On a nonlinear eigenvalue problem for generalized Laplacian in Orlicz-Sobolev spaces

Abstract

We consider a nonlinear eigenvalue problem for some elliptic equations governed by general operators including the $p$-Laplacian. The natural framework in which we consider such equations is that of Orlicz-Sobolev spaces. we exhibit two positive constants ${\lambda }_0$ and ${\lambda }_1$ with ${\lambda }_0\le {\lambda }_1$ such that ${\lambda }_1$ is an eigenvalue of the problem while any value $\lambda <{\lambda }_0$ cannot be so. By means of Harnack-type inequalities and a strong maximum principle, we prove the isolation of ${\lambda }_1$ on the right side. We emphasize that throughout the paper no ${\Delta }_2$-condition is needed.

23 pages