Critical equations with a sharp change of sign in the nonlinearity

TL;DR

This paper investigates the existence of solutions to a semilinear elliptic equation with a nonlinearity that sharply changes sign, depending on the geometry of a bounded region, using variational and topological methods.

Contribution

It establishes conditions for existence and nonexistence of solutions based on the geometry of the region, extending classical critical problem results.

Findings

Existence of solutions depends on the shape of the bounded region.

Nonexistence results are also characterized by geometric conditions.

Methods include variational techniques and topological tools.

Abstract

We establish the existence and nonexistence of entire solutions to a semilinear elliptic problem whose nonlinearity is the critical power multiplied by a function that takes the value 1 in an open bounded region and the value -1 in its complement. The existence or not of solutions depends on the geometry of the bounded region, in a way analogous to what happens with the classical critical Dirichlet problem in a bounded domain. Our methods are variational and include the use of topological tools.