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

Abstract

In this paper, we first construct multi-lump (nonlinear) bound states of the nonlinear Schrödinger equation

for sufficiently small ℏ>0, in which sense we call them “semiclassical bound states.” We assume that 1≦p<∞ forn=1,2 and 1≦p<1+4/(n−2) forn≧3, and thatV is in the class(V) _a in the sense of Kato for somea. For any finite collection {x ₁,...,x _N} of nondegenerate critical points ofV, we construct a solution of the forme ^−iEt/ℏv(x) forE<a, wherev is real and it is a small perturbation of a sum of one-lump solutions concentrated nearx ₁,...,x _N 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 2^N−1 distinct unstable bound states which may have nodes in general, and the above positive bound state is just one of them.