Homotopical linking and Morse index estimates in min-max theorems
Let \(E\) be a real Hilbert space and \(f\in C^2(E, R)\). It is shown in this paper that if \(f\) satisfies certain conditions of saddle or linking type, then the associated minimax level must necessarily contain a critical point whose Morse index satisfies an explicitly given bound (upper or lower -- depending on the minimax procedure chosen). This extends earlier results by \textit{A. C. Lazer} and \textit{S. Solimini} [Nonlinear Anal., Theory Methods Appl. 12, No. 8, 761-775 (1988; Zbl 0667.47036)] and \textit{S. Solimini} [Manuscr. Math. 63, No. 4, 421-453 (1989; Zbl 0685.58010)]. Applications are given to the problem of obtaining multiple solutions for ordinary and elliptic partial differential equations of second-order. In particular, it is shown that under suitable conditions on \(g\) and whenever \(T\) is sufficiently large, the equation \(\ddot u(t)+ g(u(t))= 0\) has at least \(k\) nonconstant solutions of period \(kT\).
scientific article; zbMATH DE number 784074
scientific article