The authors consider the nonlinear, stationary Schrödinger equation in . Assume that and are 1-periodic in , , and , is invertible. Moreover, let satisfy for for , otherwise). Then the equation has a nontrivial solution , provided that , , with some constant and .
To obtain this solution, the existence of a Palais-Smale sequence to a suitable level of the functional
is proved, i.e., there is a and such that and in . The difficulty here is that satisfies no Palais-Smale condition because of the periodicity. Moreover, the functional is strongly indefinite since 0 lies in a spectral gap of the linear operator .