Rocky Mountain Journal of Mathematics

Nontrivial solutions for Kirchhoff-type problems involving the $p(x)$-Laplace operator

Abdeljabbar Ghanmi

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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.

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Rocky Mountain J. Math., Volume 48, Number 4 (2018), 1145-1158.

First available in Project Euclid: 30 September 2018

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Primary: 35J35: Variational methods for higher-order elliptic equations 35J50: Variational methods for elliptic systems 35J60: Nonlinear elliptic equations

Nontrivial solution Kirchhoff-type problems variational method


Ghanmi, Abdeljabbar. Nontrivial solutions for Kirchhoff-type problems involving the $p(x)$-Laplace operator. Rocky Mountain J. Math. 48 (2018), no. 4, 1145--1158. doi:10.1216/RMJ-2018-48-4-1145.

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