Eigenvalue Problem for a Class of Quasilinear Elliptic Operators with Mixed Boundary Value Condition in a Variable Exponent Sobolev Space (Communications in Mathematical Research, 2024)
In this paper, we consider an eigenvalue problem for a class of nonlinear elliptic operators containing $p(·)$-Laplacian and mean curvature operator with mixed boundary conditions. More precisely, we are concerned withthe problem with the Dirichlet condition on a part of the boundary and theSteklov boundary condition on another part of the boundary. Using theLjusternik-Schnirelmann variational method, we show the existence of infinitely many positive eigenvalues of the equation. Furthermore, under some conditions, we derive that the infimum of the set of all the eigenvalues becomeszero or remains to be positive.
Keywords: Eigenvalue problem; $p(·)$-Laplacian type equation; mean curvature operator; mixed boundary value problem.