Existence and multiplicity of solutions for a prescribed mean-curvature problem with critical growth

Abstract

In this work we study an existence and multiplicity result for the following prescribed mean-curvature problem with critical growth $$ \left\{\begin{array}{rl} -\mbox{div}\biggl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\biggl) = λ|u|^{q-2}u+ |u|^{2^*-2}u & \mbox{in $Ω$} u = 0 & \mbox{on $\partial Ω$}, \end{array} \right. $$ where $\Omega$ is a bounded smooth domain of ${\mathbb{R}}^N$, $N\ge 3$ and $1<q<2$. In order to employ variational arguments, we consider an auxiliary problem which is proved to have infinitely many solutions by genus theory. A clever estimate in the gradient of the solutions of the modified problem is necessary to recover solutions of the original one.