Degenerate Nonlinear Elliptic Equations Lacking in Compactness

In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form $-\mbox{div}(a(x)\nabla_{p}u(x))=g(\lambda,x,|u|^{p-2}u)$ in $\mathbb{R}^N$, where $\nabla_{p}u=|\nabla u|^{p-2}\nabla u$ and $a(x)$ is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.