Generalized Schr\"odinger-Newton system in dimension $N\ge 3$: critical case

TL;DR

This paper investigates a nonlocal Schrödinger-Newton system, analyzing existence and nonexistence of positive solutions in various dimensions, highlighting challenges due to nonlocal critical nonlinearities.

Contribution

It extends the analysis of Schrödinger-Newton systems to higher dimensions with critical nonlinearities, providing new existence and nonexistence results.

Findings

Existence of positive solutions for N=3

Nonexistence results in certain cases for N=3

Solutions exist in higher dimensions under resonance and nonresonance conditions

Abstract

In this paper we study a system which is equivalent to a nonlocal version of the well known Brezis Nirenberg problem. The difficulties related with the lack of compactness are here emphasized by the nonlocal nature of the critical nonlinear term. We prove existence and nonexistence results of positive solutions when $N=3$ and existence of solutions in both the resonance and the nonresonance case for higher dimensions.