Weak solution of a Neumann boundary value problem with 𝑝(𝑥)-Laplacian-like operator
Mohamed El Ouaarabi, Chakir Allalou, Saïd Melliani
Abstract
Abstract In this paper, we study the existence of a weak solution for a class of Neumann boundary value problems for equations involving the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> p(x) -Laplacian-like operator. Using a topological degree theory for a class of demicontinuous operators of generalized <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>S</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (S_{+}) -type and the theory of the variable exponent Sobolev spaces, we establish the existence of a weak solution of this problem. Our results extend and generalize several corresponding results from the existing literature.