Nonlinear equations involving the square root of the Laplacian

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 ▫$Omegasubset mathbb{R}^n$▫ (▫$ngeq 2$▫) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation ▫$$ left{ begin{array}{ll} A_{1/2}u=lambda f(u) & mbox{ in } Omega\ u=0 & mbox{ on } partialOmega. end{array}right. $$▫ 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