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We consider a singularly perturbed elliptic equation Berestycki-Lions [3] found almost necessary and sufficient conditions on the nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of the potential V under possibly general conditions on f. In this paper, we prove that under the optimal conditions of Berestycki-Lions on f 2 C1, there exists a solution concentrating around topologically stable positive critical points of V, whose critical values are characterized by minimax methods. [ABSTRACT FROM AUTHOR]

For singularly perturbed Schrödinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeroes of the potentials. The results generalize some recent work of Ambrosetti-Malchiodi-Ni [3] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in [3]. [ABSTRACT FROM AUTHOR]