Groundstates and infinitely many high energy solutions to a class of nonlinear Schrödinger–Poisson systems

Abstract

We study a nonlinear Schrödinger–Poisson system which reduces to the nonlinear and nonlocal PDE $$\begin{aligned} - \Delta u+ u + \lambda ^2 \left( \frac{1}{\omega |x|^{N-2}}\star \rho u^2\right) \rho (x) u = |u|^{q-1} u \quad x \in {{\mathbb {R}}}^N, \end{aligned}$$where \(\omega = (N-2)|{\mathbb {S}}^{N-1} |,\) \(\lambda >0,\) \(q\in (1,2^{*} -1),\) \(\rho :{{\mathbb {R}}}^N \rightarrow {{\mathbb {R}}}\) 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.