On an obstacle problem for degenerate elliptic operators involving the critical Sobolev exponent (Q1001343)

Let \(\Omega\subset {\mathbb R}^N\) be a bounded domain with smooth boundary and assume that \(1<p<N\). Fix \(\Psi\in W^{1,p}(\Omega)\) such that \(\Psi\leq 0\) on \(\partial\Omega\) and denote \[ K_\Psi =\{ v\in W_0^{1,p}(\Omega);\;v\geq\Psi\;\hbox{a.e.\;on}\;\Omega\}. \] This paper deals with the following inequality problem: if \(\lambda\) is a positive parameter, find \(u\in K_\Psi\) such that \[ \int_\Omega A(x,\nabla u)(\nabla v-\nabla u)dx\geq\lambda\int_\Omega (u^+)^{p^*-1}(v-u)dx\qquad\hbox{for\;every}\;v\in K_\Psi\,, \] where \(A\) is a monotone operator and \(p^*\) denotes the critical Sobolev exponent. The solution of this problem is obtained through the local minimization of the associated energy functional, by means of Ekeland's variational principle. In the last part of this paper it is analyzed the obstacle problem involving a lower order perturbation.