Luiz F.O. Faria

Comparison and Positive Solutions for Problems with the ( p, q )-Laplacian and a Convection Term

The aim of this paper is to prove the existence of a positive solution for a quasi-linear elliptic problem involving the ( p, q )-Laplacian and a convection term, which means an expression that is not in the principal part and depends on the solution and its gradient. The solution is constructed through an approximating process based on gradient bounds and regularity up to the boundary. The positivity of the solution is shown by applying a new comparison principle, which is established here.

The Brezis-Nirenberg problem for nonlocal systems

Abstract By means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

Critical Brezis-Nirenberg problem for nonlocal systems

We deal with the existence of solutions to a critical elliptic system involving the fractional Laplacian operator. We consider the primitive of the nonlinearity interacting with the spectrum of the operator. The one side resonant case is also considered. Variational methods are used to obtain the existence, and our result improves earlier results of the authors.

Infinitely many sign-changing solutions for a class of critical elliptic systems with Neumann conditions

In this work we study the multiplicity results for a class of critical elliptic systems related to the Brezis–Nirenberg problem with the Neumann boundary condition on a ball. Our approach relies on a minimization argument for an auxiliary problem with a mixed boundary condition and on suitable estimates of the critical level for the system case.