Asymptotic behavior of constraint minimizers for the mass-critical fractional nonlinear Schrödinger equation with a subcritical perturbation

Abstract

We study the asymptotic behavior of constraint minimizers for the energy functional related to the mass-critical fractional nonlinear Schrödinger equation with a subcritical perturbation

$$\begin{aligned} I(a) = \inf \left\{ E(\phi ) \ : \ \phi \in H^s({\mathbb {R}}^N), \Vert \phi \Vert ^2_{L^2} =a \right\} , \end{aligned}$$

where $N\ge 1$, $0<s<1$, $a>0$ and

$$\begin{aligned} E(\phi ) := \frac{1}{2} \Vert (-\Delta )^{s/2} \phi \Vert ^2_{L^2} {- \frac{N}{2(N+2s)} \Vert \phi \Vert ^{2+\frac{4s}{N}}_{L^{2+\frac{4s}{N}}}} -\frac{1}{\alpha +2} \Vert \phi \Vert ^{\alpha +2}_{L^{\alpha +2}} \end{aligned}$$

with $0<\alpha <\frac{4s}{N}$. We first show that minimizer for I(a) blows up as $a \nearrow a^*:= \Vert Q\Vert ^2_{L^2}$, where Q is the unique positive radial solution to the elliptic equation

$$\begin{aligned} (-\Delta )^s Q + Q - |Q|^{\frac{4s}{N}} Q=0. \end{aligned}$$

We then give a detailed description of the blow-up behavior of minimizers as $a \nearrow a^*$.

Appendices

Appendix A. An alternative proof of the existence of minimizers

In this “Appendix,” we give an alternative simple proof of the existence of minimizers for I(a) with $0<a<a^*$.

Let $0<a<a^*$ and $(\phi _n)_{n\ge 1}$ be a minimizing sequence for I(a). We have from (3.3) that $(\phi _n)_{n\ge 1}$ is a bounded sequence in $H^s({\mathbb {R}}^N)$. Moreover, by replacing $\phi _n$ by its Schwarz symmetric rearrangement, we may assume that $\phi _n$ is radially symmetric and radially decreasing. Since $H^s_{{{\,\mathrm{rd}\,}}}({\mathbb {R}}^N) \hookrightarrow L^q({\mathbb {R}}^N)$ is compact for any $2<q<2^*$, there exists $\phi \in H^s({\mathbb {R}}^N)$ such that up to a subsequence, $\phi _n \rightarrow \phi$ weakly in $H^s({\mathbb {R}}^N)$ and strongly in $L^q({\mathbb {R}}^N)$ for any $2<q<2^*$. We will show that $\phi$ is actually a minimizer for I(a). To see this, we first claim that $\phi \ne 0$. In fact, assume by contradiction that $\phi \equiv 0$, then the weak convergence in $H^s({\mathbb {R}}^N)$ and the strong convergence in $L^q({\mathbb {R}}^N)$ for any $2<q<2^*$ imply that

$$\begin{aligned} 0 \le \liminf _{n\rightarrow \infty } \frac{1}{2} \Vert (-\Delta )^{s/2} \phi _n\Vert ^2_{L^2} = \liminf _{n\rightarrow \infty } E(\phi _n) = I(a). \end{aligned}$$

This is a contradiction since for any $0<a<a^*$, $I(a)<0$. The later follows from taking $\phi \in H^s({\mathbb {R}}^N)$ with $\Vert \phi \Vert ^2_{L^2}=a$ and using the scaling $\phi _\lambda (x) := \lambda ^{\frac{N}{2}} \phi (\lambda x)$ to get

$$\begin{aligned} I(a) \le E(\phi _\lambda )&= \frac{1}{2} \lambda ^{2s}\Vert (-\Delta )^{s/2} \phi \Vert ^2_{L^2} - \frac{N}{2(N+2s)} \lambda ^{2s} \Vert \phi \Vert ^{2+\frac{4s}{N}}_{L^{2+\frac{4s}{N}}} \nonumber \\&\quad \,\, -\frac{1}{\alpha +2} \lambda ^{\frac{N\alpha }{2}} \Vert \phi \Vert ^{\alpha +2}_{L^{\alpha +2}} <0 \end{aligned}$$

(A.1)

for $\lambda >0$ small enough.

We also have that

$$\begin{aligned} E(\phi ) \le \liminf _{n\rightarrow \infty } E(\phi _n) =I(a) \end{aligned}$$

(A.2)

and

$$\begin{aligned} E(\phi _n) = E(\phi ) + \frac{1}{2} \Vert (-\Delta )^{s/2}(\phi _n-\phi )\Vert ^2_{L^2} + o_n(1). \end{aligned}$$

(A.3)

Note that

$$\begin{aligned} E(\lambda \phi )&= \frac{1}{2} \lambda ^2 \Vert (-\Delta )^{s/2} \phi \Vert ^2_{L^2} - \frac{N}{2(N+2s)} \lambda ^{2+\frac{4s}{N}} \Vert \phi \Vert ^{2+\frac{4s}{N}}_{L^{2+\frac{4s}{N}}} - \frac{1}{\alpha +2} \lambda ^{\alpha +2} \Vert \phi \Vert ^{\alpha +2}_{L^{\alpha +2}} \\&=\lambda ^2 E(\phi ) + \frac{N}{2(N+2s)} \lambda ^2 \left( 1-\lambda ^{\frac{4s}{N}} \right) \Vert \phi \Vert ^{2+\frac{4s}{N}}_{L^{2+\frac{4s}{N}}} + \frac{1}{\alpha +2} \lambda ^2 \left( 1-\lambda ^\alpha \right) \Vert \phi \Vert ^{\alpha +2}_{L^{\alpha +2}} \end{aligned}$$

which implies that

$$\begin{aligned} E(\phi ) = \frac{1}{\lambda ^2} E(\lambda \phi ) + \frac{N}{2(N+2s)} \left( \lambda ^{\frac{4s}{N}} -1 \right) \Vert \phi \Vert ^{2+\frac{4s}{N}}_{L^{2+\frac{4s}{N}}} + \frac{1}{\alpha +2} \left( \lambda ^\alpha -1 \right) \Vert \phi \Vert ^{\alpha +2}_{L^{\alpha +2}}. \end{aligned}$$

Now set

$$\begin{aligned} \lambda := \frac{\sqrt{a}}{\Vert \phi \Vert _{L^2}} \ge 1. \end{aligned}$$

It follows from (A.3) and $\phi \ne 0$ that

$$\begin{aligned} E(\phi _n)&= E(\phi ) + \frac{1}{2} \Vert (-\Delta )^{s/2} (\phi _n- \phi )\Vert _{L^2} + o_n(1) \\&= \frac{1}{\lambda ^2} E(\lambda \phi ) + \frac{N}{2(N+2s)} \left( \lambda ^{\frac{4s}{N}} -1 \right) \Vert \phi \Vert ^{2+\frac{4s}{N}}_{L^{2+\frac{4s}{N}}} \\&\quad \mathrel {}+ \frac{1}{\alpha +2} \left( \lambda ^\alpha -1 \right) \Vert \phi \Vert ^{\alpha +2}_{L^{\alpha +2}} + \frac{1}{2} \Vert (-\Delta )^{s/2} (\phi _n- \phi )\Vert _{L^2} + o_n(1) \\&\ge \frac{\Vert \phi \Vert ^2_{L^2}}{a} I(a) + o_n(1). \end{aligned}$$

Taking the limit $n\rightarrow \infty$, we get

$$\begin{aligned} I(a) \ge \frac{\Vert \phi \Vert ^2_{L^2}}{a} I(a) \end{aligned}$$

which implies that $\Vert \phi \Vert ^2_{L^2} \ge a$ or $\lambda \le 1$ since $I(a)<0$. Thus, $\lambda =1$ or $\Vert \phi \Vert ^2_{L^2} =a$, and by (A.2), we obtain $E(\phi ) = I(a)$. This shows that $\phi$ is a minimizer for I(a). The proof is complete.

Appendix B. Asymptotic behavior of minimizers with a subcritical Choquard perturbation

The argument presented in this paper can be easily extended to the mass-critical fractional Schrödinger equation with a subcritical Choquard perturbation. More precisely, we consider

$$\begin{aligned} I(a) := \inf \left\{ E(\phi ) \ : \ \phi \in H^s({\mathbb {R}}^N), \Vert \phi \Vert ^2_{L^2} =a \right\} , \end{aligned}$$

where $N\ge 1, a>0$ and

$$\begin{aligned} E(\phi ) =\frac{1}{2} \Vert (-\Delta )^{s/2} \phi \Vert ^2_{L^2} - \frac{N}{2(N+2s)} \Vert \phi \Vert ^{2+\frac{4s}{N}}_{L^{2+\frac{4s}{N}}} - \frac{1}{2p} \Vert (I_\alpha *|\phi |^p) |\phi |^p\Vert _{L^1}. \end{aligned}$$

Here $0<\alpha <N$, $1+\frac{\alpha }{N}<p<1+ \frac{2s+\alpha }{N}$, and $I_\alpha$ is the Riesz potential defined by

$$\begin{aligned} I_\alpha (x) := \frac{\Gamma \left( \frac{N-\alpha }{2}\right) }{\Gamma \left( \frac{\alpha }{2}\right) \pi ^{\frac{N}{2}} 2^\alpha } |x|^{-N+\alpha }, \quad \forall x \ne 0. \end{aligned}$$

The existence of minimizers for I(a) was studied by Feng–Zhang [11]. More precisely, they proved that for $N\ge 2, 0<s<1$, $0<\alpha <N$, $1+\frac{\alpha }{N}<p<1+\frac{2s+\alpha }{N}$ and $0<a<a^*$, there exists at least a minimizer for I(a). The proof of this result is again based on the profile decomposition of bounded sequences in $H^s({\mathbb {R}}^N)$ similar to [32]. By the same argument as above with a suitable modification, for instance, the non-local fractional Gagliardo–Nirenberg inequality and the non-local Brezis–Lieb lemma, we can prove the following result.

Theorem B.1

Let $N\ge 1$, $0<s<1$, $0<\alpha <N$ and $1+\frac{\alpha }{N}< p<1+\frac{2s+\alpha }{N}$. Then it holds that:

• For any $0<a<a^*$, there exists at least a minimizer for I(a);

• For any $a\ge a^*$, there is no minimizer for I(a);

• If $\phi _a$ is a nonnegative minimizer for I(a) with $0<a<a^*$, then $\phi _a$ blows up as $a \nearrow a^*$ in the sense that

  $$\begin{aligned} \lim _{a \nearrow a^*} \Vert (-\Delta )^{s/2} \phi _a\Vert _{L^2} =\infty . \end{aligned}$$

  Moreover,

  $$\begin{aligned} \beta _a^{\frac{N}{2}} \phi _a(\beta _a \cdot ) \rightarrow \beta ^{\frac{N}{2}} Q(\beta \cdot ) \text { strongly in } H^s({\mathbb {R}}^N) \end{aligned}$$

  as $a \nearrow a^*$, where

  $$\begin{aligned} \beta _a:= \left( 1-\left( \frac{a}{a^*}\right) ^{\frac{2s}{N}} \right) ^{\frac{2}{4s-N\alpha }}, \quad \beta := \left( \frac{\Vert (I_\alpha *|Q|^p)|Q|^p\Vert _{L^1}}{Np a^*} \right) ^{\frac{1}{N+2s+\alpha -Np}}. \end{aligned}$$