A variational result in a domain with boundary (Q1601143)

The authors prove: Let \(F\) be a real \(C^2\) function in the closure \(\overline{\Omega}\) of a smooth bounded domain in \(\mathbb{R}^n.\) Assume that \[ \varphi:=F|_{\partial\Omega}: \partial\Omega \to \mathbb{R} \] has only two critical values, \(\max\) and \(\min.\) Denote by \(m\) the set where \(\varphi\) takes its minimum. Assume: \((i)\)~\(m\) is contractible to a point in \(\overline{\Omega};\) \((ii)\)~in some \(\alpha\)-neighborhood on \(\partial\Omega\) of \(m,\) \(m\) is not contractible to a point. Then \(F\) has a critical point in \(\Omega\). The result is extended to a domain \(\Omega\) in Hilbert space, assuming uniform continuity in some \(\beta\)-neighborhood of \(\partial\Omega\) of the Fréchet derivative \(F'.\)