Eigenvalue Problem for a Class of Nonlinear Operators Containing $p$(·)-Laplacian in a Variable Exponent Sobolev Space

Abstract

In this paper, we consider an eigenvalue problem for a class of nonlinear operators containing $p(·)-$Laplacian and mean curvature operator with mixed boundary conditions. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show that the eigenvalue problem has infinitely many eigenpairs by using the celebrated Ljusternik-Schnirelmann principle of the calculus of variation. Moreover, in a variable exponent Sobolev space, there are two cases where the infimum of all eigenvalues is equal to zero and is positive.