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Cingolani, M. Lazzo, Multiple positive solutions to nonlinear Schr¨odinger equations with competing potential functions, Journal of Diff. , 160, (2000), 118–138. [9] R. Courant, D. Hilbert. Methods of mathematical physics. Wiley, New York, 1989. [10] M. Degiovanni, A. Musesti, M. Squassina, On the regularity of solutions in the Pucci-Serrin identity, Calc. Var. PDE, in press. [11] M. del Pino, P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Cal. Var. PDE, 4, (1996), 121–137.

Let h ∈ C 2 (RN ) and let M ⊂ RN be a smooth compact nondegenerate manifold of critical points of h. Let U be a neighborhood of M and let l ∈ C 1 (RN ). Then, if h − l C 1 (U¯ ) is sufficiently small, the function l possesses at least l(M ) critical points in U . Let us suppose that Γ has a smooth manifold of critical points M . We say that M is nondegenerate (for Γ) if every x ∈ M is a nondegenerate critical point of Γ|M ⊥ . The Morse index of M is, by definition, the Morse index of any x ∈ M , as critical point of Γ|M ⊥ .

The concentration statement follows as in [3]. Clearly, this theorem shows that there is no essential difficulty in dealing with local maxima of Γ instead of minima. Moreover, when we deal with local minima (resp. maxima) of Γ, the preceding results can be improved because the number of positive solutions of (1) can be estimated by means of the category and M does not need to be a manifold. 9. Let (V-V1) and (J-J1) hold and suppose Γ has a compact set X where Γ achieves a strict local minimum (resp.

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