Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation

Abstract

We consider the nonlinear difference equations of the form $$Lu = f\left( {n,u} \right),\;n \in Z,$$ where L is a Jacobi operator given by (Lu)(n) = a(n)u(n+1)+a(n−1)u(n−1)+b(n)u(n) for n ∈ Z, {a(n)} and {b(n)} are real valued N-periodic sequences, and f(n, t) is superlinear on t. Inspired by previous work of Pankov [Discrete Contin. Dyn. Syst., 19, 419–430 (2007)] and Szulkin and Weth [J. Funct. Anal., 257, 3802–3822 (2009)], we develop a non-Nehari manifold method to find ground state solutions of Nehari–Pankov type under weaker conditions on f. Unlike the Nehari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari–Pankov manifold by using the diagonal method.