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Pairs of critical points produced by linking subsets with applications to semilinear elliptic problems. (English) Zbl 0785.58017
Let \(E\) be a real Hilbert space and \(G\in C^ 1(E,\mathbb{R})\) satisfy the Palais-Smale condition. Then \(A\subset E\) is said to be linked to \(B\subset E\) if \(A\cap B=\emptyset\) and if for any one-parameter family of homeomorphisms \(h_ t:E\to E\) with \(h_ 0=\)id and \(h_ 1=\text{const}\) there exists \(t\) with \(h_ t(A)\cap B\neq\emptyset\). The authors show that if \(A\) links \(B\) and \(B\) links \(A\) and \(\sup G(A)\leq\inf G(B)\) then there exist two critical points of \(G\). This generalizes a result of P. H. Rabinowitz [Minimax methods in critical point theory with applications to differential equations, Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)] where one of the two critical points is a given trivial solution. As an application the authors consider a semilinear elliptic equation where a priori no trivial solution is known.

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J60 Nonlinear elliptic equations
35J99 Elliptic equations and elliptic systems