Existence of a nontrivial solution to a strongly indefinite semilinear equation

Abstract:

Under general hypotheses, we prove the existence of a nontrivial solution for the equation $Lu = N(u)$, where $u$ belongs to a Hilbert space $H$, $L$ is an invertible continuous selfadjoint operator, and $N$ is superlinear. We are particularly interested in the case where $L$ is strongly indefinite and $N$ is not compact. We apply the result to the Choquard-Pekar equation \[ - \Delta u(x) + p(x)u(x) = u(x)\int _{{\mathbb {R}^3}} {\frac {{{u^2}(y)}} {{|x - y|}}dy,\qquad u \in {H^1}({\mathbb {R}^3}),\quad u \ne 0,} \] where $p \in {L^\infty }({\mathbb {R}^3})$ is a periodic function.