A three critical points theorem revisited
Let \(X\) be a reflexive Banach space, \(I \subset\mathbb R\) an interval; \(\Phi: X \to\mathbb R\) a sequentially weakly lower semicontinuous \(C^1\) functional, bounded on each bounded subset of \(X\), whose derivative admits a continuous inverse on \(X^*\); \(J: X \to\mathbb R\) a \(C^1\) functional with compact derivative. Assume that \(\lim_{\| x\| \to \infty}(\Phi(x)+\lambda J(x))=+\infty\) for all \(\lambda \in I\), and there exists \(\rho \in\mathbb R\) such that \(\sup_{\lambda \in I}\inf_{x \in X}(\Phi(x)+\lambda(J(x)+\rho))<\inf_{x \in X}\sup_{\lambda \in I}(\Phi(x)+\lambda(J(x)+\rho))\). Then there exists a subset \(A \subset I\), \(A \neq \emptyset\), and \(r>0\) with the following property: for every \(\lambda \in A\) and every \(C^1\) functional \(\Psi: X \to\mathbb R\) with compact derivative, there exists \(\delta>0\) such that, for each \(\mu \in [0,\delta]\), the equation \(\Phi'(x)+\lambda J'(x)+\mu \Psi'(x)=0\) has at least three solutions in \(X\) whose norms are less than \(r\).
scientific article; zbMATH DE number 5539186
scientific article