NORMALIZED SOLUTIONS FOR THE SCHRODINGER-POISSON SYSTEM WITH DOUBLY CRITICAL GROWTH

In this paper we are concerned with normalized solutions to the Schrodinger-Poisson system with doubly critical growth {-Delta u = phi|u|(3)u = lambda mu + mu|u|(q - 2)+ |u|(4)u, x is an element of R-3, Delta phi = |u|(5), x is an element of R-3, and prescribed mass integral(R3) |u|(2) dx = a(2), where a > 0 is a constant, mu > 0 is a parameter and 2 < q < 6. In the L-2-subcritical case, we study the multiplicity of normalized solutions by applying the truncation technique, and the genus theory, and in the L2 -supercritical case, we obtain a couple of normalized solutions by developing a fiber map. Under both cases, to recover the loss of compactness of the energy functional caused by the critical growth, we need to adopt the concentration-compactness principle. Our results complement and improve some related studies for the Schrodinger-Poisson system with nonlocal critical term in the literature.