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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.

##### MSC:
 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