POSITIVE SOLUTIONS FOR SCHRODINGER-POISSON-SLATER SYSTEM WITH COERCIVE POTENTIAL

We study the following Schrodinger-Poisson-Slater type system: (0.1) {-Delta u + V (x)u + lambda phi(x)u = vertical bar u vertical bar(p-1) u, x is an element of R-3, -Delta phi = u(2), lim(vertical bar x vertical bar ->+infinity) phi(x) = 0, where lambda > 0 is a parameter, p is an element of (1, 2), V is an element of C(R-N, R-0(+) ). If Visa constant, or p is an element of (2,5) with V is an element of L-infinity(R-3), the above system has been considered in many papers. In this paper, we are interested in the case: p is an element of (1,2) and V being a coercive potential, i.e., lim(vertical bar x vertical bar ->+infinity) V(x) = infinity. We prove that there exists lambda(0) > 0 such that system (0.1) has at least two positive solutions u(lambda)(0) and u(lambda)(1) for any lambda is an element of (0, lambda(0)). Moreover, u(lambda)(0) is a ground state (i.e., the least energy solution) which must blow up as lambda -> 0. Particularly, when p is an element of (11/7, 2) and lambda > 0 is small enough, we show that the ground state of (0.1) must be non-radially symmetric even if V (x) = V (vertical bar x vertical bar), such as V (x) = vertical bar x vertical bar(2).