Nonrelativistic limit of normalized solutions to a class of nonlinear Dirac equations

Abstract

In this paper, we investigate the nonrelativistic limit of normalized solutions to a nonlinear Dirac equation as given below:

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}-i c\sum \limits _{k=1}^3\alpha _k\partial _k u +mc^2 \beta {u}- \Gamma * (K |{u}|^\kappa ) K|{u}|^{\kappa -2}{u}- P |{u}|^{s-2}{u}=\omega {u}, \\ &{}\displaystyle \int _{\mathbb {R}^3}\vert u \vert ^2 dx =1. \end{array}\right. } \end{aligned}$$

Here, $c>0$ represents the speed of light, $m > 0$ is the mass of the Dirac particle, $\omega \in \mathbb {R}$ emerges as an indeterminate Lagrange multiplier, $\Gamma$, K, P are real-valued function defined on $\mathbb {R}^3$, also known as potential functions. Our research first confirms the presence of normalized solutions to the Dirac equation under high-speed light conditions. We then illustrate that these solutions converge to normalized ground states of nonlinear Schrödinger equations, and we also show uniform boundedness and exponential decay of these solutions. Our results form the first discussion on nonrelativistic limit of normalized solutions to nonlinear Dirac equations. This not only aids in the study of normalized solutions of the nonlinear Schrödinger equations, but also physically explains that the normalized ground states of high-speed particles and low-speed motion particles are consistent.

Appendix

Lemma 2.7 Let $P\in L^{\infty }(\mathbb {R}^3)$, then for any ${u} \in H^{1/2}(\mathbb {R}^3,\mathbb {C}^4)$, with $\Vert {u}\Vert _{L^2}=1$, ${u}=t w+u^-$, where $w=\frac{{u}_+}{\Vert {u}_+\Vert _{L^2}}$, $t=\sqrt{1-\Vert u^-\Vert _{L^2}^2}$, the following inequality holds

$$\begin{aligned} \int _{\mathbb {R}^3 }P |{u}|^s dx\ge C \int _{\mathbb {R}^3 }P |w|^sdx-C \Vert u^- \Vert _{L^2}^2\Vert w\Vert _{H^{1/2}}^2- C \Vert u^- \Vert _{H^{1/2}}^2. \end{aligned}$$

For $w=\textbf{U}_{\textrm{FW}}^{-1}\left( \begin{array}{l}v \\ 0\end{array}\right)$ with $v \in H^1\left( \mathbb {R}^3, \mathbb {C}^2\right)$ and $\Vert v\Vert _{L^2}^2=1$, we have

$$\begin{aligned} \int _{\mathbb {R}^3 } P |{u}|^sdx\ge C \int _{\mathbb {R}^3 } P |v|^sdx- C \Vert \nabla v\Vert _{L^2}^s- C \Vert u^- \Vert _{L^2}^2\Vert w\Vert _{H^{1/2}}^2- C \Vert u^- \Vert _{H^{1/2}}^2. \end{aligned}$$

Proof

Notice for $u=tw+u^-$, one has

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^3}P|u|^sdx\ge&2^{1-s}t^s\int _{\mathbb {R}^3}P|\omega |^s dx- \int _{\mathbb {R}^3}P|u^-|^sdx\\ \ge&C \int _{\mathbb {R}^3}P|\omega |^s dx- C \Vert u^-\Vert _{L^2}^2\int _{\mathbb {R}^3}P|\omega |^sdx- \int _{\mathbb {R}^3}P|u^-|^sdx\\ \ge&C \int _{\mathbb {R}^3 }P |w|^s dx- C \Vert u^- \Vert _{L^2}^2\Vert w\Vert _{H^{1/2}}^2- C \Vert u^- \Vert _{H^{1/2}}^2. \end{aligned} \end{aligned}$$

Now for $w=\textbf{U}_{\textrm{FW}}^{-1}\left( \begin{array}{l}v \\ 0\end{array}\right)$ with $v \in H^1\left( \mathbb {R}^3, \mathbb {C}^2\right)$ and $\Vert v\Vert _{L^2}^2=1$, we have

$$\begin{aligned} w(x)=\mathcal {F}^{-1}\textbf{U}^{-1}\left( \begin{array}{l}{\hat{v}} \\ 0\end{array}\right) =\left( \begin{array}{l}\mathcal {F}^{-1}[\Upsilon _-(\xi )\frac{\varvec{\sigma }\cdot \xi }{|\xi |}{\hat{v}}] \\ \mathcal {F}^{-1}[\Upsilon _+(\xi ){\hat{v}}]\end{array}\right) (x). \end{aligned}$$

Set $g(x)= \mathcal {F}^{-1}[\Upsilon _+(\xi ){\hat{v}}](x)$, $f(x)=\mathcal {F}^{-1}[1-\Upsilon _+(\xi ){\hat{v}}](x)$, then $g=v-f$ and

$$\begin{aligned} \int _{\mathbb {R}^3}P|\omega |^s dx\ge \int _{\mathbb {R}^3}P|g|^sdx \ge 2^{1-s}\int _{\mathbb {R}^3}P|v|^sdx - \int _{\mathbb {R}^3}P|f|^sdx. \end{aligned}$$

Since

$$\begin{aligned} | 1-\Upsilon _+(\xi )|=\frac{1-\Upsilon _+(\xi )^2}{1+\Upsilon _+(\xi )}=\frac{\Upsilon _-(\xi )^2}{1+\Upsilon _+(\xi )}\le \Upsilon _-(\xi )^2, \end{aligned}$$

then we obtain

$$\begin{aligned} \begin{aligned}{} & {} \Vert f\Vert ^2=\Vert (\lambda (\xi ))^{1/2}| 1-\Upsilon _+(\xi ){\hat{v}}|\Vert _{L^2}^2\le \Vert (\lambda (\xi ))^{1/2}| \Upsilon _-(\xi ){\hat{v}}|\Vert _{L^2}^2= \frac{1}{2}(\Vert v\Vert ^2-mc^2\Vert v\Vert _{L^2}^2)\\{} & {} \quad \le \frac{1}{4m}\Vert \nabla v\Vert _{L^2}^2,\end{aligned} \end{aligned}$$

which implies

$$\begin{aligned} \int _{\mathbb {R}^3 } P |{u}|^sdx\ge C \int _{\mathbb {R}^3 } P |v|^sdx- C \Vert \nabla v\Vert _{L^2}^s- C \Vert u^- \Vert _{L^2}^2\Vert w\Vert _{H^{1/2}}^2- C \Vert u^- \Vert _{H^{1/2}}^2. \end{aligned}$$

This ends the proof. $\square$

Lemma 2.8 Let $\Gamma$ satisfy $(\Gamma _1),$ then $\mathscr {F}: H^{1/2}\rightarrow L^p$ is locally Lipschitz, where $p\in [2,3]$.

Proof

It is sufficient to show the case when $p=2$ and 3. When $p=3$, for $u _1, u _2\in H^{1/2}$, one has

$$\begin{aligned} \mathscr {F}( u _1)-\mathscr {F}( u _2)=\frac{1}{2}\Gamma *\left( K \left( | u _1|^\kappa -| u _2|^\kappa \right) \right) \left( K | u _1|^{\kappa -1}+K| u _2|^{\kappa -1}\right)&\\+\frac{1}{2}\Gamma *\left( K \left( | u _1|^\kappa +| u _2|^\kappa \right) \right) \left( K| u _1|^{\kappa -1}-K| u _2|^{\kappa -1}\right) .&\end{aligned}$$

Hence for $t_i$, $s_i>0$($i=1,2$) satisfying $\frac{1}{3}=\frac{1}{t_i}+\frac{1}{s_i},$ we have

$$\begin{aligned} \begin{aligned} \Vert \mathscr {F}( u _1)-\mathscr {F}( u _2)\Vert _{L^3}\lesssim \Vert \Gamma *\left( | u _1|^\kappa -| u _2|^\kappa \right) \Vert _{L^{t_{1}}}\Vert | u _1|^{\kappa -1} +| u _2|^{\kappa -1}\Vert _{L^{s_{1}}}&\\ +\Vert \Gamma *\left( | u _1|^\kappa +| u _2|^\kappa \right) \Vert _{L^{t_{2}}}\Vert | u _1|^{\kappa -1}-| u _2|^{\kappa -1}\Vert _{L^{s_{2}} }.&\end{aligned} \end{aligned}$$

By Young inequality, for $t_i$, $r_i$ satisfying $\frac{1}{t_i}+1= \frac{14-6\kappa }{6}+ \frac{1}{r_i}$, $i=1,2$, one has

$$\begin{aligned}{} & {} \Vert \Gamma *\left( | u _1|^\kappa -| u _2|^\kappa \right) \Vert _{L^{t_{1}}}\lesssim \Vert | u _1|^\kappa -| u _2|^\kappa \Vert _{L^{r_{1}}}, \\{} & {} \Vert \Gamma *\left( | u _1|^\kappa +| u _2|^\kappa \right) \Vert _{L^{t_{2}}}\lesssim \left\| | u _1|^\kappa +| u _2|^\kappa \right\| _{L^{r_{2}}}. \end{aligned}$$

Notice

$$\begin{aligned} || u _1|^\kappa -| u _2|^\kappa |\lesssim | u _1- u _2|\cdot \left| | u _1|^{\kappa -1}+| u _2|^{\kappa -1}\right| , \end{aligned}$$

then for $r_1$, m, n satisfying $\frac{1}{r_1}=\frac{1}{m}+\frac{1}{n}$, one has

$$\begin{aligned}{} & {} \Vert \Gamma *\left( | u _1|^\kappa -| u _2|^\kappa \right) \Vert _{L^{t_{1}}}\Vert | u _1|^{\kappa -1}+| u _2|^{\kappa -1}\Vert _{L^{s_{1}}}\lesssim \Vert u _1- u _2\Vert _{L^m}\Vert | u _1|^{\kappa -1}\\{} & {} \quad +| u _2|^{\kappa -1}\Vert _{L^{s_{1}}}\Vert | u _1|^{\kappa -1}+| u _2|^{\kappa -1}\Vert _{L^n} \end{aligned}$$

where $\frac{1}{s_1}+\frac{1}{m}+\frac{1}{n}=\kappa -1$. Let $s_1=n$, $m=\max \left\{ 2,\frac{3}{\kappa -1}\right\}$, then $s_1=n\in \left[ \frac{6}{3\kappa -4},\frac{3}{\kappa -1}\right]$. Hence

$$\begin{aligned} \Vert \Gamma *\left( | u _1|^\kappa -| u _2|^\kappa \right) \Vert _{L^{t_{1}}}\Vert | u _1|^{\kappa -1}+| u _2|^{\kappa -1}\Vert _{L^{s_{1}}}\lesssim \left( \Vert u _1\Vert +\Vert u _2\Vert \right) ^{2\kappa -2}\Vert u _1- u _2\Vert . \end{aligned}$$

Similarly, for $s_2$, m, n satisfying $\frac{1}{s_2}=\frac{1}{m}+\frac{1}{n}$, one has

$$\begin{aligned}{} & {} \Vert \Gamma *\left( | u _1|^\kappa +| u _2|^\kappa \right) \Vert _{L^{t_{2}}}\Vert | u _1|^{\kappa -1}-| u _2|^{\kappa -1}\Vert _{L^{s_{2}} }\lesssim \Vert u _1- u _2\Vert _{L^m}\Vert | u _1|^\kappa \\{} & {} \quad +| u _2|^\kappa \Vert _{L^{r_{2}}}\Vert | u _1|^{\kappa -2}+| u _2|^{\kappa -2}\Vert _{L^n} \end{aligned}$$

where $\frac{1}{r_2}+\frac{1}{m}+\frac{1}{n}=\kappa -1$. Take $m=3$, $n =\frac{3}{\kappa -2}$, $r_2=\frac{3}{2\kappa -2}$, we get

$$\begin{aligned} \Vert \Gamma *\left( | u _1|^\kappa +| u _2|^\kappa \right) \Vert _{L^{t_{2}}}\Vert | u _1|^{\kappa -1}-| u _2|^{\kappa -1}\Vert _{L^{s_{2}} }\lesssim \left( \Vert u _1\Vert +\Vert u _2\Vert \right) ^{2\kappa -2}\Vert u _1- u _2\Vert . \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert \mathscr {F}( u _1)-\mathscr {F}( u _2)\Vert _{L^3}\lesssim \left( \Vert u _1\Vert +\Vert u _2\Vert \right) ^{2\kappa -2}\Vert u _1- u _2\Vert . \end{aligned}$$

When $p=2$, we can also get

$$\begin{aligned} \Vert \mathscr {F}( u _1)-\mathscr {F}( u _2)\Vert _{L^2}\lesssim \left( \Vert u _1\Vert +\Vert u _2\Vert \right) ^{2\kappa -2}\Vert u _1- u _2\Vert . \end{aligned}$$

Hence by interpolation inequality, for $p\in [2,3]$,

$$\begin{aligned} \Vert \mathscr {F}( u _1)-\mathscr {F}( u _2)\Vert _{L^p}\lesssim \left( \Vert u _1\Vert +\Vert u _2\Vert \right) ^{2\kappa -2}\Vert u _1- u _2\Vert . \end{aligned}$$

This ends the proof. $\square$

Future remark. Although the method of nonrelativistic limit has effectively addressed some of the difficulties in the normalized solution problem of the nonlinear Dirac equations, there still exist some urgent public issues that need to be resolved, such as: (1) The case where the potential function does not have a compactness assumption, (2) The super-mass-critical case of the nonlinear term, i.e., $s\in (8/3,3)$, (3) How fast the nonrelativistic limit process is.