Existence of solutions for a nonhomogeneous Dirichlet problem
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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].
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