Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 52, pp. 1-24.

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

       
Authors: Pablo Alvarez-Caudevilla (Univ. Carlos III, Madrid, Spain)
         Eduardo Colorado (Univ. Carlos III, Madrid, Spain)
         Alejandro Ortega   (Univ. Carlos III, Madrid, Spain)

Abstract: 
This article concerns the existence of positive solutions for the
 second 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
 γ>0, and up to the critical value of the exponent p, i.e. when
 1&lt;p&ltq; 2<sup>*</sup>-1, where 2<sup>*</sup>=(2N)/(N-2) is the critical Sobolev
 exponent. For p=2<sup>*</sup>-1, this leads us to a Brezis-Nirenberg type
 problem, cf. [5], 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&geq;7 for
 this nonlocal problem.

Submitted November 3, 2020. Published June 14, 2021.
Math Subject Classifications: 35G20, 35A15, 35J50, 35B38, 35J91.
Key Words: Critical point; concentration compactness principle;
           mountain pass theorem.