Let \(\underline{u} \in W^{1,\overrightarrow{p(x)}}_{0}(\Omega )\) and \(\underline{v} \in W^{1,\overrightarrow{q(x)}}_{0}(\Omega )\) be the function given in Lemma 3.1. Consider \(\widetilde{T} : W^{1,p(x)}_{0}(\Omega ) \rightarrow W^{1,p(x)}_{0}( \Omega )\) and \(\widetilde{S} : W^{1,q(x)}_{0}(\Omega ) \rightarrow W^{1,q(x)}_{0}( \Omega )\) defined by
$$ \widetilde{T}u(x):=\textstyle\begin{cases} u(x), & \text{if} \ \underline{u}(x) \leq u(x), \\ \underline{u}(x), & \text{if} \ u(x) < \underline{u}(x), \end{cases}\displaystyle \qquad \widetilde{S}v(x):=\textstyle\begin{cases} v(x), & \text{if} \ \underline{v}(x) \leq v(x), \\ \underline{v}(x), & \text{if} \ v(x) < \underline{v}(x), \end{cases} $$
the functions \({\widetilde{G}}_{u}(x,u,v) : = a(x)(\widetilde{T} u)^{\alpha (x)-1} + F_{u}(x,\widetilde{T} u , \widetilde{S} v )\), \({\widetilde{G}}_{v}(x,u,v) : = b(x)(\widetilde{S} v)^{\beta (x)-1} + F_{v}(x, \widetilde{T} u , \widetilde{S} v )\), \(u \in W^{1,\overrightarrow{p(x)}}_{0}( \Omega )\), \(v \in W^{1,\overrightarrow{q(x)}}_{0}(\Omega )\) and the problem
whose solutions are given by the critical points of the \(C^{1}\) functional
$$ L(u,v):= \int _{\Omega } \sum_{i=1}^{N} \frac{1}{p_{i}(x)} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}(x)} + \int _{\Omega } \sum_{i=1}^{N} \frac{1}{q_{i}(x)} \biggl\vert \frac{\partial v}{\partial x_{i}} \biggr\vert ^{q_{i}(x)} - \int _{\Omega } \widetilde{G}(x,u,v) ,\quad (u,v) \in W, $$
where W was defined in the proof of Theorem 1.1 and
$$ \widetilde{G}(x,s,t):= \int _{0}^{s}{\widetilde{G}}_{\tau }(x, \tau ,t) \,d \tau + \int _{0}^{t}{\widetilde{G}}_{\tau }(x,s, \tau ) \,d \tau .$$
Lemma 4.1
The Palais-Smale condition is satisfied by the functional L.
Proof
Consider \((u_{n},v_{n}) \subset W\) a sequence such that \(L^{\prime }(u_{n} , v_{n})\rightarrow 0\) and \(L(u_{n} ,v_{n})\rightarrow c\) for some \(c \in \mathbb{R}\). With respect to the first part of (i), note that \((F_{3})\) holds with \(\overline{\theta },\overline{\xi }>0\) such that \(\max \{ \frac{1}{\alpha ^{-}}, \theta \} < \overline{\theta } < \frac{1}{p^{+}_{+}}\) and \(\max \{ \frac{1}{\beta ^{-}}, \xi \} < \overline{\xi } < \frac{1}{q^{+}_{+}}\). Applying \((H)\), \((F_{1})\)-\((F_{3})\), Propositions 2.1, embedding (2.4), the boundedness of the functions \(\underline{u}\) and \(\underline{v}\) and arguing as in the proof of [1, Theorem 36] (see also inequality 3.2 of [24]), we obtain that there are constants \(C_{i}>0\), \(i=1,\ldots,4\) such that
$$\begin{aligned} C_{1} + o_{n}(1) \bigl\Vert ( u_{n} , v_{n}) \bigr\Vert \geq& {L}^{\prime }(u_{n} , v_{n}) (u_{n} ,v_{n}) - \overline{\theta } L^{\prime }(u_{n} , v_{n}) (u_{n} , 0) - \overline{\xi } L^{\prime }(u_{n} , v_{n}) (0, v_{n}) \\ \geq& C_{2} \bigl( \Vert u_{n} \Vert ^{p^{-}_{-}}_{1,\overrightarrow{p(x)}} + \Vert v_{n} \Vert ^{q^{-}_{-}}_{1,\overrightarrow{q(x)}} \bigr) -C_{3} \bigl\Vert (u_{n} , v_{n}) \bigr\Vert \\ &{}+ \int _{\{u_{n} \geq \underline{u}\}} \biggl( \overline{\theta }- \frac{1}{\alpha (x)} \biggr)a(x) {u_{n}}^{\alpha (x)} \\ &{} + \int _{\{v_{n} \geq \underline{v}\}} \biggl(\overline{\xi }- \frac{1}{\beta (x)} \biggr)b(x) {v_{n}}^{\beta (x)} \\ \geq& C_{2} \bigl( \Vert u_{n} \Vert ^{p^{-}_{-}}_{1,\overrightarrow{p(x)}} + \Vert v_{n} \Vert ^{q^{-}_{-}}_{1,\overrightarrow{q(x)}} \bigr)-C_{4} \bigl\Vert (u_{n} , v_{n}) \bigr\Vert , \end{aligned}$$
which provide that \((u_{n} , v_{n})\) is bounded in W.
With respect to the second case of (i), that is \(\beta ^{+} < q^{-}_{-}\), we have constants \(C_{i} >0\), \(i=1,\ldots,5\) with
$$\begin{aligned} C_{1} + o_{n}(1) \bigl\Vert (u_{n},v_{n}) \bigr\Vert \geq& {L}^{\prime }(u_{n} , v_{n}) (u_{n} ,v_{n}) - \overline{\theta } L^{\prime }(u_{n} , v_{n}) (u_{n} , 0) - { \xi } L^{\prime }(u_{n} , v_{n}) (0, v_{n}) \\ \geq& C_{2} \bigl( \Vert u_{n} \Vert ^{p^{-}_{-}}_{1,\overrightarrow{p(x)}} + \Vert v_{n} \Vert ^{q^{-}_{-}}_{1,\overrightarrow{q(x)}} \bigr)-C_{4} \bigl\Vert (u_{n} , v_{n}) \bigr\Vert \\ &{}- C_{5} \max \bigl\{ \Vert v_{n} \Vert ^{\beta ^{+}}_{\beta (x)}, \Vert v_{n} \Vert ^{\beta ^{-}}_{\beta (x)}\bigr\} , \end{aligned}$$
where \(\overline{\theta } >0\) was provided in the first part of the proof (i). Thus, the continuous embedding \(W^{1,\overrightarrow{q(x)}}_{0}(\Omega ) \hookrightarrow L^{\beta (x)}( \Omega )\), which is given by (2.4), implies that
$$ \begin{aligned}C_{1} + o_{n}(1) \bigl\Vert (u_{n}, v_{n}) \bigr\Vert +C_{2} \bigl\Vert (u_{n}, v_{n}) \bigr\Vert \geq{}& C_{3} \bigl( \Vert u_{n} \Vert ^{p^{-}_{-}}_{1,\overrightarrow{p(x)}} + \Vert v_{n} \Vert ^{q^{-}_{-}}_{1, \overrightarrow{q(x)}} \bigr) \\ &{} -C_{4} \max \bigl\{ \Vert v_{n} \Vert ^{\beta ^{+}}_{1, \overrightarrow{q(x)}}, \Vert v_{n} \Vert ^{\beta ^{-}}_{1, \overrightarrow{q(x)}} \bigr\} , \end{aligned} $$
for constants \(C_{i} >0\), \(i=1,\ldots,5\). Since \(\beta ^{+} < q^{-}\), we obtain that the sequence \((u_{n} , v_{n})\) is bounded in W.
Thus, for a subsequence still denoted by \((u_{n}, v_{n})\), we obtain that
$$ \textstyle\begin{cases} u_{n} \rightharpoonup u & \text{in } W^{1,\overrightarrow{p(x)}}_{0}(\Omega ), \\ u_{n}(x) \rightarrow u(x) & \text{a.e. in } \Omega , \\ u_{n} \rightarrow u & \text{in } L^{h(x)}(\Omega ) \end{cases}\displaystyle \quad \text{and} \quad \textstyle\begin{cases} v_{n} \rightharpoonup v & \text{in } W^{1,\overrightarrow{q(x)}}_{0}( \Omega ), \\ v_{n}(x) \rightarrow v(x) & \text{a.e. in } \Omega , \\ v_{n} \rightarrow v & \text{in } L^{k(x)}(\Omega ), \end{cases} $$
(4.1)
for all \(h,k \in C(\overline{\Omega })\) with \(1 < h^{-}\leq h^{+} < (p^{\star })^{-}\), \(1 < k^{-}\leq k^{+} < (q^{\star })^{-}\) and some pair \((u,v) \in W\). From Lebesgue’s Dominated Convergence Theorem and (4.1), it follows that
$$\begin{aligned} & \int _{\Omega } \biggl( \biggl\vert \frac{\partial u_{n} }{\partial x_{i}} \biggr\vert ^{p_{i}-2}\frac{\partial u_{n} }{\partial x_{i}}- \biggl\vert \frac{\partial u }{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial u }{\partial x_{i}} \biggr) \biggl( \frac{\partial u_{n} }{\partial x_{i}}- \frac{\partial u }{\partial x_{i}} \biggr) \rightarrow 0, \\ & \int _{\Omega } \biggl( \biggl\vert \frac{\partial u_{n} }{\partial x_{i}} \biggr\vert ^{q_{i}-2}\frac{\partial v_{n} }{\partial x_{i}}- \biggl\vert \frac{\partial v }{\partial x_{i}} \biggr\vert ^{q_{i}-2} \frac{\partial v }{\partial x_{i}} \biggr) \biggl( \frac{\partial v_{n} }{\partial x_{i}}- \frac{\partial v }{\partial x_{i}} \biggr) \rightarrow 0. \end{aligned}$$
Since \(p^{-}_{-} , q^{-}_{-} \geq 2 \), we have the result by the inequality (see, for instance, [27, page 97])
$$ \bigl\langle \vert x \vert ^{m-2} x - \vert y \vert ^{m-2} y , x- y \bigr\rangle \geq\frac{1}{2^{m-2}} \vert x-y \vert ^{m} $$
(4.2)
for all \(x, y \in \mathbb{R}^{N}\) and \(m\geq 2\), where \(\langle \cdot , \cdot \rangle \) denotes the usual Euclidean inner product in \(\mathbb{R}^{N}\). □
The next result provides the Mountain Pass Geometry for the functional L.
Lemma 4.2
If the hypotheses \((H)\), \((F_{1})\)-\((F_{3})\) hold, then for \(\max \{\|a\|_{{\infty }},\|b\|_{{\infty }} \}\) small enough, the claims below are true.
-
(i)
There are constants \(R,\sigma >0\) with \(R > \|(\underline{u},\underline{v})\|\) such that
$$ L(\underline{u},\underline{v}) < 0 < \sigma \leq \inf_{(u,v) \in \partial B_{R}(0)} L(u,v). $$
-
(ii)
There is \(e \in W\setminus \overline{B_{2R}(0)}\) such that \(L(e) < \sigma \).
Proof
The inequalities \(p^{-}_{-}, q^{-}_{-}>1 \) and (3.1) provide that \(L(\underline{u},\underline{v}) <0\). Consider \((u,v) \in W\) with \(\| (u,v)\| \geq 1 \). From the embeddings \(W^{1,\overrightarrow{p(x)}}_{0}(\Omega ) \hookrightarrow L^{\alpha (x)}( \Omega )\), \(W^{1,\overrightarrow{q(x)}}_{0}(\Omega ) \hookrightarrow L^{ \beta (x)}(\Omega )\) and Proposition 2.1, it follows that
$$\begin{aligned} L(u,v)\geq {}&K_{1} \bigl\Vert (u,v) \bigr\Vert ^{\iota } - K_{2} -K_{3} \bigl\Vert (u,v) \bigr\Vert \\ &{}- \Vert a \Vert _{\infty } K_{4} \bigl(\max \bigl\{ \Vert u \Vert ^{\alpha ^{+}}_{1, \overrightarrow{p(x)}}, \Vert u \Vert ^{\alpha ^{-}}_{1,\overrightarrow{p(x)}} \bigr\} \\ & {} + \max \bigl\{ \Vert u \Vert ^{r^{+}}_{1,\overrightarrow{p(x)}}, \Vert u \Vert ^{r^{-}}_{1, \overrightarrow{p(x)}} \bigr\} \bigr) \\ & {}- \Vert b \Vert _{\infty } K_{5} \bigl(\max \bigl\{ \Vert v \Vert ^{\beta ^{+}}_{1, \overrightarrow{q(x)}}, \Vert v \Vert ^{\beta ^{-}}_{1,\overrightarrow{q(x)}} \bigr\} \\ & {}+ \max \bigl\{ \Vert v \Vert ^{r^{+}}_{1,\overrightarrow{q(x)}}, \Vert v \Vert ^{r^{-}}_{1, \overrightarrow{q(x)}} \bigr\} \bigr), \end{aligned}$$
for positive constants \(K_{i}>0\), \(i=1,\ldots,5\), where \(\iota := \min \{p^{-}, q^{-}\}\). If necessary, decrease \(\max \{\| a\|_{\infty }, \| b\|_{\infty }\}\) in a such way that \(\| (\underline{u}, \underline{v}) \| <1\), which is possible applying the functions \(\varphi = \underline{u}\) and \(\psi = \underline{v}\) in the inequality (3.1) and using Lemma 2.5. Fix \(\sigma >0\) and let \(R> 1\) be a constant such that \(K_{1} R^{\iota } - K_{3} R \geq 2\sigma \). Considering \(\max \{\|a\|_{\infty },\|a\|_{\infty }\}\) small enough such that \(K_{4} \|a\|_{\infty } ( R^{\alpha ^{+}} + R^{r^{+}}) + K_{5} \|b\|_{ \infty } ( R^{\beta ^{+}} + R^{r^{+}})\leq \sigma \), it follows that \(L(u,v)\geq \sigma \) for \((u,v) \in W\) with \(\| (u,v)\|=R\), which provides (i).
With respect to (ii), note that the hypothesis \((F_{3})\) and the inequality \(\frac{1}{\theta }> p^{+}_{+}\), provide constants \(K_{i}>0\), \(i=1,\ldots,4\) and \(t>0\) large enough such that \(L(t\underline{u},0) \leq C_{1} t^{p^{+}_{+}} - C_{2}t^{\alpha ^{-}} - C_{3} t^{\frac{1}{\theta }} + C_{4} <0\) and \(\| (t\underline{u},0)\| > 2R \). □
Proof of Theorem 1.2
Let \((\underline{u}, \underline{v}), (\overline{u}, \overline{v}) \in W\) be the pairs given in Lemma 3.1. Consider \((u_{1} ,v_{1}) \in W\), the solution to the system (S) provided in Theorem 1.1, which minimizes the functional \(J{ |}_{ A} \), where J was given in (3.5) and
$$A= \bigl\{ (u,v) \in W; \underline{u}(x)\leq u(x) \leq \overline{u}(x), \underline{v}(x)\leq v(x) \leq \overline{v}(x) \text{ a.e. in } \Omega \bigr\} . $$
The Lemmas 4.1 and 4.2 provide that the hypotheses of the Mountain Pass Theorem [28, Theorem 2.1] are verified by the functional L. Therefore,
$$ c := \inf_{\gamma \in \Gamma } \max_{t \in [0,1]} L\bigl( \gamma (t)\bigr), \quad \text{where } \Gamma :=\bigl\{ \gamma \in C\bigl([0,1], W\bigr); \gamma (0)= (\underline{u},\underline{v}), \gamma (1)=e \bigr\} $$
is a critical value of L, i.e., \(L^{\prime }(u_{2} , v_{2} ) = 0\) and \(L(u_{2} , v_{2})=c\), for some \((u_{2},v_{2}) \in W\). From the definition of \(G_{u}\) and \(G_{v}\) provided in (3.4), we obtain that \(J(u,v)=L(u,v)\) for \((u,v) \in \{(w,z) \in W; 0\leq w(x)\leq \overline{u}(x), 0\leq z(x) \leq \overline{v}(x) \ \text{a.e in} \ \Omega \}\). Thus, it follows that \(J(\underline{u},\underline{v}) = L(\underline{u},\underline{v})\) and \(L(u_{1},v_{1}) = J(u_{1} , v_{1})= \inf_{(u,v) \in A} J(u,v)\). Recall that \(L(\underline{u},\underline{v}) <0\). Thus, if \(u_{2} (x)\geq \underline{u}(x)\), \(v_{2} (x)\geq \underline{v}(x)\) a.e. in Ω, then it follows that (S) has two weak solutions \((u_{1},v_{1}),(u_{2} , v_{2}) \in W\) with \(L(u_{1} , v_{1})\leq L(\underline{u} , \underline{v})<0<\sigma \leq c = L(u_{2} , v_{2})\), where \(\sigma >0\) given in Lemma 4.2.
We affirm that \(u_{2} (x)\geq \underline{u}(x)\), \(v_{2} (x)\geq \underline{v}(x)\) a.e. in Ω. In order to prove such inequality, consider the test functions \((\underline{u} - u_{2})^{+} \in W^{1,\overrightarrow{p(x)}}_{0}( \Omega )\), \((\underline{v} - v_{2})^{+} \in W^{1,\overrightarrow{q(x)}}_{0}( \Omega )\) and \(w \in [\underline{v} , \overline{v}]\), \(z \in [\underline{u} , \overline{u}]\). It follows from \((\widetilde{S})\) and (3.1) that
$$ \begin{aligned} & \int _{\Omega } \sum_{i=1}^{N} \biggl\vert \frac{ \partial u_{2}}{\partial x_{i}} \biggr\vert ^{p_{i}(x)-2} \frac{\partial u_{2}}{\partial x_{i}} \frac{ \partial (\underline{u} - u_{2})^{+}}{\partial x_{i}} + \int _{ \Omega } \sum_{i=1}^{N} \biggl\vert \frac{ \partial v_{2}}{\partial x_{i}} \biggr\vert ^{q_{i}(x)-2} \frac{\partial v_{2}}{\partial x_{i}} \frac{ \partial (\underline{v} - v_{2})^{+}}{\partial x_{i}} \\ &\quad = \int _{\{u_{2} < \underline{u }\}} a(x){\underline{u}}^{\alpha (x)-1} + F_{u}(x,u_{2} , w) + \int _{\{v_{2} < \underline{v}\}} b(x){ \underline{v}}^{\alpha (x)-1} + F_{v}(x,z , v_{2}), \\ & \int _{\Omega } \sum_{i=1}^{N} \biggl\vert \frac{ \partial \underline{u}}{\partial x_{i}} \biggr\vert ^{p_{i}(x)-2} \frac{\partial \underline{u}}{\partial x_{i}} \frac{ \partial (\underline{u} - u_{2})^{+}}{\partial x_{i}} + \int _{ \Omega } \sum_{i=1}^{N} \biggl\vert \frac{ \partial \underline{v}}{\partial x_{i}} \biggr\vert ^{q_{i}(x)-2} \frac{\partial \underline{v}}{\partial x_{i}} \frac{ \partial (\underline{v} - v_{2})^{+}}{\partial x_{i}} . \end{aligned} $$
Therefore,
$$ \begin{aligned} & \int _{\{\underline{u} > u_{2}\}} \sum^{N}_{i=1} \biggl( \biggl\vert \frac{\partial \underline{u} }{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial \underline{u} }{\partial x_{i}}- \biggl\vert \frac{\partial u_{2} }{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial u_{2} }{\partial x_{i}} \biggr) \biggl( \frac{\partial \underline{u} }{\partial x_{i}}- \frac{\partial u_{2} }{\partial x_{i}} \biggr)\leq 0, \\ & \int _{\{\underline{v} > v_{2}\}} \sum^{N}_{i=1} \biggl( \biggl\vert \frac{\partial \underline{v} }{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial \underline{v} }{\partial x_{i}}- \biggl\vert \frac{\partial v_{2} }{\partial x_{i}} \biggr\vert ^{p_{i}-2} \frac{\partial v_{2} }{\partial x_{i}} \biggr) \biggl( \frac{\partial \underline{v} }{\partial x_{i}}- \frac{\partial v_{2} }{\partial x_{i}} \biggr)\leq 0. \end{aligned} $$
(4.3)
From inequality (4.2) and (4.3), it follows that
$$ \begin{aligned} \int _{\Omega } \biggl\vert \frac{\partial }{\partial x_{i}} (\underline{u} - u_{2})^{+} \biggr\vert ^{p_{i}(x)}& =0, \\ \int _{\Omega } \biggl\vert \frac{\partial }{\partial x_{i}} (\underline{v} - v_{2})^{+} \biggr\vert ^{q_{i}(x)}& =0,\ \end{aligned} $$
for \(i=1,\ldots, N\), which provides that \(\frac{\partial }{\partial x_{i}}(\underline{u} - u_{2})^{+}(x) = \frac{\partial }{\partial x_{i}} (\underline{v} - v_{2})^{+}(x) =0 \) a.e. in Ω. Thus, it follows from (2.4) that \((\underline{u} - u_{2})^{+}(x) =(\underline{v} - v_{2})^{+}(x)=0\) a.e. in Ω, which proves the claim. □