# Existence of positive solutions for a Brezis--Nirenberg type problem involving an inverse operator

## TL;DR

This paper proves the existence of positive solutions for a nonlocal fourth-order differential equation involving inverse operators, extending classical results to higher dimensions due to the nonlocal term's influence.

## Contribution

It introduces a Brezis--Nirenberg type problem with a nonlocal linear term and establishes existence results for positive solutions depending on a parameter, extending classical dimension bounds.

## Key findings

- Existence of positive solutions for the nonlocal problem in dimensions N≥7.
- The nonlocal term alters the classical dimension threshold from N≥4 to N≥7.
- The problem includes a critical Sobolev exponent case, linking to Brezis--Nirenberg problems.

## Abstract

This paper is devoted to the existence of positive solutions for a problem related to a fourth-order differential equation involving a nonlinear term depending on a second order differential operator, $$(-\Delta)^2 u=\lambda u+ (-\Delta)|u|^{p-1}u,$$ in a bounded domain $\Omega\subset\mathbb{R}^N$, $N\geq 7$, and assuming homogeneous Navier boundary conditions. In particular, we study a second order equation involving a nonlocal term of the form, $$-\Delta u=\lambda (-\Delta)^{-1} u+|u|^{p-1}u,$$ under Dirichlet boundary conditions and we prove the existence of positive solutions depending on the positive real parameter $\lambda>0$, 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 equivalence 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 ensures the existence of solution, going from dimensions $N\geq 4$ in the classical Brezis-Nirenberg problem, to dimensions $N\geq7$ for this nonlocal problem.