Eigenvalue problems for p(x)-Kirchhoff type equations

• Main Authors: Ghasem A. Afrouzi, Maryam Mirzapour
• Format: Article
• Language: English
• Published: Texas State University 2013-11-01
• Series: Electronic Journal of Differential Equations
• Subjects: p(x)-Kirchhoff type equations; variational methods; boundary value problems
• Online Access: http://ejde.math.txstate.edu/Volumes/2013/253/abstr.html

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.