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

TL;DR

This paper proves the existence and multiplicity of solutions for a prescribed mean-curvature problem with critical growth, using variational methods and genus theory, overcoming challenges posed by the nonlinearity and critical exponent.

Contribution

It introduces a novel approach combining an auxiliary problem and gradient estimates to establish multiple solutions for a complex geometric PDE.

Findings

Infinitely many solutions for the auxiliary problem.

Existence of solutions for the original mean-curvature problem.

Application of genus theory to nonlinear PDEs with critical growth.

Abstract

In this work we study an existence and multiplicity result for the following prescribed mean-curvature problem with critical growth $$ \left\{\begin{array}{rl} -\mbox{div}\biggl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\biggl) = \lambda |u|^{q-2}u+ |u|^{2^*-2}u & \mbox{in $\Omega$} u = 0 & \mbox{on $\partial \Omega$}, \end{array} \right. $$ 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.