Existence and concentration of bound states of nonlinear Schrödinger equations with compactly supported and competing potentials

Abstract

We study the existence and concentration of solutions to the N-dimensional nonlinear Schrödinger equation −ε 2 u ε + V (x)u ε = K (x)|u ε | p−1 u ε + Q(x)|u ε | q−1 u ε with u ε (x) > 0 and u ε ∈ H 1 ‫ޒ(‬ N), where N ≥ 3, 1 < q < p < (N+2)/(N−2), and ε > 0 is sufficiently small. We take potential functions V (x) ∈ C ∞ 0 ‫ޒ(‬ N) with V (x) ≡ 0 and V (x) ≥ 0, and show that if K (x) and Q(x) are permitted to be unbounded under some necessary restrictions, then a positive solution u ε (x) exists in H 1 ‫ޒ(‬ N) when the corresponding ground energy function G(x) has local minimum points. We establish the concentration property of u ε (x) as ε tends to zero. We have removed from some previous papers the crucial restriction that the nonnegative potential function V (x) has a positive lower bound or decays at infinity like (1 + |x|) −α with 0 < α ≤ 2.

Existence and concentration of bound states for a Kirchhoff type problem with potentials vanishing or unbounded at infinity

“…Yin and Zhang considered the case that V ( x ) decays to zero faster than | x | −2 and K ( x )≥0 permitted to be unbounded. Their method and results were later generalized by Fei and Yin to the compacting potentials case. In addition, by Lyapunov‐Schmidt reduction, Cao and Peng showed that has a positive bound state with k ‐peaks under some decay or growth conditions on V ( x ) or K ( x ).…”

Multi-peak bound states for Schrödinger equations with compactly supported or unbounded potentials

“…Precisely, the authors proved a mountain pass solution to a modified problem and then obtained a type of the weak Harnack inequality, by which they proved that the solution decays at infinity at the desired speed and hence is a bound state solving the original problem. However, the solution obtained in [22] is in some sense a least energy solution and indeed one-peaked. Moreover, because of the absence of an exact estimate to the energy related to the solution, their argument cannot be adopted to look for multi-peak solutions with higher energy.…”

“…In [16], the case V (x) ∼ |x| α at infinity with α −4 was considered by a constructive argument and multi-peak bound states with prescribed number of maximum points approaching to a local minimum point of the function G(x) as ε → 0 were obtained. The case that V (x) has compact support, which is the most difficult case, was studied by Fei and Yin in [22], where it was shown that problem (1.1) admits a bound state with exact one local maximum point x ε which tends to a minimum point of the function G(x) as ε → 0. The method introduced in [22] is quite new.…”

“…The case that V (x) has compact support, which is the most difficult case, was studied by Fei and Yin in [22], where it was shown that problem (1.1) admits a bound state with exact one local maximum point x ε which tends to a minimum point of the function G(x) as ε → 0. The method introduced in [22] is quite new. Precisely, the authors proved a mountain pass solution to a modified problem and then obtained a type of the weak Harnack inequality, by which they proved that the solution decays at infinity at the desired speed and hence is a bound state solving the original problem.…”

Stationary solutions of the nonlinear Schrödinger equation with fast-decay potentials concentrating around local maxima

“…It was used by D. Bonheure together with the authors to study solutions concentrating around spheres [8]. Another penalized problem was defined by Yin Huicheng and Zhang Pingzheng [28] (see also Fei Mingwen and Yin Huicheng [15] and Ba Na, Deng Yinbin and Peng Shuangjie [6]). …”