Nonlinear eigenvalue problems for quasilinear equations in unbounded domains

We prove the existence of a solution of the nonlinear equation \documentclass{article}\pagestyle{empty}\begin{document}$$ - {\rm div}\left({a\left(x \right)|\nabla _u |^{p - 2} \nabla _u} \right) = \lambda f\left({x,u} \right) $$\end{document} in IRN and in exterior domains, respectively. We concentrate to the case when p ≥ N and the nonlinearity f(x, · ) is “superlinear” and “subcritical”.