Existence results for some quasilinear elliptic equations involving critical Sobolev exponents

Abstract

In this paper we study the existence of solutions to zero-Dirichlet-boundary-value problems for the quasilinear elliptic equation ${\rm (QE)_c}$ $- \Delta_p u - p \nabla \theta(x) \cdot \nabla u |\nabla u|^{p-2} = \lambda a(x) |u|^{p-2}u + K(x)|u|^{p^*-2}u$ in an unbounded domain $\Omega \subset {\bf R}^N$ with smooth boundary $\partial \Omega$. By using Brézis-Nirenberg's results, we prove that ${\rm (QE)_c}$ admits at least one nontrivial weak solution for positive $\lambda$ in a suitable interval.