon )\left\| z\right\| ^{\frac{qp_{N}}{ qp_{N}-p}} \\\ge & {} \left( C-\varepsilon \right) \left( \left\| w\right\| ^{p_{N}}+\left\| z\right\| ^{p_{N}}\right) +f(\left\| z\right\| ) \end{aligned}$$

    where

    $$\begin{aligned} f(t)=\varepsilon t^{p_{N}}-C(\varepsilon )t^{\frac{qp_{N}}{qp_{N}-p}}, \end{aligned}$$

    so it is always possible to have for \(t>T\)

    $$\begin{aligned} f(t)>0, \end{aligned}$$

    choosing

    $$\begin{aligned} \left\| w\right\| ^{p_{N}}+\left\| z\right\| ^{p_{N}}=R, \end{aligned}$$

    we obtain

    $$\begin{aligned} J_{n}(w,z)>\alpha \text { for }\left\| w\right\| ^{p_{N}}+\left\| z\right\| ^{p_{N}}=R. \end{aligned}$$

    \(\psi \in W_{0}^{1,(p_{i})}\left( \Omega \right) \cap L^{\infty }\left( \Omega \right)\), \(\left\| \psi \right\| _{L^{\infty }\left( \Omega \right) }=1\) that is \(0\le \psi \le 1,\) and for \(t\ge 1\)

    $$\begin{aligned} g_{n}(t\psi )\ge \frac{(t\psi )^{2}}{(1+t\psi )^{2-p}} \end{aligned}$$

    and as \(p+q>p_{N}>p_{1}\)

    $$\begin{aligned} J_{n}(t\psi ,t\psi )= & {} 2\sum \limits _{i=1}^{N}\frac{1}{p_{i}} \int \limits _{\Omega }\left| \partial _{i}t\psi \right| ^{p_{i}}-\int \limits _{\Omega }g_{n}(t\psi )(t\psi )^{q} \\\le & {} \frac{2Ct^{p_{N}}}{p_{1}}\left\| \psi \right\| _{W_{0}^{1,(p_{i})}\left( \Omega \right) }^{p_{N}}-\int \limits _{\Omega } \frac{(t\psi )^{2+q}}{(1+t\psi )^{2-p}} \\\le & {} C_{1}t^{p_{N}}-C_{2}\frac{(t)^{2+q}}{(1+t)^{2-p}} \end{aligned}$$

    since \(p+q>p_{N}\) we have that

    $$\begin{aligned} \underset{t\rightarrow +\infty }{\lim }\left( C_{1}t^{p_{N}}-C_{2}\frac{ (t)^{2+q}}{(1+t)^{2-p}}\right) =-\infty \end{aligned}$$

    and thus for large \(\overline{t}>>1\) we have

    $$\begin{aligned} J_{n}(\overline{t}\psi ,\overline{t}\psi )<0, \end{aligned}$$

    so if we choose

    $$\begin{aligned} \left( \overline{w},\overline{z}\right) =(\overline{t}\psi ,\overline{t}\psi ) \end{aligned}$$

    and for \(t\in \left[ 0,1\right]\)

    $$\begin{aligned} \overline{\gamma }(t):=\left( \overline{\gamma _{1}}(t),\overline{\gamma _{2} }(t)\right) =\left( t\overline{w},t\overline{z}\right) \end{aligned}$$

    then

    $$\begin{aligned} \overline{\gamma }\in \Gamma \end{aligned}$$

    in view of the previous computation, we obtain for \(J_{n}(w,z)>\alpha\) and \(\left\| w\right\| ^{p_{N}}+\left\| z\right\| ^{p_{N}}=R\)

    $$\begin{aligned} \underset{t\in \left[ 0,1\right] }{\max }J_{n}\left( \gamma _{1}(t),\gamma _{2}(t)\right) \ge \alpha \text { for\ }\gamma (.)=\left( \gamma _{1}(.),\gamma _{2}(.)\right) \in \Gamma \end{aligned}$$

    we finally reach that

    $$\begin{aligned} \underset{\gamma \in \Gamma }{\inf }\underset{t\in \left[ 0,1\right] }{\max } J_{n}\left( \gamma _{1}(t),\gamma _{2}(t)\right) \ge \alpha >0=\max \left\{ J_{n}\left( 0,0\right) ,J_{n}\left( \overline{w},\overline{z}\right) \right\} . \end{aligned}$$
  5. (H5)

    We assume that condition (a), (b) and (c) in (H5) Theorem 3 are fulfilled by \(J_{n}=F\) and the sequence \(\left\{ \left( w_{k},z_{k}\right) \right\} _{k}\subset Y\), we claim that \(\left\{ \left( w_{k},z_{k}\right) \right\} _{k}\) is bounded in X. As a direct consequence of hypothesis (a), one have that

    $$\begin{aligned} \sum \limits _{i=1}^{N}\frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}w\right| ^{p_{i}}+\sum \limits _{i=1}^{N}\frac{1}{p_{i}} \int \limits _{\Omega }\left| \partial _{i}z\right| ^{p_{i}}-\int \limits _{\Omega }g_{n}(w_{k+})z_{k+}^{q}\le C \end{aligned}$$

    by (b) and (c), we have that

    $$\begin{aligned} -\left\langle J_{n}^{\prime }\left( w_{k},z_{k}\right) ,\left( w_{k},z_{k}\right) \right\rangle= & {} -\sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}w_{k}\right| ^{p_{i}}-\sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}z_{k}\right| ^{p_{i}} \\&+\int \limits _{\Omega }g_{n}^{\prime }(w_{k+})z_{k+}^{q}w_{k+}+q\int \limits _{\Omega }g_{n}(w_{k+})z_{k+}^{q} \\&\le \varepsilon _{k}\left[ 2+\left\| \left( w_{k},z_{k}\right) \right\| _{X}\right] . \end{aligned}$$

    Multiplying by \(\frac{1}{p_{N}}\) both sides of the last inequality

    $$\begin{aligned}&-\int \limits _{\Omega }g_{n}(w_{k+})z_{k+}^{q}+\frac{1}{p_{N}} \int \limits _{\Omega }g_{n}^{\prime }(w_{k+})z_{k+}^{q}w_{k+}+\frac{q}{p_{N}} \int \limits _{\Omega }g_{n}(w_{k+})z_{k+}^{q}\\&\le C+\frac{\varepsilon _{k}}{ p_{N}}\left[ 2+\left\| \left( w_{k},z_{k}\right) \right\| _{X}\right] \end{aligned}$$

    we then arrive at

    $$\begin{aligned} \frac{1}{p_{N}}\int \limits _{\Omega }\left[ \frac{1}{p_{N}}g_{n}^{\prime }(w_{k+})w_{k+}+\left( \frac{q}{p_{N}}-1\right) g_{n}(w_{k+})\right] z_{k+}^{q}\le C+\frac{\varepsilon _{k}}{p_{N}}\left[ 2+\left\| \left( w_{k},z_{k}\right) \right\| _{X}\right] \end{aligned}$$

    by the properties of \(g_{n}\)

    $$\begin{aligned} \frac{p+q+p_{N}}{p_{N}}\int \limits _{\Omega }g_{n}(w_{k+})z_{k+}^{q}\le C+ \frac{\varepsilon _{k}}{p_{N}}\left[ 2+\left\| \left( w_{k},z_{k}\right) \right\| _{X}\right] \end{aligned}$$

    as \(p+q>p_{N}\)

    $$\begin{aligned} \frac{1}{p_{N}}\left\| \left( w_{k},z_{k}\right) \right\| _{X}\le C+ \frac{p+q+p_{N}}{p_{N}}\left( C+\frac{\varepsilon _{k}}{p_{N}}\left[ 2+\left\| \left( w_{k},z_{k}\right) \right\| _{X}\right] \right) \end{aligned}$$

    as \(\varepsilon _{k}\rightarrow 0\) the claim follows, that is \(\left\{ \left\| \left( w_{k},z_{k}\right) \right\| _{X}\right\} _{k}\) is bounded, and by the sequel \(\left\{ \left( w_{k},z_{k}\right) \right\} _{k}\) weakly converges in X. We will now prove strong convergence of the sequence \(\left\{ \left( w_{k},z_{k}\right) \right\} _{k}.\) First observe that if \(M_{k}\) in (b) of (H5) is such that \(M_{k}\rightarrow 0\) then immediately \(\left\{ \left( w_{k},z_{k}\right) \right\} _{k}~\) strongly converges. So we will assume without loss of generality that \(M_{k}\rightarrow l>0\) (or more precisely that \(\lim \inf M_{k}>0\)) and we have

    $$\begin{aligned}&\left\langle J_{n}^{\prime }\left( w_{k},z_{k}\right) ,\left( w_{k}-T_{h}\left( w_{k}\right) ,0\right) \right\rangle \\= & {} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}w_{k}\right| ^{p_{i}-2}\partial _{i}w_{k}\partial _{i}\left( w_{k}-T_{h}\left( w_{k}\right) \right) -\int \limits _{\Omega }g_{n}^{\prime }(w_{k+})z_{k+}^{q}\left( w_{k}-T_{h}\left( w_{k}\right) \right) \end{aligned}$$

    Using hypotheses (b), (c) and the fact that \(\left\{ w_{k}\right\} _{k}\) is bounded in \(W_{0}^{1,(p_{i})}\left( \Omega \right)\) we came to

    $$\begin{aligned}&\left\langle J_{n}^{\prime }\left( w_{k},z_{k}\right) ,\left( w_{k}-T_{h}\left( w_{k}\right) ,0\right) \right\rangle \\\le & {} \varepsilon _{k}\left[ \frac{\left\| w_{k}-T_{h}(w_{k})\right\| _{W_{0}^{1,(p_{i})}\left( \Omega \right) \cap L^{\infty }\left( \Omega \right) }}{M_{k}}+\left\| w_{k}-T_{h}(w_{k})\right\| _{W_{0}^{1,(p_{i})}\left( \Omega \right) }\right] \\\le & {} \varepsilon _{k}\left[ \frac{2M_{k}+h+C_{1}}{M_{k}}+C_{2}\right] \\\le & {} \varepsilon _{k}\left[ \frac{h+C_{1}}{M_{k}}+C_{3}\right] \\\le & {} \widetilde{\varepsilon _{k}}+h\widetilde{\varepsilon _{k}} \end{aligned}$$

    Now observe that as \(\left\{ w_{k}\right\} _{k}\) is bounded, we must have that \(w_{k+}^{p}z_{k+}^{q}\) is bounded in \(L^{\frac{\overline{p}^{*}}{p+q} }\left( \Omega \right)\)and as \(\frac{\overline{p}^{*}}{p+q}>1\), Vitali’s theorem and generelized Lebesgue theorem give

    $$\begin{aligned} \int \limits _{\Omega }g_{n}^{\prime }(w_{k+})z_{k+}^{q}w_{k+}=\int \limits _{\Omega }g_{n}^{\prime }(w)z_{k+}^{q}+ \widetilde{\varepsilon _{k}} \end{aligned}$$

    Now using the fact

    $$\begin{aligned} g_{n}^{\prime }(w_{k+})z_{k+}^{q}T_{h}(w)\le 2n^{1-q}z_{k+}^{q}h \end{aligned}$$

    and as \(2n^{1-q}z_{k+}^{q}h\) is bounded in \(L^{\frac{\overline{p}^{*}}{q} }\left( \Omega \right)\) for \(\frac{\overline{p}^{*}}{q}>1\), newly Vitali’s theorem and generelized Lebesgue theorem give that

    $$\begin{aligned} \int \limits _{\Omega }g_{n}^{\prime }(w_{k+})z_{k+}^{q}T_{h}(w)=\int \limits _{\Omega }g_{n}^{\prime }(w_{+})z_{+}^{q}T_{h}(w)+\widetilde{\varepsilon _{k}} \end{aligned}$$

    The previous computations added to hypotheses (a), (b) and (c) lead us to

    $$\begin{aligned} \left| \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}w_{k}\right| ^{p_{i}-2}\partial _{i}w_{k}\partial _{i}\left( w_{k}-T_{h}\left( w\right) \right) -\int \limits _{\Omega }g_{n}^{\prime }(w_{k+})z_{k+}^{q}\left( w_{k}-T_{h}\left( w\right) \right) \right| \nonumber \\ \le o(1)+ho(1). \end{aligned}$$
    (15)

    Due to weak convergence of \(\left\{ w_{k}\right\} _{k}\) to w in \(W_{0}^{1,(p_{i})}\left( \Omega \right)\) we have,

    $$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}w\right| ^{p_{i}-2}\partial _{i}w\partial _{i}\left( w_{k}-T_{h}\left( w\right) \right) =o(1) \end{aligned}$$

    so from (15) and the latter observation

    $$\begin{aligned}&\sum \limits _{i=1}^{N}\int \limits _{\Omega }\left( \left| \partial _{i}w_{k}\right| ^{p_{i}-2}\partial _{i}w_{k}-\left| \partial _{i}T_{h}\left( w\right) \right| ^{p_{i}-2}\partial _{i}T_{h}\left( w\right) \right) \partial _{i}\left( w_{k}-T_{h}\left( w\right) \right) \\&-\int \limits _{\Omega }g_{n}^{\prime }(w_{+})z_{+}^{q}\left( w-T_{h}\left( w\right) \right) \\&\le o(1)+ho(1), \end{aligned}$$

    using Lebesgue theorem we have

    $$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\int \limits _{\Omega }g_{n}^{\prime }(w_{+})z_{+}^{q}\left( w-T_{h}\left( w\right) \right) =0 \end{aligned}$$

    we thus obtain

    $$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left( \left| \partial _{i}w_{k}\right| ^{p_{i}-2}\partial _{i}w_{k}-\left| \partial _{i}T_{h}\left( w\right) \right| ^{p_{i}-2}\partial _{i}T_{h}\left( w\right) \right) \partial _{i}\left( w_{k}-T_{h}\left( w\right) \right) \le o(1)+ho(1). \end{aligned}$$

    By (6) and (7) we conclude that

    $$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}\left( w_{k}-T_{h}\left( w\right) \right) \right| ^{p_{i}}\le o(1)+ho(1). \end{aligned}$$
    (16)

    we also observe that

    $$\begin{aligned} \left| \partial _{i}\left( w_{k}-w\right) \right| ^{p_{i}}\le 2^{p_{i}-1}\left( \left| \partial _{i}\left( w_{k}-T_{h}\left( w\right) \right) \right| ^{p_{i}}+\left| \partial _{i}G_{h}(w)\right| ^{p_{i}}\right) \end{aligned}$$
    (17)

    so for \(h>\widetilde{h}\)

    $$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}G_{h}(w)\right| ^{p_{i}}\le o(1) \end{aligned}$$

    and by (16) and (17)

    $$\begin{aligned} \underset{k\rightarrow +\infty }{\lim \sup }\sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}\left( w_{k}-w\right) \right| ^{p_{i}}=ho(1) \end{aligned}$$

    and in conclusion

    $$\begin{aligned} w_{k}\rightarrow w\text { strongly in }W_{0}^{1,(p_{i})}\left( \Omega \right) . \end{aligned}$$

    The same reasonning applied to \(\left\{ z_{k}\right\} _{k}\) instead of \(\left\{ w_{k}\right\} _{k}\) leads us to the conclusion

    $$\begin{aligned} z_{k}\rightarrow z\text { strongly in }W_{0}^{1,(p_{i})}\left( \Omega \right) . \end{aligned}$$

    This concludes the proof of the theorem.

\(\square\)

Proposition 5

For each \(n\in \mathbb {N}\) there exists a nonnegative couple \(\left( w_{n},z_{n}\right) \in Y\) such that

$$\begin{aligned} \left\{ \begin{array}{ll} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left[ \left| \partial _{i}w_{n}\right| ^{p_{i}-2}\partial _{i}w_{n}\partial _{i}\varphi \right] =\int \limits _{\Omega }g_{n}^{\prime }(w_{n})z_{n}^{q}\varphi &{}\quad \varphi \in W_{0}^{1,(p_{i})}\left( \Omega \right) , \\ \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left[ \left| \partial _{i}z_{n}\right| ^{p_{i}-2}\partial _{i}z_{n}\partial _{i}\psi \right] =q\int \limits _{\Omega }g_{n}(w_{n})z_{n}^{q-1}\psi &{}\quad \psi \in W_{0}^{1,(p_{i})}\left( \Omega \right) , \end{array} \right. \end{aligned}$$
(18)

Proof

Let

$$\begin{aligned} \Gamma =\left\{ \text { Continuous }\gamma :\left[ 0,1\right] \rightarrow Y; \text { }\gamma (0)=(0,0)\text { and }\gamma (1)=(\overline{w},\overline{z} )\right\} \end{aligned}$$

as before

$$\begin{aligned} J_{n}(w_{n},z_{n})=\sum \limits _{i=1}^{N}\frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}w_{n}\right| ^{p_{i}}+\sum \limits _{i=1}^{N} \frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}z_{n}\right| ^{p_{i}}-\int \limits _{\Omega }g_{n}(w_{n+})z_{n+}^{q} \end{aligned}$$

and

$$\begin{aligned} c_{n}=\underset{\gamma \in \Gamma }{\inf }\underset{t\in \left[ 0,1\right] }{ \max }J_{n}\left( \gamma (t)\right) \end{aligned}$$

considering

$$\begin{aligned} (\overline{w},\overline{z})=(\overline{t}\chi ,\overline{t}\chi ) \end{aligned}$$

a similar reasonning as the one made in the proof of the previous theorem and using Theorem 3 lead us to that \(c_{n}\) is a critical point of \(J_{n}\) so there exists a \((w_{n},z_{n})\) such that for \(\left( \varphi ,\psi \right) \in Y\)

$$\begin{aligned} J_{n}(w_{n},z_{n})=c_{n}\text { and }\left\langle J_{n}^{\prime }\left( w_{n},z_{n}\right) ,\left( \varphi ,\psi \right) \right\rangle =0\text { } \end{aligned}$$

choosing firstly \(\varphi =0\) and secondly \(\psi =0\) we obtain (18).

Now as \(c_{n}>0\) and by the reasonning made in the proof of the previous theorem we must have that at least one of \(w_{n}\) and \(z_{n}\) must be non identically null. We claim that both \(w_{n}\) and \(z_{n}\) are not identically null, indeed let us assume by contradiction that \(z_{n}\equiv 0\) if we put \(\varphi =w_{n}\) in (18) we came to the conclusion that also \(w_{n}\equiv 0\) which is not possible, so necessarily both \(w_{n}\) and \(z_{n}\) are not identically equal to zero

To prove the nonnegativity of \(w_{n}\) and \(z_{n}\), we only have to take \(\varphi =w_{n}^{-}\) and \(\psi =z_{n}^{-}\) in (18). \(\square\)

4 Passage to the limit

Now we will pass to the limit, as n goes to \(+\infty\) in \(\left\{ \left( w_{n},z_{n}\right) \right\} _{n}.\)

Proposition 6

Let \(\left\{ \left( w_{n},z_{n}\right) \right\} _{n}\) be the sequence introduced in the previous proposition, then

$$\begin{aligned} w_{n}\rightharpoonup w\text { weakly in }W_{0}^{1,(p_{i})}\left( \Omega \right) , \end{aligned}$$

and

$$\begin{aligned} z_{n}\rightharpoonup z\text { weakly in }W_{0}^{1,(p_{i})}\left( \Omega \right) . \end{aligned}$$

Proof

Let

$$\begin{aligned} \Gamma =\left\{ \text { Continuous }\gamma :\left[ 0,1\right] \rightarrow Y; \text { }\gamma (0)=(0,0)\text { and }\gamma (1)=(\overline{w},\overline{z} )\right\} \end{aligned}$$

as before

$$\begin{aligned} J_{n}(w_{n},z_{n})=\sum \limits _{i=1}^{N}\frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}w_{n}\right| ^{p_{i}}+\sum \limits _{i=1}^{N} \frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}z_{n}\right| ^{p_{i}}-\int \limits _{\Omega }g_{n}(w_{n+})z_{n+}^{q} \end{aligned}$$

and

$$\begin{aligned} c_{n}=\underset{\gamma \in \Gamma }{\inf }\underset{t\in \left[ 0,1\right] }{ \max }J_{n}\left( \gamma (t)\right) \end{aligned}$$

so

$$\begin{aligned} J_{n}(w_{n},z_{n})\le \underset{t\in \left[ 0,1\right] }{\max }(t\overline{w },t\overline{z}) \end{aligned}$$

since

$$\begin{aligned} \gamma :t\mapsto (t\overline{w},t\overline{z})\in \Gamma . \end{aligned}$$

Repeating the same reasonning as the one done in the proof of Theorem 3 we arrive at

$$\begin{aligned} J_{n}(t\overline{w},t\overline{z})\le Ct^{p_{N}}-C_{2}\frac{t^{q+2}}{\left( 1+t\right) ^{2-p}} \end{aligned}$$

and so

$$\begin{aligned} J_{n}(w_{n},z_{n})\le \underset{t\in \left[ 0,1\right] }{\max }\left[ Ct^{p_{N}}-C_{2}\frac{t^{q+2}}{\left( 1+t\right) ^{2-p}}\right] =T \end{aligned}$$

we deduce that

$$\begin{aligned} \sum \limits _{i=1}^{N}\frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}w_{n}\right| ^{p_{i}}+\sum \limits _{i=1}^{N}\frac{1}{p_{i}} \int \limits _{\Omega }\left| \partial _{i}z_{n}\right| ^{p_{i}}-\int \limits _{\Omega }g_{n}(w_{n+})z_{n}^{q}\le T. \end{aligned}$$
(19)

Using \(\varphi =\) \(w_{n}\) as test function in its equation in (18) we obtain

$$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}w_{n}\right| ^{p_{i}}=\int \limits _{\Omega }g_{n}^{\prime }(w_{n})z_{n}^{q}w_{n} \end{aligned}$$

and using \(\psi =\) \(z_{n\text { }}\) as test function its equation in () we obtain

$$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}z_{n}\right| ^{p_{i}}=q\int \limits _{\Omega }g_{n}(w_{n})z_{n}^{q} \end{aligned}$$

by (19) we arrive at

$$\begin{aligned} \int \limits _{\Omega }\left[ \frac{1}{p_{N}}\int \limits _{\Omega }g_{n}^{\prime }(w_{n})w_{n}+\left( \frac{q}{p_{N}}-1\right) g_{n}(w_{n}) \right] z_{n}^{q}\le T \end{aligned}$$

by the properties of \(g_{n}\) we have that

$$\begin{aligned} \frac{p+q-p_{N}}{p_{N}}\int \limits _{\Omega }g_{n}(w_{n})z_{n}^{q}\le T \end{aligned}$$

in conclusion \(\int \limits _{\Omega }g_{n}(w_{n})z_{n}^{q}\) is bounded independently of \(n\in \mathbb {N}\), and consequently in view of (19) both \(\left\{ w_{n}\right\} _{n}\) and \(\left\{ z_{n}\right\} _{n}\) are bounded in \(W_{0}^{1,(p_{i})}\left( \Omega \right)\) which implies that

$$\begin{aligned} w_{n}\rightharpoonup w\text { weakly in }W_{0}^{1,(p_{i})}\left( \Omega \right) , \end{aligned}$$

and

$$\begin{aligned} z_{n}\rightharpoonup z\text { weakly in }W_{0}^{1,(p_{i})}\left( \Omega \right) . \end{aligned}$$

\(\square\)

Proposition 7

Neither w nor z can be identically equal to zero.

Proof

Let us assume by contradiction that \(z=0\), so \(z_{n}\rightharpoonup 0\) weakly in \(W_{0}^{1,(p_{i})}\left( \Omega \right)\) and by the sequel \(\left\{ z_{n}\right\} _{n}\) converges strongly to 0 in any \(L^{r}\left( \Omega \right)\) for every \(r<\overline{p}^{*}\), since \(p+q<\overline{p} ^{*}\) we can always choose r in such a way to have \(q<\frac{\overline{p }^{*}q}{\overline{p}^{*}-q}<r<\overline{p}^{*}\) and by the properties of \(g_{n}\) associated to Hölder inequality we have

$$\begin{aligned} \int \limits _{\Omega }g_{n}(w_{n})z_{n}^{q}\le & \, \int \limits _{\Omega }w_{n}^{p}z_{n}^{q} \\ \le & \, \left( \int \limits _{\Omega }w_{n}^{\frac{pr}{r-q}}\right) ^{\frac{1-p }{r}}\left( \int \limits _{\Omega }z_{n}^{r}\right) ^{\frac{q}{r}} \end{aligned}$$

since \(p<\overline{p}^{*}\) and by the choice of r we have \(\frac{pr}{ r-q}<\overline{p}^{*},\) and thus

$$\begin{aligned} \int \limits _{\Omega }g_{n}(w_{n})z_{n}^{q}\le C\left( \int \limits _{\Omega }z_{n}^{r}\right) ^{\frac{q}{r}} \end{aligned}$$

and as \(\left\{ z_{n}\right\} _{n}\) is assumed to converge to 0

$$\begin{aligned} \int \limits _{\Omega }g_{n}(w_{n})z_{n}^{q}\rightarrow 0. \end{aligned}$$

Choosing \(\psi =z_{n}\) in the second equation of (18), we obtain

$$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}z_{n}\right| ^{p_{i}}\rightarrow 0 \end{aligned}$$

and so \(z_{n}\rightharpoonup 0\) strongly in \(W_{0}^{1,(p_{i})}\left( \Omega \right) .\) Now choosing \(\varphi =w_{n}\) in the first equation of (18), we obtain

$$\begin{aligned} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left| \partial _{i}w_{n}\right| ^{p_{i}}= & {} \int \limits _{\Omega }g_{n}^{\prime }(w_{n})w_{n}z_{n}^{q} \\\le & {} 2\int \limits _{\Omega }w_{n}^{p}z_{n}^{q} \end{aligned}$$

and thus also \(w_{n}\rightharpoonup 0\) strongly in \(W_{0}^{1,(p_{i})}\left( \Omega \right) .\) Finally we must have in that case

$$\begin{aligned} \underset{n\rightarrow +\infty }{\lim }J_{n}(w_{n},z_{n})=0 \end{aligned}$$

which is a contradiction. With a similar reasonning we came to the same conclusion if we assume by contradiction that \(w=0.\) \(\square\)

Theorem 8

Under the extra condition \(p_{1}>2,\) w and z solve problem (1) in the sense of (8).

Proof

To prove this theorem we need to pass to the limit when n goes to \(+\infty\) in (18).

We begin by the second equation in (18), which represents the regular part of the system. As \(\left\{ w_{n}\right\} _{n}\) and \(\left\{ z_{n}\right\} _{n}\) converge strongly in \(L^{r}\left( \Omega \right)\) for every \(r<\overline{p}^{*},\) so by the properties of \(g_{n}\) and in view of \(p+q<\overline{p}^{*}\) , we obtain

$$\begin{aligned} g_{n}(w_{n})z_{n}^{q-1}\rightarrow qw^{p}z^{q-1}\text { strongly in }\left( W_{0}^{1,(p_{i})}\left( \Omega \right) \right) ^{\prime }, \end{aligned}$$

so we conclude that z verify the second equation of (8), and as \(p_{1}>2\) the anisotropic operator verify a strong maximum principle, see for instance [15], and thus \(z>0\) in \(\Omega .\)

Now we turn our attention to the first equation which represents the singular part of (1), observe that the right hand side of this equation converges almost everywhere to \(p\frac{z^{q}}{w^{1-p}}\) which is singular on the set where w vanishes. Since w is not identically equal to zero, one can always find two real numbers a,  b such that \(0<a<b\),

$$\begin{aligned} \left| \left\{ x\in \Omega ,~a<w(x)<b\right\} \right| >0 \end{aligned}$$

and

$$\begin{aligned} \left| \left\{ x\in \Omega ,~w(x)=a\right\} \right| \cup \left| \left\{ x\in \Omega ,~w(x)=b\right\} \right| =0 \end{aligned}$$

where \(\left| A\right|\) stands for the Lebesgue measure of the set A.

From the properties of \(g_{n}\) we have

$$\begin{aligned} g_{^{\prime }n}(s)\ge M\text { for }s\in \left[ a,b\right] \text { and }b> \frac{1}{n} \end{aligned}$$

thus

$$\begin{aligned} g_{^{\prime }n}(w_{n})z_{n}^{q}\ge & {} M\chi _{\left\{ x\in \Omega ,~a<w_{n}(x)<b\right\} }z_{n}^{q} \\\ge & {} M\chi _{\left\{ x\in \Omega ,~a<w_{n}(x)<b\right\} }T_{1}\left( z_{n}^{q}\right) . \end{aligned}$$

where \(\chi _A\) denotes the characteristic function of the set A.

Let \(y_{n}\) be solution to the auxiliary problem

$$\begin{aligned} \left\{ \begin{array}{ll} -Ly_{n}=-\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}y_{n}\right| ^{p_{i}-2}\partial _{i}y_{n}\right] =M\chi _{\left\{ x\in \Omega ,~a<w_{n}(x)<b\right\} }T_{1}\left( z_{n}^{q}\right) &{}\quad in~\Omega , \\ y_{n}=0 &{} \quad on~\partial \Omega . \end{array} \right. \end{aligned}$$

Observe that \(y_{n}\) always exists by classical theory as the right hand side belongs to \(L^{\infty }\left( \Omega \right) .\) The comparison principle leads us to the fact that

$$\begin{aligned} w_{n}\ge y_{n}. \end{aligned}$$

From one hand by strong maximum principle we have that

$$\begin{aligned} y_{n}\ge C_{K}\text { on every }K\subset \subset \Omega \end{aligned}$$

On the other hand, a simple modification of De Giorgi theorem [26], allows us to obtain the uniform convergence of \(\left\{ y_{n}\right\} _{n}\) to \(y\in W_{0}^{1,(p_{i})}\left( \Omega \right)\) on every \(K\subset \subset \Omega\); that is in particular

$$\begin{aligned} y_{n}(x)\ge y(x)-\varepsilon \ge C_{K}-\varepsilon \text { for all }x\in K \text { and }n\ge n_{\varepsilon }, \end{aligned}$$

which leads to

$$\begin{aligned} w_{n}\ge \widetilde{C}_{K}\text { in }K\text { for }n\ge n_{\varepsilon }. \end{aligned}$$

Now we are allowed to pass to the limit in the right hand side of the first equation of (18) as

$$\begin{aligned} g_{n}^{\prime }(s)\le 2s^{p-1} \end{aligned}$$

and using \(\varphi \in C_{0}^{1}\left( \Omega \right)\) as test function, considering \(K=\left\{ x\in \Omega \text {, }\varphi (x)>0\right\}\) we obtain

$$\begin{aligned} \left| g_{n}^{\prime }(w_{n})z_{n}^{q}\varphi \right| \le \frac{p}{ \left( \widetilde{C}_{K}\right) ^{1-p}}z_{n}^{q}\left\| \varphi \right\| _{L^{\infty }} \end{aligned}$$

and as \(q<\overline{p}^{*},\) by generelized Lebesgue theorem and Vitali’s theorem we get

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\int \limits _{\Omega }g_{n}^{\prime }(w_{n})z_{n}^{q}\varphi =p\int \limits _{\Omega }\frac{z^{q}}{w^{1-p}}, \end{aligned}$$

this associated to the weak convergence of \(\left\{ w_{n}\right\} _{n}\) to w lead us to the conlusion that w verify the first equation of (8), which ends the proof. \(\square\)

5 Concluding remarks

We give here some remarks and observations.

  1. 1.

    We can generalize all the results obtained here to the more general problem

    $$\begin{aligned} \left\{ \begin{array}{ll} -L_{(p_{i})}u=-\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\right] =p\dfrac{v^{q}}{u^{1-p}} &{}\quad in~\Omega , \\ -L_{(q_{i})}v=-\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}v\right| ^{q_{i}-2}\partial _{i}v\right] =qv^{q-1}u^{p} &{}\quad in~\Omega , \\ u>0\text { and }v>0 &{}\quad in~\Omega , \\ u=0\text { and }v=0 &{}\quad on~\partial \Omega , \end{array} \right. \end{aligned}$$

    but this will generate a huge number of indices which can be tedious for the reader.

  2. 2.

    By some minor modifications, we can obtain similar results for anisotropic-isotropic problem of the form

    $$\begin{aligned} \left\{ \begin{array}{ll} -L_{(p_{i})}u=-\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\right] =p\dfrac{v^{q}}{u^{1-p}} &{}\quad in~\Omega , \\ -\Delta _{m}v=-div\left( \left| \nabla v\right| ^{m-2}\nabla v\right) =qv^{q-1}u^{p} &{}\quad in~\Omega , \\ u>0\text { and }v>0 &{}\quad in~\Omega , \\ u=0\text { and }v=0 &{}\quad on~\partial \Omega , \end{array} \right. \end{aligned}$$
  3. 3.

    The regular-regular case corresponding to \(p>1\) and \(q>1,\) can also be studied, by the use of classical variational methods and similar existence results can be obtained.

  4. 4.

    The hypothesis \(p_{1}>2\) was introduced in the last theorem, as—to the best of our knowledge—strong maximum principle for anisotropic operator L is established only in that case.

  5. 5.

    The singular-singular case corresponding to \(p<1\) and \(q<1,\) can be studied in a similar way by introducing the following definition of solution

    $$\begin{aligned} \left\{ \begin{array}{ll} \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\partial _{i}\varphi \right] =p\int \limits _{\Omega }\dfrac{v^{q}}{u^{1-p}}\varphi &{}\quad \varphi \in C_{0}^{1}\left( \Omega \right) , \\ \sum \limits _{i=1}^{N}\int \limits _{\Omega }\left[ \left| \partial _{i}v\right| ^{p_{i}-2}\partial _{i}v\partial _{i}\psi \right] =q\int \limits _{\Omega }v^{q-1}u^{p}\psi &{}\quad \psi \in C_{0}^{1}\left( \Omega \right) , \end{array} \right. \end{aligned}$$

    and an approximation energy functional of the form

    $$\begin{aligned} J_{n}(w,z)=\sum \limits _{i=1}^{N}\frac{1}{p_{i}}\int \limits _{\Omega }\left| \partial _{i}w\right| ^{p_{i}}+\sum \limits _{i=1}^{N}\frac{1}{ p_{i}}\int \limits _{\Omega }\left| \partial _{i}z\right| ^{p_{i}}-\int \limits _{\Omega }g_{n}(w_{+})h_{n}\left( z_{+}\right) \end{aligned}$$

    with

    $$\begin{aligned} g_{n}(s)=\frac{s^{2}}{\left( s+\frac{1}{n}\right) ^{2-p}}\text { and } h_{n}(s)=\frac{s^{2}}{\left( s+\frac{1}{n}\right) ^{2-q}} \end{aligned}$$

    and all the steps can be reproduced, but not for the last theorem as we need to have \(p_{1}>2\) so the condition \(p_{N}<p+q\) cannot be fulfilled as \(p<1\), \(q<1\) and \(p_{N}>p_{1}>2.\) Thus one have to use some other techniques to prove existence of solution.

  6. 6.

    In the particular case, \(p_{1}=p_{2}=\cdots =p_{N}=P\), all the results obtained here are still vaild, but we underline that in this case we deal with an isotropic operator that absolutely doesn’t coincide with the usual P-Laplace operator. Instead we obtain a closely related operator called the orthotropic P-Laplace operator or the pseudo P-Laplace operator who has an importance on its own. We invite the reader interested in this operator to see the very recent works [10, 11] and the references therein.