Ground states of a system of nonlinear Schrödinger equations with periodic potentials

Jarosław Mederski

Published 2014-11-20Version 1

We are concerned with a system of coupled Schr\"odinger equations $$-\Delta u_i + V_i(x)u_i = \partial_{u_i}F(x,u)\hbox{ on }\mathbb{R}^N,\,i=1,2,...,K,$$ where $F$ and $V_i$ are periodic in $x$ and $0$ belongs to a spectral gap of the Schr\"odinger operator $-\Delta+V_i(x)$ for any $i=1,2,...,K$. The problem appears in nonlinear optics, where gap solitons in photonic crystals are studied. Photonic crystals have a periodic nanostructure and the term $F$ is due to the presence of nonlinear polarization. General assumptions on $F$ are imposed and we find a ground state solution being a minimizer of the energy functional associated with the system on a Nehari-Pankov manifold. If $0$ lies below all the spectra $\sigma(-\Delta+V_i)$ we admit a sign-changing behaviour of $F$ and we are able to treat problems with the mixture of self-focusing and defocusing nonlinearties. Our approach is based on a new linking-type result involving the Nehari-Pankov manifold.