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

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 $\Omega \subset {\mathbb{R}}^n$ ($n\ge 2$) and with Dirichlet zero-boundary conditions, i.e. \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=λf(u) & \mbox{ in } Ω\\ u=0 & \mbox{ on } \partialΩ. \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.