Generalized linking theorem with an application to a semilinear Schrödinger equation

Abstract

Consider the semilinear Schrödinger equation (*) $-\Delta u + V(x)u = f(x,u)$, $u\in H^1(\mathbf {R} ^N)$. It is shown that if $f$, $V$ are periodic in the $x$-variables, $f$ is superlinear at $u=0$ and $\pm\infty$ and 0 lies in a spectral gap of $-\Delta+V$, then (*) has at least one nontrivial solution. If in addition $f$ is odd in $u$, then (*) has infinitely many (geometrically distinct) solutions. The proofs rely on a degree-theory and a linking-type argument developed in this paper.