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Multiple nontrivial solutions for some fourth-order semilinear elliptic problems. (English) Zbl 1103.35027
The goal of the authors is to study the existence of multiple nontrivial solutions to the fourth order semilinear equation: $\Delta^2u+c \Delta u=f(x,u)\text{ in }\Omega\quad u|_{\partial\Omega}=\Delta u |_{\partial\Omega}=0,\tag{1}$ where $$\Omega$$ is a bounded open set in $$\mathbb R^N$$ with smooth boundary, $$\Delta^2$$ denotes the biharmonic operator, $$c\in\mathbb R$$ and $$f$$ is a given Carathéodory function. To this end they use Morse theory and local linking to find weak solutions.

##### MSC:
 35J40 Boundary value problems for higher-order elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35J35 Variational methods for higher-order elliptic equations 47J30 Variational methods involving nonlinear operators 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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