Spiked solutions for fractional Schrödinger systems with Sobolev critical exponent

Abstract

In this article, we study the following fractional critical Schrödinger system

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u_i=\mu _iu_i^3+\beta u_i\sum _{j\ne i}u_j^{2}+\lambda _iu_i &{}\text { in } \ \Omega ,\\ u_i=0 &{}\text { on } \ {\mathbb {R}}^N\setminus \Omega , \end{array}\right. } \quad i=1,2,\ldots ,m, \end{aligned}$$

where $0<s<1$, $\mu _i>0$, coupling constant $\beta$ satisfies either $-\infty <\beta \le {\bar{\beta }}$ (${\bar{\beta }}>0$ small) or $\beta \rightarrow -\infty$, $0<\lambda _i<\lambda _1^s(\Omega )$, where $\lambda _1^s(\Omega )$ is the first eigenvalue of $(-\Delta )^s$ on $\Omega$, with $\Omega$ is a smooth bounded domain in ${\mathbb {R}}^N$ with $N=4s$. Under some geometric assumptions on $\Omega$, we construct solutions which concentrate and blow up at different points as $\lambda _1,\ldots ,\lambda _m\rightarrow 0$.

Appendix

In this section, we collect several technical lemmas.

Lemma 7.1

[13] Given any $K\subseteq \Omega \subset {\mathbb {R}}^N$, there holds

$$\begin{aligned} Pw_{\delta ,\xi }:=w_{\delta ,\xi }-b_1\delta ^{s}H(x,\xi )+S_{\delta ,\xi }, \end{aligned}$$

(7.1)

with

$$\begin{aligned} \Vert S_{\delta ,\xi }\Vert _{\infty }=o(\delta ^s),\quad \left\| \frac{\partial S_{\delta ,\xi }}{\partial \delta }\right\| _{\infty }=o(\delta ^{s-1}),\quad \left\| \frac{\partial S_{\delta ,\xi }}{\partial \xi _j}\right\| _{\infty }=o(\delta ^s), \end{aligned}$$

as $\delta \rightarrow 0$, uniformly for $\xi \in K$, where $b_1$ is a positive constant. Moreover,

$$\begin{aligned}&Pw_{\delta ,\xi }:=w_{\delta ,\xi }-b_1\delta ^{s}H(x,\xi )+o(\delta ^s), \end{aligned}$$

(7.2)

$$\begin{aligned}&P\psi _{\delta ,\xi }^0:=\psi _{\delta ,\xi }^0-sb_1\delta ^{s}H(x,\xi )+o(\delta ^s), \end{aligned}$$

(7.3)

and

$$\begin{aligned} P\psi _{\delta ,\xi }^j:=\psi _{\delta ,\xi }^j-b_1\delta ^{s+1}\frac{\partial H(x,\xi )}{\partial \xi }+o(\delta ^{s+1}),\ j=1,\ldots ,N, \end{aligned}$$

(7.4)

as $\delta \rightarrow 0$, uniformly for $\xi \in K$.

Similar to the proof of [24, Lemma A.5] or [12, Lemma 4.2], we have the following result.

Lemma 7.2

It holds

$$\begin{aligned}&\langle P \Psi ^j_i, P \Psi ^h_l\rangle _{{\mathcal {C}}}=o\left( \frac{\delta _l}{\delta _i}\right) \quad \text { if } l>i, \\&\langle P\Psi ^j_i, P\Psi ^h_i\rangle _{{\mathcal {C}}}=o(1) \quad \text {if } j\ne h, \\&\langle P\Psi ^j_i, P\Psi ^j_i\rangle _{{\mathcal {C}}}=c_j(1+o(1)) \end{aligned}$$

for some positive constants $c_0$ and $c_1=\cdots =c_N$.

In the same way to [25, Lemma A.2, Lemma A.4, Lemma A.5], we have the following results.

Lemma 7.3

Take $K\subseteq \Omega \subset {\mathbb {R}}^N$. It holds

$$\begin{aligned} \int _{\Omega }w_{\delta ,\xi }^pdx= {\left\{ \begin{array}{ll} O(\delta ^{ps}), &{}\text { if } 0<p<2,\\ \alpha _{N,s}^2\omega _{N-1}\delta ^{2s}|\ln \delta |+O(\delta ^{2s}), &{}\text { if } p=2,\\ O(\delta ^{N-ps}), &{}\text { if } 2<p<4, \end{array}\right. } \end{aligned}$$

uniformly for $\xi \in K$, where $\omega _{0}=1$ and $\omega _{N-1}$ denotes the measure of the unit sphere $S^{N-1}\subset {\mathbb {R}}^N$.

Lemma 7.4

Given $p,q>0$ and $\eta >0$ small, we have

$$\begin{aligned} \int _{\Omega }w_{\delta _1,\xi _1}^pw_{\delta _2,\xi _2}^qdx \le O(\delta _2^{qs})\int _{\Omega }w_{\delta _1,\xi _1}^pdx+O(\delta _1^{ps})\int _{\Omega }w_{\delta _2,\xi _2}^qdx +O(\delta _1^{ps}\delta _2^{qs}), \end{aligned}$$

as $(\delta _1, \delta _2)\rightarrow (0,0)$, uniformly for all $\xi _1, \xi _2\in \Omega$ satisfying $|\xi _1-\xi _2|\ge 2\eta$, $dist(\xi _1, \partial \Omega )\ge 2\eta$, $dist(\xi _2, \partial \Omega )\ge 2\eta$.

Lemma 7.5

For $\eta >0$ small and all $\xi _1, \xi _2\in \Omega$ satisfying $|\xi _1-\xi _2|\ge 2\eta$, $dist(\xi _1, \partial \Omega )\ge 2\eta$, $dist(\xi _2, \partial \Omega )\ge 2\eta$. It holds:

$$\begin{aligned} \left\| (Pw_{\delta _2,\xi _2})^2(P\psi ^j_{\delta _1,\xi _1})\right\| _{\frac{4}{3}}=O(\delta _1^s\delta _2^s)\ \text { for } j=0,1,\ldots ,N, \end{aligned}$$

and

$$\begin{aligned} \left\| (Pw_{\delta _1,\xi _1})(Pw_{\delta _2,\xi _2})(P\psi ^j_{\delta _1,\xi _1})\right\| _{\frac{4}{3}}=O(\delta _1^s\delta _2^s)\ \text { for } j=0,1,\ldots ,N. \end{aligned}$$