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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
##### Keywords:
Palais Smale condition; mountain pass theorem
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