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

Afrouzi, Ghasem A. ; Mirzapour, Maryam ; Chung, Nguyen Thanh

Electronic Journal of Differential Equations, Tome 2015 (2015).

Résumé

Summary: In this article, we study the following problem with Navier boundary conditions $$\displaylines{ \Delta (|\Delta u|^{p(x)-2}\Delta u)+|u|^{p(x)-2}u =\lambda |u|^{q(x)-2}u +\mu|u|^{\gamma(x)-2}u\quad {in } \Omega,\cr u=\Delta u=0 \quad {on } \partial\Omega. }$$ 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.

Détail

Classification : 35J60, 35B30, 35B40
Keywords: $p(x)$-biharmonic, variable exponent, critical points, minimum principle, Fountain theorem, dual Fountain theorem