A fourth order elliptic equation with nonlinear boundary conditions
The authors deal with the existence of nontrivial solutions of the following fourth-order problem: \[ \begin{cases} -\Delta^2u= u\quad & \text{in } \Omega\\ -{\partial\Delta u\over\partial n}= f(x,u)\quad & \text{on }\partial \Omega,\\ \Delta u=0 \text{ or }{\partial u\over\partial n}= 0\quad & \text{on }\partial\Omega,\end{cases} \tag{1} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb R^N\), \({\partial\over \partial n}\) is the outer normal derivative and \(f: \partial \Omega\times \mathbb R\to\mathbb R\) is a given function. Under suitable assumptions on \(f(x,u)\) the authors prove the existence of infinitely many weak solutions for (1). To this end, they use abstract results from critical point theory, some topological tools and Sobolev trace inequalities.
scientific article; zbMATH DE number 1787982
scientific article