Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity
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.
scientific article