On Schrodinger Equation with Periodic Potential and Critical Sobolev Exponent

The main purpose of this paper is to establish the existence of a solution of the semilinear Schr¨odinger equation (1) −u + V (x)u = K(x)|u|2−2u + f(x, u) in RN, involving a critical Sobolev exponent 2 = 2N/(N − 2) with N 4 and a subcritical nonlinearity f : RN × R ! R. Throughout this paper it is assumed that (A) The coefficients V and K are continuous and 1-periodic functions in each variable xi, i = 1, . . . ,N. Moreover, we assume that K 0 on RN. In this case it is known that the operator − + V on L2(RN) has a purely continuous spectrum consisting of closed disjoint intervals. In this paper we consider the case: (B) 0 is in the spectral gap of the operator − + V . There are many existence results in the case K 0 on RN and we refer to the papers [4], [5], [9], [10], [13]. 1991 Mathematics Subject Classification. 35J65