CHOQUARD EQUATIONS VIA NONLINEAR RAYLEIGH QUOTIENT FOR CONCAVE-CONVEX NONLINEARITIES

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form {-Delta u + V(x)u = (I-alpha*vertical bar u vertical bar(p))vertical bar u vertical bar(p-2)u + lambda vertical bar u vertical bar(q-2)u in R-N, u is an element of H-1(R-N) where lambda > 0 N >= 3, alpha is an element of(0, N). The potential V is a continuous function and I-alpha denotes the standard Riesz potential. Assume also that 1 < q < 2, 2(alpha) < p < 2(alpha)(*) where 2(alpha) = (N + alpha)/N, 2(alpha) = (N + alpha)/(N - 2). Our main contribution is to consider a specific condition on the parameter lambda > 0 taking into account the nonlinear Rayleigh quotient. More precisely, there exists lambda* > 0 such that our main problem admits at least two positive solutions for each lambda is an element of (0, lambda*]. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter lambda* > 0 is optimal in some sense which allow us to apply the Nehari method.