Non-Nehari manifold method for asymptotically periodic Schrodinger equations

We consider the semilinear Schrodinger equation where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V (0)(x) + V (1)(x), V (0) a C(ae (N) ), V (0)(x) is 1-periodic in each of x (1), x (2), aEuro broken vertical bar, x (N) and sup[sigma(-Delta + V (0)) a (c) (-a, 0)] < 0 < inf[sigma(-Delta + V (0)) a (c) (0, a)], V (1) a C(ae (N) ) and lim(|x|-> a) 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.