Odd symmetry of ground state solutions for the Choquard system

1.   Introduction

In this paper, we study the following Choquard system:

where $u: = u(x), v: = v(x)$ in $H^1(\mathbb R^N)$ $(N\geq1)$ are real valued functions. $I_{\alpha}$: $\mathbb R^{N}\rightarrow \mathbb R$ is the Riesz potential defined at each point $x\in\mathbb R^{N}\backslash\{0\}$ by

where $\Gamma$ denotes the classical Gamma function and $*$ the convolution on the Euclidean space $\mathbb R^{N}$. When $p = q$ and $u(x) \equiv v(x)$, the system (1.1) is the following Choquard equation:

Choquard Eq (1.2) is firstly appeared in a work by Pekar describing the quantum mechanics of a polaron at rest ^[21]. In the case of $N = 3, \ \alpha = 2$ and $p = 2$, Choquard described an electron trapped in its own hole, in a certain approximation to Hartree-Fock theory of one component plasma ^[10]. In the pioneering work ^[10], Lieb first proved the existence and uniqueness of positive solutions. Later Lions ^[12,13] obtained the existence and multiplicity of solutions to (1.2). In 1996, Penrose proposed a model of self-gravitating matter, in a programme in which quantum state reduction is understood as a gravitational phenomenon ^[15]. For more existence and qualitative properties of solutions to (1.2), we refer the reader to ^{[1,14,16,18,19,20]} and references therein.

In recent years, there has been increasing attention to equations like (1.2) on the existence of positive solutions and ground states solutions. For the optimal range of parameters $\frac{N+\alpha}{N} < p < \frac{N+\alpha}{N-2}$. Here and after, the $2_*^\alpha$ denotes $\frac{N+\alpha}{N-2}$ if $N\geq 3$ and $2_*^\alpha: = \infty$ if $N = 1, \ 2$. By using the concentration-compactness and Brézis-Lieb lemmas, Moroz and Schaftingen ^[17] showed the existence of ground state solutions for (1.2). Under symmetric assumptions on $\Omega$, which is an unbounded smooth domain in $\mathbb R^N$, Clapp and Salazar ^[5] proved the existence of positive solutions of (1.2).

The Eq (1.2) with an additional perturbation of the following form:

has been studied in ^[2], where the existence of solutions is obtained for $N = 3$, $0 < \alpha < 1$, $p = 2$ and $4\leq q < 6$. When $N = 3, \ \alpha = 2, \ p = 2$ and $q\in(2, 6)$, Vaira ^[24,25] has obtained a positive radial ground state solution and further studied the nondegeneracy of the radial ground state solution for the special case $q = 3$. With the help of the mountain pass theorem and Pohožaev identity, Li et al. in ^[9] studied the Choquard Eq (1.3) with $\frac{N+\alpha}{N} < p < 2_*^\alpha$ and $2 < q < \frac{2N}{N-2}, N\geq3,$ where the existence of ground state solution of mountain pass type is obtained. Here, we would also like to mention the papers ^[7,23,30] for related topics.

For elliptic system of Choquard type, Chen and Liu ^[3] have obtained the existence of positive radial ground state solutions to

Yang et al. ^[29] have proved the existence of positive radial ground state solutions when $p$ reaches the critical exponent. A slightly general version of system of Choquard type was studied by Xu et al. ^[28], where the authors have proven that the system of Choquard type admits a nontrivial vector solution under the Schwarz symmetrization method. In ^[3,28,29], all of them are positive solutions or ground state solutions of the linear coupled type Choquard system.

But we do not see any results to (1.1) in the case of $p\neq q$. The main purpose of the present paper is to study odd symmetry of ground state solutions to (1.1). According to Fubini theorem, we obtain the following symmetry property: for every $u, v\in H^{1}(\mathbb R^N)$,

Therefore for each function $(u, v)$ in the Sobolev space $H: = H^{1}(\mathbb R^N)\times H^{1}(\mathbb R^N)$, we call $(u, v)\in H$ is a weak solution of (1.1) if for any $\varphi_{1}, \varphi_{2}\in C_{0}^{\infty}(\mathbb R^{N})$,

and

Hence there is a one-to-one correspondence between solutions of (1.1) and critical points of the following functional $\mathcal{I}$: $H\rightarrow \mathbb R$ defined by

For any $\varphi_{1}, \varphi_{2}\in C_{0}^{\infty}(\mathbb R^{N})$ and $(u, v)\in H,$ we compute the Gateaux derivative:

Then, $(u, v)\in H$ is a solution of (1.1) if and only if

In view of the Hardy-Littlewood-Sobolev inequality (see [11,Theorem 4.3]), which states that if $s\in(1, \frac{N}{\alpha})$ then for every $v \in L^s(\mathbb R^N)$, $I_\alpha\ast v\in L^\frac{Ns}{N- \alpha s}(\mathbb R^N)$ and

and of the classical Sobolev embedding, the action functional $\mathcal{I}$ is well defined and continuously differentiable whenever $\frac{N+\alpha}{N} < p, \ q < 2_*^\alpha.$ Denote

and set

We know that $\mathcal{N}_0$ is a Nahari manifold which can be a natural constraint to find critical point of the functional $\mathcal{I}$ on $H$. A nontrivial solution $(u, v)\in H$ of (1.1) is called a ground state if

Motivated by the results in ^[4], we consider the Sobolev space of odd functions

where

Define the odd Nehari manifold

and the corresponding level $c_\mathrm{odd} = \inf\limits_{\mathcal{N}_{\mathrm{odd}}}\mathcal{I}$.

Our main result is that this level $c_\mathrm{odd}$ is achieved.

Theorem 1.1. If $p, \ q\in \left(\frac{N+\alpha}{N}, 2_*^\alpha\right)$ and $p\neq q$, then there exists a solution $(u, v)\in H_\mathrm{odd}$ to the system (1.1) such that $\mathcal{I}(u, v) = c_\mathrm{odd}$.

Remark 1.2. The question we are interested in is whether the sign-changing solution of system (1.1) is odd, thus consistent with the sign-changing solution of Theorem 1.1, and it is not even known whether the sign-changing solution has axial symmetry with Theorem 1.1. We consider these issues as further research questions.

Remark 1.3. Since $\mathcal{N}_{\mathrm{odd}} = \mathcal{N}_0 \cap H_{\mathrm{odd}}\subset \mathcal{N}_0$ we have $c_{\mathrm{odd}}\ge c_0$. We not know whether $c_0 = c_\mathrm{odd}.$

Remark 1.4. Compared with related results, there are some differences and difficulties in our proofs: Firstly, by using the mountain pass theorem, the authors in ^[4] proved the existence of ground state solution. Here, we use the method of Nehari manifold (see Section 2) for our proof. Moreover, we generalize the Brézis-Lieb lemma for the nonlocal term $\int_{\mathbb R^{N}}(I_\alpha\ast|u|^{p})|v|^{q}$ (see Lemma 5).

Secondly, different from ^[6,8,26], in this paper, we have to overcome the difficulty caused by the nonlinear coupled Choquard term $\int_{\mathbb R^{N}}(I_\alpha\ast|u|^{p})|v|^{q}$. Finally, in ^[3,28,29], they all obtain the positive solutions or ground state solutions of the Choquard system with $p = q$ and $u(x) = v(x)$, and these solutions have radial symmetry. However, we prove that the positive ground state solution of the Choquard system (1.1) with $p\neq q$ and $u(x)\neq v(x)$ has odd symmetry.

The rest of the paper is organized as follows. In Section 2, we set the variational framework for system (1.1) and some preliminary results. Section 3 is devoted to showing the existence of ground state solutions to system (1.1) by using minimizing arguments on odd Nehari set and then prove the Theorem 1.1.

2.   Preliminaries and the proof Theorem 1.1

Throughout this paper, $\|u\|_{H^1}$ and $|u|_r$ denote the usual norm of $H^1(\mathbb R^N)$ and $L^r(\mathbb R^N)$ for $r > 1$, respectively. Let $\|(u, v)\|^2: = \|u\|_{H^1}^2+\|v\|_{H^1}^2.$ For convenience, $C$ and $C_{i}$ $(i = 1, 2, \ldots)$ denote (possibly different) positive constants, $\int_{\mathbb R^{N}}g$ denotes the integral $\int_{\mathbb R^{N}}g(z)dz$. The "$\rightarrow$" and "$\rightharpoonup$" denote strong convergence and weak convergence, respectively.

Lemma 2.1. If $(u, v)$ is a weak solution of (1.1), then $(u, v)$ satisfies $P(u, v) = 0,$ where

Proof. The proof is standard, so we omit the details here. □

Let $t\in\mathbb R^{+}$ and $(u, v)\in H_\mathrm{odd}$, one has

Denote

Since

we see that $h(t) > 0$ for $t > 0$ small enough and $h(t)\rightarrow -\infty$ as $t\rightarrow +\infty,$ which implies that $h(t)$ attains its maximum.

Lemma 2.2. Let $\theta_1$ and $\theta_2$ be positive constants. For $t\geq0$, we define

Then $h$ has a unique critical point which corresponds to its maximum.

Proof. We already know that $h$ has a maximum. For $t\geq0$, we compute directly that derivatives of $h$:

Since $h'(t)\rightarrow -\infty$ as $t\rightarrow +\infty$ and is positive for $t > 0$ small, we obtain that there is $t > 0$ such that $h'(t) = 0.$ The uniqueness of the critical point of $h$ follows from the fact that the equation

has a unique positive solution $\bigl(\frac{2\theta_{1}}{(p+q)\theta_{2}}\bigr)^{\frac{1}{p+q-2}}$. □

Lemma 2.3. Suppose that $u, \ v\in H^1_\mathrm{odd}(\mathbb{R}^N)\backslash\{0\}$, then there is a unique $\tilde{t}$: = $t(u, v) > 0$ such that $h$ attains its maximum at $\tilde{t}$ and

Moreover, if $\mathcal{P}(u, v) < 0$, then $\tilde{t}\in(0, 1)$.

Proof. For every $u, \ v\in H^1_\mathrm{odd}(\mathbb{R}^N)\backslash\{0\}$ and any $t > 0,$ we consider

From Lemma 2.2, $h$ has a unique critical point $\tilde{t} > 0$ corresponding to its maximum, i.e.,

and hence $\mathcal{P}(\tilde{t}u, \tilde{t}v) = 0$ and $(\tilde{t}u, \tilde{t}v)\in\mathcal{N}_\mathrm{odd}.$ If $\mathcal{P}(u, v) < 0,$ one has

then

which implies that $\tilde{t} < 1$. This finishes the proof. □

Lemma 2.4. The $\mathcal{N}_\mathrm{odd}$ is a $C^{1}$ manifold and every critical point of $\mathcal{I}|_{\mathcal{N}_\mathrm{odd}}$ is a critical point of $\mathcal{I}$ in $H_\mathrm{odd}.$

Proof. According to Lemma 2.3, we know $\mathcal{N}_\mathrm{odd}\neq\emptyset$.

Claim 1. $\mathcal{N}_\mathrm{odd}$ is bounded away from zero. For any $(u, v)\in \mathcal{N}_\mathrm{odd}$, by using $\mathcal{P}(u, v) = 0$, the semigroup property of the Riesz potential $I_\alpha = I_{\frac{\alpha}{2}}\ast I_{\frac{\alpha}{2}}$ [11,Theorem 5.9 and Corollary 5.10], Cauchy-Schwarz inequality, (1.4), Sobolev and Young inequalities, one has

Hence, there is $C' > 0$ such that $\|(u, v)\| \geq C'$. This proves that $\mathcal{N}_\mathrm{odd}$ is bounded away from zero.

Claim 2. $c_{\mathrm{odd}} > 0$. Since $\mathcal{N}_{\mathrm{odd}} = \mathcal{N}_0 \cap H_{\mathrm{odd}}\subset \mathcal{N}_0$ we have $c_{\mathrm{odd}}\ge c_0$. For any $(u, v)\in \mathcal{N}_0,$

we obtain $c_{0} > 0.$

Claim 3. The $\mathcal{N}_\mathrm{odd}$ is a $C^{1}$ manifold. Since $\mathcal{P}(u, v)$ is a $C^{1}$ functional, in order to prove $\mathcal{N}_\mathrm{odd}$ is a $C^{1}$ manifold, it suffices to prove that $\mathcal{P}'(u, v)\neq0$ for all $(u, v)\in\mathcal{N}_\mathrm{odd}$. Suppose on the contrary that $\mathcal{P}'(u, v) = 0$ for some $(u, v)\in\mathcal{N}_\mathrm{odd}$, then $(u, v)$ satisfies

and

Therefore,

which is a contradiction. Hence $\mathcal{P}'(u, v)\neq0$ for any $(u, v)\in\mathcal{N}_\mathrm{odd}$.

Claim 4. Every critical point of $\mathcal{I}|_{\mathcal{N}_\mathrm{odd}}$ is a critical point of $\mathcal{I}$ in $H_\mathrm{odd}.$ If $(u, v)$ is a critical point of $\mathcal{I}|_{\mathcal{N}_\mathrm{odd}}$, i.e., $(\mathcal{I}|_{\mathcal{N}_\mathrm{odd}})'(u, v) = 0$ and $(u, v)\in\mathcal{N}_\mathrm{odd}$. Thanks to the Lagrange multiplier rule, there exists $\rho\in \mathbb R$ such that

i.e.,

According to $\mathcal{P}'(u, v) = 0$ and its corresponding Pohožaev identity we get

by {Claim 1}, we know $\langle\mathcal{P}'(u, v), (u, v)\rangle\neq0$, we deduce $\rho = 0$, and then $\mathcal{I}'(u, v) = 0$. □

Our main tool is the following Brézis-Lieb lemma for the nonlocal term $\int_{\mathbb R^{N}}(I_{\alpha}\ast|u|^{p})|v|^{q}$.

Lemma 2.5. Let $u_{n}\rightharpoonup u$ and $v_{n}\rightharpoonup v$ in $H^{1}(\mathbb R^{N})$. If $u_{n}\rightarrow u$ and $v_{n}\rightarrow v$ a.e in $\mathbb R^{N}$, then

Proof. For $n = 1, \ 2, \ldots$, we have

Since $u_n\rightharpoonup u$ in $H^1(\mathbb{R}^N)$, by [17,Lemma 2.5] with $q = p$ and $r = \frac{2Np}{N+\alpha}$, one has

which means

Since the Riesz potential is a linear bounded map from $L^{\frac{2N}{N+\alpha}}(\mathbb R^N)$ to $L^{\frac{2N}{N-\alpha}}(\mathbb R^N)$, by the Hardy-Littlewood-Sobolev inequality (1.4), this implies that

in $L^{\frac{2N}{N-\alpha}}(\mathbb{R}^N)$. Since $v_n\rightharpoonup v$ in $H^1(\mathbb{R}^N)$, by $|v_n|^q\rightharpoonup|v|^q$ in $L^{\frac{2N}{N+\alpha}}(\mathbb{R}^N)$, we obtain

Similarly, according to $v_n\rightharpoonup v$ in $H^1(\mathbb{R}^N)$, by [17,Lemma 2.5] with $r = \frac{2Nq}{N+\alpha}$, one has

Since $|u_n-u|^p\rightharpoonup 0$ in $L^{\frac{2N}{N+\alpha}}(\mathbb{R}^N)$, by the Riesz potential is a linear bounded map from $L^{\frac{2N}{N+\alpha}}(\mathbb R^N)$ to $L^{\frac{2N}{N-\alpha}}(\mathbb R^N)$ and the Hardy-Littlewood-Sobolev inequality (1.4), this implies that $I_{\alpha}\ast|u_n-u|^p\rightharpoonup 0$ in $L^{\frac{2N}{N-\alpha}}(\mathbb{R}^N)$, we obtain

This proves the lemma. □

Lemma 2.6. If $c_{\mathrm{odd}} < 2 c_0,$ $c_\mathrm{odd}$ is achieved at some $(u, v)\in\mathcal{N}_\mathrm{odd}$.

Proof. Let $(u_{n}, v_{n})\in\mathcal{N}_\mathrm{odd}$ so that $\mathcal{I}(u_{n}, v_{n})\rightarrow c_\mathrm{odd}$. We first show that $\{(u_n, v_n)\}$ is bounded in $H_\mathrm{odd}$. For $n$ large enough, we get

Then, there exist the subsequence of $\{u_{n}\}$, $\{v_{n}\}$ (still denoted by $\{u_{n}\}$, $\{v_{n}\}$) such that $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb R^{N})$ and $v_{n}\rightharpoonup v$ in $H^{1}(\mathbb R^{N})$. This implies in particular that $\{|u_{n}|^p\}$ and $\{|v_{n}|^q\}$ are bounded in $L^{\frac{2N}{N+\alpha}} (\mathbb R^{N})$, $p, q\in(\frac{N+\alpha}{N}, 2_*^\alpha)$.

Claim 5. We claim $v\neq0$. We show that there exists $R > 0$ such that

where the set $D_R \subset \mathbb R^N$ is the infinite slab $D_R = \mathbb R^{N-1}\times[-R, R].$ Suppose by contradiction that for each $R > 0$,

Define

and

Since $(u_n, v_n)\in H_{\mathrm{odd}}$, we have $(\omega_n, \nu_n)\in H_{\mathbb R^{N-1}\times(0, \infty)}\subset H$ and $(\tilde{\omega}_n, \tilde{\nu}_n)\in H_{\mathbb R^{N-1 }\times(-\infty, 0)}\subset H$. We now compute

By definition of the region $D_R$ we have, if $\beta \in (\alpha, N)$,

By using the semigroup property of the Riesz potential $I_\alpha = I_{\frac{\alpha}{2}}\ast I_{\frac{\alpha}{2}},$ Cauchy-Schwarz inequality, (1.4), we obtain

Using (2.3) and the classical Sobolev inequality, we obtain

from which, and as $\{(u_n, v_{n})\}$ is bounded in the space $H$, we deduce

Similarly, one has

For each $n\in \mathbb N$, we fix $t_n\in(0, \infty)$ so that $(t_n \omega_n, t_n\nu_n)\in\mathcal{N}_{\mathrm{odd}}$ or, equivalently,

For every $n\in \mathbb N$, we have

By (2.3)–(2.6), in view of Lemma 2.1, we note that $\lim\limits_{n \to \infty} t_n = 1$ and thus according to (2.4) and (2.5) again we conclude

in contradiction with the assumption $c_{\mathrm{odd}} < 2c_0$ of the Lemma. We can now fix $R > 0$ such that (2.2) holds. We take a function $\eta \in C^\infty (\mathbb R^N)$ such that $\mathrm{supp}\eta\subset D_{3R/2}$, $\eta = 1$ on $D_R$, $\eta\le1$ on $\mathbb R^N$ and $\nabla\eta$ in $L^\infty (\mathbb R^N)$. By using the inequality [22,(3.4)],

Since the sequence $\{(u_n, v_n)\}$ is bounded in $H_{\mathrm{odd}}$, we deduce from (2.2) that there exists a sequence of points $\{a_n\}$ in the hyperplane $\mathbb R^{N-1}\times\{0\}$ such that

By the Rellich theorem, $v_n\rightarrow v$ in $L^{\frac{2Nq}{N+\alpha}}_{loc}(\mathbb R^N)$ and then $v\neq0.$

Claim 6. The infimum of $\mathcal{I}|_{\mathcal{N}_{\mathrm{odd}}}$ is achieved.

We claim that $(u, v)\in\mathcal{N}_{\mathrm{odd}}.$ Indeed, if $(u, v)\not\in\mathcal{N}_{\mathrm{odd}},$ we will discuss it in two cases: $\mathcal{P}(u, v) < 0$ and $\mathcal{P}(u, v) > 0$.

Case 1: $\mathcal{P}(u, v) < 0$. By Lemma 2.3, there exists $t\in(0, 1)$ such that $(tu, tv)\in \mathcal{N}_{\mathrm{odd}},$ it follows from $(u_{n}, v_{n})\in\mathcal{N}_{\mathrm{odd}}$ and Fatou's lemma that

which is a contradiction.

Case 2: $\mathcal{P}(u, v) > 0$. We define

Using Brézis-Lieb lemma [27,Lemma 1.32] and Lemma 2.5, we may obtain

and

Then

By Lemma 2.3, there exists $t_{n}\in(0, 1)$ such that $(t_{n}\xi_{n}, t_{n}\gamma_{n})\in \mathcal{N}_{\mathrm{odd}}.$ Furthermore, one has

otherwise, along a subsequence, $t_{n}\rightarrow1$ and then

which is a contradiction. For $n$ large enough, it follows from $(u_{n}, v_{n})\in \mathcal{N}_{\mathrm{odd}}$, (2.7) and (2.8) that

which is also a contradiction.

Therefore, $(u, v)\in\mathcal{N}_{\mathrm{odd}}$ and then $(u, v)$ is a minimizer of $\mathcal{I}|_{\mathcal{N}_{\mathrm{odd}}}$. □

It remains now to establish the strict inequality $c_\mathrm{odd} < 2 c_0$.

Lemma 2.7. $c_{\mathrm{odd}} < 2 c_0.$

Proof. Motivated by the Proposition 2.4 in ^[4]. We give a detailed proof. It is easy to prove that Choquard system (1.1) has a least action solution on the usual Nehari manifold. More precisely, there exists $0\neq \omega, \nu \in H^1 (\mathbb R^N) \setminus \{0\}$ such that

We take a function $\eta\in C^2_c(\mathbb R^N)$ such that $\eta = 1$ on $B_1$, $0\le\eta\le1$ on $\mathbb R^N$ and supp $\eta \subset B_2$ and we define for each $R > 0$ the function $\eta_R\in C^2_c(\mathbb R^N)$ for every $x\in \mathbb R^N$ by $\eta_R(x) = \eta(x/R)$, $\eta_R$ is even in $x$. We define the function $u_R$: $\mathbb R^N\to\mathbb R$ for each $x = (x', x_N)\in\mathbb R^N$ by

It is clear that $(u_R, v_R)\in H_{\mathrm{odd}}$. Note that

if and only if $t_R\in(0, \infty)$ satisfies

Such a $t_R$ always exists and

The proposition will follow once we have established that for some $R > 0$,

Observe that, by construction of the function $(u_R, v_R)$

For the first term, without losing generality, we may suppose $p\geq q$, one has

By the asymptotic properties of $(I_\alpha \ast|\omega|^p)|\nu|^q$ [17,Theorem 4], we obtain

so

Thus

Since

according to $(\omega, \nu)$ decays exponentially at infinity, we may obtain

Thus

By using integration by parts, we have

Thus

It follows from (2.10) and (2.11) that

The inequality (2.9) holds thus when $R$ is large enough, and the conclusion follows. □

In this work, by using a variant of Nehari constraint, we obtain the odd symmetry of ground state solutions for Choquard system. Our results can be looked on as a partial generalization to some recent ones.

Proof of Theorem 1.1. From Lemma 2.6, we have a $(u, v)\in\mathcal{N}_{\mathrm{odd}}$ such that $\mathcal{I}(u, v) = c_{\mathrm{odd}}$. By Lemma 2.4, the $(u, v)$ is a critical point of $\mathcal{I}$ and hence a solution to (1.1). We claim $u, v\neq0$. From Lemma 2.6, we already know $v\neq0$. Now we prove $u\neq0$. Indeed, if $u = 0$, then the second equation of (1.1) yields that $v = 0$, then $(u, v) = (0, 0)$, this is impossible by the {Claim 1} in Lemma 2.4. The proof is complete.

3.   Conclusions

In this work, by using a variant of Nehari constraint, we obtain the odd symmetry of ground state solutions for the Choquard system. Our results can be looked on as a partial generalization to some recent ones.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We appreciate the editor and referees for their invaluable comments. This work was supported by the Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (No. SX202203), National Natural Science Foundation of China (No. 11871152) and Key Project of Natural Science Foundation of Fujian (No. 2020J02035).

Conflict of interest

The authors declare that there are no conflicts of interest.