Abstract

We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the p(x)\(p(x)\)-Laplacian operator with Dirichlet boundary condition: Δp(x)u+V(x)|u|q(x)2u=f(x,u)in Ω,u=0 on Ω,\[ -\Delta _{p(x)}u+V(x) \vert u \vert ^{q(x)-2}u =f(x,u)\quad \text{in }\varOmega , u=0 \text{ on }\partial \varOmega , \] where Ω is a smooth bounded domain in RN\(\mathbb{R}^{N}\), V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, f(x,t)\(f(x,t)\) 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.

Details

Title
Existence of solutions for a nonhomogeneous Dirichlet problem involving p ( x ) \(p(x)\) -Laplacian operator and indefinite weight
Author
Aboubacar Marcos 1   VIAFID ORCID Logo  ; Abdou, Aboubacar 1 

 Institut de Mathématiques et de Sciences Physiques, UAC, Dangbo, Bénin 
Pages
1-21
Publication year
2019
Publication date
Oct 2019
Publisher
Hindawi Limited
ISSN
16872762
e-ISSN
16872770
Source type
Scholarly Journal
Language of publication
English
ProQuest document ID
2310422575
Copyright
Boundary Value Problems is a copyright of Springer, (2019). All Rights Reserved., © 2019. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.