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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.

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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