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In this paper, we first construct multi-lump (nonlinear) bound states of the nonlinear Schrödinger equation for sufficiently small >0, in which sense we call them semiclassical bound states. We assume that 1p forn=1,2 and 1pn–2) forn3, and thatV is in the class(V)a in the sense of Kato for somea. For any finite collection {x1,...,xN} of nondegenerate critical points ofV, we construct a solution of the forme–iEt/v(x) forEa, wherev is real and it is a small perturbation of a sum of one-lump solutions concentrated nearx1,...,xN respectively. The concentration gets stronger as 0. And we also prove these solutions are positive, and unstable with respect to perturbations of initial conditions for possibly smaller >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.

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