Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations
The author considers the following equation with real parameter \(\alpha\) in bounded domains \[ \begin{gathered} \Delta^2 u= |x|^\alpha|u|^{q-2} u\quad\text{in }\Omega,\\ u= {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(1< q<\infty\) and \(\Omega\subset\mathbb R^N\) \((N> 4)\) is a domain with smooth boundary. He is interested in the existence and nonexistence of nontrivial solutions for (1) depending on \(\alpha\), \(q\) and \(N\). To prove the existence theorem for (1) the author uses a weighted Sobolev-Poincaré type inequality, while the nonexistence results are shown by using a Pokhozhaev-type identity for fourth-order equations.
scientific article; zbMATH DE number 2046207