Geodesic
Existence results for a Dirichlet boundary value problem involving the $p(x)$-Laplacian operator
Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 2, pp. 5-13 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $Au:=-\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u)-f(x,u)$. When this problem admits a non-zero weak solution $u$, $\lambda$ is an eigenvalue of it and $u$ is an associated eigenfunction. 
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     title = {Existence results for a {Dirichlet} boundary value problem involving the $p(x)${-Laplacian} operator},
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M. Ait Hammou. Existence results for a Dirichlet boundary value problem involving the $p(x)$-Laplacian operator. Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 2, pp. 5-13. http://geodesic.mathdoc.fr/item/VMJ_2022_24_2_a0/

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