Existence and Uniqueness of Solutions for the p(x)-Laplacian Equation with Convection Term

Bin-Sheng Wang,Bin Ge,Gang-Ling Hou

doi:10.3390/math8101768

Abstract

In this paper, we consider the existence and uniqueness of solutions for a quasilinear elliptic equation with a variable exponent and a reaction term depending on the gradient. Based on the surjectivity result for pseudomonotone operators, we prove the existence of at least one weak solution of such a problem. Furthermore, we obtain the uniqueness of the solution for the above problem under some considerations. Our results generalize and improve the existing results.

Highlights

Introduction and Main ResultsThe p( x )-Laplacian operator has been used in the modelling of electrorheological fluids ([1,2,3]), in elastic mechanics ([4]), in image restoration ([5,6,7]) and in magnetostatics problems ([8]).Up to now, a large number of results have been obtained for solutions to equations related to this operator
We obtain that lim suphL(un ), un − ui = lim suphL(un ), un − ui ≤ 0. It follows that un → u because L is a mapping of type (S+ )
In order to overcome this difficulty, in this paper, we use the theory of pseudomonotone operators to obtain the existence of solutions for problem (1), formulated in the paper as Theorem 1

Summary

Introduction and Main Results

The p( x )-Laplacian operator has been used in the modelling of electrorheological fluids ([1,2,3]), in elastic mechanics ([4]), in image restoration ([5,6,7]) and in magnetostatics problems ([8]). The authors in [41] used sub-supersolution techniques with Schaefer’s fixed point theorem to prove the existence of a positive and a negative solution for problem (2). In order to overcome this difficulty, Yin, Li and Ke in [17], by using Krasnoselskii’s fixed point theorem on the cone, proved the existence of positive solutions for problem (1) under certain assumptions. Motivated by the aforementioned works, in the present paper, we consider the problem (1) in the case when the nonlinearity f satisfies the subcritical growth condition. To this end, we assume that f : Ω × R × R N 7→. We complete the proofs of Theorems 1 and 2 of this paper

Preliminaries

Proof of Theorem 1

Conclusions