Non-Nehari manifold method for asymptotically periodic Schrödinger equations

Abstract

We consider the semilinear Schrödinger equation $$\left\{ {\begin{array}{*{20}c} { - \Delta u + V(x)u = f(x,u), \mathbb{R}^N ,} \\ {u \in H^1 (\mathbb{R}^N ),} \\ \end{array} } \right.$$ where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V 0(x) + V 1(x), V 0 ∈ C(ℝN), V 0(x) is 1-periodic in each of x 1, x 2, …, x N and sup[σ(−Δ + V 0) ∩ (−∞, 0)] < 0 < inf[σ(−Δ + V 0) ∩ (0, ∞)], V 1 ∈ C(ℝN) and lim|x|→∞ V 1(x) = 0. Inspired by previous work of Li et al. (2006), Pankov (2005) and Szulkin and Weth (2009), we develop a more direct approach to generalize the main result of Szulkin and Weth (2009) by removing the “strictly increasing” condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N 0 by using the diagonal method.