Scattering threshold for the focusing Choquard equation

Abstract

It is the purpose of this note, to obtain a scattering versus finite time blow-up dichotomy for a mass super-critical and energy sub-critical Choquard equation in the energy space.

Appendix

Let us prove (2.7). Take $u\in {\mathcal {A}}^+$ and denote for simplicity the quantities

$$\begin{aligned} E_0:=\Vert \nabla u\Vert ^2,\quad \bar{E}:=\Vert \nabla \phi \Vert ^2,\quad Q:=\int (I_\alpha *|u|^p)|u|^p\,dx. \end{aligned}$$

Then, $E_0+M-\frac{1}{p}Q<m$ and $Y:=2E_0-\frac{B}{p}Q>0.$ Thus,

$$\begin{aligned} E>\left( 1-\frac{2}{B}\right) E_0. \end{aligned}$$

Then,

$$\begin{aligned} M+\left( 1-\frac{2}{B}\right) E_0<M+E=S[u]<S[\phi ]=m=\frac{2(p-1)}{B}\bar{E}. \end{aligned}$$

Thus, by Young inequality

$$\begin{aligned} 2(p-1)\left( \frac{M}{A}\right) ^{\frac{A}{2(p-1)}}\left( \frac{E_0}{B}\right) ^{\frac{B-2}{2(p-1)}}<\frac{2(p-1)}{B}\bar{E}. \end{aligned}$$

This implies via Pohozaev identities that

$$\begin{aligned} E_0[u]^{s_c}M[u]^{1-s_c}<\bar{E}[\phi ]^{s_c}M[\phi ]^{1-s_c}. \end{aligned}$$

On the other hand, we have by Young inequality

$$\begin{aligned} \frac{2(p-1)}{B}\bar{E}>S[u]\ge 2(p-1)\left( \frac{M}{A}\right) ^{\frac{A}{2(p-1)}}\left( \frac{E}{B-2}\right) ^{\frac{B-2}{2(p-1)}}. \end{aligned}$$

So

$$\begin{aligned} E^{\frac{B-2}{2(p-1)}}M^{\frac{A}{2(p-1)}}<A^{\frac{A}{2(p-1)}}(B-2)^{\frac{B-2}{2(p-1)}}\frac{\bar{E}}{B}. \end{aligned}$$

This implies via Pohozaev identities that

$$\begin{aligned} E[u]^{s_c}M[u]^{1-s_c}<E[\phi ]^{s_c}M[\phi ]^{1-s_c}. \end{aligned}$$

Reciprocally, take u satisfying (2.7). Recall that the Eq. (1.1) enjoys the scaling invariance

$$\begin{aligned} u_\lambda =\lambda ^\frac{2+\alpha }{2(p-1)}u(\lambda ^{2}.,\lambda .),\quad \lambda >0. \end{aligned}$$

Compute

$$\begin{aligned} M[u_\lambda ]= & {} \lambda ^\frac{2+\alpha }{p-1}\Vert u_0(\lambda .)\Vert ^2 =\lambda ^{-N+\frac{2+\alpha }{p-1}}M[u]:=\lambda ^aM[u];\\ E[u_\lambda ]= & {} \lambda ^\frac{2+\alpha }{p-1}E[u_0(\lambda .)] =\lambda ^{2-N+\frac{2+\alpha }{p-1}}E[u]:=\lambda ^bE[u]. \end{aligned}$$

Using Young inequality, one gets

$$\begin{aligned} 2(p-1)\left( M[u_\lambda ]+E[u_\lambda ]\right) \ge \left( \frac{M[u_\lambda ]}{A}\right) ^{\frac{A}{2(p-1)}}\left( \frac{E[u_\lambda ]}{B-2}\right) ^{\frac{B-2}{2(p-1)}}. \end{aligned}$$

Moreover, the equality is equivalent to

$$\begin{aligned} \frac{M[u_\lambda ]}{A}=\frac{E[u_\lambda ]}{B-2}\Longleftrightarrow \lambda =\lambda _0=\left( \frac{(B-2)M[u]}{AE[u]}\right) ^\frac{1}{2}. \end{aligned}$$

Compute, taking $\lambda =\lambda _0$,

$$\begin{aligned} M[u_\lambda ]+E[u_\lambda ]= & {} M[u_\lambda ](1+\frac{B-2}{A})\\= & {} \left( \frac{(B-2)M}{AE}\right) ^\frac{a}{2}M(1+\frac{B-2}{A})\\= & {} \frac{2(p-1)}{A}M^{1+\frac{a}{2}}E^{-\frac{a}{2}}\left( \frac{B-2}{A}\right) ^\frac{a}{2}\\= & {} \frac{2(p-1)}{A}\left( \frac{B-2}{A}\right) ^{-s_c}M^{1-s_c}E^{s_c}\\< & {} \frac{2(p-1)}{A}\left( \frac{B-2}{A}\right) ^{-s_c}M[\phi ]^{1-{s_c}}E[\phi ]^{s_c}. \end{aligned}$$

By Pohozaev identities, one gets

$$\begin{aligned} M[u_\lambda ]+E[u_\lambda ]< & {} \frac{2(p-1)}{A}\left( \frac{B-2}{A}\right) ^{-{s_c}}M[\phi ]^{1-{s_c}} \left( \frac{B-2}{A}M[\phi ]\right) ^{{s_c}}\\< & {} \frac{2(p-1)}{A}M[\phi ]\\< & {} S[\phi ]. \end{aligned}$$

Now, write using Proposition 2.3,

$$\begin{aligned} \frac{N}{4}{\mathcal {I}}(u)= & {} \Vert \nabla u\Vert ^{B+2-B}-\frac{B}{2p}\int (I_\alpha *|u|^p)|u|^p\,dx\\> & {} \frac{B}{2p}\left( \frac{2p}{B}\Vert \nabla u\Vert ^B\Vert \nabla \phi \Vert ^{(2-B)} \left( \frac{M[\phi ]}{M[u]}\right) ^{(2-B)\frac{1-s_c}{2s_c}}-\int (I_\alpha *|u|^p)|u|^p\,dx\right) \\> & {} \frac{B}{2p}\left( \frac{2p}{B}\Vert \nabla u\Vert ^B\Vert u\Vert ^A\Vert \nabla \phi \Vert ^{2-B}M[\phi ]^{-\frac{A}{2}}-\int (I_\alpha *|u|^p)|u|^p\,dx\right) \\> & {} \frac{B}{2p}\left( \frac{2p}{B}\Vert \nabla u\Vert ^B\Vert u\Vert ^A\left( \frac{B}{A} M[\phi ]\right) ^{\frac{2-B}{2}}M[\phi ]^{-\frac{A}{2}}-\int (I_\alpha *|u|^p)|u|^p\,dx\right) \\> & {} \frac{B}{2p}\left( C_{N,p,\alpha }\Vert \nabla u\Vert ^B\Vert u\Vert ^A-\int (I_\alpha *|u|^p)|u|^p\,dx\right) \\> & {} 0. \end{aligned}$$

This closes the proof.