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A note on nonsmooth functionals with infinitely many critical values. (English) Zbl 0798.58013
The paper deals with critical point theory for functionals with $$\varphi$$-monotone subdifferential, a class of nonsmooth functionals introduced by E. DeGiorgi, the first author, A. Marino and M. Tosques [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 75, 1-8 (1983; Zbl 0597.47045)]. The main result asserts that, if $$H$$ is a Hilbert space, $$f: H \to \mathbb{R} \cup \{+ \infty\}$$ is a lower semicontinuous function with $$\varphi$$-monotone subdifferential of order 2 and the domain of $$f$$, endowed with the graph metric, has not the homotopy type of a compact polyhedron, then $$f$$ admits a sequence $$(c_ n)$$ of critical values with $$c_ n \to +\infty$$. This theorem corresponds, in a nonsmooth setting, to a well known result of A. Marino and G. Prodi [Boll. Unione Mat. Ital., IV. Ser. 11, Suppl. Fasc. 3, 1-32 (1975; Zbl 0311.58006)].
As an application, the authors give another proof that, if $$M$$ is a complete locally closed $$p$$-convex subset of $$\mathbb{R}^ n$$, $$u_ 0,u_ 1 \in M$$ and $$M$$ is connected and noncontractible in itself, then there exist infinitely many geodesics on $$M$$ joining $$u_ 0$$ and $$u_ 1$$. Such a result was proved by A. Canino [J. Differ. Equations 75, No. 1, 118-157 (1988; Zbl 0661.34042)], provided that the loop space $$\Omega(M)$$ of $$M$$ has infinite category. This last condition was in turn satisfied, by a classical theorem of J.-P. Serre, when $$M$$ is compact and 1-connected, while an incomplete argument was given in the general case. The correct proof that $$\Omega(M)$$ has infinite category in the general case was then given by E. Fadell and S. Husseini [Nonlinear Anal., Theory Methods Appl. 17, No. 12, 1153-1161 (1991; Zbl 0756.55008)].
On the contrary, the proof of the authors is independent of the paper of E. Fadell and S. Husseini. In fact, it is well known that $$\Omega(M)$$ has not the homotopy type of a compact polyhedron, in the general case, by another classical result of J. P. Serre.
MSC:
 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable) 49J40 Variational inequalities