Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency

Abstract

We study the nonlinear Schrödinger equations:where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has "critical frequency" inf x∈R N V(x) = 0. We also assume thatIn critical frequency cases, Byeon-Wang [5,6] showed the existence of singlepeak solutions which concentrating around global minimum of V(x). Their limiting profiles-which depend on the local behavior of the potential V(x)-are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u (x) as follows: rescaled function w (y) = g( ) 2 p−1 u (g( )y + x ) converges to a least energy solution of − w + V 0 (y)w = w p , w > 0 in 0 , w ∈ H 1 0 ( 0 ). Here g( ), V 0 (x) and 0 depend on the local behaviors of V(x). IntroductionIn this paper, we consider the following nonlinear Schrödinger equations:

Multi-bump standing waves with critical frequency for nonlinear Schrödinger equations

“…If the decaying behavior of the potential V to 0 around global minimum points is regular and the corresponding limiting problems have good properties, a gluing of solutions concentrating around global minimum points has been worked out in [7,10,12,11]. Recently, without requiring the existence of limiting problems, Sato [36] were able to glue localized solutions concentrating around local minimum points when f (u) = u p , p ∈ (1, 2 * ), where 2 * = (n + 2)/(n − 2) for n 3 and 2 * = ∞ for n = 1, 2. He glues the solution via a minimization on a torus of finite codimension in a Sobolev space which depends strongly on the homogeneity of an f (u) = u p .…”

Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations with potentials vanishing at infinity

“…The works [13,14] require a non-degeneracy condition of solutions of related limiting problems. Without any non-degeneracy condition for limiting problems, Sato [34] showed the existence of a solution concentrating around any set of finitely many isolated local minimum points of V . On the other hand, Ambrosetti et al .…”

“…For our construction of multi-bump solutions concentrating around some positive critical points of V and some zero sets of potential, we do not require that V has only one type of decay rate around the zeros, which can be compared with the works of [13,14]; in [13] multibump solutions were constructed when V has a polynomial decay and in [14] multibump solutions were constructed when V has an exponential decay rate around the zero sets of potential V . Through a non-degeneracy condition for limiting problems, we combine small amplitude solutions concentrating around some zeros of V and big amplitude solutions concentrating around general positive critical points, not just local minimum points; this can be compared with [34], in which the author combined solutions concentrating only local minimum points of V .…”

Formation of Radial Patterns via Mixed Attractive and Repulsive Interactions for Schrödinger Systems

“…,Ω . We consider a minimization problem With the aid of [29,5], we argue as in [7] to have the following proposition.…”