Strongly indefinite functionals and multiple solutions of elliptic systems

Abstract:

We study existence and multiplicity of solutions of the elliptic system \[ \begin {cases} -\Delta u =H_u(x,u,v) & \text {in $\Omega $}, \\ -\Delta v =-H_v(x,u,v) & \text {in $\Omega $}, \quad u(x) = v(x) = 0 \quad \text {on $\partial \Omega $}, \end {cases} \] where $\Omega \subset \mathbb {R}^N, N\geq 3$, is a smooth bounded domain and $H\in \mathcal {C}^1(\bar {\Omega }\times \mathbb {R}^2, \mathbb {R})$. We assume that the nonlinear term \[ H(x,u,v)\sim |u|^p + |v|^q + R(x,u,v) \ \ \text {with} \ \ \lim _{|(u,v)|\to \infty }\frac {R(x,u,v)}{|u|^p+|v|^q}=0, \] where $p\in (1, \ 2^*)$, $2^*:=2N/(N-2)$, and $q\in (1, \ \infty )$. So some supercritical systems are included. Nontrivial solutions are obtained. When $H(x,u,v)$ is even in $(u,v)$, we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if $p>2$ (resp. $p<2$). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.