On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent
The authors look for solutions to the problem \[ \Delta^{2} u(x)=u^{(n+4)/(n-4)}(x),\;x\in\Omega,\qquad u(x)>0,\;x\in\Omega,\qquad u(x)=\Delta u(x)=0, \;x\in\partial\Omega,\tag{P} \] where \(\Omega\) is a bounded smooth domain of \({\mathbb R}^{n}\) and \(n\geq 5\). Denote by \(H_{k}(\Omega, Z_{2})\) the homology of dimension \(k\) with coefficient in \(\mathbb{Z}_{2}\), then the authors prove the following theorem: Theorem. If \(H_{k}(\Omega, \mathbb{Z}_{2})\not=\emptyset\) for some \(k\in {\mathbb N}\), then problem (P) has a solution. This solution is characterized to be of higher energy and higher index. The methods used in proving that the result rely on topological arguments was already introduced in [\textit{A. Bahri} and \textit{J.-M. Coron}, Commun. Pures Appl. Math. 253-294 (1988; Zbl 0649.35033)].
scientific article; zbMATH DE number 1880110