Orbital stability of standing waves for Schr\"{o}dinger type equations with slowly decaying linear potential

TL;DR

This paper investigates the existence and orbital stability of standing waves in Schrödinger equations with slowly decaying potentials, using advanced mathematical tools to cover subcritical and critical cases.

Contribution

It provides a systematic analysis of standing wave stability for Schrödinger equations with slowly decaying potentials, employing concentration compactness, Gagliardo-Nirenberg inequalities, and refined operator estimates.

Findings

Existence of standing waves established

Orbital stability proven in subcritical and critical cases

Methodology applicable to various nonlinearities

Abstract

In this paper, kinds of Schr\"{o}dinger type equations with slowly decaying linear potential and power type or convolution type nonlinearities are considered. By using the concentration compactness principle, the sharp Gagliardo-Nirenberg inequality and a refined estimate of the linear operator, the existence and orbital stability of standing waves in $L^2$-subcritical and $L^2$-critical cases are established in a systematic way.

Equations

$$\{\begin{array}{l}i{\partial }_{t}u+\Delta u+V(x)u+f(u)=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,\\ u(0,x)=u_0(x)\in H^1({\mathbb{R}}^N),x\in {\mathbb{R}}^N,\end{array}$$

$$I_{\beta }(x)=\frac{\Gamma (\frac{N-\beta }{2})}{\Gamma (\frac{\beta }{2}){\pi }^{N/2}{2}^{\beta }|x{|}^{N-\beta }}$$

$$-\Delta \phi -\frac{\gamma }{|x{|}^{\alpha }}\phi +\omega \phi =f(\phi ),x\in {\mathbb{R}}^N.$$

$$u(t,x)\to {\lambda }^{\frac{2}{p-1}}u({\lambda }^{2}t,\lambda x).$$

$$u(t,x)\to {\lambda }^{\frac{\beta +2}{2q-2}}u({\lambda }^{2}t,\lambda x)$$

$$S_{\omega }(v):=\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(|\nabla v{|}^2-\frac{\gamma }{|x{|}^{\alpha }}|v{|}^2+\omega |v{|}^2\right)dx-{\int }_{{\mathbb{R}}^N}F(v)dx,$$

$$\inf \{S_{\omega }(v):v\in H^1({\mathbb{R}}^N)\backslash \{0\},v\mathrm{is}\mathrm{a}\mathrm{solution}\mathrm{of}(\text{ref}e1.2)\}.$$

$$E(u)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(|\nabla u{|}^2-\frac{\gamma }{|x{|}^{\alpha }}|u{|}^2\right)dx-{\int }_{{\mathbb{R}}^N}F(u)dx,$$

$$A_{\rho }=\underset{u\in M_{\rho }}{\inf }E(u),M_{\rho }=\{u\in H^1({\mathbb{R}}^N):\Vert u{\Vert }_2^2=\rho \}$$

$$G_{\rho }=\{u\in H^1({\mathbb{R}}^N):u\in M_{\rho },E(u)=A_{\rho }\}.$$

$$-\Delta u-\frac{\gamma }{|x{|}^{\alpha }}u-f(u)=-\lambda u.$$

$$\underset{v\in G_{\rho }}{\inf }\Vert u_0-v{\Vert }_{H^1({\mathbb{R}}^N)}<\delta ,$$

$$\underset{v\in G_{\rho }}{\inf }\Vert u(t)-v{\Vert }_{H^1({\mathbb{R}}^N)}<ϵ$$

$$\{\begin{array}{l}i{\partial }_{t}u+\Delta u+(I_{\beta }\ast |u{|}^q)|u{|}^{q-2}u=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,\\ u(0,x)=u_0(x)\in H^1({\mathbb{R}}^N),x\in {\mathbb{R}}^N,\end{array}$$

$$\Vert u(t){\Vert }_2^2=\Vert u_0{\Vert }_2^2,E(u(t))=E(u_0),t\in (-T_{\ast },T^{\ast }).$$

$$-\Delta Q+Q=Q^p;$$

$$-\Delta W+W=(I_{\beta }\ast |W{|}^q)|W{|}^{q-2}W;$$

$$\Vert u{\Vert }_{\eta +2}^{\eta +2}\le C_{GN}(\eta )\Vert u{\Vert }_2^{2+\eta (2-N)/2}\Vert \nabla u{\Vert }_2^{\eta N/2}$$

$$C_{GN}(\eta )=\frac{2(\eta +2)}{4-(N-2)\eta }{\left(\frac{4-(N-2)\eta }{N\eta }\right)}^{N\eta /4}\frac{1}{\Vert Q_{\eta +1}{\Vert }_2^{\eta }},$$

$$ϵ{\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx-\gamma {\int }_{{\mathbb{R}}^N}\frac{|u{|}^2}{|x{|}^{\alpha }}dx\ge -\delta (ϵ,\Vert u{\Vert }_2)$$

$$\begin{array}{rl}{\int }_{{\mathbb{R}}^N}\frac{|u{|}^2}{|x{|}^{\alpha }}dx& ={\int }_{B_1(0)}\frac{|u{|}^2}{|x{|}^{\alpha }}dx+{\int }_{B_1^c(0)}\frac{|u{|}^2}{|x{|}^{\alpha }}dx\\ & \le \Vert |x{|}^{-\alpha }{\chi }_{B_1(0)}{\Vert }_r\Vert |u{|}^2{\Vert }_{r^{\prime }}+\Vert |x{|}^{-\alpha }{\chi }_{B_1^c(0)}{\Vert }_s\Vert |u{|}^2{\Vert }_{s^{\prime }}\\ & =C_1\Vert u{\Vert }_{2r^{\prime }}^2+C_2\Vert u{\Vert }_{2s^{\prime }}^2,\end{array}$$

$$\Vert u{\Vert }_{2r^{\prime }}^2\le C\Vert u{\Vert }_2^{N/r^{\prime }-(N-2)}\Vert \nabla u{\Vert }_2^{N-N/r^{\prime }}.$$

$$a^{1/q}{b}^{1/q^{\prime }}\le \frac{a}{q}+\frac{b}{q^{\prime }},a,b>0,1/q+1/q^{\prime }=1,$$

$$\begin{array}{rl}\Vert u{\Vert }_2^{N/r^{\prime }-(N-2)}\Vert \nabla u{\Vert }_2^{N-N/r^{\prime }}& =\Vert u{\Vert }_2^{N/r^{\prime }-(N-2)}\Vert \nabla u{\Vert }_2^{2\frac{N-N/r^{\prime }}{2}}\\ & \le {ϵ}_1\Vert \nabla u{\Vert }_2^2+{\delta }_1({ϵ}_1,\Vert u{\Vert }_2).\end{array}$$

$${\int }_{{\mathbb{R}}^N}|u_n{|}^{2}dx=\lambda ,$$

$$\forall ϵ>0,\exists R<\infty ,{\int }_{B_R(y_{n_k})}|u_{n_k}(x){|}^{2}dx\ge \lambda -ϵ;$$

$$\{\begin{array}{l}|u_{n_k}^{(1)}|,|u_{n_k}^{(2)}|\le |u_{n_k}|;\\ {\Vert u_{n_k}-(u_{n_k}^{(1)}+u_{n_k}^{(2)})\Vert }_p\le {\delta }_p(ϵ)\to 0\mathrm{as}ϵ\to 0\mathrm{for}\mathrm{any}2\le p<2N/(N-2{)}_{+};\\ \left|{\int }_{{\mathbb{R}}^N}|u_{n_k}^{(1)}{|}^{2}dx-\sigma \right|\le ϵ,\left|{\int }_{{\mathbb{R}}^N}|u_{n_k}^{(2)}{|}^{2}dx-(\lambda -\sigma )\right|\le ϵ;\\ \mathrm{Supp}{u}_{n_k}^{(1)}\subset B_{2R_1}(y_{n_k});\\ \mathrm{dist}(\mathrm{Supp}{u}_{n_k}^{(1)},\mathrm{Supp}{u}_{n_k}^{(1)})\to \infty \mathrm{as}k\to \infty ;\\ {\mathrm{liminf}}_{k\to \infty }{\int }_{{\mathbb{R}}^N}(|\nabla u_{n_k}{|}^2-|\nabla u_{n_k}^{(1)}{|}^2-|\nabla u_{n_k}^{(2)}{|}^2)dx\ge 0.\end{array}$$

$$\left|{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{u(x)v(y)}{|x-y{|}^{N-\beta }}dxdy\right|\le C(N,\beta ,p)\Vert u{\Vert }_p\Vert v{\Vert }_r.$$

$$C(N,\beta ,p)=C_{\beta }(N)={\pi }^{\frac{N-\beta }{2}}\frac{\Gamma (\frac{\beta }{2})}{\Gamma (\frac{N+\beta }{2})}{\left\{\frac{\Gamma (\frac{N}{2})}{\Gamma (N)}\right\}}^{-\frac{\beta }{N}}.$$

$$\Vert I_{\alpha }\ast v{\Vert }_{L^{\frac{Ns}{N-\alpha s}}}\le C\Vert v{\Vert }_{L^s},$$

Full text

2010 Mathematics Subject Classification: 35Q41, 35B35, 35J20.

Keywords: orbital stability, concentration compactness principle, standing waves, slowly decaying potential, Schrödinger type equations.

Email Addresses: lxylxf@tjcu.edu.cn (XL), lxyzhjy@tjcu.edu.cn (JZ)

Xinfu Li and Junying Zhao

School of Science, Tianjin University of Commerce, Tianjin 300134, China

1. Introduction and main results

In this paper, we consider the following nonlinear Schrödinger type equations with slowly decaying linear potential

[TABLE]

where $N\geq 1$, $u:\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{C}$ is a complex valued function, $V(x)=\gamma|x|^{-\alpha}$, $\gamma\in\mathbb{R}$, $0<\alpha<\min\{2,N\}$, and $f:\mathbb{C}\to\mathbb{C}$ is assumed to be one of the following five types:

Type 1 $f(u)=|u|^{p-1}u$ with $1<p<(N+2)/(N-2)_{+}$, where $1/(N-2)_{+}=1/(N-2)$ if $N\geq 3$ and $1/(N-2)_{+}=+\infty$ if $N=1,2$;

Type 2 $f(u)=(I_{\beta}\ast|u|^{q})|u|^{q-2}u$ with $0<\beta<N$, $1+\beta/N<q<(N+\beta)/(N-2)_{+}$ and $I_{\beta}$ being the Riesz potential defined for every $x\in\mathbb{R}^{N}\setminus\{0\}$ by

[TABLE]

with $\Gamma$ denoting the Gamma function (see [28], P.19);

Type 3 $f(u)=|u|^{p_{1}-1}u+|u|^{p_{2}-1}u$ with $1<p_{1}<p_{2}<(N+2)/(N-2)_{+}$;

Type 4 $f(u)=(I_{\beta}\ast|u|^{q_{1}})|u|^{{q_{1}}-2}u+(I_{\beta}\ast|u|^{q_{2}})|u|^{{q_{2}}-2}u$ with $0<\beta<N$ and $1+\beta/N<q_{1}<q_{2}<(N+\beta)/(N-2)_{+}$;

Type 5 $f(u)=(I_{\beta}\ast|u|^{q})|u|^{{q}-2}u+|u|^{p-1}u$ with $0<\beta<N$, $1+\beta/N<q<(N+\beta)/(N-2)_{+}$ and $1<p<(N+2)/(N-2)_{+}$.

The Schrödinger operator $i\partial_{t}+\Delta$ arises in various physical contexts such as nonlinear optics and plasma physics, see [10, 31, 35]. The nonlinearity enters due to the effect of changes in the field intensity on the wave propagation characteristics of the medium. The potential $V(x)$ can be thought of as modeling inhomogeneities in the medium. In particular, the operator $-\Delta-\frac{\gamma}{|x|}$ with Coulomb potential provides a quantum mechanical description of the Coulomb force between two charged particles and corresponds to having an external attractive long-range potential due to the presence of a positively charged atomic nucleus, see [25, 29]. For the study of Schrödinger equation with a harmonic potential, we refer the readers to Zhang [36, 37].

By a standing wave, we mean a solution of (1.1) in the form $u(t,x)=e^{i\omega t}\varphi(x)$, where $\omega\in\mathbb{R}$ is a constant and $\varphi\in H^{1}(\mathbb{R}^{N})$ satisfies the stationary equation

[TABLE]

For $f(u)$ is of Type 1, the existence of ground state to (1.2) was studied by Fukaya and Ohta [14] and Fukuizumi and Ohta [16], and the uniqueness of the positive radial solution with $\alpha=1$ was studied by Benguria and Jeanneret [2]. For $f(u)$ is of Type 2, the nonexistence, existence and uniqueness of positive solution to (1.2) were studied in [9, 21, 22, 23].

When $\gamma=0$ and $f(u)$ is of Type 1, (1.1) is invariant under the scaling transform

[TABLE]

If $p=p_{c}:=1+{4}/{N}$, the transform keeps the mass invariant and the nonlinearity is called $L^{2}$-critical. In this case, Cazenave [6] proved that the ground state solution of (1.2) is orbitally stable for all $\omega>0$ if $p<p_{c}$, while is unstable for all $\omega>0$ if $p_{c}<p<(N+2)/(N-2)$. The instability of the bound state solution with $p=p_{c}$ was proved by Weinstein [34]. When $f(u)$ is of Type 2, the transform

[TABLE]

keeps (1.1) invariant and $q=q_{c}:=1+(2+\beta)/{N}$ is the $L^{2}$-critical exponent. In this case, Cazenave and Lions [7] showed the existence and orbital stability of standing waves for $N=3,\ q=2$ and $\beta=2$. Recently, in the $L^{2}$-subcritical case, that is, $1+\beta/N<q<1+(2+\beta)/{N}$, Wang et al. [33] studied the orbital stability of standing waves to (1.1). When $f(u)$ is of Type 3, we refer to [15, 30] for the existence, orbital stability and strong instability of standing waves and [1, 8, 19, 24] for that of Type 5.

When $\gamma\neq 0$, it is easy to see that equation (1.1) does not enjoy the scaling invariance as well as the pace translation invariance. In this case, when $f(u)$ is of Type 1, Fukuizumi and Ohta studied the existence and orbital stability of standing waves for $1<p<1+4/N$ in [16] and the strong instability of standing waves for $\gamma>0$ and $1+4/N<p<(N+2)/(N-2)$ was studied by Fukaya and Ohta in [14]. The authors in [5, 11, 18, 26] studied the local well-posedness, global existence, scattering and finite time blowup of the solutions to (1.1). When $f(u)$ is of Type 2, if $\beta=2$, $q=2$, $\alpha=1$ and $\gamma>0$, Ginibre and Velo [17] obtained the global existence of the solution to (1.1). If $N=3$ and $\beta=2$, Cazenave and Lions [7] and Lions [23] established the existence and orbital stability of standing waves for $q=2$, and Cheng and Yang [9] studied that for $q>2$ and close to 2. More precisely, in (1.2), taking $\omega$ as a fixed parameter, it is known that every solution $v\in H^{1}(\mathbb{R}^{N})$ to (1.2) is a critical point of the functional $S_{\omega}$ defined by

[TABLE]

where $F(s)=\int_{0}^{s}f(\tau)d\tau$. Denote by $\mathcal{M_{\omega}^{G}}$ the set of all non-negative minimizers for

[TABLE]

For $f(u)$ is of Type 1, $1<p<1+4/N$ and $\gamma\in\mathbb{R}$, [16] proved that there exist $\omega_{0}>0$ and $\omega_{*}>\omega_{0}$ such that $\mathcal{M_{\omega}^{G}}$ is not empty for any $\omega\in(\omega_{0},\infty)$ and $e^{i\omega t}\varphi_{\omega}(x)$ is orbitally stable for any $\varphi_{\omega}(x)\in\mathcal{M_{\omega}^{G}}$ and $\omega\in(\omega_{*},\infty)$, by using a sufficient condition for orbital stability, that is, the positive definite of the operator $S^{\prime\prime}_{\omega}(\varphi_{\omega})$. Similar results were obtained by [9] for Type 2 nonlinearities, $N=3$, $\beta=2$, $q>2$ close to 2. Note that $\omega_{0}$ and $\omega_{*}$ are not given explicitly, and moreover, they did not consider the $L^{2}$-critical case. Hence, in this paper, we further discuss the existence and orbital stability of standing waves of (1.1).

Note that we may take $\omega$ as unknown in (1.2). Indeed, for any $\rho>0$, if we define

[TABLE]

[TABLE]

and

[TABLE]

Then the Lagrange multiplier theorem implies that for any $u\in G_{\rho}$, there exists $\lambda\in\mathbb{R}$ such that

[TABLE]

Hence, $e^{i\lambda t}u(x)$ is a standing wave to (1.1) with initial data $u_{0}(x)=u(x)$. One usually calls $e^{i\lambda t}u(x)$ the orbit of $u$. Moreover, if $u\in G_{\rho}$, then $e^{i\theta}u(x)\in G_{\rho}$ for any $\theta\in\mathbb{R}$. In this paper, we consider the orbital stability of $G_{\rho}$. For this, we give the following definition of orbital stability which is similar to that in [32].

Definition 1.1**.**

The set $G_{\rho}$ is said to be orbitally stable if, for any $\epsilon>0$, there exists $\delta>0$ such that for any initial data $u_{0}$ satisfying

[TABLE]

the corresponding solution $u$ to (1.1) satisfies

[TABLE]

for all $t\geq 0$.

Remark 1.2**.**

(1) Note that our definition of orbital stability is different from that in [16] for we do not know whether or not $\mathcal{M_{\omega}^{G}}$ and $G_{\rho}$ are single point sets.

(2) For the lack of scaling invariance, there is not direct connection between $\mathcal{M_{\omega}^{G}}$ and $G_{\rho}$. However, for equation

[TABLE]

there is some equivalence between $\mathcal{M_{\omega}^{G}}$ and $G_{\rho}$, see [33].

To study the orbital stability of standing waves of (1.1), we first make the following assumption on the local well-posedness.

Assumption A. Let $N\geq 1$, $V(x)=\gamma|x|^{-\alpha}$, $\gamma\in\mathbb{R}$, $0<\alpha<\min\{2,N\}$ and $f(u)$ be one of Types 1-5 such that the following local well-posedness holds for (1.1):

For any $u_{0}\in H^{1}(\mathbb{R}^{N})$, there exist $T_{*},\ T^{*}\in(0,\infty]$ and a unique solution $u(t)\in C((-T_{*},T^{*}),H^{1}(\mathbb{R}^{N}))\cap C^{1}((-T_{*},T^{*}),H^{-1}(\mathbb{R}^{N}))$ with $u(0)=u_{0}$ to (1.1). If $T^{*}<\infty$ ($T_{*}<\infty$), then $\lim_{t\uparrow T^{*}}\|u(t,x)\|_{H^{1}(\mathbb{R}^{N})}=\infty$ ($\lim_{t\downarrow-T_{*}}\|u(t,x)\|_{H^{1}(\mathbb{R}^{N})}=\infty$). Moreover, there is conservation of mass and energy, i.e.,

[TABLE]

Remark 1.3**.**

Motivated by [6], [11] and [12], in Section 2 we give a proof for Assumption A when $q\geq 2$.

Under Assumption A, by using the concentration compactness principle, the sharp Gagliardo-Nirenberg inequality and a refined estimate of the linear operator $-\Delta-\gamma|x|^{-\alpha}$, we can obtain the following theorems.

Theorem 1.4**.**

Let $N\geq 1$, $0<\alpha<\min\{2,N\}$, $f(u)=|u|^{p-1}u$, $V(x)$ and $f(u)$ satisfy Assumption A. Assume one of the following conditions hold:

(1) $\gamma>0$, $1<p<1+4/N$, $\rho>0$;

(2) $\gamma>0$, $p=1+4/N$, $0<\rho<\|Q_{p}\|_{2}^{2}$, where $Q_{p}$ is the unique positive radial solution of equation

[TABLE]

then $G_{\rho}$ is not empty and orbitally stable.

Theorem 1.5**.**

Let $N\geq 1$, $0<\alpha<\min\{2,N\}$, $\beta\in(0,N)$, $f(u)=(I_{\beta}\ast|u|^{q})|u|^{q-2}u$, $V(x)$ and $f(u)$ satisfy Assumption A. Assume one of the following conditions hold:

(1) $\gamma>0$, $1+{\beta}/{N}<q<1+(2+\beta)/N$, $\rho>0$;

(2) $\gamma>0$, $q=1+(2+\beta)/N$, $0<\rho<\|W_{q}\|_{2}^{2}$, where $W_{q}$ is a radically ground state solution of the elliptic equation

[TABLE]

then $G_{\rho}$ is not empty and orbitally stable.

Theorem 1.6**.**

Let $N\geq 1$, $0<\alpha<\min\{2,N\}$, $f(u)=|u|^{p_{1}-1}u+|u|^{p_{2}-1}u$, $V(x)$ and $f(u)$ satisfy Assumption A. Assume one of the following conditions hold:

(1) $\gamma>0$, $1<p_{1}<p_{2}<1+4/N$, $\rho>0$;

*(2) $\gamma>0$, $1<p_{1}<p_{2}=1+4/N$, $0<\rho<\|Q_{p_{2}}\|_{2}^{2}$;

then $G_{\rho}$ is not empty and orbitally stable.*

Theorem 1.7**.**

Let $N\geq 1$, $0<\alpha<\min\{2,N\}$, $\beta\in(0,N)$, $f(u)=(I_{\beta}\ast|u|^{q_{1}})|u|^{q_{1}-2}u+(I_{\beta}\ast|u|^{q_{2}})|u|^{q_{2}-2}u$, $V(x)$ and $f(u)$ satisfy Assumption A. Assume one of the following conditions hold:

(1) $\gamma>0$, $1+{\beta}/{N}<q_{1}<q_{2}<1+(2+\beta)/N$, $\rho>0$;

*(2) $\gamma>0$, $1+{\beta}/{N}<q_{1}<q_{2}=1+(2+\beta)/N$, $0<\rho<\|W_{q_{2}}\|_{2}^{2}$;

then $G_{\rho}$ is not empty and orbitally stable.*

Theorem 1.8**.**

Let $N\geq 1$, $0<\alpha<\min\{2,N\}$, $\beta\in(0,N)$, $f(u)=(I_{\beta}\ast|u|^{q})|u|^{q-2}u+|u|^{p-1}u$, $V(x)$ and $f(u)$ satisfy Assumption A. Assume one of the following conditions hold:

(1) $\gamma>0$, $1<p<1+4/N$, $1+{\beta}/{N}<q<1+(2+\beta)/N$, $\rho>0$;

(2) $\gamma>0$, $1<p<1+4/N$, $q=1+(2+\beta)/N$, $0<\rho<\|W_{q}\|_{2}^{2}$;

(3) $\gamma>0$, $p=1+4/N$, $1+{\beta}/{N}<q<1+(2+\beta)/N$, $0<\rho<\|Q_{p}\|_{2}^{2}$;

*(4) $\gamma>0$, $p=1+4/N$, $q=1+(2+\beta)/N$, $(\sqrt{\rho}/\|Q_{p}\|_{2})^{4/N}+(\sqrt{\rho}/\|W_{q}\|_{2})^{(2\beta+4)/N}<1$;

then $G_{\rho}$ is not empty and orbitally stable.*

We should point out that, among the methods used in the study of orbital stability of standing waves, the profile decomposition method plays an important role in recent studies, see [4], [13] and [38]. In [4], the authors considered a Schrödinger equation with inverse-square potential, i.e. (1.1) with $N\geq 3$, $\alpha=2$ and $\gamma<(N-2)^{2}/4$. By using the equivalence of $\|\nabla u\|_{2}^{2}$ and $\int_{\mathbb{R}^{N}}(|\nabla u|^{2}-\gamma|x|^{-2}|u|^{2})dx$, Bensouilah [3] obtained the profile decomposition of a bounded sequence in $H^{1}(\mathbb{R}^{N})$ related to the problem, and based of which, [4] studied the orbital stability of standing waves. However, there is not an equivalence between $\|\nabla u\|_{2}^{2}$ and $\int_{\mathbb{R}^{N}}(|\nabla u|^{2}-\gamma|x|^{-\alpha}|u|^{2})dx$ for $0<\alpha<\min\{2,N\}$ and we can not obtain the profile decomposition in this case. But in view of that $\int_{\mathbb{R}^{N}}(|\nabla u|^{2}-\gamma|x|^{-\alpha}|u|^{2})dx$ can be controlled by $\epsilon\|\nabla u\|_{2}^{2}$ and a function of $\|u\|_{2}^{2}$ (see Lemma 2.2), and by carefully examining the application of concentration compactness principle in the study of orbital stability of standing waves (see [7] and [23]), we can solve the problem by using the concentration compactness principle in a systematic way. In fact, the profile decomposition can be seen as another equivalent description of the concentration compactness principle, see [13].

This paper is organized as follows. In Section 2, we give some preliminaries. Section 3 is devoted to the proof of Theorems 1.4-1.8.

Notation. Throughout this paper, we use the following notation. $C>0$ stands for a constant that may be different from line to line when it does not cause any confusion. The notation $A\lesssim B$ means that $A\leq CB$ for some constant $C>0$. $L^{r}(\mathbb{R}^{N})$ with $1\leq r<\infty$ denotes the Lebesgue space with the norms $\|u\|_{r}=\left(\int_{\mathbb{R}^{N}}|u|^{r}dx\right)^{1/r}$. $H^{1}(\mathbb{R}^{N})$ is the usual Sobolev space with norm $\|u\|_{H^{1}(\mathbb{R}^{N})}=\left(\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+|u|^{2})dx\right)^{1/2}$. $B_{R}(x)$ denotes the ball in $\mathbb{R}^{N}$ centered at $x$ with radius $R$. $B_{R}^{c}(x)=\mathbb{R}^{N}\backslash B_{R}(x)$. $\chi_{B}(x)=1$ if $x\in B$, and $=0$ if $x\not\in B$.

2. Preliminaries

The following generalized Gagliardo-Nirenberg inequality can be found in [34].

Lemma 2.1**.**

Let $N\geq 1$ and $0<\eta<4/(N-2)_{+}$, then the following sharp Gagliardo-Nirenberg inequality

[TABLE]

holds for any $u\in H^{1}(\mathbb{R}^{N})$. The sharp constant $C_{GN}(\eta)$ is

[TABLE]

where $Q_{\eta+1}$ is defined in Theorem 1.4.

Next, we give a refined estimate for the linear operator $-\Delta-\gamma|x|^{-\alpha}$.

Lemma 2.2**.**

Let $N\geq 1$, $0<\alpha<\min\{2,N\}$ and $\gamma\in\mathbb{R}$. Then for any $\epsilon>0$, there exists a constant $\delta=\delta(\epsilon,\|u\|_{2})>0$ such that

[TABLE]

for any $u\in H^{1}(\mathbb{R}^{N})$.

Proof.

It obviously holds for $\gamma\leq 0$. Now we prove the lemma for $\gamma>0$. By the domain decomposition, the Hölder inequality and $0<\alpha<\min\{2,N\}$, we know

[TABLE]

where $1/r+1/r^{\prime}=1$, $1/s+1/s^{\prime}=1$, $r<N/\alpha$, $s>N/\alpha$, $N/\alpha-r$ and $s-N/\alpha$ are both sufficiently small. Hence, $r^{\prime}-N/(N-\alpha)>0$ and $N/(N-\alpha)-s^{\prime}>0$ are small. By the Gagliardo-Nirenberg inequality (Lemma 2.1), we know

[TABLE]

Noting that $|N-N/r^{\prime}-\alpha|$ is sufficiently small and $0<\alpha<2$, and by using the Young inequality

[TABLE]

we have for any $\epsilon_{1}>0$, there exists $\delta_{1}=\delta_{1}(\epsilon_{1},\|u\|_{2})$ such that

[TABLE]

The same estimates hold for $\|u\|_{2s^{\prime}}^{2}$. In view of (2.1)-(2.3), we complete the proof. ∎

The following concentration compactness principle is cited from Lemma III.1 in [23].

Lemma 2.3**.**

Let $N\geq 1$ and $\{u_{n}\}_{n=1}^{\infty}$ be a bounded sequence in $H^{1}(\mathbb{R}^{N})$ satisfying:

[TABLE]

where $\lambda>0$ is fixed. Then there exists a subsequence $\{u_{n_{k}}\}_{k=1}^{\infty}$ satisfying one of the three possibilities:

(i) (compactness) there exists $\{y_{n_{k}}\}_{k=1}^{\infty}\subset\mathbb{R}^{N}$ such that $|u_{n_{k}}(\cdot+y_{n_{k}})|^{2}$ is tight, i.e.,

[TABLE]

(ii) (vanishing) $\lim_{k\to\infty}\sup_{y\in\mathbb{R}^{N}}\int_{B_{R}(y)}|u_{n_{k}}(x)|^{2}dx=0$ for all $R<\infty$;

(iii) (dichotomy) there exists $\sigma\in(0,\lambda)$ such that for any $\epsilon>0$, there exist $k_{0}\geq 1$, $R_{1}=R_{1}(\epsilon)>0$, $\{y_{n_{k}}\}_{k=1}^{\infty}\subset\mathbb{R}^{N}$ and $u_{n_{k}}^{(1)}$, $u_{n_{k}}^{(2)}$ bounded in $H^{1}(\mathbb{R}^{N})$ satisfying for $k\geq k_{0}$:

[TABLE]

The following well-known Hardy-Littlewood-Sobolev inequality can be found in [20].

Lemma 2.4**.**

Let $N\geq 1$, $p$, $r>1$ and $0<\beta<N$ with $1/p+(N-\beta)/N+1/r=2$. Let $u\in L^{p}(\mathbb{R}^{N})$ and $v\in L^{r}(\mathbb{R}^{N})$. Then there exists a sharp constant $C(N,\beta,p)$, independent of $u$ and $v$, such that

[TABLE]

If $p=r=\frac{2N}{N+\beta}$, then

[TABLE]

Remark 2.5**.**

(1). By the Hardy-Littlewood-Sobolev inequality above, for any $v\in L^{s}(\mathbb{R}^{N})$ with $s\in(1,\frac{N}{\alpha})$, $I_{\alpha}\ast v\in L^{\frac{Ns}{N-\alpha s}}(\mathbb{R}^{N})$ and

[TABLE]

where $C>0$ is a constant depending only on $N,\ \alpha$ and $s$.

(2). By the Hardy-Littlewood-Sobolev inequality above and the Sobolev embedding theorem, we obtain

[TABLE]

for any $p\in\left[\frac{N+\beta}{N},\frac{N+\beta}{N-2}\right]$ if $N\geq 3$ and $p\in\left[\frac{N+\beta}{N},+\infty\right)$ if $N=1,2$, where $C>0$ is a constant depending only on $N,\ \beta$ and $p$.

The following generalized Gagliardo-Nirenberg inequality for the convolution problem can be found in [12] and [27].

Lemma 2.6**.**

Let $N\geq 1$, $0<\beta<N$, $1+\beta/N<p<(N+\beta)/(N-2)_{+}$, then

[TABLE]

The best constant $C(\beta,p)$ is defined by

[TABLE]

where $W_{p}$ is defined in Theorem 1.5. In particular, in the $L^{2}$-critical case, i.e., $p=1+\frac{2+\beta}{N}$, $C(\beta,p)=p\|W_{p}\|_{2}^{2-2p}$.

Remark 2.7**.**

Note that $W_{p}$ may be not unique, but the ground state has the same $L^{2}$-norm, see [12].

In the end of this section, we prove the local well-posedness in the energy space $H^{1}(\mathbb{R}^{N})$ for (1.1) when $q\geq 2$. We first recall the following result due to Cazenave (see Theorem 4.3.1 in [6]).

Theorem 2.8**.**

Let $N\geq 1$. Consider the following Cauchy problem

[TABLE]

Let $g=g_{1}+g_{2}+\cdots+g_{k}$ be such that the following assumptions hold for every $i=1,2,\cdots,k$:

(A1) $g_{i}\in C(H^{1}(\mathbb{R}^{N}),H^{-1}(\mathbb{R}^{N}))$ and there exists $G_{i}\in C^{1}(H^{1}(\mathbb{R}^{N}),\mathbb{R})$ such that $g_{i}=G_{i}^{\prime}$;

(A2) there exist $r,\rho\in[2,2N/(N-2)_{+})$ if $N\geq 2$ ($r,\rho\in[2,+\infty]$ if $N=1$) such that $g_{i}\in C(H^{1}(\mathbb{R}^{N}),L^{\rho^{\prime}}(\mathbb{R}^{N}))$ and such that for any $M<\infty$ there exists $C(M)<\infty$ such that

[TABLE]

for all $v,w\in H^{1}(\mathbb{R}^{N})$ with $\|v\|_{H^{1}(\mathbb{R}^{N})}+\|w\|_{H^{1}(\mathbb{R}^{N})}\leq M$;

*(A3) for any $v\in H^{1}(\mathbb{R}^{N})$, $\mathrm{Im}(g_{i}(v)\bar{v})=0$ a.e. in $\mathbb{R}^{N}$.

Then for any $u_{0}\in H^{1}(\mathbb{R}^{N})$, there exist $T_{*},\ T^{*}\in(0,\infty]$ and a unique solution $u(t)\in C((-T_{*},T^{*}),H^{1}(\mathbb{R}^{N}))\cap C^{1}((-T_{*},T^{*}),H^{-1}(\mathbb{R}^{N}))$ with $u(0)=u_{0}$ to (2.5). If $T^{*}<\infty$ ($T_{*}<\infty$), then $\lim_{t\uparrow T^{*}}\|u(t,x)\|_{H^{1}(\mathbb{R}^{N})}=\infty$ ($\lim_{t\downarrow-T_{*}}\|u(t,x)\|_{H^{1}(\mathbb{R}^{N})}=\infty$). Moreover, there is conservation of mass and energy:*

[TABLE]

for all $t\in(-T_{*},T^{*})$.

Applying Theorem 2.8, we obtain the local well-posedness in $H^{1}(\mathbb{R}^{N})$ for (1.1).

Theorem 2.9**.**

Let $N\geq 1$, $V(x)=\gamma|x|^{-\alpha}$, $\gamma\in\mathbb{R}$, $0<\alpha<\min\{2,N\}$, $f(u)$ be one of Types 1-5, $q\geq 2$. Then Assumption A holds.

Proof.

Denote $g(u)=\gamma|x|^{-\alpha}u+|u|^{p-1}u+(I_{\beta}\ast|u|^{q})|u|^{q-2}u:=g_{1}+g_{2}+g_{3}$. Since $0<\alpha<\min\{2,N\}$, $\gamma|x|^{-\alpha}\in L^{r}(\mathbb{R}^{N})+L^{\infty}(\mathbb{R}^{N})$ for some $r>\max\{1,N/2\}$. By Example 3.2.11 in [6], we know that $g_{1}$ and $g_{2}$ satisfy assumptions (A1)-(A3) in Theorem 2.8.

Since $2\leq q<2N/(N-2)_{+}$, it follows from (3.2.15) in [6] that

[TABLE]

By the mean value theorem,

[TABLE]

Set $\rho=\frac{2Nq}{N+\alpha}$. By Lemma 2.4, Remark 2.5 and the Hölder inequality, we know

[TABLE]

which implies that $g_{3}$ satisfies (A1)-(A3) in Theorem 2.8. The proof is complete. ∎

3. Proof of the main results

Proof of Theorem 1.4. We will prove this theorem in seven steps.

Step 1. We show that $A_{\rho}>-\infty$, that is, $A_{\rho}$ is well defined. For any $u\in M_{\rho}$, we have $\|u\|_{2}^{2}=\rho$, and by Lemmas 2.1 and 2.2, we have for any $\epsilon>0$,

[TABLE]

For $1<p<1+4/N$, we have $0<(p-1)N/2<2$. Thus, by the Young inequality, we obtain from (3.1) that

[TABLE]

by choosing $\epsilon=1/4$.

For $p=1+4/N$, we have

[TABLE]

and $C_{GN}(p-1)=(N+2)/N\|Q_{p}\|_{2}^{-4/N}$. Thus, we obtain from (3.1) that

[TABLE]

by choosing $\epsilon>0$ small enough since $\|u\|_{2}<\|Q_{p}\|_{2}$.

Step 2. For any $\rho>0$ there exists $C_{1}>0$ such that $A_{\rho}\leq-C_{1}<0$. Indeed, for any $\varphi\in M_{\rho}$ and $\lambda>0$, we define $\varphi_{\lambda}(x)=\lambda^{N/2}\varphi(\lambda x)$. It is easy to check that $\|\varphi_{\lambda}\|_{2}^{2}=\|\varphi\|_{2}^{2}=\rho$ and

[TABLE]

For $1<p<1+4/N$, we have $0<(p-1)N/2<2$. So by choosing $\lambda>0$ sufficiently small, we obtain $E(\varphi_{\lambda})<0$ for any $\gamma>0$. For $p=1+4/N$, we have $(p-1)N/2=2$. By choosing $\varphi$ such that

[TABLE]

we have $E(\varphi_{\lambda})<0$ for any $\lambda>0$ and $\gamma>0$.

Let $\{u_{n}\}_{n=1}^{\infty}\subset M_{\rho}$ be a minimizing sequence of $E$, that is,

[TABLE]

Similarly to Step 1, one can show that $\{u_{n}\}_{n=1}^{\infty}$ is bounded in $H^{1}(\mathbb{R}^{N})$. Then there exists a subsequence $\{u_{n_{k}}\}_{k=1}^{\infty}$ such that one of the three possibilities in Lemma 2.3 holds.

Step 3. The vanishing case in Lemma 2.3 does not occur. Suppose by contradiction that

[TABLE]

By Lion’s lemma, $u_{n_{k}}\to 0$ in $L^{s}(\mathbb{R}^{N})$ as $k\to\infty$ for all $s\in(2,2N/(N-2)_{+})$. Hence,

[TABLE]

by (2.1), and thus

[TABLE]

which contradicts $A_{\rho}<0$.

Step 4. We show that the dichotomy case in Lemma 2.3 does not occur.

Firstly, according to (3) in [23], we know

[TABLE]

where $A_{0}=0$, $A_{\rho-\eta}^{\infty}=\inf_{M_{\rho-\eta}}E^{\infty}(u)$ and $E^{\infty}(u)=\frac{1}{2}\|\nabla u\|_{2}^{2}-\frac{1}{p+1}\|u\|_{p+1}^{p+1}$.

Next, we claim that

[TABLE]

Indeed, choose a sequence $\{\varphi_{n}\}_{n=1}^{\infty}\subset M_{\eta}$ such that $\lim_{n\to\infty}E(\varphi_{n})=A_{\eta}$. Then $\|\sqrt{\theta}\varphi_{n}\|_{2}^{2}=\theta\eta$ and

[TABLE]

since $\theta>1$ and $p>1$. Hence the claim holds. It follows from (3.2), (3.3) and Lemma II.1 in [23], we have

[TABLE]

Finally, suppose by contradiction that (iii) in Lemma 2.3 holds. Denote

[TABLE]

Then we may assume without loss of generality that

[TABLE]

as $k\to\infty$ with $\bar{\sigma},\ \bar{\eta}\in(0,\rho)$, $|\bar{\sigma}-\sigma|\leq\epsilon$ and $|\bar{\eta}-(\rho-\sigma)|\leq\epsilon$.

By direct calculation, we obtain that

[TABLE]

where $\delta(\epsilon)\to 0$ as $\epsilon\to 0$. If $\{y_{n_{k}}\}_{k=1}^{\infty}$ is bounded, in view of (2.1), we have

[TABLE]

as $k\to\infty$ since $d_{n_{k}}\to\infty$. Similarly, if $\{y_{n_{k}}\}_{k=1}^{\infty}$ is unbounded, then

[TABLE]

By using the inequality

[TABLE]

with $r>1$ and $l\geq 2$, and in view of $u_{n_{k}}^{(1)}u_{n_{k}}^{(2)}=0$, we obtain that

[TABLE]

which implies that

[TABLE]

for any $k\geq k_{0}$, where $\delta(\epsilon)\to 0$ as $\epsilon\to 0$. By (3.5)-(3.9), we obtain

[TABLE]

Letting $\epsilon\to 0$, we obtain that $A_{\rho}\geq A_{\sigma}+A_{\rho-\sigma}^{\infty}$ or $A_{\rho}\geq A_{\sigma}^{\infty}+A_{\rho-\sigma}$, which contradicts (3.4). Hence, (iii) in Lemma 2.3 does not occur.

Step 5. From Steps 3 and 4 and Lemma 2.3, there exist a subsequence $\{u_{n_{k}}\}_{k=1}^{\infty}$ and a sequence $\{y_{n_{k}}\}_{k=1}^{\infty}\subset\mathbb{R}^{N}$ such that for all $\epsilon>0$, there exists $R(\epsilon)>0$ such that for all $k\geq 1$,

[TABLE]

Denote $\tilde{u}_{n_{k}}(\cdot)=u_{n_{k}}(\cdot+y_{n_{k}})$. Then there exists $\tilde{u}$ such that $\tilde{u}_{n_{k}}\rightharpoonup\tilde{u}$ weakly in $H^{1}(\mathbb{R}^{N})$, $\tilde{u}_{n_{k}}\to\tilde{u}$ strongly in $L_{\mathrm{loc}}^{r}(\mathbb{R}^{N})$ with $r\in[2,2N/(N-2)_{+})$, which combine (3.10) show that

[TABLE]

Thus $\int_{\mathbb{R}^{N}}|\tilde{u}(x)|^{2}dx=\lambda$, i.e., $\tilde{u}_{n_{k}}\to\tilde{u}$ strongly in $L^{2}(\mathbb{R}^{N})$. By the Gagliardo-Nirenberg inequality, $\tilde{u}_{n_{k}}\to\tilde{u}$ strongly in $L^{s}(\mathbb{R}^{N})$ for $s\in[2,2N/(N-2)_{+})$.

Next, we claim that $\{y_{n_{k}}\}_{k=1}^{\infty}$ is bounded. If it were not the case we deduce that

[TABLE]

as $k\to\infty$. Hence $A_{\rho}\geq A_{\rho}^{\infty}$. On the other hand, we know that $A_{\rho}^{\infty}$ is attained by a nontrivial function $v_{\rho}$. Indeed, Steps 1-4 hold for $\gamma=0$. Hence,

[TABLE]

By the definition of $A_{\rho}^{\infty}$, we see that $\tilde{u}$ is a minimizer of $A_{\rho}^{\infty}$. Consequently,

[TABLE]

which contradicts $A_{\rho}\geq A_{\rho}^{\infty}$. Thus, $\{y_{n_{k}}\}_{k=1}^{\infty}$ is bounded. So $u_{n_{k}}$ converges strongly to some $u$ in $L^{s}(\mathbb{R}^{N})$ for $s\in[2,2N/(N-2)_{+})$. Hence,

[TABLE]

By the definition of $A_{\rho}$, we see that $u$ is a minimizer of $A_{\rho}$, $\lim_{k\to\infty}\|\nabla u_{n_{k}}\|_{2}^{2}=\|\nabla u\|_{2}^{2}$, and hence $u_{n_{k}}\to u$ in $H^{1}(\mathbb{R}^{N})$.

Step 6. We show that under Assumption A, if $1<p<1+4/N$ and $u_{0}\in H^{1}(\mathbb{R}^{N})$ or $p=1+4/N$ and $\|u_{0}\|_{2}<\|Q_{p}\|_{2}$, then the Cauchy problem (1.1) admits a global solution $u(t)\in C(\mathbb{R},H^{1}(\mathbb{R}^{N}))\cap C^{1}(\mathbb{R},H^{-1}(\mathbb{R}^{N}))$ with $u(0)=u_{0}$.

Indeed, by Assumption A, it suffices to bound $\|\nabla u(t)\|_{2}$ in the existence time. By the conversation law, Lemmas 2.1 and 2.2, we know

[TABLE]

Similarly to Step 1, for $p=1+4/N$, we have

[TABLE]

Since $\|u(t)\|_{2}=\|u(0)\|_{2}<\|Q_{p}\|_{2}$, by choosing $\epsilon>0$ sufficiently small, the above estimate implies the boundedness of $\|\nabla u(t)\|_{2}$. For $1<p<1+4/N$, we have

[TABLE]

and we arrive at the conclusion.

Step 7. We prove that $G_{\rho}$ is orbitally stable. Suppose by contradiction that there exist sequences $\{u_{0,n}\}_{n=1}^{\infty}\subset H^{1}(\mathbb{R}^{N})$ and $\{t_{n}\}_{n=1}^{\infty}\subset\mathbb{R}^{+}$ and a constant $\epsilon_{0}>0$ such that for all $n\geq 1$,

[TABLE]

and

[TABLE]

where $u_{n}(t)$ is the solution to (1.1) with initial data $u_{0,n}$.

We claim that there exists $v\in G_{\rho}$ such that

[TABLE]

Indeed, by (3.12), there exists $\{v_{n}\}_{n=1}^{\infty}\subset G_{\rho}$ such that

[TABLE]

That $\{v_{n}\}_{n=1}^{\infty}\subset G_{\rho}$ implies that $\{v_{n}\}_{n=1}^{\infty}\subset M_{\rho}$ is a minimizing sequence of $E$. So by Steps 3-5, there exists $v\in G_{\rho}$ such that

[TABLE]

Then the claim follows from (3.14) and (3.15) immediately. Hence,

[TABLE]

By the conservation of mass and energy, we have

[TABLE]

Similarly to Step 1, $\{u_{n}(t_{n})\}_{n=1}^{\infty}$ is bounded in $H^{1}(\mathbb{R}^{N})$. Set

[TABLE]

Then $\|\tilde{u}_{n}\|_{2}^{2}=\rho$ and

[TABLE]

which implies that

[TABLE]

Hence, $\{\tilde{u}_{n}\}_{n=1}^{\infty}\subset M_{\rho}$ is a minimizing sequence of $E$, and by Steps 3-5, there exists $\tilde{v}\in G_{\rho}$ such that

[TABLE]

By the definition of $\tilde{u}_{n}$, we know

[TABLE]

(3.16) and (3.17) imply that

[TABLE]

which contradicts (3.13). The proof is complete.

Proof of Theorem 1.5. The proof follows from that of Theorem 1.4 by line to line, so we only point out the differences.

Step 1. We show that $A_{\rho}>-\infty$. For any $u\in M_{\rho}$, we have $\|u\|_{2}^{2}=\rho$, and by Lemmas 2.2 and 2.6, we have for any $\epsilon>0$,

[TABLE]

For $1+\beta/N<q<1+(\beta+2)/N$, we have $0<Nq-N-\beta<2$. Thus, by the Young inequality, we obtain from (3.18) that

[TABLE]

by choosing $\epsilon=1/4$.

For $q=1+(\beta+2)/N$, we have $Nq-N-\beta=2$, $N+\beta-Nq+2q=(2\beta+4)/N$ and $C(\beta,q)=q\|W_{q}\|_{2}^{2-2q}=q\|W_{q}\|_{2}^{-(2\beta+4)/N}$. Thus, we obtain from (3.18) that

[TABLE]

by choosing $\epsilon>0$ small enough since $\|u\|_{2}<\|W_{q}\|_{2}$.

Step 2. For any $\rho>0$ there exists $C_{1}>0$ such that $A_{\rho}\leq-C_{1}<0$. Indeed, for any $\varphi\in M_{\rho}$ and $\lambda>0$, we define $\varphi_{\lambda}(x)=\lambda^{N/2}\varphi(\lambda x)$. Then $\|\varphi_{\lambda}\|_{2}^{2}=\|\varphi\|_{2}^{2}=\rho$ and

[TABLE]

Similarly to Step 2 in the proof of Theorem 1.4, for any $1+\beta/N<q\leq 1+(\beta+2)/N$, we can choose $\varphi$ and $\lambda$ such that $E(\varphi_{\lambda})<0$.

Let $\{u_{n}\}_{n=1}^{\infty}\subset M_{\rho}$ be a minimizing sequence of $E$, that is,

[TABLE]

Then $\{u_{n}\}_{n=1}^{\infty}$ is bounded in $H^{1}(\mathbb{R}^{N})$ and there exists a subsequence $\{u_{n_{k}}\}_{k=1}^{\infty}$ such that one of the three possibilities in Lemma 2.3 holds.

Step 3. The vanishing case in Lemma 2.3 does not occur. Suppose by contradiction that

[TABLE]

Lion’s lemma implies that $u_{n_{k}}\to 0$ in $L^{s}(\mathbb{R}^{N})$ for any $s\in(2,2N/(N-2)_{+})$. Hence,

[TABLE]

according to (2.1) and (2.4), and then

[TABLE]

which contradicts $A_{\rho}<0$.

Step 4. The dichotomy case in Lemma 2.3 does not occur. Similarly, we have

[TABLE]

and

[TABLE]

where $A_{0}=0$, $A_{\rho-\eta}^{\infty}=\inf_{M_{\rho-\eta}}E^{\infty}(u)$ and $E^{\infty}(u)=\frac{1}{2}\|\nabla u\|_{2}^{2}-\frac{1}{2q}\int_{\mathbb{R}^{N}}(I_{\beta}\ast|u|^{q})|u|^{q}dx$. Consequently,

[TABLE]

Suppose by contradiction that (iii) in Lemma 2.3 holds. By using (3.8) and $u_{n_{k}}^{(1)}u_{n_{k}}^{(2)}=0$, we obtain that

[TABLE]

where $\delta(\epsilon)\to 0$ as $\epsilon\to 0$. Since $0<\beta<N$ and $1+\beta/N<q<(N+\beta)/(N-2)_{+}$, we can choose a constant $\delta>0$ sufficiently small such that

[TABLE]

which combines with $d_{n_{k}}=\mathrm{dist}\{\mathrm{Supp}\ {u_{n_{k}}^{(1)}},\mathrm{Supp}\ {u_{n_{k}}^{(2)}}\}\to\infty$ and (2.4) gives that

[TABLE]

as $k\to\infty$. Hence,

[TABLE]

Letting $\epsilon\to 0$, we obtain that $A_{\rho}\geq A_{\sigma}+A_{\rho-\sigma}^{\infty}$ or $A_{\rho}\geq A_{\sigma}^{\infty}+A_{\rho-\sigma}$, which contradicts (3.19).

Step 5. From Steps 3 and 4 and Lemma 2.3, similarly to Step 5 in the proof of Theorem 1.4, we know

[TABLE]

and then

[TABLE]

Hence, $u$ is a minimizer of $A_{\rho}$ and $u_{n_{k}}\to u$ in $H^{1}(\mathbb{R}^{N})$.

Step 6. Under Assumption A, if $1+\beta/N<q<1+(2+\beta)/N$ and $u_{0}\in H^{1}(\mathbb{R}^{N})$ or $q=1+(2+\beta)/N$ and $\|u_{0}\|_{2}<\|W_{q}\|_{2}$, then the Cauchy problem (1.1) admits a global solution $u(t)\in C(\mathbb{R},H^{1}(\mathbb{R}^{N}))\cap C^{1}(\mathbb{R},H^{-1}(\mathbb{R}^{N}))$ with $u(0)=u_{0}$.

Step 7. Similarly, $G_{\rho}$ is orbitally stable. The proof is complete.

Proof of Theorem 1.6. It can be done by small modifications of the proof of Theorem 1.4, and we omit it.

Proof of Theorem 1.7. It can be done by small modifications of the proof of Theorem 1.5, and we omit it.

Proof of Theorem 1.8. It can be done by combining the proof of Theorems 1.4 and 1.5, and we omit it.

Acknowledgements. The authors would like to express sincere thanks to the anonymous referee for his or her carefully reading the manuscript and valuable comments and suggestions. This work was supported by Tianjin Municipal Education Commission with the Grant no. 2017KJ173 “Qualitative studies of solutions for two kinds of nonlocal elliptic equations”.