# Multiple solutions of nonlinear equations involving the square root of the Laplacian

## TL;DR

This paper proves the existence of multiple bounded solutions for a class of nonlinear fractional Laplacian equations using variational methods and an extension technique.

## Contribution

It demonstrates the existence of at least three solutions for parametric fractional equations involving the square root of the Laplacian, under specific conditions.

## Key findings

- At least three bounded solutions exist for certain parameters.
- The solutions are obtained using variational methods.
- The approach employs a variant of the Caffarelli-Silvestre extension method.

## Abstract

In this paper we examine the existence of multiple solutions of 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 Dirichlet zero-boundary conditions, i.e. \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 three $L^{\infty}$-bounded weak solutions is established for certain values of the parameter $\lambda$ requiring that the nonlinear term $f$ is continuous and with a suitable growth. Our approach is based on variational arguments and a variant of Caffarelli-Silvestre's extension method.