Existence of solutions for a nonhomogeneous Dirichlet problem

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Date
2019
Journal Title
Journal ISSN
Volume Title
Publisher
Springer Open
Abstract
We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the p(x)-Laplacian operator with Dirichlet boundary condition: – p(x)u + V(x)|u| q(x)–2u = f(x, u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in RN, V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, f(x,t) is a Carathéodory function satisfying some growth conditions. Depending on the assumptions, the solutions set may consist of a bounded infinite sequence of solutions or a unique one. Our technique is based on a symmetric version of the mountain pass theorem.
Description
. A growing interest in the study of the p(x)- Laplacian operator has arisen during the last two decades, in regard to its involvement in the modelings of a large number of phenomena. One can name for instance electrorheological fluids [30, 32, 36], elastic mechanics, flows in porous media and image processing [11], curl systems emanating from electromagnetism [4, 7].
Keywords
Generalized Lebesgue–Sobolev spaces, p(x)-Laplacian operator, Symmetric mountain pass lemma
Citation
Marcos, A., Abdou, A. (2019). Existence of solutions for a nonhomogeneous Dirichlet problem involving p(x)-Laplacian operator and indefinite weight. Springer Open Journal. Pg 1-21. https://doi.org/10.1186/s13661-019-1276-z
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