Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions

Abstract

We obtain uniqueness and nondegeneracy results for ground states of Choquard equations $-\Delta u+u=\left( |x|^{-1}*|u|^{p}\right) |u|^{p-2}u$ in $\mathbb {R}^{3}$, provided that $p>2$ and p is sufficiently close to 2.

Appendices

Appendix A: Proof of Theorem 1.1

In this section, we give a short proof for the existence part of Theorem 1.1 for the sake of completeness. A complete proof of Theorem 1.1 can be found in Moroz and Van Schaftingen [17]. We start the proof with the following observation.

Lemma A.1

Assume that $5/3<p<7/3$. Then $-\infty<m(N,p)<0$ for any $N>0$.

Proof

It is a consequence of Lemmas 2.2 and 2.4. $\square$

By Lemma 2.3, we have the following conclusion.

Lemma A.2

The infimum m(N, p) is strictly decreasing with respect to N.

To prove the existence of minimizers of problem (1.4), we apply the rearrangement technique. For any given function $u\in H^{1}(\mathbb R^{3})$, we denote by $u^{*}$ the symmetric-decreasing rearrangement of u. The following properties hold for all $u\in H^{1}(\mathbb R^{3})$

$$\begin{aligned} \Vert u^{*}\Vert _{q}= & {} \Vert u\Vert _{q}\quad \forall \,2\le q\le 6;\end{aligned}$$

(A.1)

$$\begin{aligned} K(u^{*})\le & {} K(u);\end{aligned}$$

(A.2)

$$\begin{aligned} D_{p}(u^{*})\ge & {} D_{p}(u). \end{aligned}$$

(A.3)

Equality of (A.3) is attained if and only if $u(x)=u^{*}(x-x_{0})$ for some $x_{0}\in \mathbb R^{3}$. For the precise definition of $u^{*}$ and the proof of above properties, we refer to e.g. Lieb and Loss [11]. Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1

By Lemma 2.3, it suffices to prove Theorem 1.1 in the case $N=1$. Recall that we denote $m(p)=m(1,p)$. That is,

$$\begin{aligned} m(p)=\inf \left\{ E_{p}(u):u\in \mathcal{A}_{1}\right\} . \end{aligned}$$

(A.4)

Our first aim is to show that m(p) is attained. Let $\{u_{n}\}\subset \mathcal{A}_{1}$ be a minimizing sequence of problem (A.4). Consider the symmetric-decreasing rearrangement $u_{n}^{*}$ for all $n\in \mathbb N$. By (A.1) (A.2) and (A.3), we deduce that the sequence $\{u_{n}^{*}\}$ is also a minimizing sequence of problem (A.4). On the other hand, since $0<3p-5<2$, we deduce from (2.7) that

$$\begin{aligned} E_{p}(u)\ge \frac{1}{4}\int _{\mathbb R^{3}}|\nabla u|^{2}\mathrm{d}x-C_{p} \quad \forall \, u\in \mathcal{A}_{1}, \end{aligned}$$

for a constant $C_{p}>0$ depending only on p. Therefore we conclude by using above estimate that $\{u_{n}^{*}\}$ is a bounded sequence in $H^{1}(\mathbb R^{3})$. Hence there exists a function $u\in H^{1}(\mathbb R^{3})$ such that

$$\begin{aligned} u_{n}^{*}\rightharpoonup u \quad \text {in }H^{1}(\mathbb R^{3}). \end{aligned}$$

(A.5)

Moreover, we have that $u\in H_{\mathrm{rad}}^{1}(\mathbb R^{3})$ and $u\ge 0$ hold since $u_{n}^{*}$ are nonnegative radial functions for all $n\in \mathbb N$. Furthermore, by the compactness embedding $H_{\mathrm{rad}}^{1}(\mathbb R^{3})\subset L^{q}(\mathbb R^{3})$ for any $2<q<6$ (see Strauss [20]), we can assume by passing to a subsequence that

$$\begin{aligned} u_{n}^{*}\rightarrow u \quad \text { in }L^{6p/5}(\mathbb {R}^{3}). \end{aligned}$$

(A.6)

Then it follows from (A.5) that

$$\begin{aligned} \int _{\mathbb {R}^{3}}|\nabla u_{n}^{*}|^{2}\mathrm{d}x=\int _{\mathbb {R}^{3}}|\nabla v_{n}|^{2}\mathrm{d}x+\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm{d}x+o(1), \end{aligned}$$

where $v_{n}=u_{n}^{*}-u$, and it follows from (A.6) that

$$\begin{aligned} \int _{\mathbb {R}^{3}}\left( I_{2}*u_{n}^{*p}\right) u_{n}^{*p}\mathrm{d}x=\int _{\mathbb {R}^{3}}\left( I_{2}*u^{p}\right) u^{p}\mathrm{d}x+o(1) \end{aligned}$$

as $n\rightarrow \infty$. Therefore, combining above two equalities gives us that

$$\begin{aligned} E_{p}(u_{n}^{*})=K(v_{n})+E_{p}(u)+o(1) \end{aligned}$$

(A.7)

as $k\rightarrow \infty$. Since $K(v_{n})\ge 0$, we obtain that $E_{p}(u)\le m(p)<0$. In particular, we obtain that $u\not \equiv 0$.

To show that $E_{p}(u)=m(p)$, it suffices to show that $u\in \mathcal {A}_{1}$. Suppose that $u\notin \mathcal {A}_{1}$ holds. Since $u_{n}^{*}\rightharpoonup u$ in $L^{2}(\mathbb R^{3})$, we have $\Vert u\Vert _{2}\le \liminf _{n}\Vert u_{n}^{*}\Vert _{2}=1$. Hence there exists $\alpha \in (0,1)$ such that $\Vert u\Vert _{2}=\alpha$, that is, $u\in \mathcal{A}_{\alpha }$. Then $m(\alpha ,p)\le E_{p}(u)$ holds. Note that $m(\alpha ,p)>m(1,p)=m(p)$ by Lemma A.2. We obtain that

$$\begin{aligned} m(p)<m(\alpha ,p)\le E_{p}(u)\le m(p), \end{aligned}$$

which is impossible! Hence $u\in \mathcal {A}_{1}$ holds. Then we obtain $E_{p}(u)=m(p)$. This shows that u is a minimizer of problem (A.4).

Suppose now $Q\in \mathcal{A}_{1}$ is an arbitrary minimizer of problem (A.4). Then we have $Q^{*}\in \mathcal{A}_{1}$ by (A.1), which implies that $E_{p}(Q^{*})\ge E_{p}(Q)$. On the other hand, by (A.2) and (A.3) we derive $E_{p}(Q^{*})\le E_{p}(Q)$. Hence, we have $E_{p}(Q^{*})=E_{p}(Q)$, from which we infer that $Q^{*}$ is also a minimizer of problem (A.4), and that the equalities in (A.2) (A.3) are attained at $u=Q$. Therefore, there exists a point $x_{0}\in \mathbb R^{3}$ such that $Q(x)=Q^{*}(x-x_{0})$. This proves that every minimizer of problem (A.4) is a nonnegative radial function with respect to a point $x_{0}\in \mathbb R^{3}$ and symmetric-decreasing with respect to $r=|x-x_{0}|$.

Next we prove that Q is positive everywhere. It is well known that Q solves Eq. (1.1) with a positive Lagrange multiplier $\lambda >0$. Thus we obtain from Eq. (1.1) that Q satisfies

$$\begin{aligned} Q=\frac{1}{-\Delta +\lambda }\left( |x|^{-1}*Q^{p}\right) Q^{p-1}. \end{aligned}$$

As the integral kernel of $\frac{1}{-\Delta +\lambda }$ is positive everywhere and Q is nonnegative nontrivial, we infer from above formula that Q is strictly positive in $\mathbb R^{3}$.

Now applying Proposition 4.1 of Moroz and Schaftingen [17], we obtain that $Q\in W^{2,s}(\mathbb R^{3})\cap C^{\infty }(\mathbb R^{3})$ for any $s>1$. By the maximum principle, we conclude that $Q^{\prime }(|x|)<0$ for $|x|\ne 0$. The last assertion of Theorem 1.1 is covered by Theorem 4 of Moroz and Schaftingen [17]. We omit the details. The proof of Theorem 1.1 is complete. $\square$

Appendix B: Regularity of F

Recall that in Sect. 3 we denote

$$\begin{aligned} \mathbb X=L_{\mathrm{rad}}^{2}(\mathbb R^{3})\cap L_{\mathrm{rad}}^{6}(\mathbb R^{3}) \end{aligned}$$

equipped with norm $\Vert u\Vert _{\mathbb X}=\Vert u\Vert _{2}+\Vert u\Vert _{6}$. Define the map $F:\mathbb X\times \mathbb R_{+}\times [2,7/3)\rightarrow \mathbb X\times \mathbb R$ by

$$\begin{aligned} F(u,\lambda ,p)=\left( \begin{array}{l} u-{\displaystyle \frac{1}{-\Delta +\lambda }}\left( |x|^{-1}*|u|^{p}\right) |u|^{p-2}u\\ \Vert u\Vert _{2}^{2}-c_{0} \end{array}\right) , \end{aligned}$$

for $u\in \mathbb X$, $\lambda \in \mathbb R_{+}$ and $p\in [2,7/3)$. Here $c_{0}$ is a fixed constant. For simplicity, we write $I=[2,7/3)$ below. Since we will make a series of estimates, it is convenient to use the standard notation $A\lesssim B$ to denote $A\le CB$ for some constant $C>0$ that only depends on some fixed quantities. We also write $A\lesssim _{a,b,\ldots }B$ to underline that C depends on the fixed quantities $a,b,\ldots$ etc.

Lemma B.1

The map $F:\mathbb X\times \mathbb R_{+}\times I\rightarrow \mathbb X\times \mathbb R$ is $C^{1}$.

Proof

First, we prove that $F:\mathbb X\times \mathbb R_{+}\times I\rightarrow \mathbb X\times \mathbb R$ is well defined. Note that there holds

$$\begin{aligned} \left\| {\displaystyle \frac{1}{-\Delta +\lambda }}v\right\| _{H^{2}(\mathbb R^{3})}\lesssim _{\lambda }\Vert v\Vert _{2} \quad \text {for }v\in L^{2}(\mathbb R^{3}). \end{aligned}$$

(B.1)

Since $H^{2}(\mathbb R^{3})$ is continuously embedded in $L^{2}(\mathbb R^{3})\cap L^{\infty }(\mathbb R^{3})$, $H^{2}(\mathbb R^{3})$ is continuously embedded in $\mathbb X$ continuously as well. Thus we only need to show that $\left( |x|^{-1}*|u|^{p}\right) |u|^{p-2}u\in L_{\mathrm{rad}}^{2}(\mathbb R^{3})$ holds for any $u\in \mathbb X$ and $p\in I$. For notational simplicity, write

$$\begin{aligned} g(u,p)=\left( |x|^{-1}*|u|^{p}\right) |u|^{p-2}u. \end{aligned}$$

Set $q=6(2p-1)/7$. Then we have $2\le p<q<6$ since $p\in I$. By interpolation, it is elementary to compute that

$$\begin{aligned} \Vert u\Vert _{q}\le \Vert u\Vert _{\mathbb X} \end{aligned}$$

(B.2)

for $u\in \mathbb X$. Then Young’s inequality gives us that

$$\begin{aligned} \left\| |x|^{-1}*|u|^{p}\right\| _{r}\lesssim _{p}\Vert u\Vert _{q}^{p}\lesssim _{p}\Vert u\Vert _{\mathbb X}^{p} \end{aligned}$$

(B.3)

where r is given by

$$\begin{aligned} \frac{1}{r}+1=\frac{1}{3}+\frac{p}{q}. \end{aligned}$$

(B.4)

It is elementary to obtain that

$$\begin{aligned} \left\| u|^{p-2}u\right\| _{\frac{q}{p-1}}=\Vert u\Vert _{q}^{p-1}\le \Vert u\Vert _{\mathbb X}^{p-1}. \end{aligned}$$

(B.5)

Hence Combining (B.3) and (B.5) yields that

$$\begin{aligned} \Vert g(u,p)\Vert _{2}\lesssim _{p}\Vert u\Vert _{\mathbb X}^{2p-1}. \end{aligned}$$

(B.6)

Thus, by (B.6) and (B.1) we deduce that

$$\begin{aligned} \left\| {\displaystyle \frac{1}{-\Delta +\lambda }}g(u,p)\right\| _{\mathbb X}\lesssim \left\| {\displaystyle \frac{1}{-\Delta +\lambda }}g(u,p)\right\| _{H^{2}}\lesssim _{\lambda }\Vert g(u,p)\Vert _{2}\lesssim _{\lambda ,p}\Vert u\Vert _{\mathbb X}^{2p-1}. \end{aligned}$$

This proves that F is well defined.

Next we turn to the Fréchet differentiability of F. It is straightforward to verify that the second component $F_{2}$ of F is continuously Fréchet differentiable and its Fréchet derivative at u is given by $F_{2}^{\prime }(u)=2\langle u,\cdot \rangle$, where $\langle u,\cdot \rangle$ denotes the map $g\mapsto \langle u,g\rangle$. Let us now turn to consider the Fréchet differentiability the first component

$$\begin{aligned} F_{1}=u-{\displaystyle \frac{1}{-\Delta +\lambda }}g(u,p). \end{aligned}$$

We claim that $F_{1}\in C^{1}$ and its partial derivatives are given by

$$\begin{aligned} \frac{\partial F_{1}}{\partial u}=1-{\displaystyle \frac{1}{-\Delta +\lambda }}g_{u}(u,p), \quad \frac{\partial F_{1}}{\partial \lambda }=\frac{1}{\left( -\Delta +\lambda \right) ^{2}}g(u,p) \end{aligned}$$

and

$$\begin{aligned} \frac{\partial F_{1}}{\partial p}=-{\displaystyle \frac{1}{-\Delta +\lambda }}g_{p}(u,p), \end{aligned}$$

where $g_{u}(u,p)=\partial _{u}g(u,p):\mathbb X\rightarrow \mathbb X$ is given by

$$\begin{aligned} g_{u}(u,p)f=\left( |x|^{-1}*\left( p|u|^{p-2}uf\right) \right) |u|^{p-2}u+\left( |x|^{-1}*|u|^{p}\right) (p-1)|u|^{p-2}f, \end{aligned}$$

and

$$\begin{aligned} g_{p}(u,p)=\frac{\partial g(u,p)}{\partial p}=\left( |x|^{-1}*\left( |u|^{p}\log |u|\right) \right) |u|^{p-2}u+\left( |x|^{-1}*|u|^{p}\right) |u|^{p-2}u\log |u|. \end{aligned}$$

This claim follows in a standard way by using Sobolev inequalities, Hölder’s inequality and estimates such as (B.2) (B.3) (B.5) and the regularity of functions such as $t\mapsto t^{p-1}$ with $p\ge 2$. In the following we prove the claim of $\partial F_{1}/\partial u$. The claims of other two partial derivatives $\partial _{\lambda }F_{1},\partial _{p}F_{1}$ can be proved similarly.

First we prove that $\partial F_{1}/\partial u$ exists and be given as above. So we have to show that for any $h\in \mathbb X$,

$$\begin{aligned} F_{1}(u+h,\lambda ,p)-F_{1}(u,\lambda ,p)-\frac{\partial F_{1}}{\partial u}(u,\lambda ,p)h=o(1)h, \end{aligned}$$

(B.7)

where $o(1)\rightarrow 0$ as $\Vert h\Vert _{\mathbb X}\rightarrow 0$. By a direct calculation, we obtain that

$$\begin{aligned} F_{1}(u+h,\lambda ,p)-F_{1}(u,\lambda ,p)-\frac{\partial F_{1}}{\partial u}(u,\lambda ,p)h=-{\displaystyle \frac{1}{-\Delta +\lambda }}\sum _{i=1}^{3}M_{i}, \end{aligned}$$

where $M_{i}$, $i=1,2,3$, are given by

$$\begin{aligned} M_{1}= & {} \left( |x|^{-1}*\left( |u+h|^{p}-|u|^{p}-p|u|^{p-2}uh\right) \right) |u+h|^{p-2}(u+h),\\ M_{2}= & {} \left( |x|^{-1}*|u|^{p}\right) \left( |u+h|^{p-2}(u+h)-|u|^{p-2}u-(p-1)|u|^{p-2}h\right) ,\\ M_{3}= & {} \left( |x|^{-1}*p|u|^{p-2}uh\right) \left( |u+h|^{p-2}(u+h)-|u|^{p-2}u\right) , \end{aligned}$$

respectively. We always assume that $\Vert h\Vert _{\mathbb X}\le 1$ since we let $\Vert h\Vert _{\mathbb X}$ tend to zero in the end. Note that

$$\begin{aligned} \left| |u+h|^{p}-|u|^{p}-p|u|^{p-2}uh\right| \lesssim _{p}(|u|^{p-2}+|h|^{p-2})|h|^{2}, \end{aligned}$$

(B.8)

Then by (B.3) (B.8) and (B.2) we deduce that

$$\begin{aligned} \left\| |x|^{-1}*\left( |u+h|^{p}-|u|^{p}-p|u|^{p-2}uh\right) \right\| _{r}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\Vert h\Vert _{\mathbb X}^{2}, \end{aligned}$$

where r is defined as in (B.4) with $q=6(2p-1)/7$. Then combining above estimate and (B.5) as before implies that

$$\begin{aligned} \Vert M_{1}\Vert _{2}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\Vert h\Vert _{\mathbb X}^{2}. \end{aligned}$$

Similarly, since $p\ge 2$, we deduce that

$$\begin{aligned} \Vert M_{2}\Vert _{2}\lesssim _{p,\Vert u\Vert _{\mathbb X}}o(1)\Vert h\Vert _{\mathbb X}, \end{aligned}$$

where $o(1)\rightarrow 0$ as $\Vert h\Vert _{\mathbb X}\rightarrow 0$, and that

$$\begin{aligned} \Vert M_{3}\Vert _{2}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\Vert h\Vert _{\mathbb X}^{2}. \end{aligned}$$

Therefore, we obtain by combining above three estimates together that

$$\begin{aligned} \left\| F_{1}(u+h,\lambda ,p)-F_{1}(u,\lambda ,p)-\frac{\partial F_{1}}{\partial u}(u,\lambda ,p)\right\| _{\mathbb X}\lesssim _{p,\Vert u\Vert _{\mathbb X}}o(1)\Vert h\Vert _{\mathbb X}. \end{aligned}$$

This proves (B.7) and thus $\partial _{u}F_{1}$ exists and be given as claimed.

Next we prove that $\partial _{u}F_{1}$ depends continuously on $(u,\lambda ,p)$. Fix an arbitrary point $(u,\lambda ,p)\in \mathbb X\times \mathbb R_{+}\times I$ and let $\epsilon >0$. We have to find $\delta >0$ such that

$$\begin{aligned} \left\| \left( \frac{\partial F_{1}}{\partial u}(u,\lambda ,p)-\frac{\partial F_{1}}{\partial u}(\tilde{u},\tilde{\lambda },\tilde{p})\right) f\right\| _{\mathbb X}\le \epsilon \Vert f\Vert _{\mathbb X}, \end{aligned}$$

(B.9)

whenever $\Vert u-\tilde{u}\Vert _{\mathbb X}+|\lambda -\tilde{\lambda }|+|p-\tilde{p}|\le \delta$ holds for $(\tilde{u},\tilde{\lambda },\tilde{p})\in \mathbb X\times \mathbb R_{+}\times I$.

Note that

$$\begin{aligned} \left\| \left( \frac{\partial F_{1}}{\partial u}(u,\lambda ,p)-\frac{\partial F_{1}}{\partial u}(\tilde{u},\tilde{\lambda },\tilde{p})\right) f\right\| _{\mathbb X}\le & {} \left\| \frac{1}{-\Delta +\lambda }\Big (g_{u}(\tilde{u},\tilde{p})-g_{u}(u,p)\Big )f\right\| _{\mathbb X}\\&+\left\| \left( \frac{1}{-\Delta +\tilde{\lambda }}-{\displaystyle \frac{1}{-\Delta +\lambda }}\right) g_{u}(\tilde{u},\tilde{p})f\right\| _{\mathbb X}\\=: & {} J_{1}+J_{2}. \end{aligned}$$

In the following we only prove (B.9) for the first term $J_{1}$. That is, there exists $\delta >0$, such that whenever $\Vert u-\tilde{u}\Vert _{\mathbb X}+|\lambda -\tilde{\lambda }|+|p-\tilde{p}|\le \delta$ holds for $(\tilde{u},\tilde{\lambda },\tilde{p})\in \mathbb X\times \mathbb R_{+}\times I$, then

$$\begin{aligned} J_{1}\equiv \left\| \frac{1}{-\Delta +\lambda }\Big (g_{u}(\tilde{u},\tilde{p})-g_{u}(u,p)\Big )f\right\| _{\mathbb X}\le \epsilon \Vert f\Vert _{\mathbb X}. \end{aligned}$$

(B.10)

The estimate of $J_{2}$ can be derived in the same way as that of Frank and Lenzmann [9, Appendix E], where even more general operators are considered. For example, the $C^{1}$ continuity about the parameters s and $\lambda$ of the operator $\left( \left( -\Delta \right) ^{s}+\lambda \right) ^{-1}$ is proven in Frank and Lenzmann [9, Lemma E.1].

To prove (B.10), (B.1) implies that it is sufficient to prove

$$\begin{aligned} \left\| \Big (g_{u}(\tilde{u},\tilde{p})-g_{u}(u,p)\Big )f\right\| _{2}\lesssim _{\lambda ,p}\epsilon \Vert f\Vert _{\mathbb X} \end{aligned}$$

(B.11)

whenever $\Vert u-\tilde{u}\Vert _{\mathbb X}+|p-\tilde{p}|\le \delta$ holds $1>\delta >0$ small enough and $\left( \tilde{u},\tilde{p}\right) \in \mathbb X\times I$. Note that

$$\begin{aligned} \left\| \left( g_{u}(\tilde{u},\tilde{p})-g_{u}(u,p)\right) f\right\| _{2}&\le \left\| \Big (g_{u}(\tilde{u},\tilde{p})-g_{u}(u,\tilde{p})\Big )f\right\| _{2}+\left\| \Big (g_{u}(u,\tilde{p})-g_{u}(u,p)\Big )f\right\| _{2}. \end{aligned}$$

Denote

$$\begin{aligned} L_{1}=\Big (g_{u}(\tilde{u},\tilde{p})-g_{u}(u,\tilde{p})\Big )f \quad \text {and} \quad L_{2}=\Big (g_{u}(u,\tilde{p})-g_{u}(u,p)\Big )f. \end{aligned}$$

We show that

$$\begin{aligned} \Vert L_{1}\Vert _{2}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\epsilon \Vert f\Vert _{\mathbb X} \end{aligned}$$

(B.12)

and that

$$\begin{aligned} \Vert L_{2}\Vert _{2}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\epsilon \Vert f\Vert _{\mathbb X} \end{aligned}$$

(B.13)

hold. In the sequel we estimate $L_{1}$ and $L_{2}$ one by one.

First we estimate $L_{1}$. it is easy to obtain that $L_{1}=\sum \nolimits _{i=1}^{4}L_{1i},$where

$$\begin{aligned} L_{11}= & {} \left( |x|^{-1}*|u|^{\tilde{p}}\right) (\tilde{p}-1)\left( |\tilde{u}|^{\tilde{p}-2}-|u|^{\tilde{p}-2}\right) f,\\ L_{12}= & {} \left( |x|^{-1}*\left( |\tilde{u}|^{\tilde{p}}-|u|^{\tilde{p}}\right) \right) (\tilde{p}-1)|\tilde{u}|^{\tilde{p}-2}f,\\ L_{13}= & {} \left( |x|^{-1}*\tilde{p}\left( |\tilde{u}|^{\tilde{p}-2}\tilde{u}-|u|^{\tilde{p}-2}u\right) f\right) |u|^{\tilde{p}-2}u, \end{aligned}$$

and

$$\begin{aligned} L_{14}=\left( |x|^{-1}*\tilde{p}|\tilde{u}|^{\tilde{p}-2}\tilde{u}f\right) \left( |\tilde{u}|^{\tilde{p}-2}u-|u|^{\tilde{p}-2}u\right) , \end{aligned}$$

respectively. We will estimate $L_{11},L_{12}$ for instance and leave the estimates of $L_{13},L_{14}$ for the interested readers. We assume that $\tilde{p}>2$, for otherwise $L_{11}\equiv 0$, we are done. Note that $\left| |u|^{\tilde{p}-2}-|\tilde{u}|^{\tilde{p}-2}\right| \le |u-\tilde{u}|^{\tilde{p}-2}$ since $0<\tilde{p}-2<1$. Set $\tilde{q}=6(2\tilde{p}-1)/7$. Thus (B.5) gives that

$$\begin{aligned} \Vert \left( |\tilde{u}|^{\tilde{p}-2}-|u|^{\tilde{p}-2}\right) f\Vert _{\frac{\tilde{q}}{\tilde{p}-1}}\le \Vert u-\tilde{u}\Vert _{\tilde{q}}^{\tilde{p}-2}\Vert f\Vert _{\tilde{q}}\le \delta ^{\tilde{p}-2}\Vert f\Vert _{\mathbb X}. \end{aligned}$$

Combining above inequality together with (B.3) yields that

$$\begin{aligned} \Vert L_{11}\Vert _{2}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\delta ^{\tilde{p}-2}\Vert f\Vert _{\mathbb X}. \end{aligned}$$

(B.14)

Here we used the assumption $\Vert u-\tilde{u}\Vert _{\mathbb X}+|\tilde{p}-p|<\delta$, which implies that $\Vert \tilde{u}\Vert _{\mathbb X}\le \Vert u\Vert _{\mathbb X}+1$ and $\tilde{p}\le p+1$. To estimate $L_{12}$, note that $\left| |u|^{\tilde{p}}-|\tilde{u}|^{\tilde{p}}\right| \lesssim _{p}\left( |u|^{\tilde{p}-1}+|\tilde{u}|^{\tilde{p}-1}\right) |u-\tilde{u}|$. Then (B.3) implies that

$$\begin{aligned} \left\| \left( |x|^{-1}*\left( |u|^{\tilde{p}}-|\tilde{u}|^{\tilde{p}}\right) \right) \right\| _{\tilde{r}}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\Vert u-\tilde{u}\Vert _{q}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\delta \end{aligned}$$

since $\Vert u-\tilde{u}\Vert _{q}\le \Vert u-\tilde{u}\Vert _{\mathbb X}\le \delta$, where r is given by $1/\tilde{r}+1=1/3+\tilde{p}/\tilde{q}$ with $\tilde{q}=6(2\tilde{p}-1)/7$. Thus combining above inequality together with (B.5) gives us that

$$\begin{aligned} \Vert L_{12}\Vert _{2}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\delta \Vert f\Vert _{\mathbb X}. \end{aligned}$$

(B.15)

Since we can obtain similar estimates for $L_{13},L_{14}$ as above, it becomes obvious from e.g. (B.14) and (B.15) that we can choose $\delta >0$ sufficiently small such that (B.12) holds. This gives the estimate of $L_{1}$.

Next we estimate $L_{2}$. We have $L_{2}=\sum \nolimits _{i=1}^{4}L_{2i},$ where

$$\begin{aligned} L_{21}= & {} \left( |x|^{-1}*|u|^{p}\right) \left( (\tilde{p}-1)|u|^{\tilde{p}-2}-(p-1)|u|^{p-2}\right) f,\\ L_{22}= & {} \left( |x|^{-1}*\left( |u|^{\tilde{p}}-|u|^{p}\right) \right) (\tilde{p}-1)|u|^{\tilde{p}-2}f,\\ L_{23}= & {} \left( |x|^{-1}*\left( p|u|^{p-2}uf\right) \right) \left( |u|^{\tilde{p}-2}u-|u|^{p-2}u\right) , \end{aligned}$$

and

$$\begin{aligned} L_{24}=\left( |x|^{-1}*\left( \tilde{p}|u|^{\tilde{p}-2}u-p|u|^{p-2}u\right) f\right) |u|^{\tilde{p}-2}u \end{aligned}$$

respectively. We estimate the first term $L_{21}$ for instance, and leave the estimates of $L_{22},$ $L_{23}$ and $L_{24}$ for the interested readers. Note that

$$\begin{aligned} \left| (p-1)|u|^{p-2}-(\tilde{p}-1)|u|^{\tilde{p}-2}\right| \le |p-\tilde{p}||u|^{p-2}+(\tilde{p}-1)\left| |u|^{p-2}-|u|^{\tilde{p}-2}\right| . \end{aligned}$$

(B.16)

Set $q=6(2p-1)/7$. Then by (B.3), (B.5) and above inequality we deduce that

$$\begin{aligned} \Vert L_{21}\Vert _{2}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\left( |p-\tilde{p}|+\left\| |u|^{p-2}-|u|^{\tilde{p}-2}\right\| _{\frac{q}{p-2}}\right) \Vert f\Vert _{\mathbb X}. \end{aligned}$$

We estimate the second term in the bracket of above inequality as follows. We only consider the case $\tilde{p}>p$. The case $\tilde{p}<p$ can be considered similarly. By an elementary calculation, we obtain that

$$\begin{aligned} \left| |u|^{p-2}-|u|^{\tilde{p}-2}\right| \lesssim _{p,\tilde{p},M}|p-\tilde{p}||u|^{p-2} \quad \text {on }\{1/M\le |u|\le M\}, \end{aligned}$$

for any given constant $M>1$, and

$$\begin{aligned} \left| |u|^{p-2}-|u|^{\tilde{p}-2}\right| \le 2|u|^{\tilde{p}-2}\chi _{\{|u|>M\}} \quad \text {on }\{|u|>M\}, \end{aligned}$$

where $\chi _{\{|u|>M\}}(x)=1$ if $|u(x)|>M$ and $\chi _{\{|u|>M\}}(x)=0$ if $|u(x)|\le M$, and

$$\begin{aligned} \left| |u|^{p-2}-|u|^{\tilde{p}-2}\right| \le 2|u|^{p-2}\chi _{\{|u|<1/M\}} \quad \text {on }\{|u|<1/M\}. \end{aligned}$$

Thus we can obtain that

$$\begin{aligned} \left\| |u|^{p-2}-|u|^{\tilde{p}-2}\right\| _{\frac{q}{p-2}}\le & {} C\left( M,p,\Vert u\Vert _{\mathbb X}\right) |p-\tilde{p}|+C_{p}\left( \int _{\mathbb R^{3}}|u|^{\frac{\tilde{p}-2}{p-2}q}\chi _{\{|u|>M\}}\mathrm{d}x\right) ^{(p-2)/q}\\&+\left\| u\chi _{\{|u|<1/M\}}\right\| _{\mathbb X}^{p-2}. \end{aligned}$$

It is elementary to show that

$$\begin{aligned} \left( \int _{\mathbb R^{3}}|u|^{\frac{\tilde{p}-2}{p-2}q}\chi _{\{|u|>M\}}\mathrm{d}x\right) ^{(p-2)/q}\rightarrow 0 \quad \text {as } M\rightarrow \infty \,\text {and }\tilde{p}\rightarrow p \end{aligned}$$

and that

$$\begin{aligned} \left\| u\chi _{\{|u|<1/M\}}\right\| _{q}^{p-2}\rightarrow 0 \quad \text {as }M\rightarrow \infty . \end{aligned}$$

Thus for given $\epsilon >0$, we first choose $M>1$ sufficiently large such that

$$\begin{aligned} \left( \int _{\mathbb R^{3}}|u|^{\frac{\tilde{p}-2}{p-2}q}\chi _{\{|u|>M\}}\mathrm{d}x\right) ^{(p-2)/q}+\left\| u\chi _{\{|u|<1/M\}}\right\| _{\mathbb X}^{p-2}\lesssim _{p}\epsilon , \end{aligned}$$

and then fix such M and choose $\delta >0$ sufficiently small enough such that $C\left( M,\Vert u\Vert _{\mathbb X},p\right) |p-\tilde{p}|\lesssim _{p}\epsilon$. This proves that

$$\begin{aligned} \Vert L_{21}\Vert _{2}\lesssim _{p,\Vert u\Vert _{\mathbb X}}\epsilon \Vert f\Vert _{\mathbb X}. \end{aligned}$$

(B.17)

Since we can derive above estimates for $L_{22},$ $L_{23}$ and $L_{24}$, we conclude that (B.13) holds. This gives the estimate for $L_{2}$.

Finally, combining (B.1), (B.11), (B.12) and (B.13) gives us the estimate (B.10), and thus follows the continuity of $\partial _{u}F_{1}$. As we can prove similarly the continuity of the derivatives $\partial _{\lambda }F_{1},\partial _{p}F_{1}$, the proof of Lemma B.1 is complete. $\square$