ON POSITIVE MULTILUMP BOUND-STATES OF NONLINEAR SCHRODINGER-EQUATIONS UNDER MULTIPLE WELL POTENTIAL

In this paper, we first construct multi-lump (nonlinear) bound states of the nonlinear Schrödinger equation[Figure not available: see fulltext.] for sufficiently small ℏ>0, in which sense we call them "semiclassical bound states." We assume that 1≦p<∞ for n=1,2 and 1≦p<1+4/(n-2) for n≧3, and that V is in the class (V)a in the sense of Kato for some a. For any finite collection {x1,..., xN} of nondegenerate critical points of V, we construct a solution of the form e-iEt/ℏv(x) for E<a, where v is real and it is a small perturbation of a sum of one-lump solutions concentrated near x1,..., xN respectively. The concentration gets stronger as ℏ→0. And we also prove these solutions are positive, and unstable with respect to perturbations of initial conditions for possibly smaller ℏ>0. Indeed, for each such collection of critical points we construct 2N-1 distinct unstable bound states which may have nodes in general, and the above positive bound state is just one of them. © 1990 Springer-Verlag.