On interacting bumps of semi-classical states of nonlinear Schrödinger equations

Abstract

We study concentrated positive bound states of the following nonlinear Schrödinger equation:where p is subcritical. We prove that, at a local maximum point x 0 of the potential function V (x) and for arbitrary positive integer K(K > 1), there always exist solutions with K interacting bumps concentrating near x 0 . We also prove that at a nondegenerate local minimum point of V (x) such solutions do not exist.

An elementary construction of complex patterns in nonlinear Schr$ouml$dinger equations

“…A number of works have appeared in recent years dealing with the construction of multi-peaked solutions in a variety of situations, including relaxing non-degeneracy assumptions and more general nonlinearities (see, for instance, [2, 9, 10, 14 15, 16, 18, 25, 27,] and references therein). An interesting phenomenon was discovered by Kang and Wei [16]. They established the existence of positive solutions with any prescribed number of peaks clustering around each given local maximum point of the potential (in any space dimension).…”

Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations

“…Under the condition (5.2), we have uniqueness and nondegeneracy of the limiting equations. The proofs of both Theorem 5.1 and Theorem 5.2 follow from the same reduction procedure in [7] for single equations. We omit the details.…”

local uniqueness of the elliptic problem with critical exponent via Pohozaev identities

“…The use of the local Pohozaev identities not only simplifies the estimates, but it also makes it possible to study the properties of solutions with large numbers of bubbles. For more work concerning the uniqueness of solutions with the concentration phenomena, one can also refer to [7,16].…”