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2018 Nontrivial solutions for Kirchhoff-type problems involving the $p(x)$-Laplace operator
Abdeljabbar Ghanmi
Rocky Mountain J. Math. 48(4): 1145-1158 (2018). DOI: 10.1216/RMJ-2018-48-4-1145

Abstract

In this article, we study the existence of nontrivial solutions for the following $p(x)$ Kirchhoff-type problem \begin{equation} \begin{cases}\smash {\!-M\big (\textstyle \int _{\Omega }A(x,\nabla u)\,dx\big ){div}(a(x,\nabla u))} =\lambda h(x)\frac{\partial F}{\partial u} (x,u), \quad \mbox {in } \Omega \\ u=0, \quad \mbox {on } \partial \Omega , \end{cases} \end{equation} where $\Omega \subset \mathbb {R}^{n}$, $n\geq 3$, is a smooth bounded domain, $\lambda >0$, $h\in C(\Omega )$, $F:\overline {\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$ is continuously differentiable and $a, A:\Omega \times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ are continuous. The proof is based on variational arguments and the theory of variable exponent Sobolev spaces.

Citation

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Abdeljabbar Ghanmi. "Nontrivial solutions for Kirchhoff-type problems involving the $p(x)$-Laplace operator." Rocky Mountain J. Math. 48 (4) 1145 - 1158, 2018. https://doi.org/10.1216/RMJ-2018-48-4-1145

Information

Published: 2018
First available in Project Euclid: 30 September 2018

zbMATH: 06958772
MathSciNet: MR3859751
Digital Object Identifier: 10.1216/RMJ-2018-48-4-1145

Subjects:
Primary: 35J35, 35J50, 35J60

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 4 • 2018
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