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Existence results for some fourth-order nonlinear elliptic problems. (English) Zbl 0981.35016
From the introduction: Let \(\Omega\) be a bounded open set in \(\mathbb{R}^n\). We are concerned with the fourth-order semilinear elliptic boundary value problem \[ \begin{aligned} &\Delta^2 u+ c\Delta u= f(x,u)\quad\text{in }\Omega,\\ & u|_{\partial\Omega}=\Delta u|_{\partial\Omega}= 0,\end{aligned}\tag{1} \] where \(\Delta^2\) denotes the biharmonic operator and \(c\in \mathbb{R}\).
We also consider the fourth-order quasilinear elliptic boundary value problem \[ \begin{aligned} &\Delta(g_1((\Delta u)^2)\Delta u)+ c\text{ div}(g_2(|\nabla u|^2)\nabla u)= f(x,u)\quad\text{in }\Omega,\\ & u|_{\partial\Omega}=\Delta u|_{\partial\Omega} =0.\end{aligned}\tag{2} \] It is the purpose of this paper to use variational methods for the fourth-order semilinear problem and the fourth-order quasilinear problem. We show the existence of solutions of problems (1) and (2) for a more general nonlinearity \(f\) under weak assumptions.

MSC:
35J60 Nonlinear elliptic equations
35J35 Variational methods for higher-order elliptic equations
35J30 Higher-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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