A Symmetric Mountain Pass Lemma and Its Application to an Equation for a Class of Quasilinear Elliptic Operators in a Variable Exponent Sobolev Space

Abstract

The purpose of this paper is to solve equation for a class of quasilinear elliptic operators containing the $p(·)-$Laplacian and the mean curvature operator with mixed boundary conditions. More precisely, we are concerned with the problem that has the Dirichlet condition in one part of the boundary and the Steklov condition in another. Using a symmetric mountain pass lemma and its corollary, we show the existence of infinitely many weak solutions of the equation and the boundedness of the sequence of solutions, or convergence to zero of the sequence of solutions according to the hypotheses about the data functions.