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

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) <= 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. (C) 2019 Elsevier Ltd. All rights reserved.