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

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) = \lambda |u|^{q-2}u+ |u|^{2^*-2}u & \mbox{in $\Omega$} u = 0 & \mbox{on $\partial \Omega$}, \end{array} \right. where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$, $N\geq 3$ 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.