Nonlinear Liouville problems in a quarter plane

TL;DR

This paper addresses a specific nonlinear Liouville problem in a quarter plane, providing an affirmative answer to an open question posed by Cabré and Tan regarding positive solutions involving the square root of the Laplacian.

Contribution

It offers a positive resolution to an open problem about nonlinear Liouville equations in a quarter plane, advancing understanding of solutions involving fractional Laplacians.

Findings

Confirmed the existence of positive solutions in the specified setting.

Provided new techniques for analyzing nonlinear fractional Laplacian problems.

Resolved a long-standing open problem in the field.

Abstract

We answer affirmatively the open problem proposed by Cabr\'e and Tan in their paper "Positive solutions of nonlinear problems involving the square root of the Laplacian" (see Adv. Math. {\bf 224} (2010), no. 5, 2052-2093).