Electron. J. Diff. Equ., Vol. 2015 (2015), No. 158, pp. 1-8.

Existence and non-existence of solutions for a p(x)-biharmonic problem

Ghasem A. Afrouzi, Maryam Mirzapour, Nguyen Thanh Chung

Abstract:
In this article, we study the following problem with Navier boundary conditions
$$\displaylines{ \Delta (|\Delta u|^{p(x)-2}\Delta u)+|u|^{p(x)-2}u =\lambda |u|^{q(x)-2}u +\mu|u|^{\gamma(x)-2}u\quad \text{in } \Omega,\cr u=\Delta u=0 \quad \text{on } \partial\Omega. }$$
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial \Omega$, $N\geq1$. $p(x),q(x)$ and $\gamma(x)$ are continuous functions on $\overline{\Omega}$, $\lambda$ and $\mu$ are parameters. Using variational methods, we establish some existence and non-existence results of solutions for this problem.

Submitted July 22, 2014. Published June 15, 2015.
Math Subject Classifications: 35J60, 35B30, 35B40.
Key Words: p(x)-Biharmonic; variable exponent; critical points; minimum principle; fountain theorem; dual fountain theorem.

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Ghasem A. 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
Nguyen Thanh Chung
Department of Mathematics, Quang Binh University
312 Ly Thuong Kiet, Dong Hoi
Quang Binh, Vietnam
email: ntchung82@yahoo.com

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