A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system
In this work, motivated by [\textit{T. F. Wu}, Nonlinear Anal., Theory Methods Appl. 68, No.~6 (A), 1733--1745 (2008; Zbl 1151.35342)], and using recent ideas from \textit{K. J. Brown} and \textit{T. F. Wu} [J. Math. Anal. Appl. 337, No.~2, 1326--1336 (2008; Zbl 1132.35361)], we prove the existence of nontrivial nonnegative solutions to the nonlinear elliptic system \[ \begin{cases} -\Delta_pu+m(x)|u|^{p-2}u=\lambda|u|^{\gamma-2}u+\frac{\alpha}{\alpha+\beta}\;c(x)|u|^{\alpha-2}u|v|^\beta,\quad & x\in\Omega,\\ -\Delta_pv+m(x)|v|^{p-2}v=\lambda|v|^{\gamma-2}v+\frac{\alpha}{\alpha+\beta}\;c(x)|u|^{\alpha-2}u|v|^\beta,& x\in\Omega,\\ u=0=v, & x\in\partial\Omega.\end{cases} \] Here \(\Delta_p\) denotes the \(p\)-Laplacian operator defined by \(\Delta_pz=\text{div}(|\nabla z|^{p-2}\nabla z)\), \(p>2\), \(\Omega\subset \mathbb R^N\) is a bounded domain with smooth boundary, \(\alpha>1\), \(\beta>1\), \(2<\alpha+\beta<p<\gamma<p*\) \((p^*=\frac{pN}{N-p}\) if \(N>p\), \(p^*=\infty\) if \(N\leq p\)), \(\frac{\partial}{\partial n}\) is the outer normal derivative, \((\lambda,\mu)\in\mathbb R^2\setminus \{(0,0)\}\), the weight \(m(x)\) is a bounded function with \(\|m\|_\infty>0\), and \(c(x)\) is a continuous function which changes sign in \(\overline\Omega\).
scientific article; zbMATH DE number 5563048
scientific article