Existence of solutions for a nonhomogeneous Dirichlet problem - Marcos, Aboubacar; Abdou, Aboubacar (Springer Open, 2019)
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.
. 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