On a p(x) $p(x)$-biharmonic problem with Navier boundary condition

• Main Author: Zheng Zhou
• Format: Article
• Language: English
• Published: SpringerOpen 2018-09-01
• Series: Boundary Value Problems
• Subjects: p ( x ) $p(x)$ -biharmonic; Critical points theory; Variational methods
• Online Access: http://link.springer.com/article/10.1186/s13661-018-1071-2

Abstract In this paper, we study a p(x) $p(x)$-biharmonic equation with Navier boundary condition {Δp(x)2u+a(x)|u|p(x)−2u=λf(x,u)+μg(x,u)in Ω,u=Δu=0on ∂Ω. $$ \textstyle\begin{cases} \Delta^{2}_{p(x)}u+a(x)|u|^{p(x)-2}u= \lambda f(x,u)+\mu g(x,u)\quad \text{in } \Omega, \\ u=\Delta u=0 \quad \text{on } \partial\Omega. \end{cases} $$ Here Ω⊂RN $\Omega\subset\mathbb{R}^{N}$ ( N≥1 $N\geq1$) is a bounded domain with smooth boundary ∂Ω, Δp(x)2u $\Delta^{2}_{p(x)}u$ is a p(x) $p(x)$-biharmonic operator with p(x)∈C(Ω‾) $p(x) \in C(\overline{\Omega})$, p(x)>1 $p(x)>1$. λ,μ∈R $\lambda,\mu\in\mathbb{R}$, a∈L∞(Ω) $a\in L^{\infty}(\Omega)$ such that infx∈Ωa(x)=a−>0 $\inf_{x\in\Omega}a(x)=a^{-}>0$. By variational methods, we establish the results of existence and non-existence of solutions.