Standing waves with a critical frequency for nonlinear Schrodinger equations, II

Abstract

For elliptic equations of the form Deltau-V(epsilonx)u + f (u) = 0, x is an element of R-N, where the potential V satisfies (\x\ -->infinity) V(x) > inf(RN)V (x) = 0, we develop a new variational approach to construct localized bound state solutions concentrating at an isolated component of the local minimum of V where the minimum value of V can be positive or zero. These solutions give rise to standing wave solutions having a critical frequency for the corresponding nonlinear Schrodinger equations. Our method allows a fairly general class of nonlinearity f(u) including ones without any growth restrictions at large.