Existence results for a Dirichlet boundary value problem involving the p(x)-Laplacian operator
Автор: Ait Hammou Mustapha
Журнал: Владикавказский математический журнал @vmj-ru
Статья в выпуске: 2 т.24, 2022 года.
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The aim of this paper is to establish the existence of weak solutions, in W1,p(x)0(Ω), for a Dirichlet boundary value problem involving the p(x)-Laplacian operator. Our technical approach is based on the Berkovits topological degree theory for a class of demicontinuous operators of generalized (S+) type. We also use as a necessary tool the properties of variable Lebesgue and Sobolev spaces, and specially properties of p(x)-Laplacian operator. In order to use this theory, we will transform our problem into an abstract Hammerstein equation of the form v+S∘Tv=0 in the reflexive Banach space W-1,p′(x)(Ω) which is the dual space of W1,p(x)0(Ω). Note also that the problem can be seen as a nonlinear eigenvalue problem of the formAu=λu, where Au:=-Div(|∇u|p(x)-2∇u)-f(x,u). When this problem admits a non-zero weak solution u, λ is an eigenvalue of it and u is an associated eigenfunction.
Dirichlet problem, topological degree, p(x)-Laplacian operator
Короткий адрес: https://sciup.org/143178748
IDR: 143178748 | DOI: 10.46698/s8393-0239-0126-b
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