SOLUTIONS IN SPECTRAL GAPS FOR A NONLINEAR EQUATION OF SCHRODINGER TYPE

In this paper we study the existence of a nontrivial H-2(R(N)) solution for an equation of the form -DELTAu(x) + p(x) u(x) - f(x, u(x)) = lambdau(x), x is-an-element-of R(N), lambda is-an-element-of R, where p is-an-element-of L(infinity)(R(N)) is a periodic function. We assume that the operator -DELTA + p - lambda H-2(R(N)) subset-of L2(R(N)) --> L2(R(N)) is strongly indefinite and invertible and that f(x, .):R --> R is odd and satisfies some superlinear but subcritical growth conditions. We extend the class of nonlinearities which has been studied up to now. In particular, under standard technical restrictions, the existence of a solution is derived, when lim\x\ --> infinity, f(x, s) = f(s) > 0 exists for all s is-an-element-of R, if we assume that f(x, s) greater-than-or-equal-to f(s) for all s is-an-element-of R and a.e. on R(N). (C) 1994 Academic Press, Inc.