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Abstract: We glue together standing wave solutions concentrating around critical points of the potential V with different energy scales. We devise a hybrid method using simultaneously a Lyapunov–Schmidt reduction method and a variational method to glue together standing waves concentrating on local minimum points which possibly have no corresponding limiting equations and those concentrating on general critical points which converge to solutions of corresponding limiting problems satisfying a non-degeneracy condition. [Copyright &y& Elsevier]

Abstract: We study semiclassical states of nonlinear Schrödinger equations with anisotropic type potentials which may exhibit a combination of vanishing and singularity while allowing decays and unboundedness at infinity. We give existence of spike type standing waves concentrating at the singularities of the potentials. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the qualitative property of the ground state solutions for the Hénon equation. By studying a limiting equation on the upper half space , we investigate the asymptotic energy and the asymptotic profile of the ground states for the Hénon equation. The limiting problem is related to a weighted Sobolev type inequality which we establish in this paper. [Copyright &y& Elsevier]