## Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity.(English)Zbl 1155.35089

The work is dealing with a stationary nonlinear equation in the space of dimension $$N\geq 3$$, $(\hbar^2/2)\Delta v -(V({\mathbf r}) - E)v + f(v) = 0,$
where a function $$f(v)$$ is real, $$\Delta$$ is the $$N$$-dimensional Laplacian, and, in a typical case, $$f(v)=-v^3$$. This equation appears as the stationary version of the multidimensional Gross-Pitaevskii equation for Bose-Einstein condensates. It is assumed that potential $$V({\mathbf r})$$ is chosen in such a form that, in the Thomas-Fermi approximation, which corresponds to $$\hbar^2 \to 0$$, a solution reduces to a set of strongly localized noninteracting peaks, pinned to local minima of the potential. The objective of the work is to produce a rigorous proof of the fact that the solution keeps essentially the same form at small finite values of $$\hbar^2$$, by “gluing” togehter the strongly localized peaks. This is done by means of a purely variational method.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35J60 Nonlinear elliptic equations 35A15 Variational methods applied to PDEs
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