Fractional Choquard Equations with an Inhomogeneous Combined Non-linearity

Abstract

This note studies the Schrödinger–Choquard equation with an inhomogeneous combined source term and a fractional Laplacian operator

$$\begin{aligned} i\dot{u}-(-\Delta )^s u= \pm |x|^{b}|u|^{2(p-1)}u\pm (I_\alpha *|u|^{q})|u|^{q-2}u. \end{aligned}$$

Indeed, for $b\ne 0$ and $0<s<1$, one obtains a sharp threshold of global existence versus finite time blow-up dichotomy for mass-super-critical and energy sub-critical radial solutions.

Appendix

In this section, one proves Lemma 2.5. Let $\epsilon >0$ and $1+\frac{b\chi _{b>0}}{N-2s}<p< p^*$. Take $(u_n)$ a bounded sequence of $H^s_{rd}.$ Without loss of generality, one assumes that $(u_n)$ converges weakly to zero in $H^s.$ The purpose is to prove that $\Vert u_n\Vert _{L^{2p}(|x|^{b}\,\mathrm{{d}}x)}\rightarrow 0.$

• A. First case $-2s<b<0$. Using Hölder estimate, if $(q,q')$ satisfies $q'|b|<N$, one gets

  $$\begin{aligned} \int _{|x|\le \epsilon }|x|^{b}|u_n|^{2p}\,\mathrm{{d}}x\le & {} \Vert u_n\Vert _{2pq}^{2p}\Vert |x|^{b}\Vert _{q'}\\\le & {} C\Vert u_n\Vert _{2pq}^{2p}\int _0^\epsilon \frac{\mathrm{{d}}t}{t^{-q'b-N+1}}\\\le & {} C\Vert u_n\Vert _{2pq}^{2p}\epsilon ^{N+q'b}. \end{aligned}$$

  Now, since $p<p^*$, one has $N+\frac{Nb}{N-p(N-2s)}>0$. Taking $q:=\frac{N}{p(N-2s)}$ and using Sobolev injections, it follows that there exists $\mu >0$ such that

  $$\begin{aligned} \int _{|x|\le \epsilon }|x|^{b}|u_n|^{2p}\,\mathrm{{d}}x\le & {} C\Vert u_n\Vert _{H^s}^{2p}\epsilon ^{N+q'b}\\\le & {} C\epsilon ^{N+\frac{Nb}{N-p(N-2s)}}\\\le & {} C\epsilon ^\mu . \end{aligned}$$

  On the other hand, by Sobolev injections,

  $$\begin{aligned} \int _{|x|\ge \epsilon }|x|^{b}|u_n|^{2p}\,\mathrm{{d}}x \le \epsilon ^{b}\Vert u_n\Vert _{2p}^{2p}. \end{aligned}$$

  Since $\epsilon$ is arbitrary, one obtains $\int _{\mathbb R^N}|x|^{b}|u_n|^{2p}\,\mathrm{{d}}x\rightarrow 0$ as $n\rightarrow \infty .$ The proof is complete.

• B. Second case $b\ge 0$. Write, using Compact Sobolev injections

  $$\begin{aligned} \int _{|x|\le \epsilon }|x|^{b}|u_n|^{2p}\,\mathrm{{d}}x\le C\Vert u_n\Vert _{2p}^{2p}\rightarrow 0. \end{aligned}$$

  Moreover, with Strauss inequality and Rellich Theorem,

  $$\begin{aligned} \int _{\epsilon \le |x|\le \frac{1}{\epsilon }}|x|^{b}|u_n|^{2p}\,\mathrm{{d}}x\le C\Vert u_n\Vert _{L^\infty (\epsilon \le |x|\le \frac{1}{\epsilon })}^{2(p-1)}\int _{\epsilon \le |x|\le \frac{1}{\epsilon }}|u_n|^2\,\mathrm{{d}}x\rightarrow 0. \end{aligned}$$

  Now, with Strauss inequality

  $$\begin{aligned} \int _{|x|\ge \frac{1}{\epsilon }}|x|^{b}|u_n|^{2p}\,\mathrm{{d}}x&=\int _{|x|\ge \frac{1}{\epsilon }}|x|^{b-(p-1)(N-2s)}(|x|^{\frac{N-2s}{2}}|u_n|)^{2(p-1)}|u_n|^2\,\mathrm{{d}}x\\\le & {} C\int _{|x|\ge \frac{1}{\epsilon }}|x|^{b-(p-1)(N-2s)}|u_n|^2\,\mathrm{{d}}x\\\le & {} C\epsilon ^{(p-1)(N-2s)-b}. \end{aligned}$$

  The proof is achieved because ${b-(p-1)(N-2s)}<0.$