STANDING WAVES FOR A CLASS OF SCHRODINGER-POISSON EQUATIONS IN R3 INVOLVING CRITICAL SOBOLEV EXPONENTS

We are concerned with the following Schrodinger-Poisson equation with critical nonlinearity: {-epsilon(2)Delta u + V(x)u + psi u = lambda vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(4)u in R-3, -epsilon(2)Delta psi = u(2) in R-3, u > 0, u is an element of H-1(R-3), where epsilon > 0 is a small positive parameter, lambda > 0, 3 < p <= 4. Under certain assumptions on the potential V, we construct a family of positive solutions u epsilon is an element of H-1 (R-3) which concentrates around a local minimum of V as epsilon --> 0. Subcritical growth Schrodinger-Poisson equation {-epsilon(2)Delta u + V(x)u + psi u = f(u) in R-3, -epsilon(2)Delta psi = u(2) in R-3, u > 0, u is an element of H-1(R-3), has been studied extensively, where the assumption for f(u) is that f(u) similar to vertical bar u vertical bar(p-2)u with 4 < p < 6 and satisfies the Ambrosetti-Rabinowitz condition which forces the boundedness of any Palais-Smale sequence of the corresponding energy functional of the equation. The more difficult critical case is studied in this paper. As g(u) := lambda vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(4)u with 3 < p < 4 does not satisfy the Ambrosetti-Rabinowitz condition (there exists mu > 4, 0 < mu integral(u)(0) g(s) ds <= g(u)u), the boundedness of Palais-Smale sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function g(s)/s(3) is not increasing for s > 0 prevents us from using the Nehari manifold directly as usual. The main result we obtained in this paper is new.