Bound states for the Schrödinger equation with mixed-type nonlinearites

Abstract

We prove the existence results for the Schrödinger equation of the form $$-\Delta u+V(x)u=g(x,u),x\in {\mathbb{R}}^N,$$ where $g$ is superlinear and subcritical in some periodic set $K$ and linear in ${\mathbb{R}}^N\setminus K$ for sufficiently large $|u|$. The periodic potential $V$ is such that $0$ lies in a spectral gap of $-\Delta +V$. We find a solution with the energy bounded by a certain min-max level, and infinitely many geometrically distinct solutions provided that $g$ is odd in $u$.

to appear in Indiana University Mathematics Journal

References in corpus (1)

• Spectra of open waveguides in periodic media

Cited by in corpus (1)

• Generalized linking-type theorem with applications to strongly indefinite problems with sign-changing nonlinearities