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DOI: 10.46698/s8393-0239-0126-b

Existence Results for a Dirichlet Boundary Value Problem Involving the \(p(x)\)-Laplacian Operator

Ait Hammou, M.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 2.
The aim of this paper is to establish the existence of weak solutions, in \(W_0^{1,p(x)}(\Omega)\), 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 \(v+S\circ Tv=0\) in the reflexive Banach space \(W^{-1,p'(x)}(\Omega)\) which is the dual space of \(W_0^{1,p(x)}(\Omega)\). Note also that the problem can be seen as a nonlinear eigenvalue problem of the form\(Au=\lambda u,\) where \(Au:=-Div(|\nabla u|^{p(x)-2}\nabla u)-f(x,u)\). When this problem admits a non-zero weak solution \(u\), \(\lambda\) is an eigenvalue of it and \(u\) is an associated eigenfunction.
Keywords: Dirichlet problem, topological degree, \(p(x)\)-Laplacian operator.
Language: English Download the full text  
For citation: Ait Hammou, M. Existence Results for a† Dirichlet Boundary Value Problem Involving the \(p(x)\)-Laplacian Operator, Vladikavkaz Math. J., 2022, vol. 24, no. 2, pp. 5-13. DOI 10.46698/s8393-0239-0126-b
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