Positive solutions of nonlinear problems involving the square root of the Laplacian

TL;DR

This paper investigates positive solutions to nonlinear elliptic problems involving the square root of the Laplacian, establishing existence, regularity, a priori estimates, and symmetry results for solutions with power nonlinearities.

Contribution

It provides new existence, regularity, and symmetry results for nonlinear problems involving the square root of the Laplacian, extending classical theories to nonlocal operators.

Findings

Existence of positive solutions established

Regularity and a priori estimates proved

Symmetry of solutions demonstrated

Abstract

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.