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Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential. (English) Zbl 1294.35131

Summary: We consider a singularly perturbed elliptic equation \[ \varepsilon^2\Delta u - V(x) u + f(u)=0,\quad u(x)>0\text{ on }\mathbb R^N,\quad \lim\limits_{| x|\to \infty}u(x) = 0, \] where \(V(x)>0\) for any \(x \in\mathbb R^N\). The singularly perturbed problem has corresponding limiting problems \[ \Delta U-cU+f(U)=0,\quad U(x)>0\text{ on }\mathbb R^N,\quad \lim\limits_{| x| \to \infty}U(x)=0,\quad c>0. \] H. Berestycki and P.-L. Lions [Arch. Ration. Mech. Anal. 82, 313–345 (1983; Zbl 0533.35029)] found almost necessary and sufficient conditions on 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 potential \(V\) under possibly general conditions on \(f\). In this paper, we prove that under the optimal conditions of Berestycki-Lions on \(f \in C^1\), there exists a solution concentrating around topologically stable positive critical points of \(V\), whose critical values are characterized by minimax methods.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs

Citations:

Zbl 0533.35029
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