Existence of nontrivial solutions for asymptotically linear periodic Schrödinger equations

Abstract

We study the Schrödinger equation: \begin{equation} - Δu+V(x)u=f(x,u) ,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{equation} where $V$ is periodic and $f$ is periodic in the $x$-variables, 0 is in a gap of the spectrum of the operator $-\Delta +V$ and $f$ is asymptotically linear as $|u|\to +\infty .$ We prove that under some asymptotically linear assumptions for $f$, this equation has a nontrivial solution. Our assumptions for $f$ are different from the classical assumptions raised by G. B. Li and A. Szulkin in

arXiv commentCommun. Contemp. Math. 4 (2002) 763-776

References in corpus (2)

• Periodic nonlinear Schrödinger equation with application to photonic crystals
• Bifurcation into spectral gaps for a noncompact semilinear Schrödinger equation with nonconvex potential