On the three critical points theorem
The authors prove a new version of the ``three critical points theorem'', previously obtained by some authors (M. A. Krasnosel'skij, K. C. Chang, and others). The main result of the paper is the following Theorem. Let \(\varphi:X\to \mathbb{R}\) be a \(C^1\) functional defined on the Banach space \(X\), bounded from below and satisfying the Palais-Smale condition. Let \(m\) be the infimum of \(\varphi\) and assume that \(\varphi\) admits an essential critical value \(c>m\). Then either \(\varphi\) admits at least three critical points, or the set of minimum points of \(\varphi\) is noncontractible in itself. In particular, \(\varphi\) has at least three critical points. A real number \(c\) is an essential critical value for \(\varphi\) if there exists \(\varepsilon >0\) such that the sublevel \(\varphi^{c-\varepsilon}\) is not a strong deformation retract of the sublevel \(\varphi^{c+\varepsilon}\). The theorem above is then applied to the existence of nontrivial solutions of the Hammerstein equation \[ x(t)= \int_\Omega k(t,s) f(s,x(s))ds, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(k(t,s): \Omega\times \Omega\to\mathbb{R}\) is a measurable, symmetric kernel and \(f(s,u): \Omega\times \mathbb{R}\) is a Carathéodory function. The solutions of the Hammerstein equations are characterized as the critical points of a suitable functional on the Hilbert space \(L^2(\Omega)\). Using the previous abstract result, the authors show that, under suitable assumptions on the data \(k(t,s)\) and \(f(x,u)\), the Hammerstein equation admits two nontrivial solutions.
scientific article; zbMATH DE number 1218089