Gluing higher-topological-type semiclassical states for nonlinear Schrödinger equations

Abstract

In Chen and Wang (Calc Var Part Differ Equ 56:1–26, 2017), we show that, if $\epsilon >0$ is small enough, then there exists a sequence of semiclassical states of higher topological type localized at a local minimum set of the potential V for the semiclassical nonlinear Schrödinger equation

$$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1\left( {\mathbb{R}}^N\right) . \end{aligned}$$

In this paper, we consider a situation where V has multiple isolated local minimum sets. We show that as $\epsilon \rightarrow 0,$ there exist multi-bump solutions of this equation being concentrated at those given local minimum sets while at the same time each bump behaves as a higher-topological-type solution in one local minimum set as aforementioned. Thus, the multi-bump solutions given here are constructed by gluing a sum of higher-topological-type solutions localized in separated local minimal sets of the potential.

Appendix

In this appendix, we give the proof of (4.54).

Proof of (4.54)

Proof

Since $\Gamma '_{\epsilon _m}(u_m)=0$, by (2.14), $u_m\in H^1({\mathbb{R}}^N)$ satisfies

$$\begin{aligned} -\Delta u_m+V(\epsilon _m x)u_m+2g'((\int _{{\mathbb{R}}^N}\chi _{\epsilon _m} u^2_m\mathrm{d}x-1)_+)\chi _{\epsilon _m} u_m=|u_m|^{p-2}u_m\quad \text{ in }\ {\mathbb{R}}^N. \end{aligned}$$

(E.1)

Since $\Gamma _{\epsilon _m}(u_m)\le L$ and $\Gamma '_{\epsilon _m}(u_m)=0$, by Lemma 2.2, we get that there exists $\varrho >0$ independent of m such that $\Vert u_m\Vert \le \varrho$. Then, by (E.1) and the standard regularity theory of elliptic equation, we get that there exists a constant $C>0$ independent of m such that

$$\begin{aligned} \Vert u_m\Vert _{L^\infty ({\mathbb{R}}^N)}\le C. \end{aligned}$$

(E.2)

Let $\xi$ be a smooth function satisfying that $0\le \xi \le 1$ in ${\mathbb{R}}^N$, $\xi =1$ in ${\mathbb{R}}^N{\setminus} \cup ^n_{j=1}({\mathcal{M}}_j)^{5\sigma /4}$, $\xi =0$ in $\cup ^n_{j=1} ({\mathcal{M}}_j)^{\sigma }$ and $|\nabla \xi |\le 8/\sigma$ in ${\mathbb{R}}^N$. Let

$$\xi _{m}(x)=\xi (\epsilon _m x),\ x\in {\mathbb{R}}^N.$$

Multiplying both sides of equation (E.1) by $\xi ^2_m u_m$ and integrating in ${\mathbb{R}}^N$, we get that

$$\begin{aligned}&\int _{{\mathbb{R}}^N}\nabla u_m\nabla \left( \xi ^2_m u_m\right) +\int _{{\mathbb{R}}^N}V(\epsilon _m x)\xi ^2_mu^2_m +2g'\left( \left( \int _{{\mathbb{R}}^N}\chi _{\epsilon _m} u^2_m\mathrm{d}x-1\right) _+\right) \int _{{\mathbb{R}}^N}\chi _{\epsilon _m} \xi ^2_mu^2_m\nonumber \\&\quad =\int _{{\mathbb{R}}^N}\xi ^2_m|u_m|^p. \end{aligned}$$

(E.3)

Combining (E.2) and (E.3) leads to

$$\begin{aligned} \int _{{\mathbb{R}}^N}\nabla u_m\nabla (\xi ^2_m u_m)\le \int _{{\mathbb{R}}^N}\xi ^2_m|u_m|^p\le C^{p-2}\int _{{\mathbb{R}}^N}\xi ^2_mu_m^2\le C^{p-2}\int _{{\mathbb{R}}^N\setminus \cup ^n_{j=1} (({\mathcal{M}}_j)^{\sigma })_{\epsilon _m}}u_m^2. \end{aligned}$$

(E.4)

Since

$$\begin{aligned}&\int _{{\mathbb{R}}^N}\nabla u_m\nabla (\xi ^2_m u_m)\\&\quad = \int _{{\mathbb{R}}^N}\xi ^2_m|\nabla u_m|^2+2\int _{{\mathbb{R}}^N} \xi _m u_m\nabla \xi _m\nabla u_m\\&\quad \ge \int _{{\mathbb{R}}^N}\xi ^2_m|\nabla u_m|^2-2(\int _{{\mathbb{R}}^N}u^2_m|\nabla \xi _m|^2)^{1/2}(\int _{{\mathbb{R}}^N}\xi ^2_m|\nabla u_m|^2)^{1/2}\\&\quad \ge \frac{1}{2}\int _{{\mathbb{R}}^N}\xi ^2_m|\nabla u_m|^2-2\int _{{\mathbb{R}}^N}u^2_m|\nabla \xi _m|^2\nonumber \\&\quad \ge \frac{1}{2}\int _{{\mathbb{R}}^N\setminus \cup ^n_{j=1} (({\mathcal{M}}_j)^{5\sigma /4})_{\epsilon _m}}|\nabla u_m|^2-\frac{16}{\sigma ^2}\epsilon ^2_m\int _{{\mathbb{R}}^N\setminus \cup ^n_{j=1} (({\mathcal{M}}_j)^{\sigma })_{\epsilon _m}}u^2_m \end{aligned}$$

by (4.53) and (E.4), we get (4.54). $\square$