A note on nonsmooth functionals with infinitely many critical values.

*(English)*Zbl 0798.58013The 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.

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.

Reviewer: Marco Degiovanni (Brescia)