Ground state solutions for some indefinite variational problems

We consider the nonlinear stationary Schrödinger equation − u + V (x)u = f (x,u) in RN. Here f is a superlinear, subcritical nonlinearity, and we mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of − + V . Inspired by previous work of Li et al. (2006) [11] and Pankov (2005) [13], we develop an approach to find ground state solutions, i.e., nontrivial solutions with least possible energy. The approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value. Our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed. For odd f , we obtain infinitely many geometrically distinct solutions. The approach also yields new existence and multiplicity results for the Dirichlet problem for the same type of equations in a bounded domain. © 2009 Elsevier Inc. All rights reserved.