Well-Posedness of a Class of Radial Inhomogeneous Hartree Equations

Abstract

The present paper investigates the following inhomogeneous generalized Hartree equation $i\dot{u}+\Delta u=\pm {\left|u\right|}^{p-2}{\left|x\right|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b)u$, where the wave function is $u:=u(t,x):\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{C}$, with $N\ge 2$. In addition, the exponent $b>0$ gives an unbounded inhomogeneous term ${\left|x\right|}^b$ and $I_{\alpha }\approx {|·|}^{-(N-\alpha )}$ denotes the Riesz-potential for certain $0<\alpha <N$. In this work, our aim is to establish the local existence of solutions in some radial Sobolev spaces, as well as the global existence for small data and the decay of energy sub-critical defocusing global solutions. Our results complement the recent work (Sharp threshold of global well-posedness versus finite time blow-up for a class of inhomogeneous Choquard equations, J. Math. Phys. 60 (2019), 081514). The main challenge in this work is to overcome the singularity of the unbounded inhomogeneous term ${\left|x\right|}^b$ for certain $b>0$.

1. Introduction

The Schrödinger equation was formulated in 1926 by the Austrian physicist Erwin Schrödinger [1], and it was one of the real bases for the development of quantum mechanics. It is a prototypical dispersive nonlinear partial differential equation that has been since derived in many areas of physics. The nonlinear Schrödinger equation appears as a model in hydrodynamics, nonlinear optics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena; see, for instance, Newell [2] and Scott–Chu–McLaughlin [3]. Moreover, it was suggested that stable high power propagation can be achieved in a plasma by sending a preliminary laser beam that creates a channel with a reduced electron density and, thus, reduces the nonlinearity inside the channel; see Gill [4] and Liu-Tripathi [5]. In this case, the beam propagation can be modeled by the inhomogeneous nonlinear Schrödinger equation.

In this note, we consider the Cauchy problem for an inhomogeneous Schrödinger equation

$$\left\{\begin{array}{c}i\dot{u}+\Delta u=ϵ{\left|u\right|}^{p-2}{\left|x\right|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b)u;\hfill \\ u(0,·)=u_0.\hfill \end{array}\right.$$

(1)

Here and hereafter, $N\ge 2$, and the wave function u is a complex valued function of the variable $(t,x)\in \mathbb{R}\times {\mathbb{R}}^N$. The real number $ϵ=\pm 1$ refers to the focusing or defocusing regime. Moreover $b\ge 0$ gives the unbounded inhomogeneous term ${|·|}^b$. Finally, $0<\alpha <N$ and the Riesz-potential is

$$I_{\alpha }(x):=\frac{\Gamma (\frac{N-\alpha }{2})}{\Gamma (\frac{\alpha }{2}){\pi }^{\frac{N}{2}}{2}^{\alpha }}{\left|x\right|}^{-(N-\alpha )},x\in {\mathbb{R}}^N.$$

The significance of Equation (1) is to model many important physical phenomena. For example, it describes the mean-field limit of large systems of non-relativistic Bosonic molecules [6,7]. It models also the propagation of plasma electromagnetic waves [8]. The fourth space dimensional case, the said Schrödinger–Poisson equation, arises in the theory of semi-conductor and quantum mechanics [9].

The Schrödinger problem with an inhomogeneous local source term, ${\left|x\right|}^b{\left|u\right|}^{p-1}u$, for $b>0$ was treated in [10]. Indeed, some well-posedness issues were investigated using the potential-well method. Later on, this problem was revisited by the author [11] in the fractional case. Indeed, the existence of global and non-global energy solutions were discussed using the ground state threshold.

The inhomogeneous generalized Hartree problem (1) was first considered by Alharbi, G. M. et al. [12], where the authors only considered the case $b<0$. Indeed, the global existence versus the blow-up of solutions was obtained. In addition, the scattering under the ground state threshold with spherically symmetric data was proven in [13], and it was extended to the non-radial regime in [14,15].

This paper complements the recent work by Alharbi, G. M. et al. [12], in which the authors only considered Equation (1) for $b<0$. The method used in [12] consists of the division of the integrals on the unit ball of ${\mathbb{R}}^N$ and its complementary fails for $b>0$ because ${\left|x\right|}^b\notin L^r(|x|>1)$ for any $r\ge 1$. Instead, we skillfully use the Strauss estimate ${\left|x\right|}^{\frac{N-1}{2}}\left|u(x)\right|\lesssim {\parallel u\parallel }_{H^1}$ for any function $u\in H_{rd}^1({\mathbb{R}}^N)$. Consequently, we are able to prove local well-posedness in the energy spaces with a radial setting and the global existence for small data. Furthermore, the energy defocusing energy global solution decays for a long time in some Lebesgue spaces. However, in this work, the question regarding the existence of standing waves, which describes the threshold of global existence versus the finite time blow-up of energy solutions, is not discussed. This is considered in a paper in progress.

This paper is organized as follows. We introduce our main results and other components associated with our work in Section 2. In Section 3, we prove the local well-posedness of (1). In Section 4, the global well-posedness for the small data of (1) in the energy space is obtained. In Section 5, the decay of defocusing global solutions is established. Finally, a Morawetz estimate and some Gagliardo–Nirenberg inequalities are proven in the Appendix A.

We mention that C (respectively, $C_T$) denotes a constant (respectively, a constant depending on T) that may vary from line to line, and if A and B are non-negative real numbers, $A\lesssim B$ means that $A\le CB$.

Here and hereafter, we denote, for short, some Lebesgue and Sobolev spaces endowed with the classical norms

$$\begin{array}{cc}\hfill L^r& :=L^r({\mathbb{R}}^N),H^1:=H^1({\mathbb{R}}^N),H_{rd}^1:=\{f\in H^1,f(·)=f(|·|)\};\hfill \\ \hfill {\parallel ·\parallel }_r& :={\parallel ·\parallel }_{L^r},\parallel ·\parallel :={\parallel ·\parallel }_2,{\parallel ·\parallel }_{H^1}:={\left({\parallel ·\parallel }^2+{\parallel \nabla ·\parallel }^2\right)}^{\frac{1}{2}}.\hfill \end{array}$$

Finally, $T^{\ast }>0$ is the maximal existence time of an eventual solution to (1). If $x\in \mathbb{R}$, we denote by $x^{+}$ a real number close to x such that $x^{+}>x$. The critical Sobolev embedding index is denoted by $2^{\ast }:=\frac{2N}{N-2}$.

2. Background Material

This section contains the main results and some useful estimates.

2.1. Preliminary

The Hartree Equation (1) employs the scaling invariance

$$u_{\lambda }(t,x):={\lambda }^{\frac{2+2b+\alpha }{2(p-1)}}u({\lambda }^{2}t,\lambda x),\lambda >0.$$

The following critical exponent,

$$s_c:=\frac{N}{2}-\frac{2+2b+\alpha }{2(p-1)},$$

is the only one keeping invariant the Sobolev norm

$$\parallel u_{\lambda }{(t)\parallel }_{{\dot{H}}^{s_c}}={\parallel u({\lambda }^{2}t)\parallel }_{{\dot{H}}^{s_c}}.$$

Therefore, the Schrödinger problem (1) is said to be $H^s$ critical if $s=s_c$; it is $H^s$ sub-critical if $s>s_c$ and $H^s$ super-critical if $s<s_c$. Let us define the real numbers related to Equation (1).

$$\begin{array}{cc}\hfill B_s& :=\frac{N}{s}(p-1-\frac{\alpha +2b}{N}),A_s:=2p-B_s;\hfill \\ \hfill B& :=B_1,A:=A_1,p_s^c:=1+\frac{2+\alpha +2b}{N-2s};\hfill \\ \hfill p_c& :=1+\frac{\alpha }{N}+\frac{2b}{N-1},p^c:=1+\frac{2+\alpha }{N-2}+\frac{2b}{N-1}.\hfill \end{array}$$

If $a:{\mathbb{R}}^N\to \mathbb{R}$ is a smooth function, we define the variance potential and the Morawetz action as follows

$$\begin{array}{cc}\hfill V_a& :={\int }_{{\mathbb{R}}^N}a(x){\left|u(.,x)\right|}^{2}dx;\hfill \\ \hfill M_a& :=2\Im {\int }_{{\mathbb{R}}^N}\overline{u}(a_{j}u)dx=2\Im {\int }_{{\mathbb{R}}^N}\overline{u}(\nabla a\nabla u)dx.\hfill \end{array}$$

In a standard way, the repeated indices are summed, and subscripts denote the partial derivatives. Finally, let us take the differential operator

$$〈\nabla 〉u:=u+\nabla u.$$

The next sub-section contains the contribution of this note.

2.2. Main Results

The first result deals with the local well-posedness of the problem (1) in the radial energy space.

Theorem 1.

Let $N\ge 2$, $b>0$, $N<4+\alpha <4+N$ and $u_0\in H_{rd}^1$. Assume that $0<b\le 1$ satisfies $1+\frac{b+\alpha }{N}<\frac{2+\alpha }{N-2}+\frac{b}{N-1}$ and $max\{2+\frac{2b}{N-1},2+\frac{b}{N-1}+\frac{b+\alpha }{N}\}<p<p^c$ or $b>1$ and $max\{2+\frac{2b}{N-1},1+\frac{2b-1}{N-1}+\frac{\alpha }{N}\}<p<p^c$. Then, there exists $T=T(\parallel u_0{\parallel }_{H^1})>0$ and $u\in C_T(H_{rd}^1)$ to be a unique local solution to (1). Moreover,

1.

the solution satisfies the following conservation laws

$$\begin{array}{cc}\hfill Mass& :=M(u(t)):={\int }_{{\mathbb{R}}^N}{\left|u(t,x)\right|}^{2}dx=M(u_0);\hfill \\ \hfill Energy& :=E(u(t)):={\int }_{{\mathbb{R}}^N}\left({|\nabla u(t)|}^2+\frac{ϵ}{p}{\left|x\right|}^b(I_{\alpha }{\ast |·|}^b{\left|u(t)\right|}^p{)|u(t)|}^p\right)dx=E(u_0);\hfill \end{array}$$

2.

$u\in L_{loc}^q((0,T),W^{1,r})$, for all admissible pairs $(q,r)$ in the meaning of Definition 1;

3.

if $ϵ=1$, then u is global.

In view of the results stated in the above theorem, some comments are in order.

• Using Proposition 4 via the Hardy–Littlewood–Sobolev inequality, we obtain

  $${\int }_{{\mathbb{R}}^N}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^p{\left|x\right|}^{b}dx\lesssim {\parallel |·|}^b{\left|u\right|}^p{\parallel }_{\frac{2N}{\alpha +N}}^2\lesssim {\parallel u\parallel }_{H^1}^{2p},$$

  when $p_c\le p\le p^c$, and the energy is well-defined for $u\in H_{rd}^1$. Therefore, it seems that the previous theorem also holds;
• the assumption $1+\frac{b+\alpha }{N}<\frac{2+\alpha }{N-2}+\frac{b}{N-1}$ is possible, at least for low space dimensions;
• the presence of the derivative term ${\left|x\right|}^{b-1}$ gives the dichotomy $b<1$ and $b\ge 1$.

The second result deals with the local well-posedness of the Hartree problem (1) in the Sobolev space $H_{rd}^s$.

Theorem 2.

Let $N\ge 2$, $b>0$, $s\in (\frac{1}{2},1)$, $0<\alpha <N<4s+\alpha$ and $u_0\in H_{rd}^s$. Assume that $0<b\le s$ and $max\{2+\frac{2b}{N-2s},1+\frac{\alpha +b}{N}+\frac{b}{N-2s}\}<p<1+\frac{2s+2b+\alpha }{N-2s}$ or $b>s$, $N<3s+\alpha$ and $2+\frac{2b}{N-2s}<p<1+\frac{s+2b+\alpha }{N-2s}$. Then, there exists $T=T(\parallel u_0{\parallel }_{H^s})$ and $u\in C_T(H_{rd}^s)$ to be a unique maximal solution to (1). Moreover,

1.

the solution satisfies the mass conservation law;

2.

$u\in L_{loc}^q((0,T),W^{s,r})$, for all admissible pairs $(q,r)$;

3.

if $ϵ=1$, then u is global.

Remark 1.

The difference between the local existence in $H_{rd}^1$ and $H_{rd}^s$ for $0<s<1$ is due to the use of some different Strauss inequalities stated in Proposition 3.

The third result deals with the global well-posedness of the Schrödinger problem (1) for small data.

Theorem 3.

Take the assumptions of Theorem 1 and $A>0$, such that if $\parallel u_0{\parallel }_{H^1}<A$. Then, there exists $\delta :=\delta (A)>0$ such that if $\parallel e^{i·\Delta }{u}_0{\parallel }_{S^{s_c}}<\delta$, then, the solution to (1) is global and scatters in $H^1$. Moreover, it satisfies

$${\parallel u\parallel }_{S^{s_c}(\mathbb{R})}<2\parallel e^{i·\Delta }{u}_0{\parallel }_{S^{s_c}(\mathbb{R})}and\parallel 〈\nabla 〉{u\parallel }_{S(\mathbb{R})}<c{\parallel u_0\parallel }_{H^1}.$$

Let us state some comments about the above theorem.

• The Strichartz norms ${\parallel ·\parallel }_{S(\mathbb{R})}$ and ${\parallel ·\parallel }_{S^{s_c}(\mathbb{R})}$ are defined in Definition 1;
• in the defocusing regime, any local solution is global; in the focusing regime, a local solution may be non-global, but for small data, this scenario cannot happen.
• the scattering means that there exists $u_{\pm }\in H_{rd}^1$, such that

  $$\underset{t\to \pm \infty }{lim}{\parallel u-e^{i·\Delta }{u}_{\pm }\parallel }_{H^1}=0.$$

The fourth result deals with the decay of global defocusing solutions.

Theorem 4.

Let $N\ge 3$, $ϵ=1$ and $b\in [0,1]$. Take the conditions of Theorem 1. Then, the global solution to (1), denoted by $u\in C(\mathbb{R},H_{rd}^1)$, satisfies for any $2<r<2^{\ast }$,

$$\underset{t\to \infty }{lim}{\parallel u(t)\parallel }_r=0.$$

Let us give some remarks about the above result.

• the assumptions $b\le 1$ and $N\ge 3$ allow the use of the Morawetz estimate (2);
• the scattering of energy global solutions is stronger than the decay but not available in the mass-sub-critical regime;
• in a future work, the authors will investigate the energy scattering of global solutions to the Hartree problem (1).

The next variance identity is established in Appendix A.

Proposition 1.

Take $ϵ=1$ and the assumptions of Theorem 1. Let a local solution to (1) be denoted as $u\in C_T(H^1)$. Then, holds on $[0,T]$ are represented as

$$\begin{array}{ccc}\hfill V_a^{″}=M_a^{\prime }& =& 4{\int }_{{\mathbb{R}}^N}{\partial }_l{\partial }_{k}a\Re ({\partial }_{k}u{\partial }_l\overline{u})dx-{\int }_{{\mathbb{R}}^N}{\Delta }^{2}a{\left|u\right|}^{2}dx\hfill \\ & -& 2(\frac{2}{p}-1){\int }_{{\mathbb{R}}^N}{\Delta a\left|x\right|}^b{\left|u\right|}^p(I_{\alpha }{\ast |·|}^b{\left|u\right|}^p)dx\hfill \\ & -& \frac{4b}{p}{\int }_{{\mathbb{R}}^N}{x\nabla a\left|x\right|}^{b-2}{\left|u\right|}^p(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b)dx\hfill \\ & -& \frac{4}{p}(\alpha -N){\int }_{{\mathbb{R}}^N}{\left|x\right|}^b{\left|u\right|}^p\nabla a(\frac{·}{{|·|}^2}{I}_{\alpha }{\ast |·|}^b{\left|u\right|}^p)dx.\hfill \end{array}$$

Moreover, if $N\ge 3$, it follows that

$$V_{|·|}^{″}\gtrsim {\int }_{{\mathbb{R}}^N}{\left|x\right|}^{b-1}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p}dx.$$

(2)

Remark 2.

The previous Morawetz estimate is an essential tool in the proof of the decay of defocusing energy global solutions to (1).

Finally, the next Gagliardo–Nirenberg type inequalities adapted to the Schrödinger Equation (1) are established in the Appendix A.

Proposition 2.

Let $N\ge 2$, $b>0$ and $s\in (\frac{1}{2},1)$.

1.

If $p_c<p<p^c$, there exists $C_{N,p,b,\alpha }>0$, such that for any $u\in H_{rd}^1$,

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^p{\left|x\right|}^{b}dx\le C_{N,p,b,\alpha }{\parallel u\parallel }^A{\parallel \nabla u\parallel }^B;$$

2.

if $1+\frac{\alpha }{N}+\frac{2b}{N-2s}<p<p_s^c$, there exists $C_{N,p,b,\alpha ,s}>0$, such that for any $u\in H_{rd}^s$,

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^p{\left|x\right|}^{b}dx\le C_{N,p,b,\alpha ,s}{\parallel u\parallel }^{A_s}{\parallel {(-\Delta )}^{\frac{s}{2}}u\parallel }^{B_s}.$$

2.3. Tools

This sub-section collects some useful estimates. First, we recall a Hardy–Littlewood–Sobolev estimate [11,16].

Lemma 1.

Let $N\ge 1$ and $0<\nu <1<s,r,q<\infty$.

1.

If $2=\frac{1}{r}+\frac{1}{s}+\frac{\nu }{N}$, then

$${\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|x-y|}^{-\nu }f(x)g(y)dxdy\le C_{N,\nu ,s}{\parallel f\parallel }_r{\parallel g\parallel }_s,foranyf\in L^r,g\in L^s;$$

2.

if $1+\frac{\nu }{N}=\frac{1}{q}+\frac{1}{r}+\frac{1}{s}$, then

$$\parallel (I_{\nu }\ast f){g\parallel }_{r^{\prime }}\le C_{N,\nu ,s}{\parallel f\parallel }_s{\parallel g\parallel }_q,foranyf\in L^s,g\in L^q.$$

The next Gagliardo–Nirenberg inequality [17] is useful.

Lemma 2.

Let $N\ge 2$, $s\in (0,1)$ and $2\le p\le \frac{2N}{N-2s}$. Then

$${\parallel u\parallel }_p\lesssim {\parallel u\parallel }^{1-\frac{N}{s}(\frac{1}{2}-\frac{1}{p})}{\parallel {(-\Delta )}^{\frac{s}{2}}u\parallel }^{\frac{N}{s}(\frac{1}{2}-\frac{1}{p})},foranyu\in H^s.$$

The radial assumption enables us to use the next Strauss inequalities [18].

Proposition 3.

Let $N\ge 2$ and $\frac{1}{2}<s<\frac{N}{2}$. Then

1.

there is $C_N>0$, such that for any $u\in H_{rd}^1,$

$${\left|x\right|}^{\frac{N-1}{2}}\left|u(x)\right|\le C_N{\parallel \nabla u\parallel }^{\frac{1}{2}}{\parallel u\parallel }^{\frac{1}{2}},fora.ex\in {\mathbb{R}}^N;$$

2.

there is $C(N,s)>0$, such that for any $u\in {\dot{H}}^s({\mathbb{R}}^N)$,

$$\underset{x\ne 0}{sup}{\left|x\right|}^{\frac{N}{2}-s}{\left|u(x)\right|\le C(N,s)\parallel (-\Delta )}^{\frac{s}{2}}u\parallel .$$

(3)

The following inhomogeneous Gagliardo–Nirenberg-type inequalities are essential in this work [10,11].

Proposition 4.

Let $N\ge 2$, $s\in (0,1)$ and $b\ge 0$. Then

1.

if $2+\frac{2b}{N-1}\le p\le 2(\frac{N}{N-2}+\frac{b}{N-1})$, there exists $C_{N,b,p}>0$, such that for all $u\in H_{rd}^1,$

$${\int }_{{\mathbb{R}}^N}{\left|x\right|}^b{\left|u(x)\right|}^{p}dx\le C_{N,b,p}{\parallel \nabla u\parallel }^{\frac{N(p-2)-2b}{2}}{\parallel u\parallel }^{\frac{2p-(N(p-2)-2b)}{2}};$$

2.

if $2\le p+1-\frac{2b}{N-2s}\le \frac{2N}{N-2s}$, there exists $C_{N,p,b,s}>0$, such that for any $u\in H_{rd}^s$,

$${\int }_{{\mathbb{R}}^N}{\left|x\right|}^b{\left|u\right|}^{1+p}dx\le C_{N,p,b,s}{\parallel u\parallel }^{1+p-\frac{N(p-1)-2b}{2s}}{\parallel {(-\Delta )}^{\frac{s}{2}}u\parallel }^{\frac{N(p-1)-2b}{2s}}.$$

Let us recall a useful fractional chain rule [19].

Lemma 3.

Let $G\in C^1(\mathbb{C})$, $0<s\le 1$ and $1<p,p_1,p_2<\infty$, such that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}<\infty$. Thus,

$${\parallel (-\Delta )}^{\frac{s}{2}}{G(u)\parallel }_p{\lesssim \parallel (-\Delta )}^{\frac{s}{2}}{u\parallel }_{p_2}{\parallel G^{\prime }(u)\parallel }_{p_1}.$$

The Strichartz estimate [20] is an essential tool to estimate an eventual solution to (1) in Sobolev spaces.

Definition 1.

Take $N\ge 2$ and $s\in [0,1)$. A pair of real numbers $(q,r)$ is s-admissible if

$$\begin{array}{c}2\le q,r\le \infty ;\\ \frac{2N}{N-2s}\le r<2^{\ast };\\ N(\frac{1}{2}-\frac{1}{r})=\frac{2}{q}+s.\end{array}$$

Take the set ${\Gamma }_s:=\{(q,r)$, s-admissible }, $\Gamma :={\Gamma }_0$ and the Strichartz norms

$${\parallel ·\parallel }_{S^s(\mathbb{R})}:=\underset{(q,r)\in {\Gamma }_s}{sup}{\parallel ·\parallel }_{L^q(\mathbb{R},L^r)}{,\parallel ·\parallel }_{S(\mathbb{R})}:={\parallel ·\parallel }_{S^0(\mathbb{R})}.$$

Proposition 5.

Let $N\ge 2$, $s\in [0,1)$, $(q,r)\in {\Gamma }_s,(\tilde{q},\tilde{r})\in {\Gamma }_{-s}$ and a time slab $I\subset \mathbb{R}$. Then

1.

${sup}_{(q,r)\in {\Gamma }_s}\parallel e^{i·\Delta }{f\parallel }_{L^q(I,L^r)}\lesssim {\parallel f\parallel }_{{\dot{H}}^s}$;

2.

$\parallel {\int }_0^{·}{e}^{i(·-s)\Delta }{h(·,s)ds\parallel }_{L^q(I,L^r)}\lesssim {\parallel h\parallel }_{L^{{\tilde{q}}^{\prime }}(I,L^{{\tilde{r}}^{\prime }})}.$

3. Well-Posedness

In what follows, we establish the energy local well-posedness of the non-linear Hartree problem (1).

3.1. $H^1$-Theory

The proof proceeds with a standard fixed point Picard theorem. Take $T>0$ and $R>c\parallel u_0{\parallel }_{H^1}$ and the distance

$$d(v,u):={\parallel u-v\parallel }_{S_T({\mathbb{R}}^N)}:=\underset{(q,r)\in \Gamma }{sup}{\parallel u-v\parallel }_{L_T^q(L^r({\mathbb{R}}^N))},$$

on the space

$$B_T(R):=\{u\in C_T(H_{rd}^1)\bigcap _{(q,r)\in \Gamma }{L}_T^q(W^{1,r}),{\parallel u\parallel }_{S_T({\mathbb{R}}^N)}\le R\}.$$

Define

$$\varphi (u):=e^{i.\Delta }{u}_0-{\int }_0^{.}{e}^{i(.-\tau )\Delta }{\left[\right|x|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p-2}u]d\tau .$$

Using Strichartz estimates, we obtain the following for $(q,r)\in \Gamma$ and $u,v\in B_T(R)$:

$$\begin{array}{ccc}\hfill d\left(\varphi (v),\varphi (u)\right)& \lesssim & {\parallel \left|x\right|}^b(I_{\alpha }{\ast \left[\right|·|}^b{\left|u\right|}^p{-|·|}^b{\left|v\right|}^p{])\left|u\right|}^{p-1}{\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ & +& {\parallel \left|x\right|}^b(I_{\alpha }{\ast |·|}^b{\left|v\right|}^p{)(\left|u\right|}^{p-2}u-{\left|v\right|}^{p-2}v{)\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ & \lesssim & (I)+(II).\hfill \end{array}$$

Since $\alpha >0$ and $r\ge 2$, we take two positive real numbers $a,c>0$, satisfying $1+\frac{\alpha }{N}=\frac{2}{r}+\frac{1}{a}+\frac{1}{c}$. Using Hardy–Littlewood–Sobolev and Hölder estimates, we obtain

$$\begin{array}{ccc}\hfill (II)& \lesssim & {\parallel \left|x\right|}^b(I_{\alpha }{\ast |·|}^b{\left|v\right|}^p{)(\left|v\right|}^{p-2}+{\left|u\right|}^{p-2}{)(u-v)\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ & \lesssim & {\parallel \parallel \left|x\right|}^b{\left|v\right|}^p{\parallel }_c{\parallel \left|x\right|}^b{(|v|}^{p-2}+{\left|u\right|}^{p-2}{)\parallel }_a{\parallel u-v\parallel }_r{\parallel }_{L^{q^{\prime }}(0,T)}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{\parallel \left|x\right|}^b{\left|v\right|}^p{\parallel }_{L_T^{\infty }(L^c)}{\parallel \left|x\right|}^b{(|v|}^{p-2}+{\left|u\right|}^{p-2}{)\parallel }_{L_T^{\infty }(L^a)}{\parallel u-v\parallel }_{L_T^q(L^r)}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{\parallel \left|x\right|}^b{\left|v\right|}^p{\parallel }_{L_T^{\infty }(L^c)}\left(\parallel {\left|u\right|}^{p-2}{\left|x\right|}^b{\parallel }_{L_T^{\infty }(L^a)}+{\parallel \left|x\right|}^b{\left|v\right|}^{p-2}{\parallel }_{L_T^{\infty }(L^a)}\right)d(v,u).\hfill \end{array}$$

Using the Gagliardo–Nirenberg inequality in Proposition 4, we write

$$\begin{array}{c}{\parallel |·|}^b{\left|v\right|}^p{\parallel }_{L_T^{\infty }(L^c)}\lesssim {\parallel v\parallel }_{H^1}^p;\\ {\parallel |·|}^b{\left|v\right|}^{p-2}{\parallel }_{L_T^{\infty }(L^a)}\lesssim {\parallel v\parallel }_{H^1}^{p-2},\end{array}$$

under the conditions $p>2+\frac{2b}{N-1}$ and

$$\begin{array}{c}\frac{2}{p-\frac{2b}{N-1}}\le c\le 2^{\ast }\frac{1}{p-\frac{2b}{N-1}};\\ \frac{2}{p-2-\frac{2b}{N-1}}\le a\le 2^{\ast }\frac{1}{p-2-\frac{2b}{N-1}}.\end{array}$$

This gives the condition

$$\frac{1}{a}+\frac{1}{c}\in \left[\frac{N-2}{N}\left(p-\frac{2b}{N-1}-1\right),p-\frac{2b}{N-1}-1\right].$$

Moreover, the equality $\frac{1}{a}+\frac{1}{c}=1+\frac{\alpha }{N}-\frac{2}{r}$, via the admissibility condition $2\le r<2^{\ast }$, gives

$$\frac{1}{a}+\frac{1}{c}\in \left[\frac{\alpha }{N},\frac{2+\alpha }{N}\right).$$

This is satisfied if

$$\frac{\alpha }{N}<p-\frac{2b}{N-1}-1<\frac{2+\alpha }{N-2}.$$

These conditions are verified because

$$2+\frac{2b}{N-1}<p<p^c.$$

Thus,

$$\begin{array}{ccc}\hfill (II)& \lesssim & T^{1-\frac{2}{q}}{\parallel v\parallel }_{L_T^{\infty }(H^1)}^p\left({\parallel u\parallel }_{L_T^{\infty }(H^1)}^{p-2}+{\parallel v\parallel }_{L_T^{\infty }(H^1)}^{p-2}\right)d(v,u).\hfill \end{array}$$

Let us estimate the term $(I)$. Since $\alpha >0$ and $r\ge 2$, we take positive real numbers $e>0$ satisfying $1+\frac{\alpha }{N}=\frac{2}{r}+\frac{2}{e}$. Due to the Hardy–Littlewood–Sobolev and Hölder estimates, it follows that

$$\begin{array}{ccc}\hfill (I)& \lesssim & {\parallel \left|x\right|}^b(I_{\alpha }{\ast \left[\right|·|}^b{(|v|}^{p-1}+{\left|u\right|}^{p-1}{)(u-v)])|u|}^{p-1}{\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ & \lesssim & \left({\parallel \left|u\right|}^{p-1}{\left|x\right|}^b{\parallel }_e+{\parallel \left|v\right|}^{p-1}{\left|x\right|}^b{\parallel }_e\right){\parallel \left|u\right|}^{p-1}{\left|x\right|}^b{\parallel }_e{\parallel u-v\parallel }_r{\parallel }_{L^{q^{\prime }}(0,T)}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}\left({\parallel \left|u\right|}^{p-1}{\left|x\right|}^b{\parallel }_{L_T^{\infty }(L^e)}+{\parallel \left|v\right|}^{p-1}{\left|x\right|}^b{\parallel }_{L_T^{\infty }(L^e)}\right){\parallel \left|v\right|}^{p-1}{\left|x\right|}^b{\parallel }_{L_T^{\infty }(L^e)}{\parallel u-v\parallel }_{L_T^q(L^r)}.\hfill \end{array}$$

Using the Gagliardo–Nirenberg inequality in Proposition 4, we write

$$\begin{array}{c}{\parallel |·|}^b{\left|v\right|}^{p-1}{\parallel }_{L_T^{\infty }(L^e)}\lesssim {\parallel v\parallel }_{H^1}^{p-1};\\ {\parallel |·|}^b{\left|u\right|}^{p-1}{\parallel }_{L_T^{\infty }(L^e)}\lesssim {\parallel u\parallel }_{H^1}^{p-1},\end{array}$$

under the conditions $p>1+\frac{2b}{N-1}$ and

$$\begin{array}{c}\frac{2}{p-\frac{2b}{N-1}-1}\le e\le 2^{\ast }\frac{1}{p-\frac{2b}{N-1}-1}.\end{array}$$

This gives the condition

$$\frac{2}{e}\in \left[\frac{N-2}{N}\left(p-\frac{2b}{N-1}-1\right),p-\frac{2b}{N-1}-1\right].$$

Moreover, the equality $\frac{2}{e}=1+\frac{\alpha }{N}-\frac{2}{r}$, via the admissibility condition $2\le r<2^{\ast }$, gives

$$\frac{2}{e}\in \left[\frac{\alpha }{N},\frac{2+\alpha }{N}\right).$$

This is satisfied as above because

$$2+\frac{2b}{N-1}<p<p^c.$$

Thus, there exists $\delta >0$ such that

$$d\left(\varphi (v),\varphi (u)\right)\lesssim T^{\frac{1}{\delta }}{R}^{2(p-1)}d(v,u).$$

Moreover,

$${\parallel \varphi (u)\parallel }_{S_T({\mathbb{R}}^N)}\le C\parallel u_0\parallel +C{T}^{\frac{1}{\delta }}{R}^{2p-1}.$$

Now, let us estimate the term ${\parallel \nabla \left[\varphi (u)\right]\parallel }_{S_T({\mathbb{R}}^N)}$. Compute

$$\begin{array}{ccc}\hfill {|\nabla (\left|x\right|}^b(I_{\alpha }{\ast |·|}^b{\left|u\right|}^p{)|u|}^{p-2}u)|& \lesssim & {\left|\right|x|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)\nabla u|u|}^{p-2}{|+|\left|x\right|}^b(I_{\alpha }{\ast |·|}^b\Re (\nabla u\overline{u}){\left|u\right|}^{p-2}{)|u|}^{p-1}|\hfill \\ & +& {\parallel \left|x\right|}^{b-1}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p-1}{|+|\left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-1}{\left|u\right|}^p{)|u|}^{p-1}|.\hfill \end{array}$$

Taking account of the Strichartz estimates and arguing as previously, we obtain

$$\begin{array}{ccc}\hfill {\parallel \nabla \varphi (u)\parallel }_{S_T({\mathbb{R}}^N)}-C\parallel \nabla u_0\parallel & \lesssim & T^{\frac{1}{\delta }}{R}^{2p-1}+{\parallel \left|u\right|}^{p-1}{\left|x\right|}^{b-1}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)\parallel }_{S_T^{\prime }({\mathbb{R}}^N)}\hfill \\ & +& {\parallel \left|u\right|}^{p-1}{\left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-1}{\left|u\right|}^p{)\parallel }_{S_T^{\prime }({\mathbb{R}}^N)}.\hfill \end{array}$$

For the above, we discuss two regimes.

1.

First case: $b\in [0,1]$.

Take the real numbers $\gamma :={\left(\frac{N}{1-b}\right)}^{+}$ and

$$1+\frac{\alpha }{N}=\frac{1}{r}+\frac{1}{\gamma }+\frac{1}{a}+\frac{p-2}{r}+\frac{1}{r}-\frac{1}{N}$$

and $q>2$ such that $(q,r)\in \Gamma$. Therefore, according to the Hölder estimate, we obtain

$$\begin{array}{ccc}\hfill {\parallel \left|u\right|}^{p-1}{\left|x\right|}^{b-1}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)\parallel }_{L^{r^{\prime }}(|x|<1)}& \le & {\parallel |·|}^{b-1}{\parallel }_{L^{\gamma }(|x|<1)}{\parallel |·|}^b{\left|u\right|}^p{\parallel }_a{\parallel u\parallel }_r^{p-2}{\parallel u\parallel }_{\frac{Nr}{N-r}}\hfill \\ & \lesssim & {\parallel u\parallel }_{H^1}^p{\parallel u\parallel }_r^{p-2}{\parallel u\parallel }_{\frac{Nr}{N-r}},\hfill \end{array}$$

under the condition

$$\frac{1}{a}\in \left[\frac{N-2}{2N}(p-\frac{2b}{N-1}),\frac{1}{2}(p-\frac{2b}{N-1})\right].$$

The equality $\frac{1}{a}=1+\frac{1+\alpha }{N}-\frac{1}{\gamma }-\frac{p}{r}$, given via the condition $2<r<2^{\ast }$, is

$$\frac{1}{a}\in \left(1+\frac{b+\alpha }{N}-\frac{p}{2},1+\frac{b+\alpha }{N}-\frac{(N-2)p}{2N}\right).$$

This is satisfied if

$$1+\frac{b+\alpha }{N}<p-\frac{b}{N-1}<\frac{N}{N-2}(1+\frac{b+\alpha }{N}).$$

This is satisfied because $p_c<p<p^c$. Then

$$\begin{array}{ccc}\hfill {\parallel \left|x\right|}^{b-1}{\left|u\right|}^{p-1}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)\parallel }_{S_T^{\prime }(|x|<1)}& \lesssim & {\parallel \parallel u\parallel }_{H^1}^p{\parallel u\parallel }_r^{p-2}{\parallel u\parallel }_{\frac{Nr}{N-r}}{\parallel }_{L^q(0,T)}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{\parallel u\parallel }_{L_T^{\infty }(H^1)}^p{\parallel u\parallel }_{L_T^{\infty }(L^r)}^{p-2}{\parallel u\parallel }_{L_T^q(L^{\frac{Nr}{N-r}})}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{\parallel u\parallel }_{L_T^{\infty }(H^1)}^{2(p-1)}{\parallel u\parallel }_{L_T^q(W^{1,r})}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{R}^{2p-1}.\hfill \end{array}$$

Moreover, with the Hölder estimate, for $\mu :={\left(\frac{N}{1-b}\right)}^{+}$ and

$$1+\frac{\alpha }{N}=\frac{1}{r}+\frac{1}{\mu }+\frac{1}{a}+\frac{p-1}{r}+\frac{1}{r}-\frac{1}{N},$$

we have

$$\begin{array}{ccc}\hfill {\parallel \left|u\right|}^{p-1}{\left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-1}{\left|u\right|}^p{)\parallel }_{L^{r^{\prime }}(|x|<1)}& \le & {\parallel |·|}^b{\left|u\right|}^{p-1}{\parallel }_a{\parallel |·|}^{b-1}{\parallel }_{L^{\mu }(|x|<1)}{\parallel u\parallel }_r^{p-1}{\parallel u\parallel }_{\frac{Nr}{N-r}}\hfill \\ & \lesssim & {\parallel u\parallel }_{H^1}^{p-1}{\parallel u\parallel }_r^{p-1}{\parallel u\parallel }_{\frac{Nr}{N-r}},\hfill \end{array}$$

under the condition

$$\frac{1}{a}\in \left(\frac{N-2}{2N}(p-\frac{2b}{N-1}-1),\frac{1}{2}(p-\frac{2b}{N-1}-1)\right).$$

Moreover, the assumption $2\le r<2^{\ast }$ gives

$$\frac{1}{a}=1+{\left(\frac{b+\alpha }{N}\right)}^{+}-\frac{p-1}{r}\in \left(1+\frac{b+\alpha }{N}-\frac{p-1}{2},1+\frac{b+\alpha }{N}-\frac{N-2}{2N}(p-1)\right).$$

This is satisfied if

$$1+\frac{\alpha +b}{N}<p-1-\frac{b}{N-1}<1+\frac{2+b+\alpha }{N-2}.$$

The integral on $(|x|>1)$ can be estimated similarly. So, for $0<T<<1$, we obtain ${\parallel \nabla \varphi (u)\parallel }_{S_T({\mathbb{R}}^N)}<R$, and $B_T(R)$ is stable by the function $\varphi$.

2.

Second case: $b\in (1,\infty )$.

By calculus in the estimation of $(II)$ with ${\parallel |·|}^{b-1}{\left|u\right|}^{p-2}{\parallel }_{L_T^{\infty }(L^a)}$, rather than ${\parallel |·|}^b{\left|u\right|}^{p-2}$${\parallel }_{L_T^{\infty }(L^a)}$, we obtain the requested estimation under the assumptions

$$\begin{array}{c}1+\frac{\alpha }{N}=\frac{2}{r}+\frac{1}{a}+\frac{1}{c};\\ \frac{2}{p-\frac{2b}{N-1}}\le c\le 2^{\ast }\frac{1}{p-\frac{2b}{N-1}};\\ \frac{2}{p-2-\frac{2(b-1)}{N-1}}\le a\le 2^{\ast }\frac{1}{p-2-\frac{2(b-1)}{N-1}}.\end{array}$$

This gives

$$\frac{1}{a}+\frac{1}{c}\in \left(\frac{N-2}{N}(p-1-\frac{2b-1}{N-1}),p-1-\frac{2b-1}{N-1}\right)\cap \left(\frac{\alpha }{N},\frac{2+\alpha }{N}\right).$$

This is satisfied because

$$1+\frac{\alpha }{N}+\frac{2b-1}{N-1}<p<p^c.$$

Now, we need to estimate the term

$$\begin{array}{ccc}\hfill {\parallel \left|u\right|}^{p-1}{\left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-1}{\left|u\right|}^p{)\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}& \lesssim & T^{1-\frac{2}{q}}{\parallel |·|}^b{\left|u\right|}^{p-2}{\parallel }_{L_T^{\infty }(L^a)}{\parallel |·|}^{b-1}{\left|u\right|}^p{\parallel }_{L_T^{\infty }(L^c)}{\parallel u\parallel }_{L_T^q(L^r)}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{\parallel u\parallel }_{H^1}^{2p-2}{\parallel u\parallel }_{L_T^q(L^r)}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{R}^{2p-2}{\parallel u\parallel }_{L_T^q(L^r)},\hfill \end{array}$$

under the conditions

$$\begin{array}{c}1+\frac{\alpha }{N}=\frac{2}{r}+\frac{1}{a}+\frac{1}{c};\\ \frac{2}{p-2-\frac{2b}{N-1}}\le a\le 2^{\ast }\frac{1}{p-2-\frac{2b}{N-1}};\\ \frac{2}{p-\frac{2(b-1)}{N-1}}\le c\le 2^{\ast }\frac{1}{p-\frac{2(b-1)}{N-1}}.\end{array}$$

This is satisfied if

$$\frac{1}{a}+\frac{1}{c}\in \left(\frac{N-2}{N}(p-1-\frac{2b-1}{N-1}),p-1-\frac{2b-1}{N-1}\right)\cap \left(\frac{\alpha }{N},\frac{2+\alpha }{N}\right).$$

This is the same above condition.

Therefore, for $0<T<<1$, we obtain ${\parallel \nabla \varphi (\mathbf{u})\parallel }_{S_T({\mathbb{R}}^N)}<R$. Thus, $B_T(R)$ is stable by the function $\varphi$.

Then, for some $0<T<<1$, the function $\varphi$ is the contract $B_T(R)$, and its fix point is a solution to the Hartree problem (1). The uniqueness is a consequence of the previous calculus with a classical translation argument.

3.2. $H^s$-Theory

Take, for $R>c\parallel u_0{\parallel }_{H^s}$ and $0<T<<1$, the closed ball

$$B_T^{\prime }(R):=\{u\in C_T(H_{rd}^s)\bigcap _{(q,r)\in \Gamma }{L}_T^q(W^{s,r})\mathrm{s}.\mathrm{t}\underset{(q,r)\in \Gamma }{sup}{\parallel u\parallel }_{L_T^q(W^{s,r})}\le R\},$$

Repeating the above argument with the same notions of the distance d and the function $\varphi$, we have for

$$1+\frac{\alpha }{N}=\frac{2}{r}+\frac{1}{a}+\frac{1}{c}=\frac{2}{r}+\frac{2}{e},$$

$$\begin{array}{ccc}\hfill d\left(\varphi (v),\varphi (u)\right)& \lesssim & T^{1-\frac{2}{q}}{\parallel \left|x\right|}^b{\left|v\right|}^p{\parallel }_{L_T^{\infty }(L^c)}\left(\parallel {\left|u\right|}^{p-2}{\left|x\right|}^b{\parallel }_{L_T^{\infty }(L^a)}+{\parallel \left|x\right|}^b{\left|v\right|}^{p-2}{\parallel }_{L_T^{\infty }(L^a)}\right)d(v,u)\hfill \\ & +& T^{1-\frac{2}{q}}\left({\parallel \left|u\right|}^{p-1}{\left|x\right|}^b{\parallel }_{L_T^{\infty }(L^e)}+{\parallel \left|v\right|}^{p-1}{\left|x\right|}^b{\parallel }_{L_T^{\infty }(L^e)}\right){\parallel \left|v\right|}^{p-1}{\left|x\right|}^b{\parallel }_{L_T^{\infty }(L^e)}d(v,u)\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{R}^{2(p-1)}d(v,u).\hfill \end{array}$$

Here, using the Gagliardo–Nirenberg inequality in Proposition 2, we write

$$\begin{array}{c}{\parallel |·|}^b{\left|v\right|}^p{\parallel }_{L_T^{\infty }(L^c)}\lesssim {\parallel v\parallel }_{H^s}^p;\\ {\parallel |·|}^b{\left|v\right|}^{p-2}{\parallel }_{L_T^{\infty }(L^a)}\lesssim {\parallel v\parallel }_{H^s}^{p-2};\\ {\parallel |·|}^b{\left|v\right|}^{p-1}{\parallel }_{L_T^{\infty }(L^e)}\lesssim {\parallel v\parallel }_{H^s}^{p-1},\end{array}$$

under the conditions $p>2+\frac{2b}{N-2s}$ and

$$\begin{array}{c}\frac{2}{p-\frac{2b}{N-2s}}\le c\le \frac{2N}{N-2s}\frac{1}{p-\frac{2b}{N-2s}};\\ \frac{2}{p-2-\frac{2b}{N-2s}}\le a\le \frac{2N}{N-2s}\frac{1}{p-2-\frac{2b}{N-2s}};\\ \frac{2}{p-\frac{2b}{N-2s}-1}\le e\le \frac{2N}{N-2s}\frac{1}{p-\frac{2b}{N-2s}-1}.\end{array}$$

This gives the condition

$$\frac{1}{a}+\frac{1}{c}\in \left[\frac{N-2s}{N}\left(p-\frac{2b}{N-2s}-1\right),p-\frac{2b}{N-2s}-1\right].$$

Moreover, the equality $\frac{1}{a}+\frac{1}{c}=1+\frac{\alpha }{N}-\frac{2}{r}$, via the admissibility condition $2\le r<2^{\ast }$, gives

$$\frac{1}{a}+\frac{1}{c}\in \left[\frac{\alpha }{N},\frac{2+\alpha }{N}\right).$$

This is satisfied if

$$\frac{\alpha }{N}\le p-\frac{2b}{N-2s}-1\le \frac{2+\alpha }{N-2s}.$$

The above condition is verified because

$$2+\frac{2b}{N-2s}<p<p_s^c.$$

Moreover,

$$\frac{2}{e}\in \left(\frac{N-2s}{N}(p-\frac{2b}{N-2s}-1),p-\frac{2b}{N-2s}-1\right)\cap \left[\frac{\alpha }{N},\frac{2+\alpha }{N}\right),$$

which is the above condition. Thus,

$$\begin{array}{c}d\left(\varphi (v),\varphi (u)\right)\lesssim T^{\frac{1}{\delta }}{R}^{2(p-1)}d(v,u);\\ {\parallel \varphi (u)\parallel }_{S_T({\mathbb{R}}^N)}\le C\parallel u_0\parallel +C{T}^{\frac{1}{\delta }}{R}^{2p-1}.\end{array}$$

Now, let us estimate the term ${\parallel (-\Delta )}^{\frac{s}{2}}{\left[\varphi (u)\right]\parallel }_{S_T({\mathbb{R}}^N)}$. Due to the fractional chain rule in Lemma 3, we have

$$\begin{array}{ccc}& & \left|{(-\Delta )}^{\frac{s}{2}}\left({\left|u\right|}^{p-1}{\left|x\right|}^b(I_{\alpha }{\ast |·|}^b{\left|u\right|}^p)\right)\right|\hfill \\ & \lesssim & {\left|\right|x|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)(-\Delta )}^{\frac{s}{2}}{u\left|u\right|}^{p-2}{|+|\left|x\right|}^b(I_{\alpha }{\ast |·|}^b{(-\Delta )}^{\frac{s}{2}}{(|u|}^p{))\left|u\right|}^{p-1}|\hfill \\ & +& |(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|x|}^{b-s}{\left|u\right|}^{p-1}|+|{\left|u\right|}^{p-1}{\left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-s}{\left|u\right|}^p)|.\hfill \end{array}$$

Due to the Strichartz estimates and arguing as previously, we obtain

$$\begin{array}{ccc}& & {\parallel (-\Delta )}^{\frac{s}{2}}{(\varphi (u))\parallel }_{S_T({\mathbb{R}}^N)}-C\parallel {(-\Delta )}^{\frac{s}{2}}{u}_0\parallel \hfill \\ & \lesssim & T^{\frac{1}{\delta }}{R}^{2p-1}+{\parallel \left|x\right|}^{b-s}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p-1}{\parallel }_{S_T^{\prime }({\mathbb{R}}^N)}\hfill \\ & +& {\parallel \left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-s}{\left|u\right|}^p{)|u|}^{p-1}{\parallel }_{S_T^{\prime }({\mathbb{R}}^N)}.\hfill \end{array}$$

For the above, we have two cases:

1.

first case $0\le b\le s$.

Take the real numbers $\mu :={\left(\frac{N}{s-b}\right)}^{+}$ and

$$1+\frac{\alpha }{N}=\frac{1}{r}+\frac{1}{\mu }+\frac{1}{a}+\frac{p-2}{r}+\frac{1}{r}-\frac{s}{N}$$

and $q>2$, such that $(q,r)\in \Gamma$. Then, using the Hölder estimate and Proposition 2, we have

$$\begin{array}{ccc}\hfill {\parallel \left|x\right|}^{b-s}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p-1}{\parallel }_{L^{r^{\prime }}(|x|<1)}& \le & {\parallel |·|}^{b-s}{\parallel }_{L^{\mu }(|x|<1)}{\parallel |·|}^b{\left|u\right|}^p{\parallel }_a{\parallel u\parallel }_r^{p-2}{\parallel u\parallel }_{\frac{Nr}{N-sr}}\hfill \\ & \lesssim & {\parallel u\parallel }_{H^s}^p{\parallel u\parallel }_r^{p-2}{\parallel u\parallel }_{\frac{Nr}{N-sr}},\hfill \end{array}$$

under the condition

$$\frac{1}{a}\in \left(\frac{N-2s}{2N}(p-\frac{2b}{N-2s}),\frac{1}{2}(p-\frac{2b}{N-2s})\right).$$

Since $\frac{1}{a}=1+\frac{s+\alpha }{N}-\frac{1}{\mu }-\frac{p}{r}$, the admissibility condition $2<r<\frac{2N}{N-2s}$ is satisfied if

$$\frac{1}{a}\in \left(1+\frac{b+\alpha }{N}-\frac{p}{2},1+\frac{b+\alpha }{N}-\frac{p(N-2s)}{2N}\right).$$

This gives the condition

$$1+\frac{b+\alpha }{N}<p-\frac{b}{N-2s}<\frac{N}{N-2s}(1+\frac{b+\alpha }{N}).$$

This is satisfied because

$$2+\frac{2b}{N-2s}<p<p_s^c.$$

Then,

$$\begin{array}{ccc}\hfill {\parallel \left|x\right|}^{b-s}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p-1}{\parallel }_{S_T^{\prime }(|x|<1)}& \lesssim & {\parallel \parallel u\parallel }_{H^s}^p{\parallel u\parallel }_r^{p-2}{\parallel u\parallel }_{\frac{Nr}{N-sr}}{\parallel }_{L^q(0,T)}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{\parallel u\parallel }_{L_T^{\infty }(H^s)}^p{\parallel u\parallel }_{L_T^{\infty }(L^r)}^{p-2}{\parallel u\parallel }_{L_T^q(L^{\frac{Nr}{N-sr}})}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{R}^{2p-1}.\hfill \end{array}$$

Now, we estimate the term

$$\begin{array}{ccc}\hfill {\parallel \left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-s}{\left|u\right|}^p{)|u|}^{p-1}{\parallel }_{L^{r^{\prime }}(|x|<1)}& \le & {\parallel |·|}^b{\left|u\right|}^{p-1}{\parallel }_a{\parallel |·|}^{b-s}{\parallel }_{L^{\mu }(|x|<1)}{\parallel u\parallel }_r^{p-1}{\parallel u\parallel }_{\frac{Nr}{N-sr}}\hfill \\ & \lesssim & {\parallel u\parallel }_{H^s}^{p-1}{\parallel u\parallel }_r^{p-1}{\parallel u\parallel }_{\frac{Nr}{N-sr}},\hfill \end{array}$$

under the conditions

$$\begin{array}{c}1+\frac{\alpha }{N}=\frac{1}{r}+\frac{1}{\mu }+\frac{1}{a}+\frac{p-1}{r}+\frac{1}{r}-\frac{s}{N};\\ \frac{1}{a}\in \left(\frac{N-2s}{2N}(p-\frac{2b}{N-2s}-1),\frac{1}{2}(p-\frac{2b}{N-2s}-1)\right).\end{array}$$

The admissibility condition $2\le r<\frac{2N}{N-2s}$ gives

$$\frac{1}{a}\in \left(1+\frac{b+\alpha }{N}-\frac{1+p}{2},1+\frac{b+\alpha }{N}-(1+p)\frac{N-2s}{2N}\right).$$

The following condition, which is the same above, yields

$$1+\frac{b+\alpha }{N}<p-\frac{b}{N-2s}<\frac{N+b+\alpha }{N-2s}.$$

The term for $\left|x\right|>1$ can be controlled similarly. Therefore, for $0<T<<1$, we obtain ${\parallel (-\Delta )}^{\frac{s}{2}}{\varphi (\mathbf{u})\parallel }_{S_T({\mathbb{R}}^N)}<R$. Thus, $B_T^{\prime }(R)$ is stable under $\varphi$.

2.

Second case $b>s$.

Arguing as previously, via the estimates

$$\begin{array}{c}{\parallel |·|}^b{\left|u\right|}^p{\parallel }_{L_T^{\infty }(L^c)}\lesssim {\parallel u\parallel }_{H^s}^p;\\ {\parallel |·|}^{b-s}{\left|u\right|}^{p-2}{\parallel }_{L_T^{\infty }(L^a)}\lesssim {\parallel u\parallel }_{H^s}^{p-2},\end{array}$$

we have

$$\begin{array}{ccc}\hfill {\parallel \left|x\right|}^{b-s}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p-1}{\parallel }_{S_T^{\prime }({\mathbb{R}}^N)}& \lesssim & T^{1-\frac{2}{q}}{\parallel |·|}^b{\left|u\right|}^p{\parallel }_{L_T^{\infty }(L^c)}{\parallel |·|}^{b-s}{\left|u\right|}^{p-2}{\parallel }_{L_T^{\infty }(L^a)}{\parallel u\parallel }_{L_T^q(L^r)}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{R}^{2p-1}.\hfill \end{array}$$

Here, we assume the conditions $p>2+\frac{2(b-s)}{N-2s}$ and

$$\begin{array}{c}1+\frac{\alpha }{N}=\frac{2}{r}+\frac{1}{a}+\frac{1}{c};\\ \frac{2}{p-\frac{2b}{N-2s}}\le c\le \frac{2N}{N-2s}\frac{1}{p-\frac{2b}{N-2s}};\\ \frac{2}{p-2-\frac{2(b-s)}{N-2s}}\le a\le \frac{2N}{N-2s}\frac{1}{p-2-\frac{2(b-s)}{N-2s}}.\end{array}$$

This is satisfied if

$$\frac{1}{a}+\frac{1}{c}\in \left(\frac{N-2s}{N}(p-1-\frac{2b-s}{N-2s}),p-1-\frac{2b-s}{N-2s}\right)\cap \left(\frac{\alpha }{N},\frac{2s+\alpha }{N}\right).$$

This is satisfied if

$$\frac{\alpha }{N}<p-1-\frac{2b-s}{N-2s}<\frac{2s+\alpha }{N-2s}.$$

The following term needs to be estimated; therefore, we have

$$\begin{array}{ccc}\hfill {\parallel \left|x\right|}^b{\left|u\right|}^{p-1}(I_{\alpha }{\ast |·|}^{b-s}{\left|u\right|}^p{)\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}& \lesssim & T^{1-\frac{2}{q}}{\parallel |·|}^b{\left|u\right|}^{p-2}{\parallel }_{L_T^{\infty }(L^a)}{\parallel |·|}^{b-s}{\left|u\right|}^p{\parallel }_{L_T^{\infty }(L^c)}{\parallel u\parallel }_{L_T^q(L^r)}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{\parallel u\parallel }_{H^s}^{2p-2}{\parallel u\parallel }_{L_T^q(L^r)}\hfill \\ & \lesssim & T^{1-\frac{2}{q}}{R}^{2p-2}{\parallel u\parallel }_{L_T^q(L^r)},\hfill \end{array}$$

under the conditions

$$\begin{array}{c}1+\frac{\alpha }{N}=\frac{2}{r}+\frac{1}{a}+\frac{1}{c};\\ \frac{2}{p-2-\frac{2b}{N-2s}}\le a\le \frac{2N}{N-2s}\frac{1}{p-2-\frac{2b}{N-2s}};\\ \frac{2}{p-\frac{2(b-s)}{N-2s}}\le c\le \frac{2N}{N-2s}\frac{1}{p-\frac{2(b-s)}{N-2s}}.\end{array}$$

This is satisfied if

$$\frac{1}{a}+\frac{1}{c}\in \left(\frac{N-2s}{N}(p-1-\frac{2b-s}{N-2s}),p-1-\frac{2b-s}{N-2s}\right)\cap \left(\frac{\alpha }{N},\frac{2s+\alpha }{N}\right).$$

This is the same above condition. Therefore, we do not need any supplementary condition.

Therefore, for $0<T<<1$, we obtain ${\parallel (-\Delta )}^{\frac{s}{2}}{\varphi (u)\parallel }_T<R$. Thus, $B_T^{\prime }(R)$ is stable under $\varphi$.

So, for some $0<T<<10$, $\varphi$ is the contract $B_T^{\prime }(R)$ and its fix point resolves the Hartree Equation (1). The uniqueness follows by the previous calculus and a classical translation argument.

4. Global Existence

In what follows, Theorem 3 is established with a Picard fixed point argument. Take the complete space

$$X:=\left\{u\in C(\mathbb{R},H_{rd}^1){,\mathrm{s}.\mathrm{t}\parallel u\parallel }_{S^{s_c}(\mathbb{R})}<2\parallel e^{i·\Delta }{u}_0{\parallel }_{S^{s_c}(\mathbb{R})}\mathrm{a}\mathrm{n}\mathrm{d}\parallel 〈\nabla 〉{u\parallel }_{S(\mathbb{R})}<c{\parallel u_0\parallel }_{H^1}\right\},$$

under the distance

$$d(v,u):={\parallel v-u\parallel }_{S(\mathbb{R})}+{\parallel v-u\parallel }_{S^{s_c}(\mathbb{R})}.$$

Let, as previously, the integral function be

$$\varphi (u):=e^{i.\Delta }{u}_0-{\int }_0^{.}{e}^{i(.-\tau )\Delta }{\left[\right|x|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p-2}u]d\tau .$$

(4)

Take, for $0<\theta <<1$, the real numbers

$$q:=\frac{2(1+\theta )}{1+\theta s_c},r:=\frac{2N(1+\theta )}{N(1+\theta )-2(1+\theta s_c)}\mathrm{a}\mathrm{n}\mathrm{d}a:=\frac{2(1+\theta )}{1-s_c}.$$

This gives

$$(q,r)\in \Gamma ,(a,r)\in {\Gamma }^{s_c}\mathrm{a}\mathrm{n}\mathrm{d}\frac{1}{q^{\prime }}=\frac{2\theta }{a}+\frac{1}{q}.$$

Taking account of the Strichartz and Hardy–Littlewood–Sobolev estimates, it follows that

$$\begin{array}{ccc}\hfill {\parallel \varphi (u)\parallel }_{S(\mathbb{R})}& \le & c\parallel u_0{\parallel +\parallel \left|x\right|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p-1}{\parallel }_{L^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ & \le & c\parallel u_0{\parallel +\parallel \left|x\right|}^b{\left|u\right|}^{p-1-\theta }{\parallel }_{L^{\infty }(L^c)}^2{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^q(L^r)}\hfill \\ & \le & c\parallel u_0{\parallel +\parallel u\parallel }_{L^{\infty }(H^1)}^{2(p-1-\theta )}{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^q(L^r)}\hfill \\ & \le & c\parallel u_0\parallel +c{A}^{2(p-1-\theta )}{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^q(L^r)}\hfill \\ & \le & c\parallel u_0{\parallel +c\parallel u\parallel }_{S^{s_c}(\mathbb{R})}^{2\theta }{\parallel u\parallel }_{S(\mathbb{R})},\hfill \end{array}$$

(5)

under the conditions

$$\begin{array}{c}1+\frac{\alpha }{N}=\frac{2(1+\theta )}{r}+\frac{2}{c};\\ \frac{N-2}{2N}(p-1-\theta -\frac{2b}{N-1})\le \frac{1}{c}\le \frac{1}{2}(p-1-\theta -\frac{2b}{N-1}).\end{array}$$

Taking $\theta =0^{+}$, the above condition, via $2<r<2^{\ast }$, is satisfied if

$$\frac{\alpha }{2N}<\frac{1}{c}=\frac{\alpha +N}{2N}-\frac{1}{r}<\frac{2+\alpha }{2N}.$$

Equivalently

$$\frac{\alpha }{N}<p-\frac{2b}{N-1}-1<\frac{2+\alpha }{N-2}.$$

Now, taking

$$\frac{1}{\tilde{a}}=\frac{1}{a}+s_c,$$

we obtain

$$(\tilde{a},r)\in {\Gamma }^{-sc},\frac{1}{{\tilde{a}}^{\prime }}=\frac{1+2\theta }{a}$$

and with the Strichartz estimate, we obtain

$$\begin{array}{ccc}\hfill {\parallel \varphi (u)\parallel }_{S^{s_c}(\mathbb{R})}& \le & \parallel e^{i·\Delta }{u}_0{\parallel }_{S^{s_c}(\mathbb{R})}+c\parallel (I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|x|}^b{\left|u\right|}^{p-1}{\parallel }_{L^{{\tilde{a}}^{\prime }}(L^{r^{\prime }})}\hfill \\ & \le & \parallel e^{i·\Delta }{u}_0{\parallel }_{S^{s_c}(\mathbb{R})}+{c\parallel \left|x\right|}^b{\left|u\right|}^{p-1-\theta }{\parallel }_{L^{\infty }(L^c)}^2{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^a(L^r)}\hfill \\ & \le & \parallel e^{i·\Delta }{u}_0{\parallel }_{S^{s_c}(\mathbb{R})}+{c\parallel u\parallel }_{L^{\infty }(H^1)}^{2(p-1-\theta )}{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^a(L^r)}\hfill \\ & \le & \parallel e^{i·\Delta }{u}_0{\parallel }_{S^{s_c}(\mathbb{R})}+c{A}^{2(p-1-\theta )}{\parallel u\parallel }_{S^{s_c}}^{1+2\theta }.\hfill \end{array}$$

Moreover, arguing as previously, we obtain

$$\begin{array}{ccc}\hfill {\parallel \nabla \varphi (u)\parallel }_{S(\mathbb{R})}& \lesssim & \parallel \nabla u_0{\parallel +\parallel \left|x\right|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p-2}{\nabla u\parallel }_{L^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ & +& {\parallel \left|x\right|}^b(I_{\alpha }{\ast |·|}^b{\left|u\right|}^{p-1}{\nabla u)|u|}^{p-1}{\parallel }_{L^{q^{\prime }}(L^{r^{\prime }})}+{\parallel \left|u\right|}^{p-1}{\left|x\right|}^{b-1}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)\parallel }_{L^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ & +& {\parallel \left|u\right|}^{p-1}{\left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-1}{\left|u\right|}^p{)\parallel }_{L^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ & \lesssim & \parallel \nabla u_0{\parallel +\parallel u\parallel }_{S^{s_c}(\mathbb{R})}^{2\theta }{\parallel \nabla u\parallel }_{S(\mathbb{R})}+{\parallel \left|u\right|}^{p-1}{\left|x\right|}^{b-1}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)\parallel }_{L^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ & +& {\parallel \left|u\right|}^{p-1}{\left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-1}{\left|u\right|}^p{)\parallel }_{L^{q^{\prime }}(L^{r^{\prime }})}.\hfill \end{array}$$

Let us discuss two regimes.

1.

Case one: $b\in [0,1]$. Taking ${\left(\frac{N}{1-b}\right)}^{+}:=d$ and $\frac{1}{c}=\frac{1}{d}+\frac{p-1-\theta }{e}$, we obtain

$$\begin{array}{ccc}& & {\parallel \left|u\right|}^{p-1}{\left|x\right|}^{b-1}(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)\parallel }_{L^{q^{\prime }}(L^{r^{\prime }}(|x|<1))}\hfill \\ & \le & {\parallel \left|x\right|}^b{\left|u\right|}^{p-1-\theta }{\parallel }_{L^{\infty }(L^c)}{\parallel \left|x\right|}^{b-1}{\parallel }_d{\parallel u\parallel }_{L^{\infty }(L^e)}^{p-1-\theta }{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^q(L^r)}\hfill \\ & \le & {\parallel u\parallel }_{L^{\infty }(H^1)}^{2(p-1-\theta )}{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^q(L^r)}\hfill \\ & \le & c{A}^{2(p-1-\theta )}{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^q(L^r)}\hfill \\ & \le & c{A}^{2(p-1-\theta )}{\parallel u\parallel }_{S^{s_c}(\mathbb{R})}^{2\theta }{\parallel u\parallel }_{S(\mathbb{R})},\hfill \end{array}$$

under the condition $2\le e\le 2^{\ast }$, which is satisfied if

$$\frac{1}{c}\in \left(\frac{N-2}{2N}(p-1)+\frac{1-b}{N},\frac{p-1}{2}+\frac{1-b}{N}\right).$$

Coupling this condition with

$$\frac{N-2}{2N}(p-1-\theta -\frac{2b}{N-1})\le \frac{1}{c}\le \frac{1}{2}(p-1-\theta -\frac{2b}{N-1}),$$

we obtain $p>2+\frac{b}{N-1}$. Thus, the above estimate holds if

$$max\{\frac{\alpha }{N},1-\frac{b}{N-1}\}<p-\frac{2b}{N-1}-1<\frac{2+\alpha }{N-2}.$$

The estimate of the term $(|x|>1)$ follows similarly, and also, we have

$${\parallel \left|u\right|}^{p-1}{\left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-1}{\left|u\right|}^p{)\parallel }_{L^{q^{\prime }}(L^{r^{\prime }})}\le c{A}^{2(p-1-\theta )}{\parallel u\parallel }_{S^{s_c}(\mathbb{R})}^{2\theta }{\parallel u\parallel }_{S(\mathbb{R})}.$$

Therefore,

$$\parallel 〈\nabla 〉{\varphi (u)\parallel }_{S(\mathbb{R})}\lesssim \parallel u_0{\parallel }_{H^1}+{\parallel u\parallel }_{S^{s_c}(\mathbb{R})}^{2\theta }\parallel 〈\nabla 〉{u\parallel }_{S(\mathbb{R})}\lesssim {\parallel u_0\parallel }_{H^1}.$$

2.

$b>1$. Taking into account the calculus of the first case $\frac{2}{c}=\frac{1}{c_1}+\frac{1}{c_2}$, we obtain

$$\begin{array}{ccc}& & \parallel (I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|x|}^{b-1}{\left|u\right|}^{p-1}{\parallel }_{L^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ & \le & {\parallel \left|x\right|}^b{\left|u\right|}^{p-1-\theta }{\parallel }_{L^{\infty }(L^{c_1})}{\parallel \left|x\right|}^{b-1}{\left|u\right|}^{p-1-\theta }{\parallel }_{L^{\infty }(L^{c_2})}{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^q(L^r)}\hfill \\ & \le & {\parallel u\parallel }_{L^{\infty }(H^1)}^{2(p-1-\theta )}{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^q(L^r)}\hfill \\ & \lesssim & A^{2(p-1-\theta )}{\parallel u\parallel }_{L^a(L^r)}^{2\theta }{\parallel u\parallel }_{L^q(L^r)}\hfill \\ & \lesssim & A^{2(p-1-\theta )}{\parallel u\parallel }_{S^{s_c}(\mathbb{R})}^{2\theta }{\parallel u\parallel }_{S(\mathbb{R})},\hfill \end{array}$$

under the conditions

$$\begin{array}{c}1+\frac{\alpha }{N}=\frac{2(1+\theta )}{r}+\frac{2}{c};\\ \frac{N-2}{2N}(p-1-\theta -\frac{2b}{N-1})\le \frac{1}{c_1}\le \frac{1}{2}(p-1-\theta -\frac{2b}{N-1});\\ \frac{N-2}{2N}(p-1-\theta -\frac{2(b-1)}{N-1})\le \frac{1}{c_2}\le \frac{1}{2}(p-1-\theta -\frac{2(b-1)}{N-1}).\end{array}$$

Taking $\theta =0^{+}$, the above condition is satisfied if

$$\frac{\alpha }{N}<p-1-\frac{2b-1}{N-1}<\frac{2+\alpha }{N-2}.$$

Similarly, we obtain

$${\parallel \left|u\right|}^{p-1}{\left|x\right|}^b(I_{\alpha }{\ast |·|}^{b-1}{\left|u\right|}^p{)\parallel }_{L^{q^{\prime }}(L^{r^{\prime }})}\lesssim A^{2(p-1-\theta )}{\parallel u\parallel }_{S^{s_c}(\mathbb{R})}^{2\theta }{\parallel u\parallel }_{S(\mathbb{R})}.$$

Therefore,

$$\parallel 〈\nabla 〉{\varphi (u)\parallel }_{S(\mathbb{R})}\lesssim \parallel u_0{\parallel }_{H^1}+{\parallel u\parallel }_{S^{s_c}(\mathbb{R})}^{2\theta }\parallel 〈\nabla 〉{u\parallel }_{S(\mathbb{R})}\lesssim {\parallel u_0\parallel }_{H^1}.$$

(6)

In both cases, the space X is stable under the function $\varphi$, which is a contraction. The proof of the global existence follows with a classical fixed point argument. Moreover, the scattering is a direct consequence of the Duhamel Formulas (4) via (5) and (6).

5. Decay of Defocusing Global Solutions

In this section, we take the defocusing regime $ϵ=1$ and prove Theorem 4. Let $\psi \in C_0^{\infty }({\mathbb{R}}^N)$ and a weakly convergent sequence

$$\begin{array}{c}\hfill \underset{n}{sup}{\parallel {\phi }_n\parallel }_{H^1}<\infty ;\\ \hfill {\phi }_n\rightharpoonup \phi \in H_{rd}^1.\end{array}$$

Let $u_n\in C(\mathbb{R},H_{rd}^1)$ be the solution to (1) with data ${\phi }_n$ and $u\in C(\mathbb{R},H_{rd}^1)$ be the solution to (1) with data $\phi$.

• Claim.

For all $ϵ>0,$ there exists $T_{ϵ},n_{ϵ}>0$, such that

$$\parallel \psi (u_n-u){\parallel }_{L_{T_{ϵ}}^{\infty }(L^2)}<ϵ,\forall n>n_{ϵ}.$$

Let us define the functions $v_n:=\psi u_n$ and $v:=\psi u$, as well as, $v_n(0,·)=\psi {\phi }^n$ and

$$\begin{array}{ccc}\hfill i{\partial }_t{v}_n+\Delta v_n& =& \Delta \psi u_n+2\nabla \psi \nabla u_n+(I_{\alpha }{\ast |·|}^b|u_n{|}^p{)|x|}^b{\left|u_n\right|}^{p-2}{v}_n.\hfill \end{array}$$

Similarly, let us define $v(0,·)=\psi \phi$ and

$$\begin{array}{ccc}\hfill i{\partial }_{t}v+\Delta v& =& \Delta \psi u+2\nabla \psi \nabla u+(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b){\left|u\right|}^{p-2}{\left|x\right|}^{b}v.\hfill \end{array}$$

Taking $z_n:=u_n-u$ and $w_n:=v_n-v$, we have

$$\begin{array}{ccc}\hfill i{\partial }_t{w}_n+\Delta w_n& =& \Delta \psi z_n+2\nabla \psi \nabla z_n+{\left|x\right|}^b\left((I_{\alpha }{\ast |·|}^b|u_n{|}^p)|u_n{|}^{p-2}{v}_n-(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b{)|u|}^{p-2}v\right).\hfill \end{array}$$

Due to Hölder’s inequality and Sobolev embedding, for $T>0$ and $(q,r)\in \Gamma$,

$$\begin{array}{ccc}\hfill & & {\parallel |x|}^b\left((I_{\alpha }{\ast |·|}^b|u_n{|}^p)|u_n{|}^{p-2}{v}_n-(I_{\alpha }{\ast |u|}^p{|·|}^b{)|u|}^{p-2}v\right){\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ \hfill & \lesssim & \parallel (I_{\alpha }{\ast [|·|}^b|u_n{|}^p{-|·|}^b{|u|}^p])|u_n{{|}^{p-2}{v}_n\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ \hfill & +& \parallel (I_{\alpha }{\ast |u|}^p{|·|}^b)(|u_n{|}^{p-2}+{|u|}^{p-2}{)w_n\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ \hfill & \lesssim & (I_1)+(I{I}_1).\hfill \end{array}$$

Moreover, for $1+\frac{\alpha }{N}=\frac{1}{a}+\frac{p}{r}$, we have

$$\begin{array}{ccc}\hfill (I{I}_1)& :=& \parallel (I_{\alpha }{\ast |u|}^p{|·|}^b)(|u_n{|}^{p-2}+{|u|}^{p-2}{)w_n\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ \hfill & \lesssim & {\parallel |·|}^b{|u|}^p{\parallel }_{L_T^{\infty }(L^a)}\left(\parallel u_n{\parallel }_{L_T^{\infty }(L^r)}^{p-2}+{\parallel u\parallel }_{L_T^{\infty }(L^r)}^{p-2}\right){\parallel w_n\parallel }_{L_T^q(L^r)}\hfill \\ \hfill & \lesssim & T^{1-\frac{2}{q}}{\parallel w_n\parallel }_{L_T^q(L^r)},\hfill \end{array}$$

under the condition

$$\frac{1}{a}\in \left(\frac{N-2}{2N}(p-\frac{2b}{N-1}),\frac{1}{2}(p-\frac{2b}{N-1})\right).$$

The admissibility condition $2<r<2^{\ast }$ gives

$$\frac{1}{a}\in \left(1+\frac{\alpha }{N}-\frac{p}{2},1+\frac{\alpha }{N}-\frac{(N-2)p}{2N}\right).$$

Equivalently,

$$1+\frac{\alpha }{N}<p-\frac{b}{N-1}<1+\frac{2+\alpha }{N-2}.$$

Moreover, for $1+\frac{\alpha }{N}=\frac{1}{c}+\frac{p+1}{r}$, we have

$$\begin{array}{ccc}\hfill (I_1)& :=& \parallel (I_{\alpha }{\ast [|·|}^b(|u_n{|}^{p-1}+{|u|}^{p-1})z_n]|u_n{{|}^{p-2}{v}_n\parallel }_{L_T^{q^{\prime }}(L^{r^{\prime }})}\hfill \\ \hfill & \lesssim & T^{1-\frac{1}{q}}\left({\parallel |·|}^b|u_n{|}^{p-1}{\parallel }_{L_T^{\infty }(L^c)}{+\parallel |·|}^b{|u|}^{p-1}{\parallel }_{L_T^{\infty }(L^c)}\right)\parallel z_n{\parallel }_{L_T^{\infty }(L^r)}\parallel u_n{\parallel }_{L_T^{\infty }(L^r)}^{p-2}{\parallel v_n\parallel }_{L_T^{\infty }(L^r)}\hfill \\ \hfill & \lesssim & T^{1-\frac{1}{q}},\hfill \end{array}$$

under the condition

$$\frac{1}{c}\in \left(\frac{N-2}{2N}(p-\frac{2b}{N-1}-1),\frac{1}{2}(p-\frac{2b}{N-1}-1)\right).$$

The admissibility condition $2<r<2^{\ast }$ gives

$$\frac{1}{c}\in \left(1+\frac{\alpha }{N}-\frac{p+1}{2},1+\frac{\alpha }{N}-\frac{(N-2)(p+1)}{2N}\right).$$

Equivalently,

$$1+\frac{\alpha }{N}<p-\frac{b}{N-1}<1+\frac{2+\alpha }{N-2}.$$

Now, according to the Rellich theorem, for the sub-sequence, when $n⟶\infty$,

$${ϵ}_n:=\parallel \psi ({\phi }_n-\phi )\parallel ⟶0.$$

Furthermore, with the conservation laws and the properties of $\psi$,

$$\begin{array}{ccc}\hfill \parallel \nabla \psi \nabla z_n{\parallel }_{L_T^1(L^2)}+{\parallel \Delta \psi z_n\parallel }_{L_T^1(L^2)}& \lesssim & \parallel \nabla z_n{\parallel }_{L_T^1(L^2)}+{\parallel z_n\parallel }_{L_T^1(L^2)}\hfill \\ \hfill & \lesssim & \left({\parallel u\parallel }_{L^{\infty }(\mathbb{R},H^1)}+\underset{n}{sup}{\parallel u_n\parallel }_{L^{\infty }(\mathbb{R},H^1)}\right)T\hfill \\ \hfill & \le & CT.\hfill \end{array}$$

Taking account of the Strichartz estimate, it follows that

$$\begin{array}{ccc}\hfill \parallel w_n{\parallel }_{L_T^q(L^r)\cap L_T^{\infty }(L^2)}& \lesssim & {ϵ}_n+T+T^{1-\frac{1}{q}}+T^{1-\frac{2}{q}}{\parallel w_n\parallel }_{L_T^q(L^r)\cap L_T^{\infty }(L^2)}\hfill \\ & \lesssim & \frac{T^{1-\frac{1}{q}}+{ϵ}_n+T}{-T^{1-\frac{2}{q}}+1}.\hfill \end{array}$$

This established the claim.

Now, it is sufficient to prove the decay for $r:=2+\frac{4}{N}$ via an interpolation argument. Let us write a Gagliardo–Nirenberg-type estimate

$${\displaystyle {\parallel u(t)\parallel }_{2+\frac{4}{N}}^{2+\frac{4}{N}}\le C{\parallel u(t)\parallel }_{H^1}^2{\left(\underset{x}{sup}{\parallel u(t)\parallel }_{L^2(Q_1(x))}\right)}^{\frac{4}{N}}.}$$

(7)

Here, $Q_a(x)$ is the square centered at x with edge length a. With the contradiction, assume that there exists $t_n\to \infty$ and $ϵ>0$, such that

$$\parallel u(t_n){\parallel }_{2+\frac{4}{N}}>ϵ,\forall n\in \mathbb{N}.$$

According to (7) and the previous inequality, there exists $x_n\in {\mathbb{R}}^N$ and a real number (denoted also by $ϵ>0)$ satisfying

$$\parallel u(t_n){\parallel }_{L^2(Q_1(x_n))}\ge ϵ,\forall n\in \mathbb{N}.$$

(8)

Following the lines in [21], we obtain

$${\parallel u(t)\parallel }_{L^2(Q_2(x_n))}\ge \frac{ϵ}{4},\forall t\in [t_n,t_n+T],\forall n\ge n_{ϵ}.$$

Therefore, from Propositions 1 and 3, for $b\le 1$, we obtain

$$\begin{array}{ccc}\hfill \parallel u_0{\parallel }_{H^1}& \gtrsim & {\int }_0^{\infty }{\int }_{{\mathbb{R}}^N}{|x|}^{b-1}{|u|}^p(I_{\alpha }{\ast |·|}^b{|u|}^p)dxdt\hfill \\ & \gtrsim & \sum _n{\int }_{t_n}^{t_n+T}{\int }_{Q_2(x_n)}{\int }_{Q_2(x_n)}\frac{{|x|}^{b-1}{|y|}^b}{{|x-y|}^{N-\alpha }}{(|u(t,x)||u(t,y)|)}^{p}dxdydt.\hfill \end{array}$$

Due to the radial assumption and (8), $(x_n)$ is bounded. Then,

$$\begin{array}{ccc}\hfill \parallel u_0{\parallel }_{H^1}& \gtrsim & \sum _n{\int }_{t_n}^{t_n+T}{\left({\int }_{Q_2(x_n)}{|u(t,x)|}^{p}dx\right)}^{2}dt\hfill \\ & \gtrsim & \sum _n{\int }_{t_n}^{t_n+T}{\parallel u(t)\parallel }_{L^2(Q_2(x_n))}^{2p}dt\hfill \\ & \gtrsim & \sum _n{(\frac{\epsilon }{4})}^{2p}T=\infty .\hfill \end{array}$$

This contradiction achieves the proof.

6. Conclusions

Theorems 1 and 2 establish the local well-posedness, in the Sobolev space $H_{rd}^s({\mathbb{R}}^N)$ for $s\in (\frac{1}{2},1]$, of the inhomogeneous Hartree problem (1) in the sub-critical regime $s>s_c$. Moreover, according to Theorem 3, the global well-posedness holds in the energy space for small data. Furthermore, in the defocusing regime, Theorem 4 proves that the mass-sub-critical solutions vanish for a long time in some Lebesgue spaces. The main ingredients used are Strichartz-type estimates in Proposition 5, coupled with some radial Strauss inequalities in Proposition 3. In a future work, the authors will try to improve the present paper in three directions. The first one is to remove the radial assumption. The second one is to treat the critical regime: $s=s_c$. The third one is to investigate the scattering versus finite time blow-up dichotomy of focusing solutions in the spirit of [22].