On Schrodinger-Poisson systems involving concave-convex nonlinearities via a novel constraint approach

In this paper, we investigate the multiplicity of positive solutions for a class of Schrodinger-Poisson systems with concave and convex nonlinearities as follows: {-Delta u + lambda V(x)u + mu phi u = a(x)vertical bar u vertical bar(p-2)u + b(x)vertical bar u vertical bar(q-2)u in R-3, -Delta phi = u(2) in R-3, where lambda, mu > 0 are two parameters, 1 < q < 2 < p < 4, V is an element of C(R-3) is a potential well, a is an element of L-infinity(R-3) and b is an element of Lp/(p-q)(R-3). Such problem cannot be studied by applying variational methods in a standard way, since the (PS) condition is still unsolved on H-1(R-3) due to 2 < p < 4. By developing a novel constraint approach, we prove that the above problem admits at least two positive solutions.