Groundstates and infinitely many high energy solutions to a class of nonlinear Schrodinger-Poisson systems

We study a nonlinear Schrodinger-Poisson system which reduces to the nonlinear and nonlocal PDE -Delta u + u + lambda(2) (1/omega|x|(N-2) rho u(2)) rho(x)u = |u|(q-1)u x is an element of R-N where omega = (N - 2)|SN-1|, lambda > 0, q is an element of (1, 2* - 1), rho : R-N -> R is nonnegative, locally bounded, and possibly non-radial, N = 3, 4,5 and 2* = 2N/( N - 2) is the critical Sobolev exponent. In our setting rho is allowed as particular scenarios, to either (1) vanish on a region and be finite at infinity, or (2) be large at infinity. We find least energy solutions in both cases, studying the vanishing case by means of a priori integral bounds on the Palais-Smale sequences and highlighting the role of certain positive universal constants for these bounds to hold. Within the Ljusternik-Schnirelman theory we show the existence of infinitely many distinct pairs of high energy solutions, having a min-max characterisation given by means of the Krasnoselskii genus. Our results cover a range of cases where major loss of compactness phenomena may occur, due to the possible unboundedness of the Palais-Smale sequences, and to the action of the group of translations.

Automated summary

This study investigates a nonlinear Schrodinger-Poisson system, analyzing its properties and the existence of solutions with the critical Sobolev exponent being a key factor.