Existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities

1.   Introduction

In this paper, we consider the following nonlinear Schrödinger equation (NLS) with inverse-power potential, and combined general power-type nonlinearity and Choquard-type nonlinearity

where $N\geq3$, $\psi:[0, T^\star)\times \mathbb{R}^{N}\rightarrow \mathbb{C}$ is the complex valued function with $0 < T^\star\leq \infty$, $\psi_0\in H^1$, $\gamma\in(0, +\infty)$, $\alpha\in(0, 2)$, $\lambda_1\in\mathbb{R}\setminus\{0\}$ and $\lambda_2\in\mathbb{R}\setminus\{0\}$, $\frac{4}{N}\leq p < \frac{4}{N-2}$, $1+\frac{2+\beta}{N}\leq q < \frac{N+\beta}{N-2}$, $\ast$ denotes the convolution, $I_\beta:\mathbb{R}^N\rightarrow \mathbb{R}$ is the Riesz potential that defined for every $x\in \mathbb{R}^N\setminus\{0\}$ by

$\beta\in(0, N)$ and $\Gamma$ is the Gamma function.

Because of important applications of (1.1) in physics, it has received much attention both from physics (see ^[1,2,3,4,5]) and mathematics (see ^{[6,7,8,9,10,11,12,13]}), and has been widely studied for a long time. The operator $-\triangle-\frac{\gamma}{|x|}$ 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, e.g., ^[2,3,14].

We are interested in the standing wave solutions of (1.1), namely solutions of the form $\psi(t, x) = e^{i\omega t}u(x)$, where $\omega\in\mathbb{R}$ is a frequency and $u\in H^1$ is a nontrivial solution to the elliptic equation

The Eq (1.3) is variational, whose action functional is defined by

where the corresponding energy $E_\gamma(u)$ is defined by

For the evolutional type Eq (1.1), one of the important problems is to consider the stability of standing waves. Then, we recall the definition of orbital stability of set $\mathcal{M}$.

Definition 1.1. The set $\mathcal{M}$ is said to be orbitally stable if, for any $\epsilon > 0$, there exists $\delta > 0$ such that for any initial data $\psi_0$ satisfying

the corresponding solution $\psi(t)$ of (1.1) with initial data $\psi_0$ satisfies

for all $t > 0$.

In view of this definition, in order to study the stability, we require that (1.1) has a unique global solution, at least for initial data $\psi_0$ sufficiently close to $\mathcal{M}$. In the $L^2$-subcritical case, all solutions for NLS exist globally. Hence, the stability of standing waves has been studied extensively in this case, see, e.g., ^{[11,14,15,16]}.

For (1.1), when two nonlinearities are both focusing $L^2$-subcritical, i.e., $\lambda_1\in(0, +\infty)$, $\lambda_2\in(0, +\infty)$, $0 < p < 4/N$, $1+\beta/N < q < 1+(2+\beta)/N$, or when one nonlinearity is focusing $L^2$-subcritical and the other is focusing $L^2$-critical, i.e., $\lambda_1\in(0, +\infty)$, $\lambda_2\in(0, +\infty)$, $0 < p < 4/N$, $q = 1+(2+\beta)/N$ and $0 < \|\psi_0\|_{L^2} < \|Q_q\|_{L^2}$, $Q_q$ be a ground state of elliptic equation

the solution $\psi(t)$ of (1.1) with the initial data $\psi_0$ exists globally. In these cases, Li and Zhao in ^[14] used the concentration compactness principle to study the existence and orbital stability of standing waves. When one nonlinearity is focusing and $L^2$-critical, the other is defocusing and $L^2$-supercritical, all solutions of (1.1) exist globally (see Lemma 2.7). Therefore, in this case, whether there exist stable standing waves is an interesting problem. To the best of our knowledge, there are no stability results for (1.1) with a defocusing $L^2$-supercritical nonlinearity.

To this purpose, applying the idea by Cazenave and Lions in ^[17], we consider the following constrained minimization problem:

where $E_\gamma(u)$ is defined by (1.4) and

We will see later (Lemma 2.8) that the above minimizing problem is well-defined. Let us denote

Our main results are as follows:

Theorem 1.2. Let $N \geq 3$, $\gamma\in(0, +\infty)$, $\alpha\in(0, 2)$, $\beta\in(0, N)$, $\lambda_1\in(-\infty, 0)$, $\lambda_2\in(0, +\infty)$, $\frac{4}{N} < p < \frac{4}{N-2}$, $q = 1+\frac{2+\beta}{N}$. Then, there exists $\gamma_0 > 0$ sufficiently small such that $0 < \gamma < \gamma_0$, for any $\eta\in(\|Q_q\|_{L^2}^2, \infty)$, $Q_q$ be a ground state of elliptic Eq (1.5), the set $K(\eta)$ is not empty and orbitally stable.

Theorem 1.3. Let $N \geq 3$, $\gamma\in(0, +\infty)$, $\alpha\in(0, 2)$, $\beta\in(0, N)$, $\lambda_1\in(0, +\infty)$, $\lambda_2\in(-\infty, 0)$, $p = \frac{4}{N}$, $1+\frac{2+\beta}{N} < q < \frac{N+\beta}{N-2}$. Then, there exists $\gamma_0 > 0$ sufficiently small such that $0 < \gamma < \gamma_0$, for any $\eta\in(\|W_p\|_{L^2}^2, \infty)$, where $W_p$ is the ground state of the following equation:

Then, the set $K(\eta)$ is not empty and orbitally stable.

Since the proof of Theorem 1.2 and Theorem 1.3 is similar, we only prove Theorem 1.2.

2.   Preliminaries

In this section, we recall some preliminary results that will be used later.

Lemma 2.1. (^[17], Lemma 7.6.1) Let $1\leq p < \infty$. If $\alpha < N$ is such that $0\leq \alpha\leq p$, then $\frac{|u(\cdot)|^p}{|\cdot|^\alpha}\in L^1$ for every $u\in W^{1, p}(\mathbb{R}^N)$. Furthermore,

for every $u\in W^{1, p}(\mathbb{R}^N)$.

Lemma 2.2. (^[14], Lemma 2.2) Let $N\geq 3$, $\alpha\in(0, 2)$, $\gamma\in\mathbb{R}$. Then for any $\epsilon > 0$, there exists a constant $\delta = \delta(\epsilon, \|u\|_{L^2}^2) > 0$ such that

for any $u\in H^1$.

Proof. It obviously holds for $\gamma\leq0$. Now, we use (2.1) to prove the Lemma for $\gamma > 0$. According to (2.1) and the Young inequality, we have

we arrive at the conclusion.

Lemma 2.3. (^[18]) Let $N \geq 3$, $\beta\in(0, N)$, and $q$, $q' > 1$ be constants such that

Assume that $u\in L^q$ and $v\in L^{q'}$, then there exists a sharp constant $C(N, \beta, q)$ independent of $u$ and $v$, such that

By the Hardy-Littlewood-Sobolev inequality above and the Sobolev embedding theorem, we obtain

for any $q\in [1+\frac{\beta}{N}, \frac{N+\beta}{N-2}]$, $C > 0$ is a constant depending only on $N$, $\beta$ and $q$.

Lemma 2.4. (^[12,19,20]) Let $N\geq 3$, $\beta\in(0, N)$, $1+\frac{\beta}{N} < q < \frac{N+\beta}{N-2}$, then for all $u\in H^1$,

the best constant $C(\beta, q)$ is defined by

where $Q_q$ is the ground state of elliptic Eq (1.5). In particular, in the $L^2$-critical case, i.e., $q = 1+\frac{2+\beta}{N}$, $C(\beta, q) = q\|Q_q\|_{L^2}^{2-2q}$. Moreover, the following Poho$\check{z}$aev's identities hold true:

Lemma 2.5. (^[17]) Let $N\geq3$ and $\{u_n\}$ be a bounded sequence in $H^1$ satisfying:

where $\mu > 0$ is fixed. Then there exists a subsequence $\{u_{n_k}\}$ satisfying one of the three possibilities:

(1) $(compactness)$ there exists $\{y_{n_k}\}\subset\mathbb{R}^N$ such that $|u_{n_k}(\cdot+y_{n_k})|^2$ is tight, i.e., for all $\epsilon > 0$, there exists $R < \infty$, such that

(2) $(vanishing)$ $\lim\limits_{k\rightarrow\infty}\sup\limits_{y\in\mathbb{R}^N}\int_{B_R(y)}|u_{n_k}(x)|^2dx = 0$ for all $R < \infty$;

(3) $(dichotomy)$ there exists $\sigma\in(0, \mu)$ such that for any $\epsilon > 0$, there exist $k_0\geq1, \; \{y_{n_k}\}\subset\mathbb{R}^N$ and $u_{n_k}^{(1)}, \; u_{n_k}^{(2)}$ bounded in $H^1$ satisfying for $k\geq k_0$:

Lemma 2.6. Let $N \geq 3$, $\lambda_1\in(-\infty, 0)$, $\lambda_2\in(0, +\infty)$, $\frac{4}{N} < p < \frac{4}{N-2}$, $q = 1+\frac{2+\beta}{N}$, or $N \geq 3$, $\lambda_1\in(0, +\infty)$, $\lambda_2\in(-\infty, 0)$, $p = \frac{4}{N}$, $1+\frac{2+\beta}{N} < q < \frac{N+\beta}{N-2}$. The initial data $\psi_0\in H^1$, there exists $T = T(\|\psi_0\|_{H^{1}})$ such that (1.1) admits a unique solution $\psi\in C([0, T], H^{1})$. Let $[0, T^\star)$ be the maximal time interval on which the solution $\psi$ is well-defined, if $T^\star < \infty$, then $\|\psi(t)\|_{H^{1}}\rightarrow\infty$ as $t\uparrow T^\star$. Moreover, for all $0\leq t < T^\star$, the solution $\psi(t)$ satisfies the following conservation of mass and energy:

Lemma 2.7. Let $N \geq 3$, $\gamma\in(0, +\infty)$, $\alpha\in(0, 2)$, $\beta\in(0, N)$, $\lambda_1\in(-\infty, 0)$, $\lambda_2\in(0, +\infty)$, $\frac{4}{N} < p < \frac{4}{N-2}$, $q = 1+\frac{2+\beta}{N}$, or $N \geq 3$, $\gamma\in(0, +\infty)$, $\alpha\in(0, 2)$, $\lambda_1\in(0, +\infty)$, $\lambda_2\in(-\infty, 0)$, $p = \frac{4}{N}$, $1+\frac{2+\beta}{N} < q < \frac{N+\beta}{N-2}$, then the solution $\psi(t)$ of (1.1) with $\psi_0$ exists globally.

Proof. We prove the first case firstly. By the Hardy-Littlewood-Sobolev and the Young inequalities, we have

where $\theta = \frac{(p+2)(Nq-N-\beta)}{Nqp}$. Under the conservation laws, we deduce that

where $\frac{2\lambda_1}{p+2}+\epsilon_1 < 0$ for $\lambda_1 < 0$ and $\epsilon_1 > 0$ small sufficiently. By choosing $\epsilon = \frac{1}{2}$, we deduce from (2.7) that

which implies the boundedness of $\|\nabla \psi(t)\|_{L^2}$.

In the following, we prove the second case. By the Gagliardo-Nirenberg and the Young inequalities, we have

Under the conservation laws, we get

By choosing $\epsilon = \frac{1}{2}$, $\epsilon_2 = \frac{1}{4}$, we deduce from (2.10) that

which implies the boundedness of $\|\nabla \psi(t)\|_{L^2}$.

And we arrive at the conclusion.

Lemma 2.8. Let $N \geq 3$, $\gamma\in(0, +\infty)$, $\alpha\in(0, 2)$, $\beta\in(0, N)$, $\lambda_1\in(-\infty, 0)$, $\lambda_2\in(0, +\infty)$, $\frac{4}{N} < p < \frac{4}{N-2}$, $q = 1+\frac{2+\beta}{N}$, and $\eta\in(\|Q_q\|_{L^2}^2, \infty)$, $Q_q$ be the ground state of elliptic Eq (1.5), there exists $\gamma_0 > 0$ sufficiently small such that $0 < \gamma < \gamma_0$. Then, there exist $\bar{u}\in H^1$ such that $G_\eta = E_\gamma(\bar{u})$.

Proof. We proceed in four steps.

Step 1. For any $\eta\in(\|Q_q\|_{L^2}^2, \infty)$, $G_\eta = \inf\limits_{u\in A(\eta)}E_\gamma(u)$ is well-defined and $G_\eta < 0$.

We deduce from (2.2) and (2.6) that

by choosing $\epsilon$ and $\epsilon_1$ sufficiently small. Therefore, $E_\gamma(u)$ is bounded from below on $A(\eta)$, that is, $G_\eta$ is well defined.

In the following, we show that $G_\eta < 0$ for all $\eta\in(\|Q_q\|_{L^2}^2, \infty)$. For $q = 1+\frac{2+\beta}{N}$, we have

We can obtain from (2.2) and (2.4) that

by choosing $\epsilon$ and $\lambda_2$ both sufficiently small, for any $u\in H^1$ and $\|u\|_{L^2}^2 = \eta\leq\|Q_q\|_{L^2}^2$, we can not judge that whether $E_\gamma(u) < 0$ or $E_\gamma(u) > 0$.

In fact, for $\eta > \|Q_q\|_{L^2}^2$, we set $v = \mu Q_q$, $\mu = \frac{\sqrt{\eta}}{\|Q_q\|_{L^2}} > 1$, $\|v\|_{L^2}^2 = \eta$. Let $v_\lambda = \lambda^{\frac{N}{2}}v(\lambda x)$ for $\lambda > 0$, $\|v_\lambda\|_{L^2}^2 = \|v\|_{L^2}^2 = \eta$. It follows from (2.5) that

where $2 < 2q$. Hence, we can deduce from (2.14) that

for $\lambda > 0$ sufficiently small and $2 < \frac{Np}{2}$. Therefore, we can obtain from (2.15) that $G_\eta < 0$ for all $\eta\in(\|Q_q\|_{L^2}^2, \infty)$.

Step 2. $\eta\in(\|Q_q\|_{L^2}^2, \infty)\mapsto G_\eta$ is a continuous mapping.

For any $\eta\in(\|Q_q\|_{L^2}^2, \infty)$, let $\eta_n\in(\|Q_q\|_{L^2}^2, \infty)$ such that $\eta_n\rightarrow \eta$ as $n$ large enough. From the definition of $G_{\eta_n}$, for any $\varepsilon > 0$ sufficiently small, let $u_n\in A(\eta_n)$ such that

(2.12) implies that $\{u_n\}$ is bounded in $H^1$. We set $\rho_n: = \sqrt{\frac{\eta}{\eta_n}}u_n$, $\rho_n\in A(\eta)$, we have

On the other hand, given a minimizing sequence $\{v_n\}\subset A(\eta)$ for $E_\gamma$, we have

Set $y_n: = \sqrt{\frac{\eta_n}{\eta}}v_n$, $y_n\in A(\eta_n)$, we have

which, together with (2.16), gives that

Step 3. For $\eta \in(\|Q_q\|_{L^2}^2, \infty)$, we have $G_\eta < G_{\eta_1}+G_{\eta-\eta_1}$ for all $\eta_1\in(\|Q_q\|_{L^2}^2, \eta)$.

Let $\{u_n\}$ be a minimizing sequence for (1.6) such that $E_\gamma(u_n)\rightarrow G_\eta$. Every minimizing sequence for (1.6) is bounded in $H^1$ and bounded from below in $L^{\frac{4}{N+\beta}+2}$. Since $G_\eta < 0$, we have $E_\gamma(u_n)\leq\frac{G_\eta}{2}$ for $n$ large enough. It follows from the definition of $E_\gamma(u_n)$ and (2.6) that

we set $-\frac{G_\eta}{2} = C^1$. There exists a constant $\tau$ such that $\lim\limits_{n\rightarrow \infty}\|\nabla u_n\|_{L^2}\geq\tau > 0$. Otherwise, if $\lim\limits_{n\rightarrow \infty}\|\nabla u_n\|_{L^2} = 0$, by the Gagliardo-Nirenberg inequality, we have $\lim\limits_{n\rightarrow \infty}\|u_n\|_{L^{p+2}}^{p+2} = 0$, which, together with (2.1) and (2.4), yields that $0 > G_\eta = \lim\limits_{n\rightarrow \infty}E_\gamma(u_n) = 0$, which is impossible. Hence, the minimizing problem (1.6) can be rewritten as

Set $C^0 = (\frac{2}{N-\alpha})^{\alpha} > 0$, we can obtain from (2.1) that

and it follows easily that

for $0 < \gamma < \gamma_0$ and $\gamma_0 > 0$ small sufficiently, we set $\tau^{\alpha}\left(\tau^{2-\alpha}-\gamma_0 C^0\eta^{\frac{2-\alpha}{2}}\right) = C^2$. For $t\in(1, \infty)$, set $\tilde{u}(x): = u(t^{-\frac{1}{N}}x)$, $\|\tilde{u}\|_{L^2}^2 = t\|u\|_{L^2}^2$. We have

Consequently,

for all $\eta_1 \in(\|Q_q\|_{L^2}^2, \eta)$.

Step 4. Now, let us apply the concentration compactness principle in $H^1$ to the minimizing sequence $\{u_n\}$. There exists a subsequence $\{u_{n_k}\}$ such that one of the three possibilities in Lemma 2.5 holds.

First, we prove that the vanishing cannot occur.

Suppose by contradiction that

by Lion's Lemma, we have $u_{n_k}\rightarrow0$ in $L^m$ as $k\rightarrow\infty$ for all $m\in(2, \frac{2N}{N-2})$. Hence,

By the domain decomposition, the Hölder inequality and $\alpha\in(0, 2)$, we have

where $\frac{1}{a}+\frac{1}{a'} = 1$, $\frac{1}{b}+\frac{1}{b'} = 1$. $a < N/\alpha$, $b > N/\alpha$, $N/\alpha-a$ and $b-N/\alpha$ are both sufficiently small. By (2.23) and the Sobolev inequality, we have

It follows from (2.3) that

Thus,

which contradicts $G_\eta < 0$ in step 1. Therefore, the vanishing cannot occur.

Subsequently, we prove that the dichotomy does not occur.

Suppose by contradiction that the dichotomy can occur. Then there exist a constant $\xi\in(0, \eta)$ and two bounded sequences $\{u_{n_k}^{(1)}\}, \{u_{n_k}^{(2)}\}\subset H^1$ such that

Similarly, we have

and

Indeed, let $u_{n_k} = u_{n_k}^{(1)}+u_{n_k}^{(2)}+v_{n_k}$, in view of $u_{n_k}^{(1)}u_{n_k}^{(2)} = 0$, by the direct calculation, we obtain that

where $\delta(\epsilon)\rightarrow 0$ as $\epsilon\rightarrow 0$. By (2.25)–(2.28), we obtain that

which combine with (2.19) and (2.24), we get

which is a contradiction with (2.22). Hence, the dichotomy cannot occur.

Finally, we have ruled out both vanishing and dichotomy, then we deduce that there exists a sequence $\{y_{n_k}\}\subset \mathbb{R}^N$ such that for all $\epsilon > 0$, there exists $R(\epsilon) > 0$ such that for all $k\geq1$

Denote $\hat{u}_{n_k}(x) = u_{n_k}(x+y_{n_k})$, we assume that $\{y_{n_k}\}$ is bounded, then there exists a $\hat{u}$ such that, up to a subsequence,

(2.29), together with (2.30), implies that

Thus $\int_{\mathbb{R}^{N}} |\hat{u}(x)|^2dx = \mu$, i.e., $\hat{u}_{n_k}\rightarrow\hat{u}$ strongly in $L^2$. By the Gagliardo-Nirenberg inequality, $\hat{u}_{n_k}\rightarrow\hat{u}$ strongly in $L^m$ for $m \in [2, \frac{2N}{N-2})$. Then (2.30) can be rewritten as

In the following, we claim that $\{y_{n_k}\}$ is bounded. Suppose by contradiction that $\{y_{n_k}\}$ is unbounded. For

by the domain decomposition and $\alpha\in(0, 2)$, $R > 0$ is fixed, we know

On $B_R(0)$, we have

which, together with (2.31), shows that $I_1\rightarrow0$ as $k\rightarrow\infty$. On the other hand, on $B_R^c(0)$, by the Cauchy convergence of improper integrals and $\hat{u}\in L^2$, for any $\epsilon > 0$, we have

which, together with (2.31), yields that

then we have

Consequently, $I_2\rightarrow0$ as $k\rightarrow\infty$. Therefore, we have

Denote

and

We know that $G_\eta^0$ is attained by a nontrivial function $w$, that is, $G_\eta^0 = \inf\limits_{w\in A(\eta)}E_0(w)$. Moreover, the above steps hold for $\gamma = 0$. Thus,

Hence, $G_\eta\geq G_\eta^0$. By the definition of $G_\eta$, we have

which contradicts $G_\eta\geq G_\eta^0$. Hence, $\{y_{n_k}\}$ is bounded. We may assume, going if necessary to a subsequence, $\lim\limits_{k\rightarrow\infty}y_{n_k} = \hat{y}$ for some $\hat{y}\in \mathbb{R}^N$. Consequently, we have

from (2.31) and $\lim\limits_{k\rightarrow\infty}y_{n_k} = \hat{y}$ for some $\hat{y}\in \mathbb{R}^N$, we have

We define $\bar{u}(x) = \hat{u}(x-\hat{y})$. Consequently,

By the definition of $G_\eta$, we see that $\bar{u}$ is a minimizer of $G_\eta$, $\lim\limits_{k\rightarrow\infty}\|\nabla u_{n_k}\|_{L^2}^2 = \|\nabla \bar{u}\|_{L^2}^2$, and hence $u_{n_k}\rightarrow \bar{u}$ in $H^1$.

3.   Proof of Theorem 1.2

Proof. By Lemma 2.7, we see that the solution $\psi$ of (2.1) exists globally. Suppose by contradiction that there exist sequences $\{u_{0, n}\}\subset H^1$ and $\{t_n\}\subset\mathbb{R}^+$ and a constant $\epsilon > 0$ such that for all $n\geq1$,

where $u_{n}(t)$ is the solution to (1.1) with initial data $u_{0, n}$. From (3.1), there exists $\{z_n\}\subset K(\eta)$ such that

and there exists $z\in K(\eta)$ such that

Hence, we get

By the conservation of mass and energy, we have

then $\{u_{n}(t_n)\}$ is bounded in $H^1$.

Set $\breve{u}_n = \frac{\sqrt{\eta}u_{n}(t_n)}{\|u_{n}(t_n)\|_{L^2}}$, $\|\breve{u}\|_{L^2}^2 = \eta$, we deduce that

which implies that

Hence, $\{\breve{u}_n\}\subset A(\eta)$ is a minimizing sequence of $E_\gamma$. There exists $\breve{z}\in K(\eta)$ such that

by the definition of $\breve{u}_n$, we know

We can get that

which contradicts (3.2). Hence, $K(\eta)$ is orbitally stable. This completes the proof.

Conflict of interest

The authors declare that they have no competing interests.