On the sub-supersolution method for p(x)-Laplacian equations

Abstract

This paper deals with the sub-supersolution method for the p(x)-Laplacian equations. A sub-supersolution principle for the Dirichlet problems involving the p(x)-Laplacian is established. It is proved that the local minimizers in the C 1 topology are also local minimizers in the W 1,p(x) topology for given energy functionals. A strong comparison theorem for the p(x)-Laplacian equations is presented. Some applications of the abstract theorems obtained in this paper to the eigenvalue problems for the p(x)-Laplacian equations are given.

Existence of Positive Solutions of Nonlocal p(x)-Kirchhoff Evolutionary Systems via Sub-Super Solutions Concept

“…In Refs. [7][8][9][10][11][12], Fan et al studied the regularity of solutions for differential equations with nonstandard p(x)-growth conditions.…”

Existence and multiplicity of positive solutions for a class of p ( x )-Kirchhoff type equations

“…In this article we will use the following sub-supersolution principle, the proof of which is based on the well known fixed point theorem for the increasing operator on the order interval (see e.g., [45]) and is similar to that given in [12] for Dirichlet problems involving the p(x)-Laplacian. …”

“…The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [7][8][9][10][11] for an overview of and references on this subject, and to [12][13][14][15][16] for the study of the variable exponent equations and the corresponding variational problems.…”

Existence and multiplicity of solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary conditions

“…The study of various mathematical problems with variable exponent are interesting in applications and raise many difficult mathematical problems. We refer the readers to [17][18][19][20][21][22][23] for the study of p(x)-Laplacian equations and the corresponding variational problems. Corrêa and Figueiredo [13] presented several sufficient conditions for the existence of positive solutions to a class of nonlocal boundary value problems of the p-Kirchhoff type equation.…”