Existence of Ground State Solutions for the Kirchhoff–Choquard Equation With Critical Exponent and External Coulomb Potential

Abstract

We study the critical Kirchhoff–Choquard type equation $- (a + b \int_{R^{3}} | \nabla u |^{2} d x) △ u + (β - \frac{μ}{| x |}) u = (I_{α} * | u |^{q}) | u |^{q - 2} u, u \in H^{1} (R^{3})$ where $α \in (0, 3)$ , $b \geq 0$ , $a$ , $β$ and $μ$ are positive constants, $I_{α}$ denotes the Riesz potential and $q = \frac{3 + α}{3}$ is the lower critical Hardy–Littlewood–Sobolev exponent. By means of the compactness-concentration principle and the squeezing energy inequality, we obtain the existence of ground state solutions of Nehari–Pohoz̆aev type for the above problem for sufficiently small $μ > 0$ .

Keywords

Kirchhoff–Choquard type equation Coulomb potential critical exponent

1 Introduction

This article is dedicated to studying the following Kirchhoff–Choquard type equation:

- (a + b \int_{R^{3}} | \nabla u |^{2} d x) △ u + (β - \frac{μ}{| x |}) u = (I_{α} * | u |^{q}) | u |^{q - 2} u, u \in H^{1} (R^{3})

(1)

where

α \in (0, 3)

b \geq 0

a

β

and

μ

are positive constants,

q = 3 + α / 3

is the lower critical Hardy–Littlewood–Sobolev exponent in the sense of the Hardy–Littlewood–Sobolev inequality and

I_{α}

is the Riesz potential defined by the following equation:

\begin{aligned} I_{α} (x) = \frac{Γ (\frac{3 - α}{2})}{Γ (\frac{α}{2}) π^{3 / 2} 2^{α} | x |^{3 - α}}, x \in R^{3} ∖ {0} \end{aligned}

When

μ \neq 0

, the term

- μ / | x |

is the so-called external Coulomb potential for Helium, as discussed in works such as Lieb & Simon (1977). The Coulomb potential plays a significant role in various fields such as quantum mechanics, nuclear physics, molecular physics and so on Sierka et al. (2003) and Vincent & Phatak (1974).

Problem (1) is related to the stationary analogue of the following equation:

\begin{aligned} ρ \frac{\partial^{2} u}{\partial t^{2}} - (\frac{P_{0}}{h} + \frac{E}{2 L} \int_{0}^{L} {| \frac{\partial u}{\partial x} |}^{2} d x) \frac{\partial^{2} u}{\partial x^{2}} = 0 \end{aligned}

which was proposed by Kirchhoff by Kirchhoff (1883) as an extension of the classical D’ Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. More mathematical and physical background and applications of such problems can be found by Arosio & Panizzi (1996) and Chipot & Lovat (1997) and the references therein.

After the pioneering work of Lions (1978) introducing the abstract framework, the following model problem:

- (a + b \int_{R^{3}} | \nabla u |^{2} d x) △ u + V (x) u = f (x, u), x \in R^{N}

(2)

has been widely studied by many authors under variant conditions on

V (x)

and

f (x, u)

, for example, see Deng et al. (2015), Gao et al. (2024), Guo (2015), He et al. (2021), He (2012), Li et al. (2012), Li & Ye (2014), Pucci et al. (2016), Tang & Chen (2017), Xiang et al. (2016) and the references therein. Among them, Li & Ye (2014) proved that (2), with special forms

V = 1

and

f (x, u) = | u |^{p - 2} u

for

3 < p < 6

, has a ground state solution by using the minimizing argument on the Nehari–Pohoz̆aev manifold. Later, by using Jeanjean (1999) monotonicity approach on a Nehari–Pohoz̆aev manifold, Guo (2015) generalized the results of Li & Ye (2014) for a general nonlinearity. In situations involving nonlocal nonlinearities, Lü (2014) investigated Kirchhoff-type problems with convolution nonlinearity:

- (a + b \int_{R^{3}} | \nabla u |^{2} d x) △ u + V (x) u = (I_{α} * | u |^{q}) | u |^{q - 2} u, u \in H^{1} (R^{3})

(3)

where

q \in (2, 3 + α)

V (x) = 1 + μ g (x)

μ > 0

is a parameter and

g (x)

is a nonnegative steep potential well function. By means of the Nehari manifold and the concentration compactness principle, Lü proved the existence of ground state solutions for (3) if the parameter

μ

is large enough. It is worth pointing out that the same result is not available in the case where

q \in (\frac{3 + α}{3}, 2]

, even when

g (x) = 0

, since both the mountain pass theorem and the Nehari manifold argument do not work. But using Nehari-Pohoz̆aev manifold with some strong assumptions on the potential

V (x)

, (Chen & Liu (2019)) obtained a ground state solution for the complete range

q \in (\frac{3 + α}{3}, 3 + α)

. More results on the Kirchhoff-type problems can be found by Chen et al. (2020), Chu & Liu (2024), Liang et al. (2021), Ricceri (2024), Sun et al. (2023), Tian & Zhang (2024) and Yin et al. (2022).

When $a = 1$ and $b = 0$ , equation (3) reduces to the well-known Choquard–Pekar equation

- Δ u + V (x) u = (I_{α} * | u |^{q}) | u |^{q - 2} u, x \in R^{N}

(4)

For

N \geq 3

α \in (0, N)

. By the Sobolev embedding theorem,

H^{1} (R^{N}) \subset L^{2 N p / N + α} (R^{N})

if and only if

p \in [N + α / N, N + α / N - 2]

. Usually,

N + α / N - 2

(or

N + α / N

) is called the upper (or lower) critical exponent with respect to the Hardy–Littlewood–Sobolev inequality (see Lieb & Loss (2001)). For the case where

V (x) \equiv 1

, equation (4) is known to have a ground state solution if and only if

q \in (N + α / N, N + α / N - 2)

(see Moroz & Van Schaftingen (2013)). Recently, Guo & Tang (2020) studied (4) with

V (x) = ω - β / | x |

and

q = N + α / N - 2

. With the help of the concentration compactness principle and the Pohoz̆aev manifold methods, they obtained the existence of ground states for

β > 0

sufficiently small.

To the best of our knowledge, few prior results exist regarding ground state solutions for the Kirchhoff equation with a Coulomb potential and the lower critical Hardy–Littlewood–Sobolev exponent. It’s worth noting that the Coulomb potential is crucial for guaranteeing the existence of nontrivial solutions. In fact, as demonstrated by Moroz and Van Schaftingen by Moroz & Van Schaftingen (2013), it is not difficult to see that (1) has no nontrivial solution when $μ = 0$ . In this article, we investigate the existence of ground state solutions of Nehari–Pohoz̆aev type for (1).

Denote the standard norms of $H^{1} (R^{3})$ and $L^{p} (R^{3})$ with $p \in (1, + \infty)$ by $‖ \cdot ‖$ and $| \cdot |_{p}$ , respectively. Clearly, the weak solutions of (1) correspond to the critical points of the energy functional defined in $H^{1} (R^{3})$ by

J (u) = \frac{a}{2} | \nabla u |_{2}^{2} + \frac{β}{2} | u |_{2}^{2} + \frac{b}{4} | \nabla u |_{2}^{4} - \frac{μ}{2} \int_{R^{3}} \frac{u^{2}}{| x |} d x - \frac{1}{2 q} \int_{R^{3}} (I_{α} * | u |^{q}) | u |^{q} d x

(5)

For any

u \in H^{1} (R^{3}) ∖ {0}

u_{t}

is defined as follows:

\begin{aligned} u_{t} (x) = {\begin{aligned} \sqrt{t} u (\frac{x}{t}), & t > 0 \\ 0, & t = 0 \end{aligned} \end{aligned}

By a direct calculation, we have the following equation:

J (u_{t}) = \frac{a t^{2}}{2} | \nabla u |_{2}^{2} + \frac{β t^{4}}{2} | u |_{2}^{2} + \frac{b t^{4}}{4} | \nabla u |_{2}^{4} - \frac{μ t^{3}}{2} \int_{R^{3}} \frac{u^{2}}{| x |} d x - \frac{t^{4 q}}{2 q} \int_{R^{3}} (I_{α} * | u |^{q}) | u |^{q} d x

(6)

From the Hölder inequality and the Hardy inequality, we have the following equation:

\int_{R^{3}} \frac{u^{2}}{| x |} d x \leq {(\int_{R^{3}} \frac{u^{2}}{| x |^{2}} d x)}^{\frac{1}{2}} {(\int_{R^{3}} u^{2} d x)}^{\frac{1}{2}} \leq 2 | \nabla u |_{2} | u |_{2}, \forall u \in H^{1} (R^{3})

(7)

By a standard argument,

J \in C^{1} (H^{1} (R^{3}), R)

and

\begin{aligned} ⟨ J^{'} (u), v ⟩ & = a \int_{R^{3}} \nabla u \nabla v d x + β \int_{R^{3}} u v d x + b | \nabla u |_{2}^{2} \int_{R^{3}} \nabla u \nabla v d x \\ - μ \int_{R^{3}} \frac{u v}{| x |} d x - \int_{R^{3}} (I_{α} * | u |^{q}) | u |^{q - 2} u v d x \end{aligned}

The associated Pohoz̆aev functional is

P (u) = \frac{a}{2} | \nabla u |_{2}^{2} + \frac{3}{2} β | u |_{2}^{2} + \frac{b}{2} | \nabla u |_{2}^{4} - μ \int_{R^{3}} \frac{u^{2}}{| x |} d x - \frac{3}{2} \int_{R^{3}} (I_{α} * | u |^{q}) | u |^{q} d x

Set

M = {u \in H^{1} (R^{3}) ∖ {0} : G (u) = \frac{1}{2} ⟨ J^{'} (u), u ⟩ + P (u) = 0}

, where

G (u) = a | \nabla u |_{2}^{2} + 2 β | u |_{2}^{2} + b | \nabla u |_{2}^{4} - \frac{3}{2} μ \int_{R^{3}} \frac{u^{2}}{| x |} d x - 2 \int_{R^{3}} (I_{α} * {| u |}^{q}) | u |^{q} d x

(8)

For

α \in (0, 3)

a > 0

and

β > 0

, let

c (a, α, β) := 8 \sqrt{α (3 + 2 α) a β} / 3 (3 + 4 α)

Now, we are ready to state the main result of this article.

Theorem 1.1

Let $q = 3 + α / 3$ , $α \in (0, 3)$ , $a > 0$ , $b \geq 0$ , $β > 0$ and $0 < μ < c (a, α, β)$ . Then (1) has a ground state solution $\bar{u} \in H^{1} (R^{3})$ .

To verify our results, we must address two key challenges. Firstly, when $b > 0$ , establishing the weak sequential continuity of $J^{'}$ is particularly difficult due to the nonlocal nature of the equation. In general, we cannot directly conclude that $| \nabla u_{n} |_{2}^{2} \to | \nabla \bar{u} |_{2}^{2}$ from $u_{n} ⇀ \bar{u}$ in $H^{1} (R^{3})$ . We will address this issue by using the energy squeezing inequality (see Lemma 2.2). Secondly, the nonlocal term $\int_{R^{3}} (I_{α} * | u |^{q}) | u |^{q} d x$ in the energy functional $J$ is invariant under the dilations of the form $t^{3 / 2} u (t \cdot)$ , which is a noncompact group action on $H^{1} (R^{3})$ , making it more difficult to prove that the Palais–Smale sequence ${u_{n}}$ is non-vanishing. To restore compactness, we will leverage the slow decay nature of the Coulomb potential at infinity to reduce the energy level (see Lemma 2.8).

2 Proof of Main Result

In the following, we assume that all the conditions in Theorem 1.1 hold. It is easy to check that for all $t > 0$

\begin{aligned} f (t) := (t^{4 q} - 1) - 2 q (t^{2} - 1) \geq 0, g (t) := (t^{4 q} - 1) - q (t^{4} - 1) \geq 0 \end{aligned}

and

\begin{aligned} h (t) := 3 (t^{4 q} - 1) - 4 q (t^{3} - 1) \geq 0 \end{aligned}

Lemma 2.1
The following inequality holds:
$\frac{f (t) g (t)}{h^{2} (t)} \geq \frac{(q + α + 1) (q + α - 1)}{9 (q + α)^{2}}, \forall t \in (0, 1) \cup (1, + \infty)$
(9)

Proof.
Let $s > 1$ . Apply the Cauchy–Schwarz inequality, we have the following equation:
$\begin{aligned} \int_{1}^{s} ω^{q + α - 1} d ω \leq {(\int_{1}^{s} ω^{q + α} d ω)}^{\frac{1}{2}} {(\int_{1}^{s} ω^{q + α - 2} d ω)}^{\frac{1}{2}} \end{aligned}$
Then
$\begin{aligned} (q + α + 1) (q + α - 1) (s^{q + α + 2} - s^{2})^{2} \leq (q + α)^{2} (s^{q + α + 2} - s) (s^{q + α + 2} - s^{3}) \end{aligned}$
So, we see for all $x \in R$
$\begin{aligned} (s^{q + α + 2} - s^{3}) x^{2} + \sqrt{(q + α + 1) (q + α - 1)} (s^{q + α + 2} - s^{2}) x + \frac{(q + α)^{2}}{4} (s^{q + α + 2} - s) \geq 0 \end{aligned}$
For each $t > 1$ , we get
$\begin{aligned} (\int_{1}^{t} (s^{q + α + 2} - s^{3}) d s) x^{2} + \sqrt{(q + α + 1) (q + α - 1)} (\int_{1}^{t} (s^{q + α + 2} - s^{2}) d s) x \\ + \frac{(q + α)^{2}}{4} (\int_{1}^{t} (s^{q + α + 2} - s) d s) \geq 0 \end{aligned}$
which implies that
$\begin{aligned} (q + α + 1) (q + α - 1) {(\frac{t^{4 q} - 1}{4 q} - \frac{t^{3} - 1}{3})}^{2} \leq (q + α)^{2} (\frac{t^{4 q} - 1}{4 q} - \frac{t^{4} - 1}{4}) (\frac{t^{4 q} - 1}{4 q} - \frac{t^{2} - 1}{2}) \end{aligned}$
This shows that (9) holds for $t > 1$ . We can prove that it also holds for $0 < t < 1$ in the same way.
Lemma 2.2
There exists $θ \in (0, 1)$ such that
$J (u) \geq J (u_{t}) + \frac{1 - t^{4 q}}{4 q} G (u) + \frac{(1 - θ) a f (t)}{4 q} | \nabla u |_{2}^{2}, \forall t > 0, u \in H^{1} (R^{3})$
(10)

Proof.
Since $β > 0$ and $0 < μ < c (a, α, β)$ , there exists $0 < θ < 1$ such that $μ^{2} < 64 θ a β α (2 α + 3) / (4 α + 3)^{2}$ , then by (9), we have $μ^{2} h^{2} (t) < 8 θ a β f (t) g (t) .$ In view of (5) to (8), we see
$\begin{aligned} 4 q (J (u) - J (u_{t}) - \frac{1 - t^{4 q}}{4 q} G (u)) \\ = a f (t) | \nabla u |_{2}^{2} + 2 β g (t) | u |_{2}^{2} + b g (t) | \nabla u |_{2}^{4} - \frac{μ}{2} h (t) \int_{R^{3}} \frac{u^{2}}{| x |} d x \\ \geq a f (t) | \nabla u |_{2}^{2} + 2 β g (t) | u |_{2}^{2} - μ h (t) | \nabla u |_{2} | u |_{2} \\ \geq (1 - θ) a f (t) | \nabla u |_{2}^{2} \end{aligned}$
This implies (10) holds.

From Lemma 2.2, we have the following corollary.
Corollary 2.3
$J (u) = max_{t > 0} J (u_{t}), \forall u \in M$ .
Lemma 2.4
For any $u \in H^{1} (R^{3}) ∖ {0}$ , there exists a unique $t > 0$ such that $u_{t} \in M$ .
Proof.
The proof is similar to Lemma 2.7 by Tang & Chen (2017), so we omit it.
Lemma 2.5
$m := {inf}_{u \in M} J (u) = inf_{u \in H^{1} (R^{3}) ∖ {0}} max_{t > 0} J (u_{t}) > 0$ .
Proof.
By virtue of Corollary 2.3 and Lemma 2.4, we have
$\begin{aligned} inf_{u \in M} J (u) = inf_{u \in H^{1} (R^{3}) ∖ {0}} max_{t > 0} J (u_{t}) \end{aligned}$
Since $G (u) = 0, \forall u \in M$ , we see from (7) and the Hardy–Littlewood–Sobolev inequality that
$\begin{aligned} a | \nabla u |_{2}^{2} + 2 β | u |_{2}^{2} \leq \frac{3}{2} μ \int_{R^{3}} \frac{u^{2}}{| x |} d x + 2 \int_{R^{3}} (I_{α} * | u |^{q}) | u |^{q} d x \leq C_{1} (a | \nabla u |_{2}^{2} + 2 β | u |_{2}^{2}) + C_{2} | u |_{2}^{2 q} \end{aligned}$
where $C_{1} = 2 \sqrt{2 α (2 α + 3)} / 4 α + 3 < 1$ . Hence ${inf}_{u \in M} | u |_{2} > 0$ . In view of (7), we have that
$\begin{aligned} J (u) & = J (u) - \frac{1}{4 q} G (u) \\ \geq \frac{1}{8 q} (2 (2 q - 1) a | \nabla u |_{2}^{2} + 4 (q - 1) β | u |_{2}^{2} - \frac{16 \sqrt{(q - 1) (2 q - 1) a β}}{3} | \nabla u |_{2} | u |_{2}) \\ \geq \frac{1}{4 q} (1 - \frac{2 \sqrt{2}}{3}) ((2 q - 1) a | \nabla u |_{2}^{2} + 2 (q - 1) β | u |_{2}^{2}) \end{aligned}$
(11)
which implies $inf_{u \in M} J (u) > 0$ due to ${inf}_{u \in M} | u |_{2} > 0$ .

Following Jeanjean (1997), we define the map $Ψ : R \times H^{1} (R^{3}) \to H^{1} (R^{3})$ for $t \in R$ , $u \in H^{1} (R^{3})$ and $x \in R^{3}$ by
$\begin{aligned} Ψ (t, u) (x) = u (e^{- t} x) \end{aligned}$
We also define the family of paths
$\begin{aligned} Γ = {γ \in C ([0, 1], R \times H^{1} (R^{3})) : γ (0) = (0, 0), (J \circ Ψ) (γ (1)) < 0} \end{aligned}$
and the minimax level
$\begin{aligned} \tilde{C} = inf_{γ \in Γ} max_{t \in [0, 1]} (J \circ Ψ) (γ (t)) \end{aligned}$

Lemma 2.6
$\tilde{C} = m$ .
Proof.
For any $u \in M$ , it is easy to see that there exist $T > 1$ such that $J (u_{T}) < 0$ . For $t \in [0, 1]$ , we set $γ (t) = (0, u_{T t})$ , then $γ \in Γ$ . In view of Corollary 2.3, one has
$\begin{aligned} max_{t \in [0, 1]} (J \circ Ψ) (γ (t)) = max_{t \in [0, 1]} J (u_{T t}) = max_{t > 0} J (u_{t}) = J (u) \end{aligned}$
which leads to $\tilde{C} \leq m$ . We claim that $J (u) < 0$ implies $G (u) < 0$ . In fact, If $J (u) < 0$ , then by (11), we have
$\begin{aligned} G (u) \leq 4 q J (u) - (1 - \frac{2 \sqrt{2}}{3}) ((2 q - 1) a | \nabla u |_{2}^{2} + 2 (q - 1) β | u |_{2}^{2}) < 0 \end{aligned}$
Next, we show that $\tilde{C} \geq m$ . For any $γ \in Γ$ , we define $φ (t) := G (Ψ (γ (t))), t \in [0, 1]$ . Since $J (Ψ (γ (1))) < 0$ , one has $φ (1) = G (Ψ (γ (1))) < 0$ . It follows from (7) and the Hardy–Littlewood–Sobolev inequality that
$\begin{aligned} G (u) & \geq a | \nabla u |_{2}^{2} + 2 β | u |_{2}^{2} - 3 μ | \nabla u |_{2} | u |_{2} - C_{1} | u |_{2}^{2 q} \\ \geq (1 - \frac{2 \sqrt{2 α (2 α + 3)}}{4 α + 3}) (a | \nabla u |_{2}^{2} + 2 β | u |_{2}^{2}) - C_{1} ‖ u ‖^{2 q} \\ \geq C_{2} ‖ u ‖^{2} - C_{1} ‖ u ‖^{2 q} \end{aligned}$
Thus, $G (u) > 0$ for $‖ u ‖ > 0$ sufficiently small. Hence, $Λ := {t \in (0, 1) : φ (t) = 0} \neq \emptyset$ . Let $t^{} = sup Λ$ . Then $Ψ (γ (t^{})) \in M$ and ${Ψ (γ (t)) : t \in [0, 1]} \cap M \neq \emptyset$ . Thus, $max_{t \in [0, 1]} (J \circ Ψ) (γ (t)) \geq m$ .

Next, we will construct a Palais–Smale sequence for the functional $J$ at the level $\tilde{C}$ that satisfies asymptotically the Pohoz̆aev identity.
Lemma 2.7
There exists a bounded sequence ${u_{n}} \in H^{1} (R^{3})$ such that $J (u_{n}) \to \tilde{C}, J^{'} (u_{n}) \to 0$ and $G (u_{n}) \to 0$ .
Proof.
For every $t \in R$ and $v \in H^{1} (R^{3})$ , the functional $J \circ Ψ$ is computed as follows:
$\begin{aligned} (J \circ Ψ) (t, v) = \frac{a}{2} e^{4 t} | \nabla v |_{2}^{2} + \frac{β}{2} e^{8 t} | v |_{2}^{2} + \frac{b}{4} e^{8 t} | \nabla v |_{2}^{4} - \frac{μ}{2} e^{6 t} \int_{R^{3}} \frac{v^{2}}{| x |} d x - \frac{1}{2 q} e^{8 q t} \int_{R^{3}} (I_{α} * | v |^{q}) | v |^{q} d x \end{aligned}$
Clearly, $J \circ Ψ \in C^{1} (R \times H^{1} (R^{3}), R)$ . For every $\frac{1}{n} \in (0, \frac{\tilde{C}}{2})$ , there exists $γ_{n} \in Γ$ such that
$\begin{aligned} sup_{s \in [0, 1]} (J \circ Ψ) (γ_{n} (s)) < \tilde{C} + \frac{1}{n} \end{aligned}$
Set ${\tilde{γ}}_{n} (s) := (0, Ψ (γ_{n} (s)))$ . Then ${\tilde{γ}}_{n} \in Γ$ , $Ψ ({\tilde{γ}}_{n} (s)) = Ψ (γ_{n} (s))$ and
$\begin{aligned} sup_{s \in [0, 1]} (J \circ Ψ) ({\tilde{γ}}_{n} (s)) < \tilde{C} + \frac{1}{n} \end{aligned}$
By the general minimax principle (Willem, 1996: Theorem 2.8), there exists a sequence $(t_{n}, v_{n}) \in R \times H^{1} (R^{3})$ such that
$\begin{aligned} \tilde{C} - \frac{2}{n} \leq (J \circ Ψ) (t_{n}, v_{n}) \leq \tilde{C} + \frac{2}{n} \end{aligned}$

$\begin{aligned} | t_{n} | \leq dist ((t_{n}, v_{n}), (0, Ψ (γ_{n}))) \leq \frac{2}{\sqrt{n}} \end{aligned}$

$\begin{aligned} ‖ (J \circ Ψ)^{'} (t_{n}, v_{n}) ‖ \leq \frac{4}{\sqrt{n}} \end{aligned}$
Therefore, $t_{n} \to 0$ and $(J \circ Ψ) (t_{n}, v_{n}) \to \tilde{C}$ . A direct calculation shows that for every $(h, w) \in R \times H^{1} (R^{3})$
$⟨ (J \circ Ψ)^{'} (t_{n}, v_{n}), (h, w) ⟩ = ⟨ J^{'} (Ψ (t_{n}, v_{n})), Ψ (t_{n}, w) ⟩ + 2 h G (Ψ (t_{n}, v_{n}))$
(12)
Taking $h = 1$ , $w = 0$ in (12), we get
$\begin{aligned} G (Ψ (t_{n}, v_{n})) \to 0 as n \to \infty \end{aligned}$
Denote $u_{n} = Ψ (t_{n}, v_{n})$ , we have that
$\begin{aligned} J (u_{n}) \to \tilde{C} and G (u_{n}) \to 0 as n \to \infty \end{aligned}$
For any $v \in H^{1} (R^{3})$ , set $w (x) = v (e^{t_{n}} x)$ , $h = 0$ in (12), we obtain, by $t_{n} \to 0$ ,
$\begin{aligned} o (1) ‖ v ‖ = o (1) ‖ v (e^{t_{n}} \cdot) ‖ = ⟨ (J \circ Ψ)^{'} (t_{n}, v_{n}), (0, v (e^{t_{n}} \cdot)) ⟩ = ⟨ J^{'} (u_{n}), Ψ (t_{n}, w) ⟩ = ⟨ J^{'} (u_{n}), v ⟩ \end{aligned}$
Thus, $J^{'} (u_{n}) \to 0$ in $H^{- 1} (R^{3})$ . Next, We show that the sequence ${u_{n}}$ is bounded in $H^{1} (R^{3})$ . Note that $μ < \frac{8 \sqrt{α (3 + 2 α) a β}}{3 (3 + 4 α)} = \frac{8 \sqrt{(q - 1) (2 q - 1) a β}}{3 (4 q - 3)}$ . In view of (7), we see that
$\begin{aligned} \tilde{C} + o (1) = J (u_{n}) - \frac{1}{4 q} G (u_{n}) \\ = \frac{2 q - 1}{4 q} a | \nabla u_{n} |_{2}^{2} + \frac{q - 1}{2 q} β | u_{n} |_{2}^{2} + \frac{q - 1}{4 q} b | \nabla u_{n} |_{2}^{4} - \frac{4 q - 3}{8 q} μ \int_{R^{3}} \frac{u_{n}^{2}}{| x |} d x \\ \geq \frac{2 q - 1}{4 q} a | \nabla u_{n} |_{2}^{2} + \frac{q - 1}{2 q} β | u_{n} |_{2}^{2} - \frac{2 \sqrt{(q - 1) (2 q - 1) a β}}{3 q} | \nabla u_{n} |_{2} | u_{n} |_{2} \\ \geq (1 - \frac{2 \sqrt{2}}{3}) (\frac{2 q - 1}{4 q} a | \nabla u_{n} |_{2}^{2} + \frac{q - 1}{2 q} β | u_{n} |_{2}^{2}) \end{aligned}$
Hence, ${u_{n}}$ is bounded in $H^{1} (R^{3})$ .□

Use $T_{α}$ to denote the best constant defined by the following equation:
$\begin{aligned} T_{α} := inf {\int_{R^{3}} | u |^{2} d x : u \in H^{1} (R^{3}), \int_{R^{3}} (I_{α} * | u |^{\frac{3 + α}{3}}) | u |^{\frac{3 + α}{3}} d x = 1} \end{aligned}$
The constant $T_{α}$ is achieved by $U_{δ} := C (\frac{δ^{\frac{1}{2}}}{δ + | x |^{2}})^{\frac{3}{2}}$ , where $C > 0$ is a fixed constant and $δ$ is a parameter (see (Lieb & Loss, 2001)). In the following, we will give an appropriate estimate on the ground state energy level $m$ , which will play a significant role in the concentration-compactness argument to restore the compactness.
Lemma 2.8
$m < C_{} := \frac{α}{2 (α + 3)} (β T_{α})^{\frac{α + 3}{α}}$
Proof.
We use $U_{δ}$ to estimate $m$ . By Corollary 2.3 and Lemma 2.4, there exists a unique $t_{δ}$ such that $(U_{δ})_{t_{δ}} \in M$ and $J ((U_{δ})_{t_{δ}}) = {s u p}_{t > 0} J ((U_{δ})_{t})$ . Thus
$\begin{aligned} a | \nabla U_{δ} |_{2}^{2} + 2 β t_{δ}^{2} | U_{δ} |_{2}^{2} + b t_{δ}^{2} | \nabla U_{δ} |_{2}^{4} - \frac{3 μ}{2} t_{δ} \int_{R^{3}} \frac{| U_{δ} |^{2}}{| x |} d x - 2 t_{δ}^{4 q - 2} \int_{R^{3}} (I_{α} | U_{δ} |^{q}) | U_{δ} |^{q} d x = 0 \end{aligned}$
(13)
Obviously, $\int_{R^{3}} | U_{δ} |^{2} d x = \int_{R^{3}} | U_{1} |^{2} d x = T_{α}$ and $\int_{R^{3}} (I_{α} * | U_{δ} |^{q}) | U_{δ} |^{q} d x = \int_{R^{3}} (I_{α} * | U_{1} |^{q}) | U_{1} |^{q} d x$ . By a direct calculation, we have that
$\begin{aligned} | \nabla U_{δ} |_{2}^{2} = C δ^{\frac{3}{2}} \int_{R^{3}} \frac{| x |^{2}}{(δ + | x |^{2})^{5}} d x = C \frac{1}{δ} \int_{R^{3}} \frac{| x |^{2}}{(1 + | x |^{2})^{5}} d x = O_{+} (\frac{1}{δ}) \end{aligned}$
and
$\begin{aligned} \int_{R^{3}} \frac{| U_{δ} |^{2}}{| x |} d x = C δ^{\frac{3}{2}} \int_{R^{3}} \frac{1}{{(δ + | x |^{2})}^{3} | x |} d x = C \frac{1}{δ^{\frac{1}{2}}} \int_{R^{3}} \frac{1}{{(1 + | x |^{2})}^{3} | x |} d x = O_{+} (\frac{1}{δ^{\frac{1}{2}}}) \end{aligned}$
By (13), we know that $t_{δ}$ is bounded as $δ \to \infty$ . Let $δ \to \infty$ , we obtain $t_{δ} \to (β T_{α})^{\frac{1}{4 (q - 1)}}$ . For $t > 0$ , let $φ (t) = \frac{β}{2} t^{4} T_{α} - \frac{1}{2 q} t^{4 q}$ . It is easy to see that $φ (t)$ attains its maximum at $t_{0} = (β T_{α})^{\frac{3}{4 α}}$ and $max_{t > 0} φ (t) = (\frac{1}{2} - \frac{1}{2 q}) t_{0}^{4 q} = \frac{α}{2 (α + 3)} (β T_{α})^{\frac{α + 3}{α}}$ . We deduce that for sufficiently large $δ > 0$
$\begin{aligned} J ((U_{δ})_{t_{δ}}) & = (\frac{β T_{α}}{2} t_{δ}^{4} - \frac{1}{2 q} t_{δ}^{4 q}) + \frac{a}{2} t_{δ}^{2} | \nabla U_{δ} |_{2}^{2} + \frac{b}{4} t_{δ}^{4} | \nabla U_{δ} |_{2}^{4} - \frac{μ}{2} t_{δ}^{3} \int_{R^{3}} \frac{| U_{δ} |^{2}}{| x |} d x \\ \leq max_{t > 0} φ (t) + O_{+} (\frac{1}{δ}) + O_{+} (\frac{1}{δ^{2}}) - O_{+} (\frac{1}{δ^{\frac{1}{2}}}) < \frac{α}{2 (α + 3)} (β T_{α})^{\frac{α + 3}{α}} \end{aligned}$
The result then follows.
Proof of Theorem 1.1. Proof of Theorem 1.1:

By Lemma 2.7, there exists a bounded $(P S)_{\tilde{C}}$ sequence ${u_{n}} \subset H^{1} (R^{3})$ for the functional $J$ with $G (u_{n}) \to 0$ . Let $δ = {lim}_{n \to \infty} {sup}_{y \in R^{3}} \int_{B_{1} (y)} {| u_{n} |}^{2} d x$ . We claim that $δ > 0$ . Otherwise, if $δ = 0$ , by Lion’s lemma, we have $u_{n} \to 0$ in $L^{s} (R^{3})$ , $\forall s \in (2, 6)$ . We assume that $‖ u_{n} ‖ \leq M$ . For any $ε > 0$ , let $R_{ε} > \frac{M^{2}}{ε}$ , then $\int_{| x | \geq R_{ε}} \frac{u_{n}^{2}}{| x |} d x \leq \frac{1}{R_{ε}} | u_{n} |_{2}^{2} \leq \frac{M^{2}}{R_{ε}} < ε$ . Let $\frac{1}{3} < λ < 1$ . By Hölder inequality and the Rellich embedding theorem, we deduce that for sufficiently large $n$

\begin{aligned} \int_{| x | \leq R_{ε}} \frac{u_{n}^{2}}{| x |} d x & \leq {(\int_{| x | \leq R_{ε}} | u_{n} |^{6 λ} d x)}^{\frac{1}{3 λ}} {(\int_{| x | \leq R_{ε}} \frac{1}{| x |^{\frac{3 λ}{3 λ - 1}}} d x)}^{\frac{3 λ - 1}{3 λ}} \\ \leq | u_{n} |_{6 λ}^{2} {(4 π \int_{0}^{R_{ε}} ρ^{\frac{3 λ - 2}{3 λ - 1}} d ρ)}^{\frac{3 λ - 1}{3 λ}} = | u_{n} |_{6 λ}^{2} {R_{ε}}^{\frac{2 λ - 1}{λ}} {(\frac{4 π (3 λ - 1)}{3 (2 λ - 1)})}^{\frac{3 λ - 1}{3 λ}} < ε \end{aligned}

Then

\int_{R^{3}} \frac{u_{n}^{2}}{| x |} d x \to 0

n \to \infty

. Let

σ_{n} = (\int_{R^{3}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x)^{\frac{1}{2 q}}

. Then

\int_{R^{3}} (I_{α} * | \frac{u_{n}}{σ_{n}} |^{q}) | \frac{u_{n}}{σ_{n}} |^{q} d x = 1

. By the definition of

T_{α}

, one has

| u_{n} |_{2}^{2} \geq T_{α} σ_{n}^{2}

. We may assume that

| u_{n} |_{2}^{2} \to ξ

and

\int_{R^{3}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x = σ_{n}^{2 q} \to η

. Thus,

ξ \geq T_{α} η^{\frac{1}{q}}

. In view of

G (u_{n}) \to 0

, we see

\begin{aligned} 2 β | u_{n} |_{2}^{2} \leq \frac{3}{2} μ \int_{R^{3}} \frac{u_{n}^{2}}{| x |} d x + 2 \int_{R^{3}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x + o (1) \end{aligned}

Let

n \to \infty

, we get

η \geq (β T_{α})^{\frac{α + 3}{α}}

. Note that

\begin{aligned} J (u_{n}) + o (1) & = J (u_{n}) - \frac{1}{4 q} G (u_{n}) \geq \frac{α β}{2 (α + 3)} | u_{n} |_{2}^{2} - \frac{4 α + 3}{8 (α + 3)} μ \int_{R^{3}} \frac{u_{n}^{2}}{| x |} d x \end{aligned}

Let

n \to \infty

, one has

\begin{aligned} m \geq \frac{α β}{2 (α + 3)} ξ \geq \frac{α β}{2 (α + 3)} T_{α} η^{\frac{1}{q}} \geq \frac{α}{2 (α + 3)} (β T_{α})^{\frac{α + 3}{α}} \end{aligned}

which contradicts Lemma 2.8. Therefore, there exists a sequence

{y_{n}} \subset R^{3}

such that

\int_{B_{1} (y_{n})} | u_{n} |^{2} d x \geq \frac{δ}{2} > 0

(14)

Set

v_{n} (\cdot) = u_{n} (\cdot + y_{n})

, then

‖ v_{n} ‖ = ‖ u_{n} ‖

. Assume that

u_{n} ⇀ \bar{u}

H^{1} (R^{3})

v_{n} ⇀ \bar{v}

H^{1} (R^{3})

. We have by (14) that

\bar{v} \neq 0

. We claim that

\bar{u} \neq 0

. Arguing by contradiction. If

\bar{u} = 0

, the Rellich embedding theorem implies that

\int_{R^{3}} \frac{u_{n}^{2}}{| x |} d x = o (1)

. Since

J^{'} (u_{n}) \to 0

and

G (u_{n}) \to 0

, we have the following equations:

o (1) = ⟨ J^{'} (u_{n}), u_{n} ⟩ = a | \nabla u_{n} |_{2}^{2} + b | \nabla u_{n} |_{2}^{4} + β | u_{n} |_{2}^{2} - \int_{R^{3}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x + o (1)

(15)

and

o (1) = G (u_{n}) = a | \nabla u_{n} |_{2}^{2} + b | \nabla u_{n} |_{2}^{4} + 2 β | u_{n} |_{2}^{2} - 2 \int_{R^{3}} (I_{α} * | u_{n} |^{q}) | u_{n} |^{q} d x + o (1)

(16)

Combining (15) with (16), we obtain

a | \nabla u_{n} |_{2}^{2} + b | \nabla u_{n} |_{2}^{4} = o (1)

and

| \nabla u_{n} |_{2}^{2} = o (1)

. So

| \nabla v_{n} |_{2}^{2} = o (1)

and

\bar{v} = 0

, which is a contradiction. Thus

\bar{u} \neq 0

. For the case

b = 0

, it is easy to obtain that

J^{'} (\bar{u}) = 0

. We may assume that

b > 0

and

| \nabla u_{n} |_{2}^{2} \to A

. By the weak lower semi-continuity, we have

| \nabla \bar{u} |_{2}^{2} \leq A

. We claim that

| \nabla \bar{u} |_{2}^{2} = A

. Otherwise, if

| \nabla \bar{u} |_{2}^{2} < A

, define

\begin{aligned} J_{A} (u) := \frac{a + b A}{2} | \nabla u |_{2}^{2} + \frac{β}{2} | u |_{2}^{2} - \frac{μ}{2} \int_{R^{3}} \frac{u^{2}}{| x |} d x - \frac{1}{2 q} \int_{R^{3}} (I_{α} * | u |^{q}) | u |^{q} d x, \forall u \in H^{1} (R^{3}) \end{aligned}

From

| \nabla u_{n} |_{2}^{2} \to A

and

J^{'} (u_{n}) \to 0

, we know

J_{A}^{'} (\bar{u}) = 0

. Hence

\begin{aligned} P_{A} (\bar{u}) = \frac{a + b A}{2} | \nabla \bar{u} |_{2}^{2} + \frac{3 β}{2} | \bar{u} |_{2}^{2} - μ \int_{R^{3}} \frac{{\bar{u}}^{2}}{| x |} d x - \frac{3}{2} \int_{R^{3}} (I_{α} * | \bar{u} |^{q}) | \bar{u} |^{q} d x = 0 \end{aligned}

Therefore

\begin{aligned} 0 & = \frac{1}{2} ⟨ J_{A}^{'} (\bar{u}), \bar{u} ⟩ + P_{A} (\bar{u}) = (a + b A) | \nabla \bar{u} |_{2}^{2} + 2 β | \bar{u} |_{2}^{2} - \frac{3}{2} μ \int_{R^{3}} \frac{{\bar{u}}^{2}}{| x |} d x - 2 \int_{R^{3}} (I_{α} * | \bar{u} |^{q}) | \bar{u} |^{q} d x \\ = G (\bar{u}) + b A | \nabla \bar{u} |_{2}^{2} - b | \nabla \bar{u} |_{2}^{4} > G (\bar{u}) \end{aligned}

(17)

On the other hand, the Rellich embedding theorem shows that

\int_{R^{3}} \frac{u_{n}^{2}}{| x |} d x \to \int_{R^{3}} \frac{{\bar{u}}^{2}}{| x |} d x

n \to \infty

. From the weak lower semi-continuity and (10), we have that

\begin{aligned} \tilde{C} & = J (u_{n}) - \frac{1}{4 q} G (u_{n}) + o (1) \\ = \frac{2 q - 1}{4 q} a | \nabla u_{n} |_{2}^{2} + \frac{q - 1}{2 q} β | u_{n} |_{2}^{2} + \frac{q - 1}{4 q} b | \nabla u_{n} |_{2}^{4} - \frac{4 q - 3}{8 q} μ \int_{R^{3}} \frac{u_{n}^{2}}{| x |} d x + o (1) \\ \geq \frac{2 q - 1}{4 q} a | \nabla \bar{u} |_{2}^{2} + \frac{q - 1}{2 q} β | \bar{u} |_{2}^{2} + \frac{q - 1}{4 q} b | \nabla \bar{u} |_{2}^{4} - \frac{4 q - 3}{8 q} μ \int_{R^{3}} \frac{{\bar{u}}^{2}}{| x |} d x \\ = J (\bar{u}) - \frac{1}{4 q} G (\bar{u}) \geq J ({\bar{u}}_{\bar{t}}) - \frac{{\bar{t}}^{4 q}}{4 q} G (\bar{u}) \geq \tilde{C} - \frac{{\bar{t}}^{4 q}}{4 q} G (\bar{u}) \end{aligned}

(18)

which leads to

G (\bar{u}) \geq 0

, a contradiction to (17). So

| \nabla u_{n} |_{2}^{2} \to A = | \nabla \bar{u} |_{2}^{2}

. Then by

J^{'} (u_{n}) \to 0

, we have

J^{'} (\bar{u}) = 0

, and the Pohoz̆aev identity implies

G (\bar{u}) = 0

. Moreover, we see from (18) that

J (\bar{u}) = \tilde{C}

, which implies that

\bar{u}

is the ground state solution of (1).

Footnotes

Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (nos. 12101273, 12171486), the Science and Technology Innovation Program of Hunan Province (no. 2024RC3021), the Young Backbone Teachers Project of Hunan Province and Natural Science Foundation for Excellent Young Scholars of Hunan Province (no. 2023JJ20057).

Ethical Considerations

If statement present, text would go here.

Consent to Participate

If statement present, text would go here.

Consent for Publication

If statement present, text would go here.

Author Contributions/CRediT

All authors contribute equally to this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is partially supported by the National Natural Science Foundation of China (nos. 12101273, 12171486), the Science and Technology Innovation Program of Hunan Province (no. 2024RC3021), the Young Backbone Teachers Project of Hunan Province and Natural Science Foundation for Excellent Young Scholars of Hunan Province (no. 2023JJ20057).

Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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