Nonlinear equations involving the square root of the Laplacian

TL;DR

This paper investigates the existence and non-existence of solutions to fractional Laplacian equations with zero boundary conditions, using variational methods and the Caffarelli-Silvestre extension to identify conditions for multiple solutions.

Contribution

It establishes the existence of multiple solutions for large parameters in fractional Laplacian equations with specific nonlinearities, employing a novel variational approach and extension technique.

Findings

Multiple solutions exist for large mbda values.

Solutions are bounded in L^ty norm.

The method combines variational principles with the Caffarelli-Silvestre extension.

Abstract

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega. \end{array}\right. \end{equation*} The existence of at least two non-trivial $L^{\infty}$-bounded weak solutions is established for large value of the parameter $\lambda$ requiring that the nonlinear term $f$ is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method.