Solutions of nonlinear problems involving p(x)-Laplacian operator

Abstract

In the present paper, by using variational principle, we obtain the existence and multiplicity of solutions of a nonlocal problem involving p(x)-Laplacian. The problem is settled in the variable exponent Sobolev space W ,p(x) (Ω), and the main tools are the Mountain-Pass theorem and Fountain theorem.

A nonlinear eigenvalue problem with <inline-formula><tex-math id="M1">\begin{document}$ p(x) $\end{document}</tex-math></inline-formula>-growth and generalized Robin boundary value condition

“…The experimental research has been done mainly in the USA, for instance in NASA laboratories. These relevant applications motivate the growing interest of mathematicians for the study of nonlinear problems with p(x)-growth conditions; we refer, e.g., to [1,2,12,20,22,25,28] and the references therein.…”

Existence Results for Steklov Problem With Nonlinear Boundary Condition

“…The interest in studying such problems was stimulated by their application in mathematical physics, more precisely in elastic mechanics [25], electrorheological fluids and stationary thermo-rheological viscous flows of non-Newtonian fluids, image processing [8,12,19,20] and the mathematical description of the processes filtration of an idea barotropic gas through a porous medium [3,7]. Many results have been obtained on this kind of problems, for instance, we here cite [1,4,10,13,14,16,18,21,22].…”

On a nonlocal Neumann problem in Orlicz-Sobolev spaces

“…If we additionally consider M(t) = 1, then equation (1.1) becomes the p(x)-Laplace equation, a generalization of p-Laplace equation given by −div(|∇u| p−2 ∇u) = f (x, u), 1 < p < N. Therefore, equation (1.1) has the capacity particularly to generalize the problems involving variable exponents. This kind of problems have been extensively studied by many authors over the past twenty years due to its significant role in many fields of mathematics; see, e.g., [1]- [17] and the references therein.…”