Existence of positive solutions for Brezis-Nirenberg type problems involving an inverse operator

Abstract

This article concerns the existence of positive solutions for thesecond order equation involving a nonlocal term $$ -\Delta u=\gamma (-\Delta)^{-1} u+|u|^{p-1}u, $$ under Dirichlet boundary conditions. We prove the existence of positive solutions depending on the positive real parameter \(\gamma>0\), and up to the critical value of the exponent \(p\), i.e. when \(1<p\leq 2^*-1\), where \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent. For \(p=2^*-1\), this leads us to a Brezis-Nirenberg type problem, cf. \cite{BN}, but, in our particular case, the linear term is a nonlocal term. The effect that this nonlocal term has on the equation changes the dimensions for which the classical technique based on the minimizers of the Sobolev constant, that ensures the existence of positive solution, going from dimensions \(N\geq 4\) in the classical Brezis-Nirenberg problem, to dimensions \(N\geq7\) for this nonlocal problem. For more information see https://ejde.math.txstate.edu/Volumes/2021/52/abstr.html