Ground State Solutions for Choquard Equation With Indefinite Potential and Critical Exponential Growth

Abstract

In this article, we study the following Choquard equation, ${\begin{cases} - Δ u + V (x) u = (I_{α} * F (u)) f (u), x \in R^{2}, \\ u \in H^{1} (R^{2}), \end{cases}$ where $I_{α}$ is the Riesz potential, $f : R^{2} \to R$ possessing critical exponential growth at infinity, $V$ is 1-periodic in $x_{1}$ , $x_{2}$ and zero lies in the gap of spectrum of $- Δ + V$ . We derive some detailed estimates to deal with difficulties arising from the strongly indefinite feature and the appearance of convolution term. Using a suitable variational framework based on the generalized Nehari manifold method, we reduce the indefinite problem to a definite case and succeed in finding a bounded Palais–Smale sequence. With the help of a proper auxiliary equation, we obtain a fine threshold to control the minimax level for above critical problem which allows us to restore the compactness. Existence of the ground state solutions is obtained via the concentration compactness argument and some delicate analyses.

Keywords

critical exponential growth strongly indefinite problem Trudinger–Moser inequality Choquard equation

1. Introduction and Main Results

In this article, we study the existence of ground state solutions for the following equation with Trudinger–Moser type nonlinearity:

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

(1.1)

where

F (t) := \int_{0}^{t} f (s) d s

α \in (0, 2)

and

I_{α}

is the Riesz potential defined by

I_{α} (x) = \frac{A_{α}}{| x |^{2 - α}}, A_{α} = \frac{Γ (\frac{2 - α}{2})}{2^{α} π Γ (\frac{α}{2})}, x \in R^{2} ∖ {0} .

For the case $α = 1$ and $f (u) = u$ , Equation (1.1) reduces to the following classical Choquard type equation,

- Δ u + V (x) u = (\frac{1}{| x |} * | u |^{2}) u, x \in R^{3} .

(1.2)

In the context of quantum mechanics, Equation (1.2) was first introduced by Fröhlich (1937) and Pekar (1954) to model quantum polaron. It is also used to describe the phenomenon of an electron trapped in its own hole, which is connected to one component plasma approximation in the Hartree–Fock theory. As mentioned by Lieb (1976), Equation (1.2) can also be used to research the steady states of the one component plasma approximation in the Hartree–Fock theory. More physical background of Choquard equation, see Ghergu and Taliaferro (2016); Penrose (1998).

If $inf_{x \in R^{N}} V (x) > 0$ , then the spectrum $σ (- Δ + V) > 0$ , problem (1.2) and the related problem have been extensively studied in the recent literature. The early existence of radially symmetry solutions were established by Lions (1980) via the minimization method and symmetric critical point theorem. Later, Ma and Zhao (2010) first studied the classification of solutions for Equation (1.2) and obtained that all the positive solutions must be radially symmetric and monotonically decreasing around some fixed points. Qin et al. (2021) established the existence of ground state solution and infinitely many solutions for (1.6) using the generalized Nehari manifold method. For further study to this case, we refer to Moroz and Schaftingen (2015a, 2015b).

For the case $N \geq 3$ , various results of problem were obtained, such as, existence and multiplicity of solutions, sign-changing solutions, semiclassical solutions, and so on, see Ghimenti and Schaftingen (2016); Ma and Zhao (2010); Moroz and Schaftingen (2015b); Qin et al. (2021); Xia and Wang (2019); Yuan et al. (2022) for the Sobolev subcritical or critical case. In the situation $N = 2$ , problem (1.1) presents a special case that is the nonlinearity is allowed to be critical exponential growth at infinity. Particularly, the nonlinearity $f$ satisfies the following conditions involving subcritical and critical exponential growth at infinity, (F1)

$f \in C (R, R)$ , and

lim_{| t | \to + \infty} \frac{| f (t) |}{e^{β t^{2}}} = 0, for all β > 0;

(1.3)

(F 1^{'})

$f \in C (R, R)$ , and there exists $β_{0} > 0$ such that

lim_{| t | \to + \infty} \frac{| f (t) |}{e^{β t^{2}}} = 0, for all β > β_{0},

(1.4)

and

lim_{| t | \to + \infty} \frac{| f (t) |}{e^{β t^{2}}} = + \infty, for all β < β_{0} .

(1.5)

Note that $H^{1} (R^{2}) ↪ L^{s} (R^{2})$ for all $s \in [2, \infty)$ , but $H^{1} (R^{2}) ⊈ L^{\infty} (R^{2})$ . In order to deal with the problem with critical exponential growth, Cao (1992) initially established the well-known Trudinger–Moser (see also Adimurthi Adachi & Tanaka, 2000, and Cassani et al., 2014). This inequality can be expressed as follows:

Lemma 1.1

(i) If $β > 0$ and $u \in H^{1} (R^{2})$ , then

\int_{R^{2}} (e^{β u^{2}} - 1) d x < \infty;

(ii) if

u \in H^{1} (R^{2}), ‖ \nabla u ‖_{2}^{2} \leq 1, ‖ u ‖_{2} \leq M < \infty

, and

β < 4 π

, then there exists a positive constant

C (M, β)

, which depends only on

M

and

β

, such that

\int_{R^{2}} (e^{β u^{2}} - 1) d x \leq C (M, β) .

Since $σ (- Δ + V) \cap R^{+} \neq \emptyset$ , then problem (1.2) turns to the indefinite case. In $R^{3}$ , Buffoni et al. (1993) applied the reduction method to get a nontrivial solution for (1.2). In Qin et al. (2021), the authors considered more general type equation of (1.2) with $N \geq 3$ and obtained the existence of nontrivial solution and infinitely many solutions via dual mountain pass theorem and some new techniques. In this article, we focus on the following strongly indefinite case when 0 lies in a gap of the spectrum $σ (- Δ + V)$ , namely, (V0)

$V \in C (R^{2}, R)$ $V (x)$ is 1-periodic in $x_{1}$ and $x_{2}$ , and

sup [σ (- Δ + V) \cap (- \infty, 0)] < 0 < inf [(σ (- Δ + V) \cap (0, \infty))] .

When (V0) holds, note that the operator $- Δ + V$ has a purely continuous spectrum consisting of closed disjoint intervals, (1.2) turns to be a strongly indefinite problem. As far as we know, there are only a few papers solved this case. In present article (Ackermann, 2004), Ackermann investigated the following related problem with (1.2), which has attracted a lot of scholars’ attention,

- Δ u + V (x) u = (W * F (u)) f (u), x \in R^{N}, N \geq 2.

(1.6)

By supposing the following Ambrosetti–Rabinowitz type condition (F3),

(F 3)

There exists $μ > 1$ such that

f (t) t \geq μ F (t) > 0, \forall t \in R ∖ {0},

they obtained the existence of nontrivial solution and infinitely many geometrically distinct solutions for Equation (1.6) employing the generalized linking theorem, such theorem was introduced by Kryszewski and Szulkin (1998) and was extended by Li and Szulkin (2002) and Ding and Lee (2006) later. In recent years, Qin et al. (2021) generalized and improved the above results by using some delicate analyses and introducing some general assumptions on

W

and

f

. Moreover, they applied the Lusternik–Schnirelmann theory and deformation arguments to show multiple results for problem (1.6).

When $N = 2$ , let us recall some related works for the planar Choquard Equation (1.1) with critical exponential growth. To the best of our knowledge, Alves et al. (2016) first studied the nonlocal Choquard equation involving critical exponential growth in $R^{2}$ . For the case $V (x) \geq V_{0} > 0$ , they employed the mountain pass theorem to obtain the nontrivial solutions of (1.1) with $β_{0} = 4 π$ . They verified that mountain pass level is less than $(2 + α) / 8$ by using (F3) and the following conditions, $(F_{1})$

$f (t) = 0$ for $t \leq 0, 0 \leq f (t) \leq C e^{4 π t^{2}}, \forall t \geq 0$ ;

(F_{6})

$lim_{t \to \infty} \frac{t f (t) F (t)}{e^{8 π t^{2}}} = l with l > inf_{ρ > 0} \frac{α (1 + α) (2 + α)^{2}}{16 π^{2} ρ^{2 + α}} e^{\frac{(2 + α)}{4} V_{ρ} ρ^{2}}$ , where $V_{ρ} := sup_{| x | \leq ρ} V (x)$ .

Without the Ambrosetti–Rabinowitz type condition (F3), Battaglia and Schaftingen (2017) employed a scaling trick to obtain a Pohožaev–Palais–Smale sequence of the energy function. They established the existence of ground state solution for problem (1.1) with $V \equiv 1$ via the variational method. Particularly, the nonlinear $f$ satisfies the following conditions, $(BL)$

$F \in C (R^{2}, R)$ and there exists $t_{0} \in R$ such that $F (t_{0}) \neq 0$ ;

(F_{1}^{'})

for each $θ > 0$ , there exists $C_{θ} > 0$ such that

| f (t) | \geq C_{θ} min {1, | t |^{\frac{α}{2}}} e^{θ | t |^{2}} .

For quasilinear equations of

N

-Laplacian type with critical exponential growth, we refer to recent papers (Chen et al., 2024; Hu et al., 2025; Liu et al., 2024) where some existence results were obtained by developing some delicate analyses and using some general conditions on

f

. For other related works, we draw readers’ attention to papers (Li & Zhang, 2024; Liang et al., 2024; Lin et al., 2024; Sun et al., 2024) and the references therein.

Consider the case $V$ is sign-changing satisfying (V0), as far as we know, there are only two papers on the literature (cf. Gao et al., 2022; Qin & Tang, 2021) concerning the strongly indefinite problem (1.1), see also Alves and Germano (2018); Chen and Tang (2021); do Ó and Ruf (2006); Zhang et al. (2025) for the Schrödinger equation. Recently, Qin and Tang (2021) established a proper Cerami sequence for problem (1.1) with condition $(V 0)$ by employing a direct method involving the approaching argument and diagonal method. Taking advantage of this inequality and the following growth condition $(F 6^{'})$ and using some delicate estimates involving the Moser’s functions, they determined a fine threshold to control the minimax-level. Finally, the existence of nontrivial solutions for problem (1.1) was established provided that $f$ satisfies $(F 1^{'}), (F 3), (F_{4})$ and the following conditions. $(F_{2})$

$| f (t) | = o (| t |^{α / 2})$ as $| t | \to 0$ .

(F 5)

there exist $M_{0} > 0$ and ${\tilde{t}}_{0} > 0$ such that

F (t) \leq M_{0} | f (t) |, \forall | t | \geq {\tilde{t}}_{0};

(F 6^{'})

$\underset{t \to \infty}{lim inf} \frac{f (t)}{e^{β_{0} t^{2}}} = κ > \frac{\sqrt{α (1 + α)} (2 + α)}{\sqrt{2 π A_{α}} ρ^{1 + α / 2}} e^{4 (2 + α) π (1 + ρ)^{2} B_{0}^{2} / (2 + ρ)}$ , where $ρ > 0$ satisfies $2 (2 + α) π ρ^{2} B_{0}^{2} < 1$ and $B_{0} > 0$ is an embedding constant.

Later, the above result was extended and improved by Gao et al. (2022). They introduced the following condition instead of $(F 6^{'})$ to restore the compactness and proved the existence of nontrivial solution for (1.1). $(F_{6}^{'})$

$\underset{t \to \infty}{lim inf} \frac{f (t)}{e^{β_{0} t^{2}}} = κ > \frac{\sqrt{α (1 + α)} (2 + α)}{\sqrt{2 π A_{α}} ρ^{1 + α / 2}} e^{6 (2 + α) π B_{0}^{2}}$ , where $ρ \in (0, 4 / ‖ V ‖_{\infty})$ satisfies $2 (2 + α) (6 + ρ) ρ π B_{0}^{2} < 1$ .

Motivated by the above mentioned works, we shall study the existence of the ground state solution for problem (1.1) with sign-changing potential $V$ satisfying (V0) and critical exponential nonlinearity $f$ . It is worth noting that the generalized Linking theorem, since the Fréchet derivative of the energy functional is no longer weakly sequentially continuous. We shall employ more techniques and detailed estimates to conquer the difficulties arising from the presence of the convolution terms. To overcome the difficulty caused by the strongly indefinite feature of the problem, we shall intend to make use of the method of generalized Nehari manifold developed by Szulkin and Weth (2010). Taking advantage of the following condition $(F 6)$ (or $(F 6^{'})$ ), we find a fine threshold (see Lemma 3.6) to control the minimax-level of the Palais–Smale sequence. With the help of which the compactness for the critical exponential case can be restored and the boundedness of the Palais–Smale sequences can be obtained by the following monotonicity condition (F4). By using the concentration compactness argument, we succeed in showing that the sequence is nonvanishing. In order to obtain the existence of ground state solution for (1.1), we employ the energy comparison argument, which allows us to establish some relationships of critical levels between the original problem (1.1) and an auxiliary problem. Precisely, we make use of following assumptions: $(F 2)$

$| f (t) | = o (| t |)$ as $| t | \to 0$ ;

(F 4)

$t \mapsto f (t) / | t |$ is strictly increasing on $(- \infty, 0)$ and $(0, + \infty)$ ;

(F 6)

there exists $p > 1 + \frac{α}{2}$ such that

F (t) \geq K^{1 / 2} t^{p}, \forall t \geq 0,

where

K > 0

is defined by (3.11);

Now we are ready to introduce the main results as follows,

Theorem 1.2

Let (V0), (F1), (F2), (F4) and (F5) be satisfied. Then problem (1.1) has a ground state solution $\bar{u} \in H^{1} (R^{2}) ∖ {0}$ such that

Φ (\bar{u}) = inf_{N} Φ > 0,

where

N

is the Nehari–Pankov manifold defined later by (2.10)

Theorem 1.3

Let (V0), $(F 1^{'})$ , (F2)-(F5) and $(F 6)$ (or $(F 6^{'})$ ) be satisfied. Then problem (1.1) has a ground state solution $\tilde{u} \in H^{1} (R^{2}) ∖ {0}$ such that

Φ (\tilde{u}) = inf_{N} Φ > 0

Remark 1.4

Particularly, the technical growth conditions $(F 6^{'}), (F_{6})$ and $(F_{6}^{'})$ are used to estimate the threshold to control the minimax level for the critical problem. With the help of monotonicity condition $(F 4)$ , we establish the existence of ground state solution in Theorems 1.2 and 1.3 and those results extend and improve the related ones of Qin and Tang (2021, Theorem 1.3) and Gao et al. (2022, Theorem 1.2). Moreover, it is interesting to obtain the existence of ground state solutions for (1.1) by weaken $(F 2)$ to $(F_{2})$ and without the Ambrosetti–Rabinowitz type condition.

Throughout the article, we make use of the following notations:

$C_{1}, C_{2}, \dots$ denote positive constants possibly different in different places;

For any $x \in R^{N}$ and $r > 0, B_{r} (x) := {y \in R^{N} : | y - x | < r}$ ;

$L^{s} (R^{N}) (1 \leq s < \infty)$ denotes the Lebesgue space with the norm $‖ u ‖_{s} = (\int_{R^{N}} | u |^{s} d x)^{1 / s}$ .

We organize this article as follows. In Section 2, we introduce the variational setting and give some preliminary results. In Section 3, we establish the existence of ground state solutions of (1.1) for the subcritical case and obtain a fine threshold of the minimax-level for the energy functional restricted on the Nehari–Pankov manifold. Theorem 1.3 will be proved in the last section.

2. Variational Setting and Preliminaries

By employing the generalized Nehari manifold method, Ekeland variational principle and some delicate analyses, we obtain a proper Palais–Smale sequence of $Φ$ in this section.

Following the same argument of Qin and Tang (2021, Section 2), we can define an inner product

(u, v) = {(| - Δ + V |^{1 / 2} u, | - Δ + V |^{1 / 2} v)}_{L^{2}}, u, v \in E := H^{1} (R^{2}),

(2.1)

and the corresponding norm

‖ u ‖ = {‖ | - Δ + V |^{1 / 2} u ‖}_{2}, u \in E,

(2.2)

where

(\cdot, \cdot)_{L^{2}}

represents the inner product of

L^{2} (R^{2})

‖ \cdot ‖_{s}

stands for the norm of

L^{s} (R^{2})

. In addition, one has the following orthogonal decomposition

E = E^{-} \oplus E^{+}

, where orthogonality is with respect to both

(\cdot, \cdot)_{L^{2}}

and

(\cdot, \cdot)

. By (V0),

E^{-}

is infinite dimensional.

Since the embedding of $E ↪ L^{s} (R^{2})$ is continuous for all $s \in [2, + \infty)$ , there exists $γ_{s} > 0$ such that

‖ u ‖_{s} \leq γ_{s} ‖ u ‖, \forall u \in E .

(2.3)

Set $β > β_{0} > 0$ , by $(F 1) (or (F 1^{'}))$ and $(F 2)$ , for any given $ε > 0$ , there exists $C_{ε} > 0$ such that

| f (t) | \leq ε | t | + C_{ε} (e^{β t^{2}} - 1), \forall t \in R,

(2.4)

and

| F (t) | \leq ε | t |^{2} + C_{ε} | t | (e^{β t^{2}} - 1), \forall t \in R .

(2.5)

Using (2.5) and Lemma 1.1, we can define the energy functional of problem (1.1) as follows,

Φ (u) = \frac{1}{2} \int_{R^{2}} (| \nabla u |^{2} + V (x) u^{2}) d x - \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] F (u) d x,

(2.6)

and there holds

⟨ Φ^{'} (u), v ⟩ = \int_{R^{2}} (\nabla u \cdot \nabla v + V (x) u v) d x - \int_{R^{2}} [I_{α} * F (u)] f (u) v d x, \forall u, v \in E .

(2.7)

Moreover, by (2.1) and (2.2), we deduce that

Φ (u) = \frac{1}{2} (‖ u^{+} ‖^{2} - ‖ u^{-} ‖^{2}) - \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] F (u) d x, \forall u = u^{-} + u^{+} \in E^{-} \oplus E^{+} = E,

(2.8)

and

⟨ Φ^{'} (u), u ⟩ = ‖ u^{+} ‖^{2} - ‖ u^{-} ‖^{2} - \int_{R^{2}} [I_{α} * F (u)] f (u) u d x, \forall u \in E .

(2.9)

Obviously, the solutions of problem (1.1) are critical points of

Φ

To get the ground state solution of (1.1), we define the following Nehari–Pankov manifold

N := {u \in E ∖ E^{-} : ⟨ Φ^{'} (u), u ⟩ = ⟨ Φ^{'} (u), ϕ ⟩ = 0, \forall ϕ \in E^{-}},

(2.10)

which is first introduced by Pankov (2005) and has been widely used in recent papers (de Figueiredo et al., 2020; Guan et al., 2024; Szulkin & Weth, 2009).

In the following remark, we shall show that if $u \neq 0$ is a critical point of $Φ$ , then $u \in N$ .

Remark 2.1

If $u \in E ∖ {0}$ and $Φ^{'} (u) = 0$ . By $(F 3)$ , we obtain $t f (t) > F (t)$ for any $t \in R ∖ {0}$ . Thus,

Φ (u) = Φ (u) - \frac{1}{2} ⟨ Φ^{'} (u), u ⟩ = \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] [f (u) u - F (u)] d x > 0.

On the other hand, if

u \in E^{-}

, we have

Φ (u) = - ‖ u ‖^{2} - \int_{R^{2}} [I_{α} * F (u)] F (u) d x \leq 0.

Consequently, if

u \in E ∖ {0}

is a critical point of

Φ

, then

u \in E ∖ E^{-}

. Moreover,

u \in N

For $u \in E ∖ E^{-}$ , set

\hat{E} (u) := E^{-} \oplus R^{+} u,

where

R^{+} := {t \in R : t \geq 0}

. Then

\hat{E} (u) = E^{-} \oplus R^{+} u^{+} = \hat{E} (u^{+}) .

The following lemmas play an important role in our proofs.

Lemma 2.2.

(Cauchy–Schwarz type inequality Mattner, 1997). For any $g, h \in L_{l o c}^{1} (R^{2})$ , the following inequality holds,

\int_{R^{2}} (I_{α} * | g |) | h | d x \leq {(\int_{R^{2}} (I_{α} * | g |) | g | d x)}^{\frac{1}{2}} {(\int_{R^{2}} (I_{α} * | h |) | h | d x)}^{\frac{1}{2}} .

(2.11)

Lemma 2.3.

(Hardy–Littlewood–Sobolev inequality Lieb, 1976). Let $s, r > 1$ and $0 < μ < 2$ with $1 / s + 1 / r = 4 - μ / 2,$ $g \in L^{s} (R^{2})$ and $h \in L^{r} (R^{2})$ . There exists a sharp constant $C (μ, s, r)$ , independent of $g, h$ , such that

\int_{R^{2}} (I_{2 - μ} * g) h d x \leq C (μ, s, r) ‖ g ‖_{s} ‖ h ‖_{r} .

(2.12)

In particular,

\int_{R^{2}} (I_{α} * g) h d x \leq C_{0} ‖ g ‖_{4 / (2 + α)} ‖ h ‖_{4 / 2 + α} .

(2.13)

Lemma 2.4

de Figueiredo et al. (2004, Lemma 2.4). The following inequality holds,

s t \leq {\begin{cases} (e^{t^{2}} - 1) + s (\log s)^{\frac{1}{2}}, & t \geq 0, s \geq e^{\frac{1}{4}}; \\ (e^{t^{2}} - 1) + \frac{1}{2} s^{2}, & t \geq 0, 0 \leq s \leq e^{\frac{1}{4}} . \end{cases}

Lemma 2.5

Suppose that $(V 0), (F 1) (o r (F 1^{'})), (F 2)$ and $(F 4)$ are satisfied. Then for any $u \in N$ , there holds

Φ (u) > Φ (w), \forall w \in \hat{E} (u) ∖ {u} .

(2.14)

Moreover,

u

is the unique maximum point of

Φ |_{\hat{E} (u)}

Proof.

For any $w \in \hat{E} (u)$ may be written as $w := t u + v, v \in E^{-}, t \geq 0$ . Define

\tilde{f} (t) := [I_{α} * F (t)] f (t) and \tilde{F} (t) = \int_{0}^{t} \tilde{f} (s) d s .

This, together with (2.6) and (2.7) yields that

\begin{aligned} Φ (u) - Φ (w) & = Φ (u) - Φ (t u + v) \\ = \frac{1 - t^{2}}{2} (‖ u^{+} ‖^{2} - ‖ u^{-} ‖^{2}) + t ⟨ v, u^{-} ⟩ + \frac{1}{2} \int_{R^{2}} [I_{α} * F (t u + v)] F (t u + v) d x \\ + \frac{1}{2} ‖ v ‖^{2} - \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] F (u) d x \\ = \frac{1 - t^{2}}{2} ⟨ Φ^{'} (u), u ⟩ - t ⟨ Φ^{'} (u), v ⟩ - t \int_{R^{2}} [I_{α} * F (u)] f (u) v d x \\ - \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] F (u) d x + \frac{1 - t^{2}}{2} \int_{R^{2}} [I_{α} * F (u)] f (u) u d x \\ + \frac{1}{2} \int_{R^{2}} [I_{α} * F (t u + v)] F (t u + v) d x + \frac{1}{2} ‖ v ‖^{2} \\ = \frac{1}{2} ‖ v ‖^{2} + \frac{1 - t^{2}}{2} ⟨ Φ^{'} (u), u ⟩ - t ⟨ Φ^{'} (u), v ⟩ \\ + \int_{R^{2}} [\tilde{f} (u) (\frac{1 - t^{2}}{2} u - t v) - \tilde{F} (u) + \tilde{F} (t u + v)] d x . \end{aligned}

(2.15)

Similarly as Szulkin and Weth (2009, Lemma 2.2), since

t \geq 0

and

(t - 1) u + v \neq 0

, it follows from (F4) that

g (t) := \tilde{f} (u) [\frac{1 - t^{2}}{2} u - t v] - \tilde{F} (u) + \tilde{F} (t u + v) > 0,

and

g (t) = 0

(t - 1) u + v = 0

. Note that

u \in N

, together with (2.15), we obtain that

Φ (u) > Φ (w), for u \neq w .

w \in \hat{E} (u)

Φ (w) = Φ (u)

, it is not difficult to show that

w = u

Lemma 2.6

Assume that $(V 0), (F 1) (o r (F 1^{'})), (F 2)$ and $(F 5)$ hold. Let $V \subset E^{+} ∖ {0}$ be any compact set, then for each $u \in V$ there exists $R > 0$ such that

Φ (w) \leq 0, i f w \in \hat{E} (u) ∖ Λ_{R} (0),

where

Λ_{R} (0) = {u \in E : ‖ u ‖ < R}

Proof.

Arguing indirectly, suppose that for some $u_{0} \in V$ , there exists a sequence ${u_{n}} \subset V$ , such that ${w_{n}} \subset \hat{E} (u_{n})$ with $‖ w_{n} ‖ \to \infty$ and $Φ (w_{n}) \geq 0$ as $n \to \infty$ for $n \in N$ . Without loss of generality, pass to subsequence, we assume that $u_{n} \to u_{0} \in E^{+}$ and $‖ u_{0} ‖ = 1$ . Let $w_{n} = w_{n}^{-} + s_{n} u_{n}, s_{n} \geq 0$ and ${\tilde{w}}_{n} = w_{n} / ‖ w_{n} ‖$ . Then

{\tilde{w}}_{n} = \frac{w_{n}^{-}}{‖ w_{n} ‖} + \frac{s_{n}}{‖ w_{n} ‖} u_{n} =: {\tilde{w}}_{n}^{-} + t_{n} u_{n},

and

‖ {\tilde{w}}_{n} ‖^{2} = ‖ {\tilde{w}}_{n}^{-} ‖^{2} + t_{n}^{2} = 1.

By virtue of (2.6), we have

\begin{aligned} 0 \leq \frac{Φ (w_{n})}{‖ w_{n} ‖^{2}} & = \frac{t_{n}^{2}}{2} - \frac{1}{2} ‖ {\tilde{w}}_{n}^{-} ‖^{2} - \frac{1}{2 ‖ w_{n} ‖^{2}} \int_{R^{2}} [I_{α} * F (w_{n})] F (w_{n}) d x \\ = \frac{t_{n}^{2}}{2} - \frac{1}{2} ‖ {\tilde{w}}_{n}^{-} ‖^{2} - \frac{1}{2} \int_{R^{2}} \int_{R^{2}} I_{α} (x - y) \frac{F (w_{n} (x))}{‖ w_{n} ‖} \frac{F (w_{n} (y))}{‖ w_{n} ‖} d x d y \end{aligned}

(2.16)

\begin{aligned} \leq \frac{t_{n}^{2}}{2} - \frac{1}{2} ‖ {\tilde{w}}_{n}^{-} ‖^{2} . \end{aligned}

Thus

t_{n}^{2} \geq ‖ {\tilde{w}}_{n}^{-} ‖^{2}

, it is easy to show that

t_{n} \in [1 / \sqrt{2}, 1]

for

n \in N

large enough. Passing to a subsequence, we may assume that

t_{n} \to t > 0, {\tilde{w}}_{n}^{-} ⇀ {\tilde{w}}^{-} \in E^{-}

E

and

{\tilde{w}}_{n}^{-} \to {\tilde{w}}^{-}

a.e. on

R^{2}

. Then

{\tilde{w}}_{n} ⇀ \tilde{w} := {\tilde{w}}^{-} + t u_{0}

E

and

{\tilde{w}}_{n} \to \tilde{w} \neq 0

a.e. on

R^{2}

. Choose

x_{0} \in R^{2}

and

r_{1} \in (0, x_{0} / 2)

such that

B_{r_{1}} (x_{0}) \subset R^{2}

and

| \tilde{w} (x) | > 0

for

x \in B_{r_{1}} (x_{0})

. It follows from (F5), (2.16) and Fatou’s lemma that

\begin{aligned} 0 & \leq \underset{n \to \infty}{lim sup} [\frac{t_{n}^{2}}{2} - \frac{1}{2} ‖ {\tilde{w}}_{n}^{-} ‖^{2} - \frac{1}{2} \int_{R^{2}} \int_{R^{2}} I_{α} (x - y) \frac{F (w_{n} (x))}{‖ w_{n} ‖} \frac{F (w_{n} (y))}{‖ w_{n} ‖} d x d y] \\ \leq \frac{t^{2}}{2} - \underset{n \to \infty}{lim inf} \frac{1}{2} \int_{R^{2}} \int_{R^{2}} I_{α} (x - y) \frac{F (w_{n} (x))}{‖ w_{n} ‖} \frac{F (w_{n} (y))}{‖ w_{n} ‖} d x d y \\ \leq \frac{t^{2}}{2} - \frac{1}{2} \int_{R^{2}} \int_{R^{2}} I_{α} (x - y) \underset{n \to \infty}{lim inf} [\frac{F (w_{n} (x))}{| w_{n} (x) |} | {\tilde{w}}_{n} (x) | \frac{F (w_{n} (y))}{| w_{n} (y) |} | {\tilde{w}}_{n} (y) |] d x d y \\ \leq \frac{t^{2}}{2} - \frac{1}{2} \int_{B_{r_{1} (x_{0})}} \int_{B_{r_{1} (x_{0})}} I_{α} (x - y) \underset{n \to \infty}{lim inf} [\frac{F (w_{n} (x))}{| w_{n} (x) |} | {\tilde{w}}_{n} (x) | \frac{F (w_{n} (y))}{| w_{n} (y) |} | {\tilde{w}}_{n} (y) |] d x d y \\ \to - \infty, \end{aligned}

(2.17)

which is a contradiction.

Lemma 2.7

Suppose that $(V 0), (F 1) (o r (F 1^{'})), (F 2)$ and $(F 4)$ are satisfied. Then (i)

there exists $ρ > 0$ such that

c := inf_{N} Φ \geq κ := inf_{S_{ρ}^{+}} Φ > 0,

where

S_{ρ}^{+} := {u \in E^{+}, ‖ u ‖ = ρ};

(ii)

$‖ u^{+} ‖ \geq max {\sqrt{2 c}, ‖ u^{-} ‖}$ .

Proof.

(i)

According to Lemma 2.5, for every $u \in N$ , there exists $t_{0} > 0$ such that $t_{0} u^{+} \in \hat{E} (u) \cap S_{ρ}^{+}$ . Thus $inf_{N} Φ \geq κ$ . By virtue of $(F 1^{'})$ (or (F1)) and (F2), there exist $β > β_{0}$ and $C_{1} > 0$ such that

| F (t) | \leq | t |^{(2 + α) / 2} + C_{1} (e^{β t^{2}} - 1) | t |^{3}, \forall t \in R .

(2.18)

It follows from Lemma 1.1-(ii) that there exists

C_{2} > 0

such that

\int_{R^{2}} (e^{4 β α^{- 1} ‖ u ‖^{2} (u / ‖ u ‖)^{2}} - 1) d x \leq C_{2}, \forall ‖ u ‖ \leq \sqrt{\frac{α π}{2 β}} .

(2.19)

Using (2.18) and (2.19) and the Hölder inequality, we have

\begin{aligned} \int_{R^{2}} | F (u) |^{\frac{4}{2 + α}} d x & \leq \int_{R^{2}} {[| u |^{\frac{(2 + α)}{2}} + C_{1} (e^{β u^{2}} - 1) | u |^{3}]}^{\frac{4}{2 + α}} d x \\ \leq 2 \int_{R^{2}} | u |^{2} + {C_{1}}^{\frac{4}{2 + α}} {(e^{β u^{2}} - 1)}^{\frac{4}{2 + α}} | u |^{\frac{12}{2 + α}} d x \\ \leq 2 {‖ u ‖_{2}^{2} + {C_{1}}^{\frac{4}{2 + α}} {(\int_{R^{2}} {(e^{β u^{2}} - 1)}^{\frac{4}{α}} d x)}^{\frac{α}{2 + α}} ‖ u ‖_{6}^{\frac{12}{2 + α}}} \\ \leq 2 {‖ u ‖_{2}^{2} + {C_{1}}^{\frac{4}{2 + α}} {(\int_{R^{2}} e^{4 β α^{- 1} ‖ u ‖^{2} (u / ‖ u ‖)^{2}} - 1 d x)}^{\frac{α}{2 + α}} ‖ u ‖_{6}^{\frac{12}{2 + α}}} \\ \leq 2 (‖ u ‖_{2}^{2} + C_{2} ‖ u ‖_{6}^{\frac{12}{2 + α}}), \forall ‖ u ‖ \leq \sqrt{\frac{α π}{2 β}} . \end{aligned}

(2.20)

It follows from (2.13) and (2.20) that

\begin{aligned} \int_{R^{2}} (I_{α} * F (u)) F (u) d x & \leq C_{0} {(\int_{R^{2}} | F (u) |^{\frac{4}{2 + α}} d x)}^{\frac{2 + α}{2}} \\ \leq C_{0} {[2 ‖ u ‖_{2}^{2} + C_{2} ‖ u ‖_{6}^{\frac{12}{2 + α}}]}^{\frac{2 + α}{2}} \\ \leq 4 C_{0} ‖ u ‖_{2}^{2 + α} + 2 {C_{2}}^{\frac{2 + α}{2}} C_{0} ‖ u ‖_{6}^{6}, \forall ‖ u ‖ \leq \sqrt{\frac{α π}{2 β}} . \end{aligned}

(2.21)

Hence, for any

u \in E^{+}

, we obtain from (2.6) and (2.21) that

\begin{aligned} Φ (u) & = \frac{1}{2} ‖ u ‖^{2} - \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] F (u) d x \\ \geq \frac{1}{2} ‖ u ‖^{2} - C_{3} (‖ u ‖^{2 + α} + ‖ u ‖^{6}), \forall ‖ u ‖ \leq \sqrt{\frac{α π}{2 β}} . \end{aligned}

Therefore, there exists

0 < ρ \leq \sqrt{α π / 2 β}

small enough such that

Φ (u) > 0

for

u \in S_{ρ}^{+}

(ii)

For $u \in N$ , we obtain from (i) and (2.8) that

0 < c \leq \frac{1}{2} (‖ u^{+} ‖^{2} - ‖ u^{-} ‖^{2}) - \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] F (u) d x \leq \frac{1}{2} (‖ u^{+} ‖^{2} - ‖ u^{-} ‖^{2}) .

Hence, we conclude that

‖ u^{+} ‖ \geq max {\sqrt{2 c}, ‖ u^{-} ‖}

Lemma 2.8

Suppose that $(V 0), (F 1) (o r (F 1^{'})), (F 2)$ and $(F 4)$ are satisfied. Then for any $u \in E ∖ E^{-}$ , the set $\hat{E} (u)$ intersects $N$ at exactly one point $\hat{m} (u)$ , which is the unique global maximum point of $Φ |_{\hat{E} (u)}$ .

Proof.

Since $\hat{E} (u) = \hat{E} (u^{+})$ , by using Lemma 2.5 it is necessary to show that $N \cap \hat{E} (u^{+}) \neq \emptyset$ for any $u \in E ∖ E^{-}$ . Without loss of generality, we assume that $u \in E^{+}$ and $‖ u ‖ = 1$ . It follows from Lemma 2.6 that there exists $R > 0$ such that $Φ (w) \leq 0$ for $w \in \hat{E} (u^{+}) ∖ Λ_{R} (0)$ . According to Lemma 2.7-(i), $Φ (t u) > 0$ for $t > 0$ small enough. Consequently,

0 < sup_{\hat{E} (u)} Φ < \infty .

Let

w_{n} = t_{n} u + v_{n} \in \hat{E} (u)

be a maximizing sequence for

sup_{\hat{E} (u)} Φ

, where

t_{n} > 0, v_{n} \in E^{-}

. It is not difficult to show that the following function,

Φ (w_{n}) = \frac{t_{n}^{2}}{2} ‖ u ‖^{2} - \frac{1}{2} ‖ v_{n} ‖^{2} - \int_{R^{2}} [I_{α} * F (w_{n})] F (w_{n}) d x

is weakly upper semicontinuous on

\hat{E} (u)

and

Φ (w_{0}) = sup_{\hat{E} (u)} Φ > 0

for some

u_{0} \in \hat{E} (u) ∖ {0}

. Therefore

w_{0}

is a critical point of

Φ |_{\hat{E} (u)}

, which implies that

w_{0} \in N

. This, together with Lemma 2.5, we conclude that

w_{0} \in N \cap \hat{E} (u)

is the unique global maximum of

Φ |_{\hat{E} (u)}

Define

\hat{m} : E ∖ E^{-} \mapsto N \cap \hat{E} (u) and m := \hat{m} |_{S^{+}} : S^{+} \mapsto N,

where

S^{+} := S \cap E^{+} = {u \in E^{+} : ‖ u ‖ = 1}

. Inspired by Szulkin and Weth (2009, Chapter 4), we obtain from Lemmas 2.6 and 2.8 that the mapping

\hat{m}

is continuous and the mapping

m

is a homeomorphism between

S^{+}

and

N

According to Lemmas 2.6 and 2.7, we can deduce the following lemma.

Lemma 2.9

Assume that $(V 0), (F 1) (o r (F 1^{'})), (F 2)$ and $(F 5)$ hold. There exists $δ > 0$ such that $‖ \hat{m} {(u)}^{+} ‖ \geq δ$ for all $u \in E ∖ E^{-}$ , and for each compact subset $W \subset E ∖ E^{-}$ there exists a constant $C_{W} > 0$ such that $‖ \hat{m} (u) ‖ \leq C_{W}$ for all $u \in W$ .

Proof.

For any $u \in E ∖ E^{-}$ , we have $\hat{m} (u) \in N$ . Choose $δ := \sqrt{2 c}$ , it follows from Lemma 2.7-(ii) that $‖ \hat{m} {(u)}^{+} ‖ \geq δ$ . Since $\hat{m} (u) = \hat{m} (u^{+} / ‖ u^{+} ‖)$ , we can assume without loss generality that $W \subset S^{+}$ . Similar as Lemma 2.6, we obtain that there exists a positive constant $C_{W}$ such that

Φ (w) \leq 0, on w \in \hat{E} (u) ∖ Λ_{C_{W}} (0), for all u \in W .

(2.22)

In view of Lemma 2.7-(i), we have

Φ (\hat{m} (u)) \geq c > 0

for all

u \in E ∖ E^{-}

. This, together with (2.22) implies that

‖ \hat{m} (u) ‖ \leq C_{W}

for all

u \in W

By virtue of the monotonicity condition (F4), Lemmas 2.5 and 2.8, we have the following useful lemma.

Lemma 2.10

Suppose that $(V 0), (F 1) (o r (F 1^{'})), (F 2)$ and $(F 4)$ are satisfied. Then

inf_{u \in N} Φ (u) = inf_{v \in E ∖ E^{-}} max_{w \in \hat{E} (v)} Φ (w) .

(2.23)

Proof.

By virtue of Lemma 2.5, we have $Φ (v) = sup_{w \in \hat{E} (v)} Φ (w)$ for any $v \in N$ . Thus

Φ (v) \geq inf_{v \in E ∖ E^{-}} max_{w \in \hat{E} (v)} Φ (w) .

For any

v \in E ∖ E^{-}

, it follows from Lemma 2.8 that there exists a unique

\hat{m} (v) \in N \cap \hat{E} (v)

such that

max_{w \in \hat{E} (v)} Φ (w) = Φ (\hat{m} (v)) \geq inf_{u \in N} Φ (u),

which implies that

inf_{v \in E ∖ E^{-}} max_{w \in \hat{E} (v)} Φ (w) \geq inf_{u \in N} Φ (u) .

Thus (2.23) holds.

For any $u \in E$ , let

E (u) = E^{-} \oplus R u = E^{-} \oplus R u^{+} .

Obviously,

S^{+}

is a

C^{1}

-submanifold of

E^{+}

and the tangent space of

S^{+}

as follows,

T_{u} (S^{+}) = {v \in E^{+} : (v, u) = 0} .

Then for any

u \in S^{+}

, we have

E = T_{u} (S^{+}) \oplus R u \oplus E^{-} = T_{u} (S^{+}) \oplus E (u) .

Define

Ψ : S^{+} \mapsto R, Ψ (u) := Φ (m (u)), for u \in S^{+} .

By virtue of Lemmas 2.8 and 2.9, we obtain the following statement (see Szulkin & Weth, 2010, Corollary 4.3).

Lemma 2.11

The following properties hold: (i)

$Ψ \in C^{1} (S^{+}, R)$ and

⟨ Ψ^{'} (u), w ⟩ = ‖ m {(u)}^{+} ‖ ⟨ Φ^{'} (m (u)), w ⟩, \forall w \in T_{u} (S^{+});

(ii)

If ${w_{n}} \subset S^{+}$ is a Palais–Smale sequence for $Ψ$ , then ${m (w_{n})} \subset N$ is a Palais–Smale sequence for $Φ$ . If ${v_{n}} \subset N$ is a bounded Palais–Smale sequence for $Φ$ , then ${m^{- 1} (v_{n})} \subset S^{+}$ is a Palais–Smale sequence for $Ψ$ ;

(iii)

$w$ is a critical point of $Ψ$ if and only if $m (w) \in N$ is a nontrivial critical point of $Φ$ ;

(iv)

$inf_{S^{+}} Ψ = inf_{N} Φ$ .

By Lemmas 2.7-(i) and 2.11, $inf_{S^{+}} Ψ = inf_{N} Φ = c > 0$ . Noting that $S^{+}$ is a regular $C^{1}$ -submanifold of $E^{+}$ , it follows from the Ekeland variational principle (see Ekeland, 1974, Theorem 5.1) that there exists a Palais–Smale sequence ${w_{n}} \subset S^{+}$ such that

Ψ (w_{n}) \to c > 0 and Ψ^{'} (w_{n}) \to 0, as n \to \infty .

(2.24)

Let

u_{n} = m (w_{n}) \subset N

. Then, applying (2.24) and Lemma 2.11, we have

Φ (u_{n}) \to c > 0 and Φ^{'} (u_{n}) \to 0, as n \to \infty .

(2.25)

3. Estimates of Minimax Level

In this section, we first consider the subcritical growth case and show the Theorem 1.2. Note that, it is more difficult to restore the compactness for the critical case by the strongly indefinite features of the problem (1.1). We shall estimate a fine threshold value of the energy functional by using delicate estimates and the technical condition (F6) (or $(F 6^{'})$ ).

In the following lemma, we shall verify the boundedness of the (PS) sequence when the nonlinearity has subcritical growth satisfying (F1).

Lemma 3.1
Assume that (V0), (F1), (F2) and (F4) hold. Then ${u_{n}}$ satisfying (2.25) is bounded in $E$ .
Proof.
To prove the boundedness of ${u_{n}}$ , arguing by contradiction, suppose that $‖ u_{n} ‖ \to \infty$ . Let $v_{n} = u_{n} / ‖ u_{n} ‖$ , then $‖ v_{n} ‖ = 1$ , $‖ v_{n} ‖_{2} \leq γ_{2}$ and $‖ \nabla v_{n} ‖_{2} \leq ν_{0}$ . Passing to a subsequence, we may assume that $v_{n} ⇀ v$ in $E$ , $v_{n} \to v$ in $L_{loc}^{s} (R^{2})$ for all $2 \leq s < \infty$ and $v_{n} \to v$ a.e. on $R^{2}$ . If
$δ := \underset{n \to \infty}{lim sup} sup_{y \in R^{2}} \int_{B_{1} (y)} | v_{n}^{+} |^{2} d x = 0,$
then by Lions’ concentration compactness principle (Willem, 1996, Lemma 1.21), $v_{n}^{+} \to 0$ in $L^{s} (R^{2})$ for $2 < s < \infty$ . Note that $u_{n} \in N$ and $t_{n} u_{n}^{+} \in \hat{E} (u_{n})$ , we deduce from Lemma 2.5 that
$Φ (u_{n}) \geq Φ (t_{n} u_{n}^{+}), \forall t_{n} \geq 0.$
(3.1)
Choose $R > [2 (1 + c)]^{1 / 2} > 0$ , where $c$ is given by Lemma 2.7. Set $β \in (0, α π / (R γ_{2} ν_{0})^{2})$ . In view of (F1) and (F2), there exists $C_{1} > 0$ such that
$| F (t) | \leq \frac{1}{{C_{0}}^{1 / 2} (2 R γ_{2})^{(2 + α) / 2}} | t |^{(2 + α) / 2} + C_{1} | t |^{3} (e^{β t^{2}} - 1), \forall t \in R .$
(3.2)
(3.2), Lemma 1.1-(ii) and Hölder inequality lead to
$\begin{aligned} \int_{R^{2}} | F (R v_{n}^{+}) |^{\frac{4}{2 + α}} d x \\ \leq \int_{R^{2}} {[\frac{1}{{C_{0}}^{1 / 2} (2 R γ_{2})^{(2 + α) / 2}} | R v_{n}^{+} |^{\frac{(2 + α)}{2}} + C_{1} (e^{β R^{2} {v_{n}^{+}}^{2}} - 1) | R v_{n}^{+} |^{3}]}^{\frac{4}{2 + α}} d x \\ \leq 2 \int_{R^{2}} [\frac{1}{{C_{0}}^{2 / (2 + α)} (2 R γ_{2})^{2}} | R v_{n}^{+} |^{2} + {C_{1}}^{\frac{4}{2 + α}} {(e^{β R^{2} {v_{n}^{+}}^{2}} - 1)}^{\frac{4}{2 + α}} | R v_{n}^{+} |^{\frac{12}{2 + α}}] d x \\ \leq 2 {\frac{1}{{C_{0}}^{2 / (2 + α)} (2 γ_{2})^{2}} ‖ v_{n}^{+} ‖_{2}^{2} + {C_{1}}^{\frac{4}{2 + α}} R^{\frac{12}{2 + α}} {(\int_{R^{2}} {(e^{β R^{2} {v_{n}^{+}}^{2}} - 1)}^{\frac{4}{α}} d x)}^{\frac{α}{2 + α}} ‖ v_{n}^{+} ‖_{6}^{\frac{12}{2 + α}}} \\ \leq 2 {\frac{1}{4 {C_{0}}^{2 / (2 + α)}} + {C_{1}}^{\frac{4}{2 + α}} R^{\frac{12}{2 + α}} ‖ v_{n}^{+} ‖_{6}^{\frac{12}{2 + α}} {(\int_{R^{2}} (e^{4 β R^{2} {ν_{0}}^{2} α^{- 1} ‖ v_{n}^{+} ‖^{2} (v_{n}^{+} / ν_{0} ‖ v_{n}^{+} ‖)^{2}} - 1) d x)}^{\frac{α}{2 + α}}} \\ \leq \frac{1}{2 {C_{0}}^{2 / (2 + α)}} + C_{2} ‖ v_{n}^{+} ‖_{6}^{\frac{12}{2 + α}} \\ \leq \frac{1}{2 {C_{0}}^{2 / (2 + α)}} + o (1) . \end{aligned}$
(3.3)
Let $t_{n} = R / ‖ u_{n} ‖$ . By (2.6), (2.24), (3.3) and Lemma 2.3, we have
$\begin{aligned} c + o (1) & = Φ (u_{n}) \geq Φ (t_{n} u_{n}^{+}) = Φ (R v_{n}^{+}) \\ \geq \frac{R^{2}}{2} ‖ v_{n}^{+} ‖^{2} - \frac{1}{2} \int_{R^{2}} [I_{α} * F (R v_{n}^{+})] F (R v_{n}^{+}) d x \\ \geq \frac{R^{2}}{2} - \frac{C_{0}}{2} {(\int_{R^{2}} | F (R v_{n}^{+}) |^{\frac{4}{2 + α}} d x)}^{\frac{2 + α}{2}} \\ \geq \frac{R^{2}}{2} - \frac{C_{0}}{2} {(\frac{1}{2 C_{0}^{2 / (2 + α)}})}^{\frac{2 + α}{2}} + o (1) \\ > c + \frac{3}{4} + o (1), \end{aligned}$
(3.4)
which is a contradiction. This shows that $δ > 0$ . The rest of the proof is standard, so we omit.
Lemma 3.2 (Qin and Tang (2021, Lemma 3.1))

Assume that (F1) and (F2) hold. Let $u_{n} ⇀ u$ in $E$ , then for every $v \in E$ ,

lim_{n \to \infty} \int_{R^{2}} [I_{α} * F (u_{n})] f (u_{n}) v d x = \int_{R^{2}} [I_{α} * F (u)] f (u) v d x .

(3.5)

Proof of Theorem 1.2. Proof of Theorem 1.2.

Applying in Lemmas 2.11 and 3.1, we deduce that there exists a sequence ${u_{n}} \subset N$ satisfying (2.25) and $‖ u_{n} ‖ \leq C_{1}$ for some $C_{1} > 0$ . Similar as the proof of Qin and Tang (2021, Lemma 3.5), we can show $u_{n} ⇀ \bar{u} \neq 0$ and $Φ^{'} (\bar{u}) = 0$ . It follows from Remark 2.1 and Lemma 2.7 that $\bar{u} \in N$ and $Φ (\bar{u}) \geq c$ . On the other hand, by using (F3), (2.6), (2.7), (2.25), Lemma 3.2 and Fatou’s lemma, we have

\begin{aligned} c & = lim_{n \to \infty} [Φ (u_{n}) - \frac{1}{2} ⟨ Φ^{'} (u_{n}), u_{n} ⟩] \\ = lim_{n \to \infty} \frac{1}{2} \int_{R^{2}} [I_{α} * F (u_{n})] [f (u_{n}) u_{n} - F (u_{n})] d x \\ \geq \frac{1}{2} \int_{R^{2}} lim_{n \to \infty} {[I_{α} * F (u_{n})] [f (u_{n}) u_{n} - F (u_{n})]} d x \\ = \frac{1}{2} \int_{R^{2}} [I_{α} * F (\bar{u})] [f (\bar{u}) \bar{u} - F (\bar{u})] d x \\ = Φ (\bar{u}) - \frac{1}{2} ⟨ Φ^{'} (\bar{u}), \bar{u} ⟩ = Φ (\bar{u}) . \end{aligned}

Hence,

Φ (\bar{u}) = c = inf_{N} Φ > 0

. This completes the proof.

Inspired by de Figueiredo et al. (1995), we define Moser type function $w_{n} (x)$ supported in $B_{ρ}$ as follows:

w_{n} (x) = \frac{1}{\sqrt{2 π}} {\begin{aligned} \sqrt{\log n}, & 0 \leq | x | \leq ρ / n; \\ \frac{\log (ρ / | x |)}{\sqrt{\log n}}, & ρ / n \leq | x | \leq ρ; \\ 0, & | x | \geq ρ . \end{aligned}

(3.6)

By Lemma 2.10, similarly as Qin and Tang (2021, Lemma 4.6), we can find a fine threshold to control the minimax level under the growth condition $(F 6^{'})$ .

Lemma 3.3 (Qin and Tang (2021, Lemma 4.6))

Assume that (V0), $(F 1^{'})$ , (F2), (F3) and $(F 6^{'})$ hold. Then there exists $\bar{n} \in N$ such that

c \leq sup_{u \in \hat{E} (w_{\bar{n}})} Φ (u) < \frac{(2 + α) π}{2 β_{0}} .

(3.7)

Consider the following auxiliary equation (i.e. $f (t) = | t |^{p - 1}$ with $p > 1 + \frac{α}{2}$ ),

- Δ v + V (x) v = (I_{α} * | v |^{p}) | v |^{p - 2} v, x \in R^{2} .

(3.8)

Note that the following corresponding energy functional is well defined,

I (v) = \frac{1}{2} \int_{R^{2}} [| \nabla v |^{2} + V (x) v^{2}] d x - \frac{1}{2 p} \int_{R^{2}} (I_{α} * | v |^{p}) | v |^{p} d x .

Let

H (v) = (I_{α} * | v |^{p}) | v |^{p} .

Thus,

\begin{aligned} H (\frac{1}{R} v) = \frac{1}{R^{2 p}} [I_{α} * | v |^{p}] | v |^{p} = \frac{1}{R^{2 p}} H (v) . \end{aligned}

(3.9)

Define the following Nehari–Pankov manifold for the energy functional $I$ ,

M := {v \in E ∖ E^{-} : ⟨ I^{'} (v), v ⟩ = ⟨ I^{'} (v), η ⟩ = 0, \forall η \in E^{-}} .

Similar to the proof of Theorem 1.2, we can show that (3.8) has a ground state solution

v \in H^{1} (R^{2}) ∖ {0}

and

c_{1} = inf_{v \in M} I (v) .

(3.10)

Choose $K > 0$ such that

K > {(\frac{2 β_{0} c_{1}}{(2 + α) π})}^{p - 1} .

(3.11)

Consider the following equation

- Δ v + V (x) v = K (I_{α} * | v |^{p}) | v |^{p - 2} v, x \in R^{2} .

(3.12)

Note that the corresponding energy functional

I_{K} (v) = \int_{R^{2}} | \nabla v |^{2} + V (x) v^{2} d x - \frac{K}{2 p} \int_{R^{2}} (I_{α} * | v |^{p}) | v |^{p} d x

I_{K} (v_{K}) = (\frac{1}{2} - \frac{1}{2 p}) K \int_{R^{2}} H (v_{K}) d x = (\frac{1}{2} - \frac{1}{2 p}) K^{\frac{1}{1 - p}} \int_{R^{2}} H (v) d x = K^{\frac{1}{1 - p}} I (v), v \in K,

where

K := {u \in E ∖ {0} : I^{'} (u) = 0} \neq \emptyset .

Similarly, we define

c_{K} = inf_{v \in M_{K}} I_{K} (v),

(3.13)

where

M_{K} := {v \in E ∖ E^{-} : ⟨ I_{K}^{'} (v), v ⟩ = ⟨ I_{K}^{'} (v), η ⟩ = 0, \forall η \in E^{-}} .

Therefore

c_{K} = K^{\frac{1}{1 - p}} c_{1} .

(3.14)

By virtue of Lemmas 2.5 and 2.7, we can show the following lemma.

Lemma 3.4

$(i)$

$I_{K} (v) = max_{\hat{E} (v)} I_{K} (w),$ for every $v \in M_{K}$ ;

(ii)

For any $v \in E ∖ E^{-}$ , there exist unique constant $t > 0$ and $η \in E^{-}$ such that $t v + η \in M_{K} \cap \hat{E} (v)$ .

Lemma 3.5

$inf_{N} Φ (v) \leq c_{K}$ .

Proof.

Suppose that $v = v^{+} + v^{-}$ is a solution of (3.12) satisfying $I_{K} (v) = c_{K}$ . By remark 2.1, we have $v^{+} \neq 0$ . In view of Lemma 2.8, there exist $t > 0$ and $η \in E^{-}$ such that $t v^{+} + η \in \hat{E} (v) \cap N$ . Note that $v \in M_{K}$ , it follows from Lemma 3.4 that

I_{K} (v) \geq I_{K} (t v^{+} + η) .

From (F6), we obtain

c_{K} = I_{K} (v) \geq I_{K} (t v^{+} + η) \geq Φ (t v^{+} + η) \geq inf_{N} Φ .

Then

c_{K} = K^{\frac{1}{1 - p}} c_{1} \geq inf_{N} Φ .

It follows from (3.11) and (3.14) that

c_{K} < \frac{(2 + α) π}{2 β_{0}} .

This, together with Lemmas 2.7, 2.10 and Lemma 3.3 (or Lemma 3.5), we can get the following lemma.

Lemma 3.6

Assume that (V0), $(F 1^{'})$ , (F2), (F3) and (F6) (or $(F 6^{'})$ ) hold. Then

0 < c = inf_{u \in N} Φ (u) < \frac{(2 + α) π}{2 β_{0}} .

(3.15)

4. Critical Case

In this section, we consider the critical exponential growth case and find a ground state solution for problem (1.1). In order to achieve our global, we shall employ the energy comparison method and the concentration compactness argument.

Lemma 4.1
Assume that (V0), $(F 1^{'})$ , (F2) and (F3) hold. Then ${u_{n}}$ satisfying (2.25) is bounded in $E$ .
Proof.
From (F3), (2.8), (2.9) and (2.25), we have
$\begin{aligned} c + o (1) & = Φ (u_{n}) - \frac{1}{2} ⟨ Φ^{'} (u_{n}), u_{n} ⟩ \\ = \frac{1}{2} \int_{R^{2}} [I_{α} * F (u_{n})] [f (u_{n}) u_{n} - F (u_{n})] d x \\ \geq \frac{μ - 1}{2 μ} \int_{R^{2}} [I_{α} * F (u_{n})] f (u_{n}) u_{n} d x \\ \geq \frac{μ - 1}{2} \int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x . \end{aligned}$
(4.1)
It follows from (2.9) and (2.25) that
$o (1) = ⟨ Φ^{'} (u_{n}), u_{n} ⟩ = ‖ u_{n}^{+} ‖^{2} - ‖ u_{n}^{-} ‖^{2} - \int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x,$
(4.2)
and
$o (1) = ⟨ Φ^{'} (u_{n}), u_{n}^{-} ⟩ = - ‖ u_{n}^{-} ‖^{2} - \int_{R^{2}} [I_{α} * F (u_{n})] f (u_{n}) u_{n}^{-} d x .$
(4.3)
By virtue of (4.2) and (4.3), one has
$‖ u_{n}^{+} ‖^{2} - ‖ u_{n}^{-} ‖^{2} \leq \frac{2 μ c}{μ - 1} + o (1) .$
(4.4)
To prove the boundedness of $u_{n}$ , arguing by contradiction, suppose that $‖ u_{n} ‖ \to \infty$ as $n \to \infty$ . Let $v_{n} = u_{n} / ‖ u_{n} ‖$ , then $‖ v_{n} ‖^{2} = 1$ and $‖ v_{n}^{-} ‖^{2} \leq 1$ . In view of $(F 1^{'})$ and $(F 2)$ , there exist $β > β_{0} > 0$ and $C_{1} > 0$ such that
$| f (t) | \leq C_{1} e^{β t^{2}}, \forall t \geq 0.$
(4.5)
Moreover, there exists $C_{2} > 0$ such that
$f^{2} (t) \leq C_{2} f (t) t, when \frac{f (t)}{C_{1}} \leq e^{\frac{1}{4}} .$
(4.6)
Set
$\begin{aligned} A_{n} := {x \in R^{2} : \frac{f (u_{n})}{C_{1}} \leq e^{\frac{1}{4}}}, B_{n} := {x \in R^{2} : \frac{f (u_{n})}{C_{1}} \geq e^{\frac{1}{4}}} . \end{aligned}$
Applying Lemma 2.4 with $t = | v_{n}^{-} |, s = f (u_{n}) / C_{1}$ , it follows form (4.3) that
$\begin{aligned} ‖ v_{n}^{-} ‖^{2} & = - \frac{1}{‖ u_{n} ‖} \int_{R^{2}} [I_{α} * F (u_{n})] f (u_{n}) v_{n}^{-} d x + o (1) \\ \leq \frac{C_{1}}{‖ u_{n} ‖} \int_{R^{2}} [I_{α} * F (u_{n})] \frac{| f (u_{n}) |}{C_{1}} | v_{n}^{-} | d x + o (1) \\ \leq \frac{C_{1}}{‖ u_{n} ‖} {\int_{R^{2}} [I_{α} * F (u_{n})] (e^{| v_{n}^{-} |^{2}} - 1) d x + \frac{1}{2 C_{1}^{2}} \int_{A_{n}} [I_{α} * F (u_{n})] (f (u_{n}))^{2} d x \\ + \int_{B_{n}} [I_{α} * F (u_{n})] \frac{f (u_{n})}{C_{1}} {(\log \frac{f (u_{n})}{C_{1}})}^{\frac{1}{2}} d x} + o (1) \\ := \frac{C_{1}}{‖ u_{n} ‖} (I_{1} + I_{2} + I_{3}) + o (1) . \end{aligned}$
(4.7)
From (2.11), (2.13), (4.1) and Lemma 1.1-(ii), we deduce that
$\begin{aligned} I_{1} & = \int_{R^{2}} [I_{α} * F (u_{n})] (e^{| v_{n}^{-} |^{2}} - 1) d x \\ \leq {[\int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x]}^{\frac{1}{2}} {[\int_{R^{2}} (I_{α} * (e^{| v_{n}^{-} |^{2}} - 1)) (e^{| v_{n}^{-} |^{2}} - 1) d x]}^{\frac{1}{2}} \\ \leq {(\frac{2 c}{μ - 1})}^{\frac{1}{2}} ‖ e^{| v_{n}^{-} |^{2}} - 1 ‖_{4 / (2 + β)} \\ \leq {(\frac{2 c}{μ - 1})}^{\frac{1}{2}} C_{3} . \end{aligned}$
(4.8)
In view of (4.1), (4.5) and (4.6), we have
$\begin{aligned} I_{2} & = \frac{1}{2 C_{1}^{2}} \int_{A_{n}} [I_{α} * F (u_{n})] (f (u_{n}))^{2} d x \\ \leq \frac{C_{2}}{2 C_{1}^{2}} \int_{R^{2}} [I_{α} * F (u_{n})] f (u_{n}) u_{n} d x \\ \leq \frac{μ c}{μ - 1} \frac{C_{2}}{C_{1}^{2}}, \end{aligned}$
(4.9)
and
$\begin{aligned} I_{3} & = \int_{B_{n}} [I_{α} * F (u_{n})] \frac{f (u_{n})}{C_{1}} {(\log \frac{f (u_{n})}{C_{1}})}^{\frac{1}{2}} d x \\ \leq C_{4} \int_{B_{n}} [I_{α} * F (u_{n})] f (u_{n}) u_{n} d x \\ \leq \frac{2 μ c}{μ - 1} C_{4} . \end{aligned}$
(4.10)
Then it follows from (3.15) and (4.7) to (4.10) that $‖ v_{n}^{-} ‖ = o (1)$ . Note that $‖ v_{n}^{+} ‖^{2} + ‖ v_{n}^{-} ‖^{2} = 1$ , together with (4.4) yields that
$‖ v_{n}^{-} ‖^{2} \geq \frac{1}{2} + o (1) .$
This is a contradiction. Thus ${u_{n}}$ is bounded in $E$ .

As in Qin and Tang (2021, Lemma 4.8), we can establish the following auxiliary lemma, which allows us to prove the weak to $weak $ continuity of the Fréchet derivative of the functional $Φ$ for the critical exponential growth case.
Lemma 4.2
Assume that $(F 1^{'})$ , (F2), (F3) and (F5) hold. Let $u_{n} ⇀ \bar{u}$ in $E$ and
$\int_{R^{2}} [I_{α} F (u_{n})] f (u_{n}) u_{n} d x \leq K_{0},$
(4.11)
for some constant $K_{0} > 0$ . Then for every $ϕ \in C_{0}^{\infty} (R^{2})$ ,
$lim_{n \to \infty} \int_{R^{2}} [I_{α} * F (u_{n})] f (u_{n}) ϕ d x = \int_{R^{2}} [I_{α} * F (\bar{u})] f (\bar{u}) ϕ d x .$
(4.12)
Proof of Theorem 1.3. Proof of Theorem 1.3.

By virtue of (2.25), Lemmas 2.11 and 3.6, there exists a bounded Palais–Smale sequence ${u_{n}} \subset N$ for $Φ$ with $c = inf_{N} Φ$ , that is

Φ (u_{n}) \to c, ‖ Φ^{'} (u_{n}) ‖ \to 0, as n \to \infty,

(4.13)

and

‖ u_{n} ‖ \leq C_{1}

. It follows from (F3), (3.15) and (4.1) that

\int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x \leq \int_{R^{2}} [I_{α} * F (u_{n})] f (u_{n}) u_{n} d x \leq C_{2} .

(4.14)

δ := \underset{n \to \infty}{lim sup} sup_{y \in R^{2}} \int_{B_{1} (y)} | u_{n} |^{2} d x = 0,

then by Lions’ concentration compactness principle (Willem, 1996, Lemma 1.21),

u_{n} \to 0

L^{s} (R^{2})

for

2 < s < \infty

. For any given

ε > 0

, set

M_{ε} > M_{0} C_{2} / ε

. Then by

(F 3)

and (4.14), we have

\begin{aligned} \int_{| u_{n} | \geq M_{ε}} [I_{α} * F (u_{n})] F (u_{n}) d x & \leq M_{0} \int_{| u_{n} | \geq M_{ε}} [I_{α} * F (u_{n})] f (u_{n}) d x \\ \leq \frac{M_{0}}{M_{ε}} \int_{| u_{n} | \geq M_{ε}} [I_{α} * F (u_{n})] f (u_{n}) u_{n} d x \\ < ε . \end{aligned}

(4.15)

Choosing

N = N (ε) \in (0, 1),

it follows from

(F 2)

, (2.3), (2.11), (2.13) and (4.14) that

\begin{aligned} \int_{| u_{n} | \leq N_{ε}} [I_{α} * F (u_{n})] F (u_{n}) d x \\ \leq \int_{| u_{n} | \leq N_{ε}} [I_{α} * F (u_{n})] f (u_{n}) u_{n} d x \\ \leq \frac{ε}{C_{1}^{2} γ_{8 / (2 + α)}^{2}} \int_{| u_{n} | \leq N_{ε}} [I_{α} * F (u_{n})] | u_{n} |^{2} d x \\ \leq \frac{ε}{C_{1}^{2} γ_{8 / (2 + α)}^{2}} {(\int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x)}^{\frac{1}{2}} {(\int_{R^{2}} [I_{α} * | u_{n} |^{2}] | u_{n} |^{2} d x)}^{\frac{1}{2}} \\ \leq \frac{\sqrt{C_{2} C_{0}} ε}{C_{1}^{2} γ_{8 / (2 + α)}^{2}} ‖ u_{n} ‖_{8 / (2 + α)}^{2} \\ = o (1) . \end{aligned}

(4.16)

(F 1^{'}), (F 2)

, (2.11), (2.13) and (4.14), we get

\begin{aligned} \int_{N \leq | u_{n} | \leq M} [I_{α} * F (u_{n})] F (u_{n}) d x \\ \leq \int_{N \leq | u_{n} | \leq M} [I_{α} * F (u_{n})] f (u_{n}) u_{n} d x \\ \leq C_{3} \int_{N \leq | u_{n} | \leq M} [I_{α} * F (u_{n})] | u_{n} |^{3} d x \\ \leq C_{3} {(\int_{N \leq | u_{n} | \leq M} [I_{α} * F (u_{n})] F (u_{n}) d x)}^{\frac{1}{2}} {(\int_{N \leq | u_{n} | \leq M} [I_{α} * | u_{n} |^{3}] | u_{n} |^{3} d x)}^{\frac{1}{2}} \\ \leq C_{4} ‖ u_{n} ‖_{12 / (2 + β)}^{3} \\ = o (1) . \end{aligned}

(4.17)

Since

ε > 0

is arbitrary, it follows from (4.15) to (4.17) that

\int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x = o (1) .

(4.18)

From (2.6), (2.7) and (4.18), one has

\begin{aligned} \int_{R^{2}} | \nabla u_{n} |^{2} + V (x) u_{n}^{2} d x & = 2 c + \int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x + o (1) \\ = 2 c + o (1) := \frac{(2 + α) π}{β_{0}} (1 - 3 ϵ) + o (1), \end{aligned}

(4.19)

and

\begin{aligned} \int_{R^{2}} | \nabla u_{n} |^{2} + V (x) u_{n}^{2} d x & = ⟨ Φ^{'} (u_{n}), u_{n} ⟩ + \int_{R^{2}} [I_{α} * F (u_{n})] f (u_{n}) u_{n} d x \\ = \int_{R^{2}} [I_{α} * F (u_{n})] f (u_{n}) u_{n} d x + o (1), \end{aligned}

(4.20)

where

ε > 0

is small enough.

Set

U_{n} = {[(2 + α) π (1 - 3 ϵ)]}^{\frac{1}{2}} \frac{u_{n}}{‖ u_{n} ‖} .

It follows from

(F 1^{'})

that there exist constants

C_{ϵ}, C_{5} > 0

such that

f (t) \leq C_{ϵ} e^{β_{0} (1 + ϵ) t^{2}}, \forall t \geq 0,

and

f (t) \leq C_{5} t when \frac{f (t)}{C_{ϵ}} \leq e^{\frac{1}{4}} .

Applying Lemma 2.4 with

s = f (u_{n}) / C_{ϵ}

and

t = U_{n}

, and combining Lemma 1.1-(ii) with (2.11), (2.13) and (4.18), we have

\begin{aligned} {[(2 + α) π (1 - 3 ϵ)]}^{\frac{1}{2}} ‖ u_{n} ‖ \\ = \int_{R^{2}} [I_{α} * F (u_{n})] f (u_{n}) U_{n}^{-} d x + o (1) \\ = C_{ϵ} \int_{R^{2}} [I_{α} * F (u_{n})] \frac{f (u_{n})}{C_{ϵ}} U_{n}^{-} d x + o (1) \\ \leq C_{ϵ} \int_{R^{2}} [I_{α} * F (u_{n})] (e^{(U_{n}^{-})^{2}} - 1) d x + \frac{C_{ϵ}}{2} \int_{\frac{f (u_{n})}{C_{ϵ}} \leq e^{\frac{1}{4}}} [I_{α} * F (u_{n})] {(\frac{f (u_{n})}{C_{ϵ}})}^{2} d x \\ + C_{ϵ} \int_{\frac{f (u_{n})}{C_{ϵ}} \geq e^{\frac{1}{4}}} (I_{α} * F (u_{n})) \frac{f (u_{n})}{C_{ϵ}} {(\log \frac{f (u_{n})}{C_{ϵ}})}^{\frac{1}{2}} d x + o (1) \\ \leq C_{ϵ} {[\int_{R^{2}} (I_{α} * F (u_{n})) F (u_{n}) d x]}^{\frac{1}{2}} {[\int_{R^{2}} (I_{α} * (e^{(U_{n}^{-})^{2}} - 1)) (e^{(U_{n}^{-})^{2}} - 1) d x]}^{\frac{1}{2}} \\ + \frac{C_{0}}{2 C_{ϵ}} {[\int_{\frac{f (u_{n})}{C_{ϵ}} \leq e^{\frac{1}{4}}} (I_{α} * F (u_{n})) F (u_{n}) d x]}^{\frac{1}{2}} {[\int_{\frac{f (u_{n})}{C_{ϵ}} \leq e^{\frac{1}{4}}} [I_{α} * (f (u_{n}))^{2}] (f (u_{n}))^{2} d x]}^{\frac{1}{2}} \\ + {[β_{0} (1 + ϵ)]}^{\frac{1}{2}} \int_{\frac{f (u_{n})}{C_{ϵ}} \geq e^{\frac{1}{4}}} (I_{α} * F (u_{n})) f (u_{n}) u_{n} d x + o (1) \\ \leq C_{0} C_{ϵ} {[\int_{R^{2}} (I_{α} * F (u_{n})) F (u_{n}) d x]}^{\frac{1}{2}} {[\int_{R^{2}} {(e^{(U_{n}^{-})^{2}} - 1)}^{\frac{4}{2 + α}} d x]}^{\frac{2 + α}{4}} \\ + \frac{C_{0} C_{5}^{4}}{2 C_{ϵ}} {[\int_{\frac{f (u_{n})}{C_{ϵ}} \leq e^{\frac{1}{4}}} (I_{α} * F (u_{n})) F (u_{n}) d x]}^{\frac{1}{2}} {[\int_{\frac{f (u_{n})}{C_{ϵ}} \leq e^{\frac{1}{4}}} | u_{n} |^{8 / (2 + α)} d x]}^{\frac{2 + α}{4}} \\ + {[β_{0} (1 + ϵ)]}^{\frac{1}{2}} \int_{R^{2}} (I_{α} * F (u_{n})) f (u_{n}) u_{n} d x + o (1) \\ \leq C_{0} C_{ϵ} {[\int_{R^{2}} (I_{α} * F (u_{n})) F (u_{n}) d x]}^{\frac{1}{2}} {[\int_{R^{2}} (e^{4 π (1 - 4 ϵ) {(\frac{u_{n}^{-}}{‖ u_{n} ‖})}^{2}} - 1) d x]}^{\frac{2 + α}{4}} \\ + \frac{C_{0} C_{5}^{4}}{2 C_{ϵ}} {[\int_{R^{2}} (I_{α} * F (u_{n})) F (u_{n}) d x]}^{\frac{1}{2}} ‖ u_{n} ‖_{8 / (2 + α)}^{2} \\ + {[β_{0} (1 + ϵ)]}^{\frac{1}{2}} \int_{R^{2}} (I_{α} * F (u_{n})) f (u_{n}) u_{n} d x + o (1) \\ = {[β_{0} (1 + ϵ)]}^{\frac{1}{2}} \int_{R^{2}} (I_{α} * F (u_{n})) f (u_{n}) u_{n} d x + o (1) . \end{aligned}

(4.21)

It follows from (4.19) to (4.21) that

‖ u_{n} ‖^{2} = \frac{(2 + α) (1 - 3 ϵ) (1 + ϵ) π}{β_{0}} .

(4.22)

Choose $q \in (1, 2)$ such that

\frac{(1 + ϵ) (1 - 3 ϵ) q}{1 - ϵ} < 1.

(4.23)

(F 1^{'})

, there exists

C_{6} > 0

such that

| f (t) | \leq C_{6} [e^{β_{0} (1 + ϵ) t^{2}} - 1], \forall | t | \geq 1.

(4.24)

It follows from (4.22) to (4.24) and Lemma 1.1-(ii) that

\begin{aligned} \int_{| u_{n} | \geq 1} | f (u_{n}) |^{\frac{4 q}{2 + α}} d x & \leq C_{6} \int_{| u_{n} | \geq 1} {[e^{β_{0} (1 + ϵ) u_{n}^{2}} - 1]}^{\frac{4 q}{2 + α}} d x \\ \leq C_{6} \int_{R^{2}} [\exp (\frac{4 β_{0} (1 + ϵ) q ‖ u_{n} ‖^{2}}{2 + α} (\frac{u_{n}^{2}}{‖ u_{n} ‖^{2}})) - 1] d x \\ \leq C_{6} \int_{R^{2}} [\exp ((1 + ϵ)^{2} (1 - 3 ϵ) 4 π q (\frac{u_{n}^{2}}{‖ u_{n} ‖^{2}})) - 1] d x \\ \leq C_{7} . \end{aligned}

(4.25)

Set

q^{'} = q / (q - 1)

. Then

q^{'} \in (2, + \infty)

. Using (2.11), (2.13), (4.14), (4.25) and Hölder inequality, we can show

\begin{aligned} \int_{| u_{n} | \geq 1} [I_{α} * F (u_{n})] f (u_{n}) u_{n} d x \\ \leq {[\int_{R^{2}} (I_{α} * F (u_{n})) F (u_{n}) d x]}^{\frac{1}{2}} {[\int_{R^{2}} (I_{α} * f (u_{n}) u_{n} χ_{| u_{n} \geq 1 |}) f (u_{n}) u_{n} χ_{| u_{n} \geq 1 |} d x]}^{\frac{1}{2}} \\ \leq \sqrt{C_{2} C_{0}} {[\int_{| {\tilde{u}}_{n} | \geq 1} | f (u_{n}) |^{\frac{4}{2 + α}} | u_{n} |^{\frac{4}{2 + α}} d x]}^{\frac{2 + α}{4}} \\ \leq \sqrt{C_{2} C_{0}} {[\int_{| u_{n} | \geq 1} | f (u_{n}) |^{\frac{4 q}{2 + α}} d x]}^{\frac{2 + α}{4 q}} {[\int_{R^{2}} | u_{n} |^{\frac{4 q'}{2 + α}} d x]}^{\frac{2 + α}{4 q'}} \\ \leq C_{13} ‖ u_{n} ‖_{\frac{4 q'}{2 + α}} = o (1) . \end{aligned}

(4.26)

By (2.8), (2.9), (2.25), (4.18) and (4.26), one has

\begin{aligned} c + o (1) & = Φ (u_{n}) - \frac{1}{2} ⟨ Φ^{'} (u_{n}), u_{n} ⟩ \\ = \frac{1}{2} \int_{R^{2}} [I_{α} * F (u_{n})] [f (u_{n}) u_{n} - F (u_{n})] d x < ϵ + o (1) . \end{aligned}

(4.27)

This contradiction shows that

δ > 0

Going if necessary to a subsequence, we may assume that there exists ${y_{n}} \subset Z^{2}$ such that $\int_{B_{1 + \sqrt{2}} (y_{n})} | u_{n} |^{2} d x > \frac{δ}{2}$ . Let us define ${\tilde{u}}_{n} (x) = u_{n} (x + y_{n})$ so that

\int_{B_{1 + \sqrt{2}} (0)} | {\tilde{u}}_{n} |^{2} d x > \frac{δ}{2} .

(4.28)

Since

V (x)

is 1-periodic in

x

, we get

‖ {\tilde{u}}_{n} ‖ = ‖ u_{n} ‖

and

Φ ({\tilde{u}}_{n}) \to c, Φ^{'} ({\tilde{u}}_{n}) \to 0, as n \to \infty .

(4.29)

Passing to a subsequence, we have

{\tilde{u}}_{n} ⇀ \tilde{u}

E

{\tilde{u}}_{n} \to \tilde{u}

L_{l o c}^{s} (R^{2})

for all

2 \leq s < \infty

and

{\tilde{u}}_{n} \to \tilde{u}

a.e. on

R^{2}

. Thus, (4.28) implies that

\tilde{u} \neq 0

. It follows from (F3), Lemma 3.3 and Fatou’s lemma that

\begin{aligned} c & = lim_{n \to \infty} [Φ ({\tilde{u}}_{n}) - \frac{1}{2} ⟨ Φ^{'} ({\tilde{u}}_{n}), {\tilde{u}}_{n} ⟩] \\ \geq \underset{n \to \infty}{lim inf} \frac{1}{2} \int_{R^{2}} [I_{α} * F ({\tilde{u}}_{n})] [f ({\tilde{u}}_{n}) {\tilde{u}}_{n} - F ({\tilde{u}}_{n})] d x \\ \geq \frac{1}{2} \int_{R^{2}} \underset{n \to \infty}{lim inf} [I_{α} * F ({\tilde{u}}_{n})] [f ({\tilde{u}}_{n}) {\tilde{u}}_{n} - F ({\tilde{u}}_{n})] d x \\ = Φ (\tilde{u}) - \frac{1}{2} ⟨ Φ^{'} (\tilde{u}), \tilde{u} ⟩ \\ = Φ (\tilde{u}) . \end{aligned}

Note that

\tilde{u} \in N

, we have

c \leq Φ (\tilde{u})

. Thus,

\tilde{u}

is a ground state solution of (1.1).

Footnotes

Authors’ Contributions

All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.

Funding

The authors 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 (No. 12171486), the Science and Technology Innovation Program of Hunan Province (No. 2024RC3021), the Young Backbone Teachers Project of Hunan Province, the Natural Science Foundation for Excellent Young Scholars of Hunan Province (No. 2023JJ20057) and the Fundamental Research Funds for the Central Universities of Central South University (No. 2024ZZTS0121).

Declaration of Conflicting Interests

The authors 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.

ORCID iDs

Lizhen Lai

Dongdong Qin

Qingfang Wu

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