Multiple solutions for nonlinear elliptic systems with changing-sign weights. (English) Zbl 1240.35154

Chen, Huyuan; Yang, Jianfu

Differ. Integral Equ. 22, No. 3-4, 239-250 (2009).

In this paper we consider the existence of solutions \(u,v\in H^{1}_{0}(\Omega )\) of nonlinear elliptic system \(-\Delta u=\mu u+| u| ^{p-1}u+\alpha Q(x)| u| ^{\alpha -2}| v| ^{\beta }u\), \(-\Delta v=\nu v+| v| ^{q-1}v+\beta Q(x)| u| ^{\alpha }| v| ^{\beta -2}v\), where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^N\), \(N\geq 2\) and \(1<p,q\leq 2^{*}-1,\;2^{*}=2N/(N-2)\) if \(N\geq 3\) and \(2^{*}=\infty \) if \(N=2.\) \(Q(x)\) is a sign-changing function. By variation methods, we show that the problem has at least three nontrivial solutions.

Reviewer: From the introduction.

MSC:

35J57	Boundary value problems for second-order elliptic systems
35J50	Variational methods for elliptic systems
35B32	Bifurcations in context of PDEs
35J61	Semilinear elliptic equations

Keywords:

multiple solutions; nonlinear elliptic system; sign-changing weight; variation method