Electron. J. Diff. Equ., Vol. 2013 (2013), No. 253, pp. 1-10.

Eigenvalue problems for p(x)-Kirchhoff type equations

Ghasem A. Afrouzi, Maryam Mirzapour

Abstract:
In this article, we study the nonlocal $p(x)$-Laplacian problem
$$\displaylines{
 -M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\Big)
 \hbox{div}(|\nabla u|^{p(x)-2}\nabla u)= \lambda|u|^{q(x)-2}u \quad
 \text{ in } \Omega,\cr
 u=0 \quad \text{on } \partial\Omega,
 }$$
By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions.

Submitted August 17, 2013. Published November 20, 2013.
Math Subject Classifications: 35J60, 35J35, 35J70.
Key Words: p(x)-Kirchhoff type equations; variational methods; boundary value problems.

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Ghasem Alizadeh Afrouzi
Department of Mathematics, Faculty of Mathematical Sciences
University of Mazandaran, Babolsar, Iran
email: afrouzi@umz.ac.ir
Maryam Mirzapour
Department of Mathematics, Faculty of Mathematical Sciences
University of Mazandaran, Babolsar, Iran
email mirzapour@stu.umz.ac.ir

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