Three solutions for a class of Neumann boundary value systems driven by a \((p_1,\ldots ,p_n)\)-Laplacian operator (Q2904267)

The authors consider the existence of at least three weak solutions for the following quasilinear elliptic system : NEWLINENEWLINE\[NEWLINE\begin{cases} -\Delta _{p_i}u_i + a_i(x) | u _i | ^{p_i-2}u_i &= \lambda F_{u_i}(x,u_1,\ldots ,u_n) + \mu G_{u_i} (x,u_1,\ldots ,u_n) \text{ in } \Omega , NEWLINE\\ \qquad\qquad\qquad\qquad\quad \frac{\partial u_i}{\partial \nu }&=0 \text{ on }\quad \partial \Omega ,\end{cases}\tag{1} NEWLINE\]NEWLINE NEWLINEfor \(i=1,\ldots ,n\), where \(\Omega \subset {\mathbb R}^n\, (n\geq 1)\) is a non-empty bounded open set with a smooth boundary \(\partial \Omega \), \(\Delta _{p_i} u_i = \mathrm{div} (| \nabla u_i| ^{p_i-2} \nabla u_i)\) is the \(p_i\)-Laplacian operator, \(p_i>n\), \(a_i \in L^{\infty }(\Omega )\) with \( \inf _{\Omega }a_i>0\) for \(1\leq i \leq n\), \(\lambda \) and \(\mu \) are positive parameters, \(F,G : \Omega \times {\mathbb R}^n\to {\mathbb R}\) are measurable functions with respect to \(x\in \Omega \), and for every \((t_1,\ldots ,t_n)\in {\mathbb R}^n\) \(F\), \(G\) are \(C^1\) with respect to \((t_1,\ldots ,t_n) \in {\mathbb R}^n\) for a.e. \(x\in \Omega \), \(F_{u_i}\) and \(G_{u_i}\) denote the partial derivatives of \(F\) and \(G\) with respect to \(u_i\), and \(\nu \) is the outer unit normal to \(\partial \Omega \). The authors assume thatNEWLINENEWLINE\((F_1)\) for every \(M>0\) and every \(1\leq i \leq n\), NEWLINE\[NEWLINE \sup _{| (t_1,\ldots ,t_n)| \leq M} | F_{u_i} (x,t_1,\ldots ,t_n)| \in L^1(\Omega ). NEWLINE\]NEWLINENEWLINENEWLINE\((F_2)\) \(F(x,0,\ldots ,0)=0\) for a.e. \(x\in \Omega \).NEWLINENEWLINE\((G)\) for every \(M>0\) and every \(1\leq i \leq n\), NEWLINE\[NEWLINE \sup _{| (t_1,\ldots ,t_n)| \leq M} | G_{u_i} (x,t_1,\ldots ,t_n)| \in L^1(\Omega ). NEWLINE\]NEWLINENEWLINENEWLINEThen they get the following theorem.NEWLINENEWLINE\textbf{Theorem} Assume that \((F_1), (F_2), (G)\) are satisfied, and there exist a positive constant \(r\) and a function \(w=(w_1,\ldots ,w_n)\in X\) (a reflexive real Banach space) such thatNEWLINENEWLINE(i) \(\sum _{i=1}^n \frac{\| w_i \| _{p_i}^{p_i}}{p_i}\geq r\),NEWLINENEWLINE(ii) \(\displaystyle{ (r\Pi _{i=1}^{n} p_i) \frac{\int _{\Omega } F(x,w_1,\ldots ,w_n)dx }{\sum _{i=1}^n (\Pi _{j=1,j\neq i}^n p_j) \| w_i \| _{p_i}^{p_i} } } \qquad {- \int _{\Omega } \sup _{(t_1,\ldots ,t_n) \in K(cr)} F(x,t_1,\ldots ,t_n)dx >0 } \) where \(K(\gamma ) = \{ (t_1,\ldots ,t_n) \in {\mathbb R}^n ; \sum _{i=1}^n \frac{| t_i | ^{p_i}}{p_i}\leq \gamma \}\),NEWLINENEWLINE(iii) \(\displaystyle{\limsup _{(| t_1 | ,\ldots, | t_n| )\to (+\infty ,\ldots ,+ \infty )} \frac{F(x,t_1,\ldots ,t_n)}{\sum _{i=1}^n| t_i | ^{p_i}/p_i} \leq 0}\). Then, setting NEWLINE\[NEWLINE \Lambda = \left( \frac{ \sum _{i=1}^n (\Pi _{j=1,j\neq i}^np_i ) \| w_i\| ^{p_i}_{p_i}}{(\Pi _{i=1}^n p_i) \int _{\Omega }F(x,w_1,\ldots ,w_n)dx } , \frac{r}{ \int _{\Omega }\sup _{(t_1,\ldots ,t_n) \in K(cr)} F(x,t_1,\ldots ,t_n)dx } \right), NEWLINE\]NEWLINE for each compact interval \([a,b]\subset \Lambda \), there exists \(\rho >0\) with the following property: for every \(\lambda \in [a,b]\), there exists \(\delta >0\) such that, for each \(\mu \in [0,\delta ]\), system (1) admits at least three weak solutions in \(X\) whose norms are less than \(\rho \).