On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential (Q1813317)

The author considers nonlinear Schrödinger equations with an additional linear potential \(V\) of class \((V)_ a\) in the sense of Kato and an attractive power law nonlinearity. He proves the existence of special solutions of type \(\exp(-iEt/h)\cdot v(x)\) (called multi-lump bound states) for each finite collection of nondegenerate critical points of \(V\); here \(v(x)\) is a real small perturbation of a sum of one-lump solutions concentrated near one critical point resp. of the potential \(V\). This generalizes results of \textit{A. Floer} and \textit{A. Weinstein} on one- lump solutions for bounded potentials [J. Funct. Anal. 69, 397-408 (1986; Zbl 0613.35076)]. The crucial technical point is a new estimate of the norm of a certain Fredholm inverse which is not needed in the one-lump case.