A Note on Inhomogeneous Coupled Schrödinger Equations

Abstract

It is the purpose of this note to study the inhomogeneous coupled Schrödinger system

$$\begin{aligned} i\dot{u}_j +\Delta u_j\pm |x|^b\left( \displaystyle \sum _{k=1}^{m}a_{jk}|u_k|^p\right) |u_j|^{p-2}u_j=0. \end{aligned}$$

Indeed, in the stationary case, the existence of ground states is proved and a sharp inhomogeneous Gagliardo–Nirenberg-type inequality is established. In the evolving case, the existence of global/non-global focusing solutions is investigated using the potential well method. Finally, the decay of defocusing global solutions is studied.

Appendix

This section is devoted to prove Lemma 2.17 and Proposition 2.20.

1.1 Compact Sobolev Injection

This subsection contains a proof of Lemma 2.17. Take $(u_n)$ a bounded sequence of $H_{rd}^1.$ Without loss of generality, one assumes that $(u_n)$ converges weakly to zero in $H_{rd}^1.$ Our purpose is to prove that $\Vert u_n\Vert _{L^{2p}(|x|^b\mathrm{d}x)}\rightarrow 0.$ Take $\epsilon >0$ and write

$$\begin{aligned} \int _{\mathbb {R}^N}|x|^b|u_n|^{2p}\,\mathrm{d}x= \left( \int _{|x|\le \epsilon }+ \int _{\epsilon \le |x|\le \frac{1}{\epsilon }}+\int _{|x|\ge \frac{1}{\epsilon }} \right) |x|^b|u_n|^{2p}\,\mathrm{d}x. \end{aligned}$$

By Strauss inequality and compact Sobolev injections,

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

On the other hand, with strauss inequality, via the fact that $-b+(N-1)(p-1)>0,$ one gets

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

Now, by the Rellich theorem, it follows that

$$\begin{aligned} \int _{\epsilon \le |x|\le \frac{1}{\epsilon }}|u_n|^{2}\,\mathrm{d}x\rightarrow 0\quad \text{ as }\quad n\rightarrow +\infty . \end{aligned}$$

Using Strauss inequality, one gets that

$$\begin{aligned} \int _{\epsilon \le |x|\le \frac{1}{\epsilon }}|x|^b|u_n|^{2p}\,\mathrm{d}x= & {} \int _{\epsilon \le |x|\le \frac{1}{\epsilon }}\left( |x|^{\frac{N-1}{2}}|u_n|\right) ^{2p-2}|x|^{b- (N-1)(p-1)}|u_n|^2\,\mathrm{d}x\\\le & {} C\Vert u_n\Vert _{H^1}^{2p-2}\int _{\epsilon \le |x|\le \frac{1}{\epsilon }}|x|^{b- (N-1)(p-1)}|u_n|^2\,\mathrm{d}x\\\le & {} C\Vert u_n\Vert _{H^1}^{2p-2} \left( \frac{1}{\epsilon }\right) ^{b- (N-1)(p-1)}\int _{\epsilon \le |x|\le \frac{1}{\epsilon }}|u_n|^2\,\mathrm{d}x\\\le & {} C\Vert u_n\Vert _{H^1}^{2p}{\epsilon }^{-b+(N-1)(p-1)}\int _{\epsilon \le |x|\le \frac{1}{\epsilon }}|u_n|^2\,\mathrm{d}x \rightarrow 0 \quad \text{ as }\quad n\rightarrow +\infty . \end{aligned}$$

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

1.2 Morawetz Estimate

This subsection is devoted to prove Proposition 2.20 about some classical Morawetz estimates satisfied by energy global solutions to the inhomogeneous coupled Schrödinger problem (1.1). Let us give an auxiliary result.

Lemma 10.1

Let $N\ge 3, b\ge 0$ and $1+\frac{b}{N-1}<p <p^*$ and $u\in C(\mathbb {R},\mathcal {H}_{rd})$ be a global solution to (1.1). Take $a:\mathbb {R}^N\rightarrow \mathbb {R}$ be a convex smooth function and define the real function

$$\begin{aligned} V:=\sum _{j=1}^m\int _{\mathbb {R}^N}a(x)|u_j(.,x)|^2\,\mathrm{d}x. \end{aligned}$$

Then, the following equality holds

$$\begin{aligned} V''= & {} -\int _{\mathbb {R}^N}\Delta ^2a|u|^2\,\mathrm{d}x+4\sum _{j=1}^m\int _{\mathbb {R}^N}\partial _l\partial _ka\mathfrak {R}(\partial _ku_j\partial _l\bar{u}_j)\,\mathrm{d}x\\&-\frac{2b}{p}\sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N}\nabla a.x|x|^{b-2}|u_k|^p|u_j|^{p}\,\mathrm{d}x\\&+2\left( 1-\frac{1}{p}\right) \sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N}\Delta a |x|^b|u_k|^p|u_j|^p\,\mathrm{d}x. \end{aligned}$$

Remark 10.2

The summation convention when indexes are repeated is used.

Proof

Denote the source term

$$\begin{aligned} \mathcal {N}:=-|x|^b\left( \displaystyle \sum _{k=1}^m a_{jk}|u_k|^p\right) |u_j|^{p-2}u_j. \end{aligned}$$

Multiplying Eq. (1.1) by $2\bar{u}$ and examining the imaginary parts, one gets

$$\begin{aligned} \partial _t (|u_j|^2) =-2\mathfrak {I}(\bar{u}_j \Delta u_j). \end{aligned}$$

Thus,

$$\begin{aligned} V'= & {} -2\int _{\mathbb {R}^N}a\mathfrak {I}(\bar{u}_j\Delta u_j)\,\mathrm{d}x\\= & {} 2\mathfrak {I}\int _{\mathbb {R}^N}(\partial _ka\partial _ku_j)\bar{u}_j \,\mathrm{d}x. \end{aligned}$$

Compute,

$$\begin{aligned} \partial _t\mathfrak {I}(\partial _k u_j\bar{u}_j)= & {} \mathfrak {I}(\partial _k\dot{u}_j\bar{u}_j)+\mathfrak {I}(\partial _k u_j\bar{\dot{u}_j})\\= & {} \mathfrak {R}(i\dot{u}_j\partial _k\bar{u}_j)-\mathfrak {R}(i\partial _k \dot{u}_j\bar{u}_j)\\= & {} \mathfrak {R}(\partial _k\bar{u}_j(-\Delta u_j-\mathcal {N}))-\mathfrak {R}(\bar{u}_j\partial _k(-\Delta u_j-\mathcal {N}))\\= & {} \mathfrak {R}(\bar{u}_j\partial _k\Delta u_j-\partial _k\bar{u}_j\Delta u_j)+\mathfrak {R}(\bar{u}_j\partial _k\mathcal {N}-\partial _k\bar{u}_j\mathcal {N}). \end{aligned}$$

Recall the identity

$$\begin{aligned} \frac{1}{2}\partial _k\Delta (|u_j|^2)-2\partial _l\mathfrak {R}(\partial _{k}u_j\partial _l\bar{u}_j)=\mathfrak {R}(\bar{u}_j\partial _k\Delta u_j-\partial _k\bar{u}_j\Delta u_j). \end{aligned}$$

Then,

$$\begin{aligned}&\int _{\mathbb {R}^N}\partial _ka\mathfrak {R}(\bar{u}_j\partial _k\Delta u_j-\partial _k\bar{u}_j\Delta u_j)\,\mathrm{d}x\\&\quad =\int _{\mathbb {R}^N}\partial _ka\left( \frac{1}{2}\partial _k\Delta (|u_j|^2)-2\partial _l\mathfrak {R}(\partial _ku_j\partial _l\bar{u}_j)\right) \,\mathrm{d}x\\&\quad =-\frac{1}{2}\int _{\mathbb {R}^N}\Delta ^2a|u_j|^2\,\mathrm{d}x+2\int _{\mathbb {R}^N}\partial _l\partial _ka\mathfrak {R}(\partial _ku_j\partial _l\bar{u}_j)\,\mathrm{d}x. \end{aligned}$$

On the other hand,

$$\begin{aligned} \int _{\mathbb {R}^N}\partial _ka\mathfrak {R}(\bar{u}_j\partial _k\mathcal {N}-\partial _k\bar{u}_j\mathcal {N})\,\mathrm{d}x= & {} \int _{\mathbb {R}^N}\partial _ka\mathfrak {R}(\partial _k[\bar{u}_j\mathcal {N}]-2\partial _k\bar{u}_j\mathcal {N})\,\mathrm{d}x\\= & {} \int _{\mathbb {R}^N}\left( -\Delta a\bar{u}_j\mathcal {N}-2\mathfrak {R}(\partial _ka\partial _k\bar{u}_j\mathcal {N})\right) \,\mathrm{d}x\\= & {} \sum _{k=1}^ma_{jk}\int _{\mathbb {R}^N}\Delta a |x|^b|u_k|^p|u_j|^p\,\mathrm{d}x-2\int _{\mathbb {R}^N}\mathfrak {R}(\partial _ka\partial _k\bar{u}_j\mathcal {N})\,\mathrm{d}x\\= & {} \sum _{k=1}^ma_{jk}\int _{\mathbb {R}^N}\Delta a |x|^b|u_k|^p|u_j|^p\,\mathrm{d}x\\&+\frac{2}{p} \displaystyle \sum _{k=1}^ma_{jk}\int _{\mathbb {R}^N}|x|^b\partial _i a\partial _i(|u_j|^p)|u_k|^p\,\mathrm{d}x. \end{aligned}$$

Moreover, with a symmetry argument

$$\begin{aligned} (A)&:=\sum _{k,j=1}^ma_{jk}\int _{\mathbb {R}^N}|x|^b\partial _i a\partial _i(|u_j|^p)|u_k|^p\,\mathrm{d}x\\&:=\frac{1}{2}\sum _{k,j=1}^ma_{jk}\int _{\mathbb {R}^N}|x|^b\partial _i a\partial _i(|u_j|^p|u_k|^p)\,\mathrm{d}x\\&=-\frac{1}{2}\sum _{k,j=1}^ma_{jk}\int _{\mathbb {R}^N}div(|x|^b\nabla a)|u_k|^p|u_j|^{p}\,\mathrm{d}x\\&=-\frac{1}{2}\sum _{k,j=1}^ma_{jk}\int _{\mathbb {R}^N}\left( \Delta a|x|^b|u_k|^p|u_j|^{p}\,\mathrm{d}x+b\nabla a.x|x|^{b-2}|u_k|^p|u_j|^{p}\right) \,\mathrm{d}x. \end{aligned}$$

Then,

$$\begin{aligned} V''= & {} -\int _{\mathbb {R}^N}\Delta ^2a|u|^2\,\mathrm{d}x+4\sum _{j=1}^m\int _{\mathbb {R}^N}\partial _l\partial _ka\mathfrak {R}(\partial _ku_j\partial _l\bar{u}_j)\,\mathrm{d}x\\&+2\sum _{k,j=1}^ma_{jk}\int _{\mathbb {R}^N}\Delta a |x|^b|u_k|^p|u_j|^p\,\mathrm{d}x+\frac{4}{p} \displaystyle \sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N}|x|^b\partial _i a\partial _i(|u_j|^p)|u_k|^p\,\mathrm{d}x\\= & {} -\int _{\mathbb {R}^N}\Delta ^2a|u|^2\,\mathrm{d}x+4\sum _{j=1}^m\int _{\mathbb {R}^N}\partial _l\partial _ka\mathfrak {R}(\partial _ku_j\partial _l\bar{u}_j)\,\mathrm{d}x\\&+2\sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N}\Delta a |x|^b|u_k|^p|u_j|^p\,\mathrm{d}x\\&-\frac{2}{p}\sum _{j,k=1}^ma_{jk}\left( \int _{\mathbb {R}^N}\Delta a|x|^b|u_k|^p|u_j|^{p}\,\mathrm{d}x+\int _{\mathbb {R}^N}b\nabla a.x|x|^{b-2}|u_k|^p|u_j|^{p}\,\mathrm{d}x\right) \\= & {} -\int _{\mathbb {R}^N}\Delta ^2a|u|^2\,\mathrm{d}x+4\sum _{j=1}^m\int _{\mathbb {R}^N}\partial _l\partial _ka\mathfrak {R}(\partial _ku_j\partial _l\bar{u}_j)\,\mathrm{d}x\\&-\frac{2b}{p}\sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N}\nabla a.x|x|^{b-2}|u_k|^p|u_j|^{p}\,\mathrm{d}x\\&+2\left( 1-\frac{1}{p}\right) \sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N}\Delta a |x|^b|u_k|^p|u_j|^p\,\mathrm{d}x. \end{aligned}$$

$\square$

Now, one proves the Morawetz estimate.

Proof of Proposition 2.20

Take the choice $a:=|\cdot |$. Then,

$$\begin{aligned} \nabla a=\frac{.}{|\cdot |},\quad \Delta a=\frac{N-1}{|\cdot |} \end{aligned}$$

and

$$\begin{aligned} -\Delta ^2a=\left\{ \begin{array}{ll} 4\pi (N-1)\delta _0,&{}\quad \text{ if }\quad N=3;\\ \frac{(N-1)(N-3)}{|\cdot |^3},&{}\quad \text{ if }\quad N\ge 4. \end{array} \right. \end{aligned}$$

Moreover, since a is convex, one gets

$$\begin{aligned} \partial _l\partial _ka\mathfrak {R}(\partial _ku_j\partial _l\bar{u}_j)\ge 0. \end{aligned}$$

Applying the previous lemma and using the fact that $p>1+\frac{b}{N-1}$, it follows that

$$\begin{aligned} V''\ge & {} -\frac{2b}{p}\sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N}\nabla a.x|x|^{b-2}|u_k|^p|u_j|^{p}\,\mathrm{d}x\\&+2\left( 1-\frac{1}{p}\right) \sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N}\Delta a |x|^b|u_k|^p|u_j|^p\,\mathrm{d}x\\\ge & {} -\frac{2b}{p}\sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N}|x|^{b-1}|u_k|^p|u_j|^{p}\,\mathrm{d}x+2(N-1)\left( 1-\frac{1}{p}\right) \\&\times \sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N} |x|^{b-1}|u_k|^p|u_j|^p\,\mathrm{d}x\\\ge & {} \left( -\frac{2b}{p}+2(N-1)\left( 1-\frac{1}{p}\right) \right) \sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N} |x|^{b-1}|u_k|^p|u_j|^p\,\mathrm{d}x\\ > rsim & {} \sum _{j,k=1}^ma_{jk}\int _{\mathbb {R}^N} |x|^{b-1}|u_k|^p|u_j|^p\,\mathrm{d}x. \end{aligned}$$

Integrating in time and using the defocusing sign, one gets

$$\begin{aligned} \sqrt{E(0)M(0)} > rsim & {} \Vert u\Vert _{L^\infty (\mathbb {R},L^2)}\Vert \nabla u(t)\Vert _{L^\infty (I,L^2)}\\ > rsim & {} \Vert V'\Vert _{L^\infty (\mathbb {R})}\\ > rsim & {} \sum _{j,k=1}^ma_{jk}\int _\mathbb {R}\int _{\mathbb {R}^N} |x|^{b-1}|u_k|^p|u_j|^p\,\mathrm{d}t\,\mathrm{d}x. \end{aligned}$$

This completes the proof. $\square$