Multiplicity of solutions for Schrodinger-Poisson system with critical exponent in R3

In this paper, we study the following Schrodinger-Poisson system with critical exponent {-Delta u - k(x)phi u = lambda h(x)vertical bar u vertical bar(p-2)u + s(x)vertical bar u vertical bar(4)u; x epsilon R-3, -Delta phi = k(x)u(2), x epsilon R-3, where 1 < p < 2 and lambda > 0: Under suitable conditions on k, h and s, we show that there exists lambda > 0 such that the above problem possesses infinitely many solutions with negative energy for each lambda epsilon (0, lambda*). Moreover, we prove the existence of infinitely many solutions with positive energy. The main tools are the concentration compactness principle, Z(2) index theory and Fountain Theorem. These results extend some existing results in the literature.