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Existence of solutions for fourth-order PDEs with variable exponents. (English) Zbl 1182.35086
Summary: We study the following problem with Navier boundary conditions
\[ \begin{aligned}\Delta _{p(x)}^2u=\lambda |u|^{p(x)-2}u+f(x,u)\quad &\text{in }\Omega,\\ u=\Delta u=0\quad &\text{on }\partial \Omega. \end{aligned} \]
Where \(\Omega \) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial \Omega \), \(N\geq 1\), \(\Delta _{p(x)}^2u:=\Delta (|\Delta u| ^{p(x)-2}\Delta u) \), is the \(p(x)\)-biharmonic operator, \(\lambda \leq 0\), \(p\) is a continuous function on \(\overline{\Omega } \) with \(\inf_{x\in \overline{\Omega}} p(x)>1\) and \(f:\Omega \times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function. Using the Mountain Pass Theorem, we establish the existence of at least one solution of this problem. Especially, the existence of infinite many solutions is obtained.

MSC:
35G30 Boundary value problems for nonlinear higher-order PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35A15 Variational methods applied to PDEs
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