Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 253, pp. 1-10.
Title: Eigenvalue problems for p(x)-Kirchhoff type equations
Authors: Ghasem A. Afrouzi (Univ. of Mazandaran, Babolsar, Iran)
Maryam Mirzapour (Univ. of Mazandaran, Babolsar, Iran)
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.
A corrigendum was posted on August 27, 2014. It modifies inequality (3.1),
and deletes assumption (M2). This is necessary because there is no function
satisying the original assumptions (M1) and (M2). See the last page
of this article.