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

Abstract

In this work we study an existence and multiplicity of solutions for the prescribed mean-curvature problem with critical growth,-div (del u/root 1+vertical bar del u vertical bar(2)) = lambda vertical bar u vertical bar(q-2) u + vertical bar u vertical bar(2*-2)u in Omegau = 0 on partial derivative Omega,where Omega is a bounded smooth domain of R-N, N >= 3 and 1 < q < 2. 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 problem.