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.


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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