Positive solutions of fractional Schrödinger-Poisson systems involving critical nonlinearities with potential. (Q2076583)

This paper deals with the existence of multiple positive solutions for a class of fractional Schrödinger-Poisson systems involving sign-changing potential and critical nonlinearities defined on \(\mathbb{R}^N\). Roughly speaking, under suitable assumptions, the main results ensure the existence of at least \(cat_{M_\delta}(M)+1\) distinct positive solutions for the system investigated, where \textit{cat} indicates the Ljusternik-Schnirelmann category and \(M=\{z\in\mathbb{R}^N:g(z)=\max_{x\in\mathbb{R}^N}g(x)=1\}\) is a non-empty compact set related to the the weighted potential \(g(x)\) of the critical nonlinearity term. In addition, some important topics of independent interest are also developed, as for instance, the technique adopted to find the minimum of the associated energy functional constrained on the Nehari manifold, the concentration-compactness result for fractional Schrödinger equations and a new analytical technique to derive precise estimates for the \((P-S)\) sequences.