Uniqueness and symmetry of ground states for higher-order equations

Abstract

We establish uniqueness and radial symmetry of ground states for higher-order nonlinear Schrödinger and Hartree equations whose higher-order differentials have small coefficients. As an application, we obtain error estimates for higher-order approximations to the pseudo-relativistic ground state. Our proof adapts the strategy of Lenzmann (Anal PDE 2:1–27, 2009) using local uniqueness near the limit of ground states in a variational problem. However, in order to bypass difficulties from lack of symmetrization tools for higher-order differential operators, we employ the contraction mapping argument in our earlier work (Choi et al. 2017. arXiv:1705.09068) to construct radially symmetric real-valued solutions, as well as improving local uniqueness near the limit.

Appendices

Appendix A: Nonlinear estimates

We show the nonlinear estimates which are used in the contraction mapping argument.

Lemma A.1

(Nonlinear estimates) Let $u\in H^1$ be real-valued. For any $\eta >0$, there exists $\delta _0>0$, depending on $\Vert u\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}$ and $\eta$, such that if $0<\delta \le \delta _0$ and

$$\begin{aligned} \Vert r\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}, \Vert \tilde{r}\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}\le \delta , \end{aligned}$$

then

$$\begin{aligned} \left\| {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right\| _{L^2({\mathbb {R}}^d;{\mathbb {R}})}\le \eta \Vert r\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})} \end{aligned}$$

and

$$\begin{aligned} \left\| \left( {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right) -\left( {\mathcal {N}}'(u+\tilde{r})-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(\tilde{r})\right) \right\| _{L^2({\mathbb {R}}^d;{\mathbb {R}})}\le \eta \Vert r-\tilde{r}\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}. \end{aligned}$$

The above lemma follows from the multilinear estimates.

Lemma A.2

(Multilinear estimates) We have

$$\begin{aligned} \left\| \left( \frac{1}{|x|}*(\phi _1\phi _2)\right) \phi _3\right\| _{L^2({\mathbb {R}}^3;{\mathbb {R}})}\lesssim \prod _{j=1}^3\Vert \phi _j\Vert _{H^1({\mathbb {R}}^3;{\mathbb {R}})}. \end{aligned}$$

Moreover, if $d=1,2$ and $k\in {\mathbb {N}}$ or if $d=3$ and $k=1$, then

$$\begin{aligned} \left\| \prod _{j=1}^{2k+1}\phi _j\right\| _{L^2({\mathbb {R}}^d;{\mathbb {R}})}\lesssim \prod _{j=1}^{2k+1}\Vert \phi _j\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}. \end{aligned}$$

Proof

By the Hölder, Young’s and Sobolev inequalities, we prove that

$$\begin{aligned} \left\| \left( \frac{1}{|x|}*(\phi _1\phi _2)\right) \phi _3\right\| _{L^2({\mathbb {R}}^3;{\mathbb {R}})}&\le \left\| \left( \frac{1}{|x|}*(\phi _1\phi _2)\right) \right\| _{L^9({\mathbb {R}}^3;{\mathbb {R}})}\Vert \phi _3\Vert _{L^{18/7}({\mathbb {R}}^3;{\mathbb {R}})}\\&\lesssim \Vert \phi _1\phi _2\Vert _{L^{9/7}({\mathbb {R}}^3;{\mathbb {R}})}\Vert \phi _3\Vert _{L^{18/7}({\mathbb {R}}^3;{\mathbb {R}})}\\&\lesssim \prod _{j=1}^3\Vert \phi _j\Vert _{L^{18/7}({\mathbb {R}}^3;{\mathbb {R}})}\lesssim \prod _{j=1}^3\Vert \phi _j\Vert _{H^1({\mathbb {R}}^3;{\mathbb {R}})} \end{aligned}$$

and similarly,

$$\begin{aligned} \left\| \prod _{j=1}^{2k+1}\phi _j\right\| _{L^2({\mathbb {R}}^d;{\mathbb {R}})}\le \prod _{j=1}^{2k+1}\Vert \phi _j\Vert _{L^{2(2k+1)}({\mathbb {R}}^d;{\mathbb {R}})}\lesssim \prod _{j=1}^{2k+1}\Vert \phi _j\Vert _{H^1({\mathbb {R}}^d;{\mathbb {R}})}. \end{aligned}$$

Proof of Lemma A.1

Suppose that $\Vert r\Vert _{H^1},\Vert \tilde{r}\Vert _{H^1}\le \Vert u\Vert _{H^1}$. For the Hartree nonlinearity, by algebra, we write

$$\begin{aligned} {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)=\left( \frac{1}{|x|}*r^2\right) u+2\left( \frac{1}{|x|}*(ur)\right) r+\left( \frac{1}{|x|}*r^2\right) r \end{aligned}$$

and

$$\begin{aligned}&\left( {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right) -\left( {\mathcal {N}}'(u+\tilde{r})-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(\tilde{r})\right) \\&\quad =\left( \frac{1}{|x|}*\left( (r+\tilde{r})(r-\tilde{r})\right) \right) u+2\left( \frac{1}{|x|}*\left( u(r-\tilde{r})\right) \right) r+2\left( \frac{1}{|x|}*(u\tilde{r})\right) (r-\tilde{r})\\&\quad +\left( \frac{1}{|x|}*\left( (r+\tilde{r})(r-\tilde{r})\right) \right) r+\left( \frac{1}{|x|}*\tilde{r}^2\right) (r-\tilde{r}) \end{aligned}$$

Thus, by the multilinear estimate (Lemma A.2),

$$\begin{aligned} \left\| {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right\| _{L^2}\le C\Big (\Vert u\Vert _{H^1}+\Vert r\Vert _{H^1}\Big )\Vert r\Vert _{H^1}^2\le 2C\delta \Vert u\Vert _{H^1}\Vert r\Vert _{H^1} \end{aligned}$$

and

$$\begin{aligned}&\left\| \left( {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right) -\left( {\mathcal {N}}'(u+\tilde{r})-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(\tilde{r})\right) \right\| _{L^2}\\&\quad \le C\Big (\Vert u\Vert _{H^1}+\Vert r\Vert _{H^1}+\Vert \tilde{r}\Vert _{H^1}\Big )\Big (\Vert r\Vert _{H^1}+\Vert \tilde{r}\Vert _{H^1}\Big )\Vert r-\tilde{r}\Vert _{H^1}\\&\quad \le 6C\delta \Vert u\Vert _{H^1}\Vert r-\tilde{r}\Vert _{H^1}. \end{aligned}$$

Then, taking $\delta _0=\eta \min \{\frac{1}{6C\Vert u\Vert _{H^1}},\Vert u\Vert _{H^1}\}$, we prove the lemma for the Hartree nonlinearity.

Similarly, for the polynomial nonlinearity, by the multilinear estimate (Lemma A.2),

$$\begin{aligned} \left\| {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right\| _{L^2}&=\left\| \sum _{j=2}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} u^{2k+1-j} r^j\right\| _{L^2}\\&\le \sum _{j=2}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} \left\| u^{2k+1-j} r^j\right\| _{L^2}\\&\le C\sum _{j=2}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} \Vert u\Vert _{H^1}^{2k+1-j} \Vert r\Vert _{H^1}^j\\&\le C_k\delta \Vert u\Vert _{H^1}^{2k-1}\Vert r\Vert _{H^1} \end{aligned}$$

and

$$\begin{aligned}&\left\| \left( {\mathcal {N}}'(u+r)-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(r)\right) -\left( {\mathcal {N}}'(u+\tilde{r})-{\mathcal {N}}'(u)-{\mathcal {N}}_{u}^+(\tilde{r})\right) \right\| _{L^2}\\&\quad =\left\| \sum _{j=3}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} u^{2k+1-j} (r-\tilde{r})(r^{j-1}+r^{j-2}\tilde{r}+\cdots +\tilde{r}^{j-1})\right\| _{L^2}\\&\quad \le \sum _{j=3}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} \left\| u^{2k+1-j} (r-\tilde{r})(r^{j-1}+r^{j-2}\tilde{r}+\cdots +\tilde{r}^{j-1})\right\| _{L^2}\\&\quad \le C\sum _{j=3}^{2k+1} \begin{pmatrix} 2k+1 \\ j \\ \end{pmatrix} \Vert u\Vert _{H^1}^{2k+1-j} \Vert r-\tilde{r}\Vert _{H^1}\left( \Vert r\Vert _{H^1}^{j-1}+\Vert r\Vert _{H^1}^{j-2}\Vert \tilde{r}\Vert _{H^1}+\cdots +\Vert \tilde{r}\Vert _{H^1}^{j-1}\right) \\&\quad \le C_k\delta \Vert u\Vert _{H^1}^{2k-1}\Vert r-\tilde{r}\Vert _{H^1} \end{aligned}$$

for some constant $C_k>0$. Then, taking $\delta _0=\eta \min \{\frac{1}{2C_k\Vert u\Vert _{H^1}^{2k-1}},\Vert u\Vert _{H^1}\}$, we complete the proof of the lemma for the polynomial nonlinearity. $\square$

Appendix B: Uniform lower bound for higher-order operators in (1.3)

Lemma B.1

(Uniform lower bound for higher-order operators in (1.3)) For any $\xi \in {\mathbb {R}}^3$, we have

$$\begin{aligned} \sum _{j=1}^{2k-1}\frac{(-1)^{j-1}\alpha _j}{m^{2j-1}c^{2j-2}}|\xi |^{2j}\ge \frac{|\xi |^2}{2m}, \end{aligned}$$

where $a_j=\frac{(2j-2)!}{j!(j-1)! 2^{2j-1}}$.

Proof

By change of variables $\frac{\xi }{m}\mapsto \xi$, it suffices to prove the lemma assuming $m=1$. The inequality is trivial when $k=1$. Suppose that $k\ge 2$. Splitting the positive and the negative terms and then applying the Cauchy–Schwarz inequality for the negative terms, we obtain

$$\begin{aligned} \sum _{j=1}^{2k-1}\frac{(-1)^{j-1}\alpha _j}{c^{2j-2}}|\xi |^{2j}&=\sum _{j=1}^k\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}-\sum _{j=1}^{k-1}\frac{\alpha _{2j}}{c^{4j-2}}|\xi |^{4j}\\&\ge \sum _{j=1}^k\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}\\&-\frac{1}{2}\sum _{j=1}^{k-1} \left\{ \frac{(\alpha _{2j})^2}{\alpha _{2j-1}\alpha _{2j+1}}\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}+\frac{\alpha _{2j+1}}{c^{4j}}|\xi |^{4j+2}\right\} . \end{aligned}$$

Since $\frac{(\alpha _{2j})^2}{\alpha _{2j-1}\alpha _{2j+1}}=\cdots =\frac{(4j-3)(2j+1)}{(4j-1)2j}\le 1$ for all $j\ge 1$, it is bounded below from

$$\begin{aligned}&\sum _{j=1}^k\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}-\frac{1}{2}\sum _{j=1}^{k-1} \left\{ \frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}+\frac{\alpha _{2j+1}}{c^{4j}}|\xi |^{4j+2}\right\} \\&\quad = \sum _{j=1}^k\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}-\frac{1}{2}\sum _{j=1}^{k-1}\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}-\frac{1}{2}\sum _{j=2}^k\frac{\alpha _{2j-1}}{c^{4j-4}}|\xi |^{4j-2}\\&\quad =\frac{1}{2}|\xi |^2+\frac{\alpha _{2k-1}}{2c^{4k-4}}|\xi |^{4k-2}\ge \frac{1}{2}|\xi |^2. \end{aligned}$$

$\square$