Ground state solutions for some indefinite variational problems

Abstract

We consider the nonlinear stationary Schrödinger equation − Δ u + V ( x ) u = f ( x , u ) in R N . 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 − Δ + 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.