Schrödinger–Poisson system with zero mass and convolution nonlinearity in R 2

Abstract

In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass $\begin{matrix} \{\begin{matrix} - Δ u + ϕ u = (I_{α} * F (u)) f (u), & x \in R^{2}, \\ Δ ϕ = u^{2}, & x \in R^{2}, \end{matrix} \end{matrix}$ where $α \in (0, 2)$ , $I_{α} : R^{2} \to R$ is the Riesz potential, $f \in C (R, R)$ is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.

Keywords

Planar Schrödinger–Poisson system Logarithmic convolution potential zero mass

1. Introduction

This paper is concerned with the following planar Schrödinger–Poisson system with zero mass and convolution nonlinearity: $\begin{matrix} (1.1) & \{\begin{matrix} - Δ u + ϕ u = (I_{α} * F (u)) f (u), & x \in R^{2}, \\ Δ ϕ = u^{2}, & x \in R^{2}, \end{matrix} \end{matrix}$ where $α \in (0, 2)$ , $I_{α} : R^{2} \to R$ is the Riesz potential defined by $\begin{matrix} (1.2) & I_{α} (x) = \frac{Γ (\frac{2 - α}{2})}{Γ (\frac{α}{2}) 2^{α} π | x |^{2 - α}} : = \frac{A_{α}}{| x |^{2 - α}}, x \in R^{2} ∖ {0}, \end{matrix}$ $F (t) = \int_{0}^{t} f (s) d s$ and $f \in C (R, R)$ has subcritical exponential growth in the sense of Trudinger–Moser, that is

$f \in C (R, R)$ and ${lim}_{| t | \to \infty} \frac{| f (t) |}{e^{β t^{2}}} = 0$ for all $β > 0$ .

The solution ϕ of the Poisson equation in (1.1) can be solved by $ϕ = Γ * u^{2}$ , where ∗ is the convolution in $R^{2}$ , Γ is the fundamental solution of the Laplacian given by $Γ (x) = \frac{1}{2 π} ln | x |$ With this formal inversion, system (1.1) is converted into an equivalent integro-differential equation $\begin{matrix} (1.3) & - Δ u + (Γ * u^{2}) u = (I_{α} * F (u)) f (u), x \in R^{2} . \end{matrix}$ Denote by $ϕ_{u} (x) = (Γ * u^{2}) (x)$ . Then at least formally, the energy functional associated with (1.3) is $\begin{matrix} Φ (u) = \frac{1}{2} \int_{R^{2}} | \nabla u |^{2} d x + \frac{1}{4} \int_{R^{2}} ϕ_{u} u^{2} d x - \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] F (u) d x . \end{matrix}$ If u is a critical point of Φ, then the pair $(u, ϕ_{u})$ is a weak solution of (1.1). For the sake of simplicity, in many cases we just say u, instead of $(u, ϕ_{u})$ , is a weak solution of (1.1).

The attention on this kind of problems has increased in the recent period, starting from the paper of Stubbe [28] where firstly a system of this type was proposed. Actually, in dimension three, the system is well known and there is a wide literature on it, see for example [3,4,7,16,26,29,31] for the subcritical growth case; [8,18,35–37] for the critical growth case.

Note that $\int_{R^{2}} ϕ_{u} u^{2} d x$ is not well-defined on the natural Sobolev space $H^{1} (R^{2})$ due to the appearance of the sign-changing and unbounded logarithmic integral kernel $ln | x |$ . This is one of the reasons why planar Schrödinger–Poisson system is definitely less known and studied, although its peculiar features and significant differences arising by a comparison with the three-dimensional Schrödinger–Poisson system are of great interest. To overcome this difficulty, Stubbe [28] introduced a smaller Hilbert space $\begin{matrix} (1.4) & X = {u \in H^{1} (R^{2}) : \int_{R^{2}} [V (x) + ln (2 + | x |)] u^{2} d x < \infty}, \end{matrix}$ where the variational method works well. Based on the variational framework developed by Stubbe [28], Cingolani and Weth [15] considered the following equation $\begin{matrix} (1.5) & - Δ u + V (x) u + ϕ_{u} (x) u = b | u |^{q - 2} u, x \in R^{2}, \end{matrix}$ where $b ⩾ 0$ and $q ⩾ 4$ . In particular, when $V (x)$ is periodic, the existence of ground state solutions was obtained by using the Nehari manifold argument and a surprisingly strong compactness condition for Cerami sequences introduced in [15, Proposition 3.1]. In the case where $V (x)$ is a positive constant, they also obtained the existence of non-radial solutions which have arbitrarily many nodal domains. Generalizations to the case $2 < q < 4$ of existence and multiplicity results were showed in Du and Weth [17] when $V (x)$ is a positive constant. Later, the above results are extended partly in [9–12] to the more general case that $b | u |^{q - 2} u$ in (1.5) is replaced by more general nonlinearity $g (x, u)$ , where $V ≢ 0$ and $V (x)$ is periodic or axially symmetric. Recently, Wen–Chen–Radulescu [33] studied the following equation with zero mass $\begin{matrix} (1.6) & - Δ u + ϕ_{u} (x) u = a (x) g (u), x \in R^{2}, \end{matrix}$ where $a \in C^{1} (R^{2}, R)$ is axially symmetric and g is superquadratic at the origin, growing no more than a power. Using a minimizing argument, they obtained the existence of ground state solutions under some suitable monotonicity assumptions on a and g. This zero mass case was also considered by Chen–Tang [13] if g has critical exponential growth in the sense of Trudinger–Moser.

Recently, using the non-Nehari manifold method (see Tang [32,32]), Chen–Pan [14] and Shen [27] proved that the following equation with a non-negative potential and a convolution nonlinearity: $\begin{matrix} (1.7) & - Δ u + V (x) u + ϕ u = (I_{α} * F (u)) f (u), x \in R^{2} \end{matrix}$ has a ground state solution if $f \in C (R, R)$ has critical exponential growth and satisfies the following monotonicity condition:

$f (t) / | t |$ is nondecreasing on $(- \infty, 0) \cup (0, + \infty)$

and other technical assumptions. Moreover, by minimizing the energy functional on the Nehari–Pohoz˘aev manifold (see), Shen [27] also obtained the existence of a ground state solution for (1.7) with

V (x) \equiv 1

if f has polynomial growth and satisfies the following monotonicity condition:

there exists a constant $p > 1$ such that $[4 p f (t) t - (2 + α) F (t)] / t | t |^{(4 p + α - 2) / 4}$ is nondecreasing on both $(- \infty, 0)$ and $(0, + \infty)$ .

It is worth pointing out that the approach used in [14,27] is only valid for (1.7) with

V (x) ⩾ 0

and

{lim inf}_{| x | \to \infty} V (x) > 0

, it does not work any more for (1.7) with

V (x) \equiv 0

, namely the zero mass case. Now a natural question is whether (1.7) has a nontrivial solution when

V (x) \equiv 0

. To the best of our knowledge, there is no related result in this case.

In the present paper, we will give a positive answer to the above question. Different from those in the previous papers [14,27], we shall use some new ideas and skills to establish the existence of nontrivial solutions and ground state solutions for (1.1). Before stating our results, we first introduce the following assumptions:

$F (t) = o (| t |^{(α + 6) / 4})$ as $| t | \to 0$ ;

${inf}_{t \neq 0} \frac{F (t)}{| t |^{(α + 6) / 4}} > - \infty$ ;

there exists a constant $μ > (α + 6) / 4$ such that $\begin{matrix} f (t) t - μ F (t) ⩾ 0, \forall t \in R; \end{matrix}$

$[4 f (t) t - (2 + α) F (t)] / t | t |^{(2 + α) / 4}$ is nondecreasing on both $(- \infty, 0)$ and $(0, + \infty)$ .

Note that (F4) and (F5) are weaker than the assumption (M2) used in [27]. Our results are as follows.

Theorem 1.1.
Assume that (F1)–(F4) hold. Then system ( 1.1 ) has a nontrivial axially symmetric solution.
Theorem 1.2.
Assume that (F1)–(F3) and (F5) hold. Then system ( 1.1 ) has a ground state solution.

The paper is organized as follows. In Section 2, we give the variational setting and preliminaries. We give the proofs of Theorem 1.1 and Theorem 1.2 in Sections 3 and Section 4, respectively.

Throughout the paper, we make use of the following notations:
$L^{s} (R^{2})$ ( $1 ⩽ s < \infty$ ) denotes the Lebesgue space with the norm $‖ u ‖_{s} = {(\int_{R^{2}} | u |^{s} d x)}^{1 / s}$ ;

For any $x \in R^{2}$ and $r > 0$ , $B_{r} (x) : = {y \in R^{2} : | y - x | < r}$ and $B_{r} = B_{r} (0)$ ;

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

2. Variational framework and preliminaries

As in [11], we define the following symmetric bilinear forms $\begin{array}{l} (2.1) & (u, v) \mapsto A_{1} (u, v) = \frac{1}{2 π} \int_{R^{2}} \int_{R^{2}} ln (2 + | x - y |) u (x) v (y) d x d y, \\ (2.2) & (u, v) \mapsto A_{2} (u, v) = \frac{1}{2 π} \int_{R^{2}} \int_{R^{2}} ln (1 + \frac{2}{| x - y |}) u (x) v (y) d x d y, \\ (2.3) & (u, v) \mapsto A_{0} (u, v) = A_{1} (u, v) - A_{2} (u, v) = \frac{1}{2 π} \int_{R^{2}} \int_{R^{2}} ln | x - y | u (x) v (y) d x d y, \end{array}$ where the definition is restricted, in each case, to measurable functions $u, v : R^{2} \to R$ such that the corresponding double integral is well defined in Lebesgue sense. Noting that $0 ⩽ ln (1 + r) ⩽ r$ for $r ⩾ 0$ , it follows from the Hardy–Littlewood–Sobolev inequality (see [21] or [23, page 98]) that $\begin{matrix} (2.4) & | A_{2} (u, v) | ⩽ \frac{1}{π} \int_{R^{2}} \int_{R^{2}} \frac{1}{| x - y |} | u (x) v (y) | d x d y ⩽ K_{0} ‖ u ‖_{4 / 3} ‖ v ‖_{4 / 3} \end{matrix}$ with a constant $C_{1} > 0$ . Using (2.1), (2.2) and (2.3), we define the functionals as follows: $\begin{array}{l} I_{1} : H^{1} (R^{2}) \to [0, \infty], I_{1} (u) = A_{1} (u^{2}, u^{2}) = \frac{1}{2 π} \int_{R^{2}} \int_{R^{2}} ln (2 + | x - y |) u^{2} (x) u^{2} (y) d x d y, \\ \begin{matrix} I_{2} : L^{8 / 3} (R^{2}) \to [0, \infty), I_{2} (u) & = A_{2} (u^{2}, u^{2}) \\ = \frac{1}{2 π} \int_{R^{2}} \int_{R^{2}} ln (1 + \frac{2}{| x - y |}) u^{2} (x) u^{2} (y) d x d y, \end{matrix} \\ I_{0} : H^{1} (R^{2}) \to R \cup {\infty}, I_{0} (u) = A_{0} (u^{2}, u^{2}) = \frac{1}{2 π} \int_{R^{2}} \int_{R^{2}} ln | x - y | u^{2} (x) u^{2} (y) d x d y . \end{array}$ Here $I_{2}$ only takes finite values on $L^{8 / 3} (R^{2})$ . Indeed, (2.4) implies $\begin{matrix} (2.5) & | I_{2} (u) | ⩽ K_{0} ‖ u ‖_{8 / 3}^{4}, \forall u \in L^{8 / 3} (R^{2}) . \end{matrix}$

We define, for any measurable function $u : R^{2} \to R$ $\begin{matrix} (2.6) & ‖ u ‖_{*}^{2} = \int_{R^{2}} ln (2 + | x |) u^{2} (x) d x \in [0, \infty] . \end{matrix}$ Let $\begin{matrix} ‖ u ‖^{2} : = \int_{R^{2}} (| \nabla u |^{2} + u^{2}) d x, \forall u \in X . \end{matrix}$ Following the idea of Chen and Tang [11], we study (1.1) in a natural constraint $E : = X \cap H_{as}^{1}$ with $\begin{matrix} H_{as}^{1} = {u \in H^{1} (R^{2}) : u (x) : = u (x_{1}, x_{2}) = u (| x_{1} |, | x_{2} |), \forall x \in R^{2}}, \end{matrix}$ since critical points of the functional Φ restricted to E are true critical points in X, where X is defined in (1.4). Then $‖ u ‖_{E} : = {(‖ u ‖^{2} + ‖ u ‖_{*}^{2})}^{1 / 2}$ is a norm on X. Moreover, it follows from Rellich’s Theorem (see [25, Theorem XIII.65]) that the embedding $X ↪ L^{s} (R^{2})$ is compact for $s \in [2, \infty)$ . Noting that $\begin{matrix} (2.7) & ln (2 + | x - y |) ⩽ ln (2 + | x | + | y |) ⩽ ln (2 + | x |) + ln (2 + | y |), \forall x, y \in R^{2}, \end{matrix}$ we have $\begin{array}{l} | A_{1} (u v, w z) | & ⩽ \frac{1}{2 π} \int_{R^{2}} \int_{R^{2}} [ln (2 + | x |) + ln (2 + | y |)] | u (x) v (x) | | w (y) z (y) | d x d y \\ (2.8) & ⩽ \frac{1}{2 π} ‖ u ‖_{*} ‖ v ‖_{*} ‖ w ‖_{2} ‖ z ‖_{2} + \frac{1}{2 π} ‖ u ‖_{2} ‖ v ‖_{2} ‖ w ‖_{*} ‖ z ‖_{*}, \forall u, v, w, z \in X . \end{array}$

According to [15, Lemma 2.2], we have $I_{0}$ , $I_{1}$ and $I_{2}$ are of class $C^{1}$ on X, $I_{0} = I_{1} - I_{2}$ and $\begin{matrix} (2.9) & ⟨ I_{i}^{'} (u), v ⟩ = 4 A_{i} (u^{2}, u v), \forall u, v \in X, i = 0, 1, 2 . \end{matrix}$

The following inequality about the convolution $A_{1}$ on E comes from [11], which is very crucial for the proofs of our theorems.

Lemma 2.1 ([11]).

The following inequality holds $\begin{matrix} (2.10) & A_{1} (u^{2}, v^{2}) ⩾ \frac{1}{8 π} ‖ u ‖_{2}^{2} ‖ v ‖_{*}^{2}, \forall u, v \in E . \end{matrix}$

From Lemma 2.4, we have the following corollary.

Corollary 2.2.
The following inequality holds $\begin{matrix} (2.11) & I_{1} (u) ⩾ \frac{1}{8 π} ‖ u ‖_{2}^{2} ‖ u ‖_{*}^{2}, \forall u \in E . \end{matrix}$

To give the variational framework, we now present the following three lemmas.
Lemma 2.3 (Hardy–Littlewood–Sobolev inequality [22]).

Let $s, r > 1$ and $0 < μ < 2$ with $\frac{1}{s} + \frac{1}{r} = \frac{4 - μ}{2}$ , $g \in L^{s} (R^{2})$ and $h \in L^{r} (R^{2})$ . There exists a sharp constant $C (μ, s, r)$ , independent of g, h, such that $\begin{matrix} (2.12) & \int_{R^{2}} (I_{2 - μ} * g) h d x ⩽ C (μ, s, r) ‖ g ‖_{s} ‖ h ‖_{r} . \end{matrix}$ In particular, $\begin{array}{l} (2.13) & \int_{R^{2}} (I_{α} * g) h d x ⩽ C_{0} ‖ g ‖_{4 / (2 + α)} ‖ h ‖_{4 / (2 + α)}, \\ (2.14) & \int_{R^{2}} (I_{(2 + α) / 2} * g) h d x ⩽ C_{1} ‖ g ‖_{1} ‖ h ‖_{4 / (2 + α)}, \end{array}$ where $C_{0} : = C (2 - α, 4 / (2 + α), 4 / (2 + α))$ and $C_{1} : = C ((2 - α) / 2, 1, 4 / (2 + α))$ .

Lemma 2.4 (Cauchy–Schwarz type inequality [1] and [24, Section 5]).

For $g, h \in L_{loc}^{1} (R^{2})$ , there holds $\begin{matrix} (2.15) & \int_{R^{2}} (I_{α} * | g |) | h | d x ⩽ {[\int_{R^{2}} (I_{α} * | g |) | g | d x \int_{R^{2}} (I_{α} * | h |) | h | d x]}^{\frac{1}{2}} . \end{matrix}$

Lemma 2.5 (Trudinger–Moser inequality [2,5,6]).

i) If $α > 0$ and $u \in H^{1} (R^{2})$ , then $\begin{matrix} \int_{R^{2}} (e^{α u^{2}} - 1) d x < \infty; \end{matrix}$ ii) if $u \in H^{1} (R^{2})$ , $‖ \nabla u ‖_{2}^{2} ⩽ 1$ , $‖ u ‖_{2} ⩽ M < \infty$ , and $α < 4 π$ , then there exists a constant $C (M, α)$ , which depends only on M and α, such that $\begin{matrix} \int_{R^{2}} (e^{α u^{2}} - 1) d x ⩽ C (M, α) . \end{matrix}$

By virtue of (F1) and (F2), we can choose $β > 0$ such that for any given $ε > 0$ , there exists $C_{ε} > 0$ such that $\begin{matrix} (2.16) & | f (t) | ⩽ ε | t |^{α / 2} + C_{ε} (e^{β t^{2}} - 1), \forall t \in R . \end{matrix}$ Consequently, $\begin{matrix} (2.17) & | F (t) | ⩽ ε | t |^{(2 + α) / 2} + C_{ε} | t | (e^{β t^{2}} - 1), \forall t \in R . \end{matrix}$ According to (2.17), Lemma 2.5 and Lemma 2.3, we can demonstrate that the energy functional $\begin{matrix} (2.18) & Φ (u) = \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{1}{4} I_{0} (u) - \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] F (u) d x \end{matrix}$ associated with (1.1) is of class $C^{1} (X, R)$ , and $\begin{matrix} (2.19) & ⟨ Φ^{'} (u), v ⟩ = \int_{R^{2}} \nabla u \cdot \nabla v d x + A_{0} (u^{2}, u v) - \int_{R^{2}} [I_{α} * F (u)] f (u) v d x, \forall u, v \in X . \end{matrix}$ Hence, the solutions of (1.1) are the critical points of the reduced functional Φ.

Let $G = {id, - id, {as}_{1}, {as}_{2}}$ , where ${as}_{i}$ is the reflection at the coordinate axis $x_{i}$ for $i = 1, 2$ . Then G is a transform group. The action of G on $H^{1} (R^{2})$ is defined by $\begin{matrix} g u (x) : = u (g^{- 1} x) . \end{matrix}$ Then it is easy to verify that for any $g_{1}, g_{2}, g \in G$ and $u \in H^{1} (R^{2})$ , $\begin{array}{l} (id) u = u, (g_{1} g_{2}) u = g_{1} (g_{2} u), u \mapsto g u is linear, ‖ g u ‖ = ‖ u ‖; \\ Fix (G) : = {u \in H^{1} (R^{2}) : g u = u, \forall g \in G} = H_{as}^{1} (R^{2}) . \end{array}$ Under (A1), (F1) and (F2), we have $Φ (g u) = Φ (u)$ for all $g \in G$ and $u \in H^{1} (R^{2})$ . Therefore, in view of [34, Theorem 1.28], we derive the following lemma, which shows that E is a natural constraint of X.

Lemma 2.6.
Assume that (F1) and (F2) hold. If u is a critical point of Φ restricted to E, then u is a critical point of Φ on X.

3. Nontrivial solutions

To prove the existence of nontrivial solutions, we shall use

Similar to [17, Lemma 3.2], we will apply the general minimax principle [20, Proposition 2.8], which is a somewhat stronger variant of [34, Theorem 2.8], to obtain a Cerami sequence for the functional Φ with $J (u_{n}) \to 0$ . This idea goes back to Jeanjean [19].

Let us define the functional: $\begin{array}{rcl} J (u) & = & 2 ⟨ Φ^{'} (u), u ⟩ - P (u) \\ = & 2 ‖ \nabla u ‖_{2}^{2} + I_{0} (u) - \frac{1}{8 π} ‖ u ‖_{2}^{4} \\ (3.1) & - \frac{1}{2} \int_{R^{2}} (I_{α} * F (u)) [4 f (u) u - (α + 2) F (u)] d x, \forall u \in E . \end{array}$

Lemma 3.1.
Assume that (F1)–(F3) hold. Then there exists a sequence ${u_{n}} \subset E$ satisfying $\begin{matrix} (3.2) & Φ (u_{n}) \to c > 0, {‖ Φ^{'} (u_{n}) ‖}_{E^{}} (1 + ‖ u_{n} ‖_{E}) \to 0 and J (u_{n}) \to 0, \end{matrix}$ where* $\begin{matrix} c : = inf_{γ \in Γ} max_{t \in [0, 1]} Φ (γ (t)), Γ : = {γ \in C ([0, 1], E) : γ (0) = 0, Φ (γ (1)) < 0} . \end{matrix}$
Proof.
First, we prove that $0 < c < \infty$ . In view of the Gagliardo–Nirenberg inequality, one has $\begin{matrix} (3.3) & ‖ u ‖_{s}^{s} ⩽ C_{s} ‖ u ‖_{2}^{2} ‖ \nabla u ‖_{2}^{s - 2} for s > 2, \end{matrix}$ where $C_{s} > 0$ is a constant determined by s. Then it follows from (3.3) and Young inequality that $\begin{array}{l} (3.4) & ‖ u ‖_{8 / 3}^{4} ⩽ C_{8 / 3}^{3 / 2} ‖ u ‖_{2}^{3} ‖ \nabla u ‖_{2} ⩽ C_{8 / 3}^{3 / 2} (\frac{2}{3} ‖ u ‖_{2}^{9 / 2} + \frac{1}{3} ‖ \nabla u ‖_{2}^{3}), \\ (3.5) & \begin{matrix} ‖ u ‖_{(α + 6) / (α + 2)}^{(α + 6) / 2} & ⩽ C_{(α + 6) / (α + 2)}^{(α + 2) / 2} ‖ u ‖_{2}^{α + 2} ‖ \nabla u ‖_{2}^{(2 - α) / 2} \\ ⩽ C_{(α + 6) / (α + 2)}^{(α + 2) / 2} (\frac{α + 2}{4} ‖ u ‖_{2}^{4} + \frac{2 - α}{4} ‖ \nabla u ‖_{2}^{2}) \end{matrix} \end{array}$ and $\begin{matrix} (3.6) & ‖ u ‖_{6}^{6} ⩽ C_{6} ‖ u ‖_{2}^{2} ‖ \nabla u ‖_{2}^{4} ⩽ C_{6} (\frac{1}{3} ‖ u ‖_{2}^{6} + \frac{2}{3} ‖ \nabla u ‖_{2}^{6}) . \end{matrix}$ By (F1) and (F2), fix $β > 0$ , for any $ε > 0$ , there exists a constant $C_{ε} > 0$ such that $\begin{matrix} (3.7) & | F (t) | ⩽ ε | t |^{(α + 6) / 4} + C_{ε} (e^{β t^{2}} - 1) | t |^{3}, \forall t \in R . \end{matrix}$ Then from (3.7), the Hölder inequality and Lemma 2.5-ii), we have $\begin{array}{l} \int_{R^{2}} {| F (u) |}^{4 / (2 + α)} d x \\ ⩽ \int_{R^{2}} {[ε | u |^{(α + 6) / 4} + C_{ε} (e^{β u^{2}} - 1) | u |^{3}]}^{4 / (2 + α)} d x \\ ⩽ 2 \int_{R^{2}} [ε^{4 / (2 + α)} | u |^{(α + 6) / (α + 2)} + C_{ε}^{4 / (2 + α)} {(e^{β u^{2}} - 1)}^{4 / (2 + α)} | u |^{12 / (2 + α)}] d x \\ ⩽ 2 {ε^{4 / (2 + α)} ‖ u ‖_{(α + 6) / (α + 2)}^{(α + 6) / (α + 2)} + C_{ε}^{4 / (2 + α)} {[\int_{R^{2}} {(e^{β u^{2}} - 1)}^{\frac{4}{α}} d x]}^{\frac{α}{2 + α}} ‖ u ‖_{6}^{12 / (2 + α)}} \\ ⩽ 2 {ε^{4 / (2 + α)} ‖ u ‖_{(α + 6) / (α + 2)}^{(α + 6) / (α + 2)} + C_{ε}^{4 / (2 + α)} {[\int_{R^{2}} (e^{4 β α^{- 1} ‖ u ‖^{2} {(u / ‖ u ‖)}^{2}} - 1) d x]}^{\frac{α}{2 + α}} ‖ u ‖_{6}^{12 / (2 + α)}} \\ ⩽ 2 {ε^{4 / (2 + α)} ‖ u ‖_{(α + 6) / (α + 2)}^{(α + 6) / (α + 2)} + C_{ε}^{4 / (2 + α)} {[C (1, 2 π)]}^{α / (2 + α)} ‖ u ‖_{6}^{12 / (2 + α)}}, \\ (3.8) & \forall ‖ u ‖ ⩽ \sqrt{α π / 2 β} . \end{array}$ Then (2.13) and (3.8) give $\begin{array}{l} \int_{R^{2}} [I_{α} * F (u)] F (u) d x \\ ⩽ C_{0} {(\int_{R^{2}} {| F (u) |}^{4 / (2 + α)} d x)}^{\frac{2 + α}{2}} \\ ⩽ C_{0} {2 ε^{4 / (2 + α)} ‖ u ‖_{(α + 6) / (α + 2)}^{(α + 6) / (α + 2)} + 2 C_{ε}^{4 / (2 + α)} {[C (1, 2 π)]}^{α / (2 + α)} ‖ u ‖_{6}^{12 / (2 + α)}}^{\frac{2 + α}{2}} \\ (3.9) & ⩽ 8 C_{0} ε^{2} ‖ u ‖_{(α + 6) / (α + 2)}^{(α + 6) / 2} + 8 C_{ε}^{2} {[C (1, 2 π)]}^{α / 2} ‖ u ‖_{6}^{6}, \forall ‖ u ‖ ⩽ \sqrt{α π / 2 β} . \end{array}$ Setting $\begin{matrix} ε = ε_{0}^{1 / 2}, ε_{0} : = \frac{ln 2}{256 π C_{0}} C_{(α + 6) / (α + 2)}^{- (α + 2) / 2} \end{matrix}$ in (3.7)–(3.9), then (3.5), (3.6) and (3.9) yield $\begin{array}{l} \int_{R^{2}} [I_{α} * F (u)] F (u) d x \\ ⩽ \frac{ln 2}{64 π} C_{(α + 6) / (α + 2)}^{- (α + 2) / 2} ‖ u ‖_{(α + 6) / (α + 2)}^{(α + 6) / 2} + C_{1} ‖ u ‖_{6}^{6} \\ (3.10) & ⩽ \frac{ln 2}{64 π} (‖ u ‖_{2}^{4} + ‖ \nabla u ‖_{2}^{2}) - C_{2} (‖ u ‖_{2}^{6} + ‖ \nabla u ‖_{2}^{6}), \forall ‖ u ‖ ⩽ \sqrt{α π / 2 β} . \end{array}$ Hence, it follows from (2.5), (2.18), (2.11), (3.4) and (3.10) that $\begin{array}{l} Φ (u) & = \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{1}{4} [I_{1} (u) - I_{2} (u)] - \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] F (u) d x \\ ⩾ \frac{1}{2} ‖ \nabla u ‖_{2}^{2} + \frac{ln 2}{32 π} ‖ u ‖_{2}^{4} - \frac{K_{0}}{8 π} ‖ u ‖_{8 / 3}^{4} - \frac{ln 2}{64 π} (‖ u ‖_{2}^{4} + ‖ \nabla u ‖_{2}^{2}) - C_{2} (‖ u ‖_{2}^{6} + ‖ \nabla u ‖_{2}^{6}) \\ ⩾ \frac{ln 2}{64 π} (‖ u ‖_{2}^{4} + ‖ \nabla u ‖_{2}^{2}) - C_{3} (‖ u ‖_{2}^{9 / 2} + ‖ \nabla u ‖_{2}^{3}) - C_{2} (‖ u ‖_{2}^{6} + ‖ \nabla u ‖_{2}^{6}), \\ (3.11) & \forall u \in E, ‖ u ‖ ⩽ \sqrt{α π / 2 β} . \end{array}$ Let $ρ (u) : = ‖ \nabla u ‖_{2}^{2} + ‖ u ‖_{2}^{4}$ . Then (3.11) gives $\begin{matrix} Φ (u) ⩾ \frac{ln 2}{64 π} ρ (u) - C_{3} (ρ^{9 / 8} (u) + ρ^{3 / 2} (u)) - C_{2} (ρ^{3 / 2} (u) + ρ^{3} (u)), \forall u \in E, ρ (u) ⩽ α π / 2 β . \end{matrix}$ Therefore, there exist $κ_{0} > 0$ and $0 < ρ_{0} ⩽ α π / 2 β$ such that $\begin{matrix} (3.12) & Φ (u) ⩾ κ_{0}, \forall u \in S : = {u \in E : ‖ \nabla u ‖_{2}^{2} + ‖ u ‖_{2}^{4} = ρ_{0}} . \end{matrix}$ Note that for any fixed $u \in E$ with $u \neq 0$ , $\begin{array}{rcl} I_{0} (t^{2} u_{t}) & = & \frac{t^{4}}{2 π} \int_{R^{2}} \int_{R^{2}} ln | x - y | u^{2} (t x) u^{2} (t y) d (t x) d (t y) \\ = & \frac{t^{4}}{2 π} \int_{R^{2}} \int_{R^{2}} (ln | t x - t y | - ln t) u^{2} (t x) u^{2} (t y) d (t x) d (t y) \\ = & \frac{t^{4}}{2 π} \int_{R^{2}} \int_{R^{2}} (ln | x - y | - ln t) u^{2} (x) u^{2} (y) d x d y \\ (3.13) & = & t^{4} I_{0} (u) - \frac{t^{4} ln t}{2 π} ‖ u ‖_{2}^{4}, \forall t > 0 . \end{array}$ Combining (2.18) with (3.13), one has $\begin{array}{rcl} Φ (t^{2} u_{t}) & = & \frac{t^{4}}{2} ‖ \nabla u ‖_{2}^{2} + \frac{t^{4}}{4} I_{0} (u) - \frac{t^{4} ln t}{8 π} ‖ u ‖_{2}^{4} \\ (3.14) & - \frac{1}{2 t^{2 + α}} \int_{R^{2}} I_{α} * F (t^{2} u) F (t^{2} u) d x, \forall t > 0 . \end{array}$ By (F1)–(F3), there exists a constant $Λ > 0$ such that $\begin{matrix} (3.15) & F (t^{2} u) ⩾ - Λ {| t^{2} u |}^{(α + 6) / 4}, \forall t > 0 . \end{matrix}$ Then, it follows from (3.14) and (3.15) that $\begin{matrix} (3.16) & lim_{t \to 0} Φ (t^{2} u_{t}) = 0, sup_{t > 0} Φ (t^{2} u_{t}) < \infty, Φ (t^{2} u_{t}) \to - \infty as t \to + \infty . \end{matrix}$ Now, we choose $T > 0$ large enough such that $w : = T^{2} u_{T} \in {u \in E : ‖ \nabla u ‖_{2}^{2} + ‖ u ‖_{2}^{4} > ρ_{0}}$ and $Φ (w) < 0$ . Let $γ_{T} (t) = {(t T)}^{2} u_{t T}$ for $t \in [0, 1]$ . Then $γ_{T} \in C ([0, 1], E)$ such that $γ_{T} (0) = 0$ , $Φ (γ_{T} (1)) < 0$ and ${max}_{t \in [0, 1]} Φ (γ_{T} (t)) < \infty$ . This shows that $Γ \neq \emptyset$ and $c < \infty$ . By the continuity of $γ (t)$ and the intermediate value theorem, there exists $t_{γ} \in (0, 1)$ such that $ρ (γ (t_{γ})) = ρ_{0}$ . Jointly with (3.12), we have $\begin{matrix} sup_{t \in [0, 1]} Φ (γ (t)) ⩾ Φ (γ (t_{γ})) ⩾ κ_{0} > 0, \forall γ \in Γ, \end{matrix}$ which yields $\begin{matrix} (3.17) & \infty > c = inf_{γ \in Γ} max_{t \in [0, 1]} Φ (γ (t)) ⩾ κ_{0} > 0 . \end{matrix}$ By using the same way as [9, Lemma 2.2], we can deduce that there exists a sequence ${u_{n}}$ satisfying (3.2). □
Lemma 3.2.
Assume that (F1)–(F4) hold. Let ${u_{n}} \subset E$ be a sequence satisfying ( 3.2 ). Then ${u_{n}}$ is bounded in E.
Proof.
First, we prove that ${u_{n}}$ is bounded in $H^{1} (R^{2})$ . By (F4), (2.18), (3.1) and (3.2), we have $\begin{array}{l} c + o (1) & = Φ (u_{n}) - \frac{1}{4} J (u_{n}) \\ (3.18) & = \frac{1}{32 π} ‖ u_{n} ‖_{2}^{4} + \frac{1}{2} \int_{R^{2}} [I_{α} * F (u_{n})] [f (u_{n}) u_{n} - \frac{α + 6}{4} F (u_{n})] d x \\ = \frac{1}{32 π} ‖ u_{n} ‖_{2}^{4} + \frac{1}{2} \int_{R^{2}} [I_{α} * F (u_{n})] [f (u_{n}) u_{n} - μ F (u_{n})] d x \\ + \frac{4 μ - (α + 6)}{8} \int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x \\ (3.19) & ⩾ \frac{1}{32 π} ‖ u_{n} ‖_{2}^{4} + \frac{4 μ - (α + 6)}{8} \int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x . \end{array}$ Then it follows from (2.5), (2.18), (3.2) and (3.19) that $\begin{array}{l} \frac{1}{2} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{1}{4} I_{1} (u_{n}) & = Φ (u_{n}) + \frac{1}{4} I_{2} (u_{n}) + \frac{1}{2} \int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x \\ ⩽ [1 + \frac{4}{4 μ - (α + 6)}] c + \frac{K_{0}}{8 π} C_{8 / 3}^{3 / 2} ‖ u_{n} ‖_{2}^{3} ‖ \nabla u_{n} ‖_{2} + o (1) \\ (3.20) & ⩽ C_{4} ‖ \nabla u_{n} ‖_{2} + [1 + \frac{4}{4 μ - (α + 6)}] c + o (1), \end{array}$ which implies $\begin{matrix} (3.21) & ‖ \nabla u_{n} ‖_{2}^{2} ⩽ C_{5}, I_{1} (u_{n}) ⩽ C_{6} . \end{matrix}$ From (3.19) and (3.21), we then derive that $‖ u_{n} ‖ ⩽ M_{0}$ for some constant $M_{0} > 0$ . To end the proof of this lemma, it remains to show the boundedness of ${‖ u_{n} ‖_{}}$ . If $δ_{0} : = {lim sup}_{n \to \infty} ‖ u_{n} ‖_{2} = 0$ , the from the Gagliardo–Nirenberg inequality: $\begin{matrix} (3.22) & ‖ u_{n} ‖_{s}^{s} ⩽ C_{s} ‖ u_{n} ‖_{2}^{2} ‖ \nabla u_{n} ‖_{2}^{s - 2} \end{matrix}$ we derive that $u_{n} \to 0$ in $L^{s} (R^{2})$ for $s ⩾ 2$ . Then (2.5) yields that $\begin{matrix} (3.23) & | I_{2} (u_{n}) | ⩽ K_{0} ‖ u_{n} ‖_{8 / 3}^{4} = o (1) . \end{matrix}$ Choosing $β \in (0, 1 / M_{0}^{2})$ , from (F1) and (F2), we know that there exists a constant $M_{1} > 0$ such that $\begin{matrix} (3.24) & f (t) t ⩽ | t |^{(α + 6) / 4} + M_{1} (e^{β t^{2}} - 1) | t |^{3}, \forall t \in R . \end{matrix}$ It follows from (3.24) and Lemma 2.5-ii) that $\begin{array}{l} \int_{R^{2}} {| f (u_{n}) u_{n} |}^{4 / (2 + α)} d x \\ ⩽ \int_{R^{2}} {[| u_{n} |^{(α + 6) / 4} + M_{1} (e^{β u_{n}^{2}} - 1) | u_{n} |^{3}]}^{4 / (2 + α)} d x \\ ⩽ 2 \int_{R^{2}} [| u_{n} |^{(α + 6) / (α + 2)} + M_{1}^{4 / (2 + α)} {(e^{β u_{n}^{2}} - 1)}^{4 / (2 + α)} | u_{n} |^{12 / (2 + α)}] d x \\ ⩽ 2 {‖ u_{n} ‖_{(α + 6) / (α + 2)}^{(α + 6) / (α + 2)} + M_{1}^{4 / (2 + α)} {[\int_{R^{2}} {(e^{β u_{n}^{2}} - 1)}^{\frac{4}{α}} d x]}^{\frac{α}{2 + α}} ‖ u_{n} ‖_{6}^{12 / (2 + α)}} \\ (3.25) & ⩽ 2 [‖ u_{n} ‖_{(α + 6) / (α + 2)}^{(α + 6) / (α + 2)} + C_{7} ‖ u_{n} ‖_{6}^{12 / (2 + α)}] \\ (3.26) & = o (1) . \end{array}$ Then (2.13), (2.15), (3.19) and (3.26) give $\begin{array}{l} | \int_{R^{2}} [I_{α} F (u_{n})] f (u_{n}) u_{n} d x | \\ ⩽ {(\int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x)}^{\frac{1}{2}} {(\int_{R^{2}} [I_{α} * f (u_{n}) u_{n}] f (u_{n}) u_{n} d x)}^{\frac{1}{2}} \\ (3.27) & ⩽ C_{8} {‖ f (u_{n}) u_{n} ‖}_{4 / (2 + α)} = o (1) . \end{array}$ Similarly, we have $\begin{matrix} (3.28) & | \int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x | = o (1) . \end{matrix}$ Thus it follows from (2.18), (2.19), (3.2), (3.17), (3.27) and (3.28) that $\begin{array}{l} κ_{0} + o (1) & ⩽ c + o (1) = Φ (u_{n}) - \frac{1}{2} ⟨ Φ^{'} (u_{n}), u_{n} ⟩ \\ = - \frac{1}{4} I_{1} (u_{n}) + \frac{1}{4} I_{2} (u_{n}) + \frac{1}{2} \int_{R^{2}} [I_{α} * F (u)] [f (u_{n}) u_{n} - F (u_{n})] d x \\ (3.29) & ⩽ o (1) . \end{array}$ This contradiction shows that $δ_{0} > 0$ . Jointly with Corollary 2.2, we have ${‖ u_{n} ‖_{}}$ is bounded, and so ${u_{n}}$ is bounded in E. □
Proof of Theorem 1.1.
Applying Lemmas 3.1 and 3.2, we deduce that there exists a sequence ${u_{n}} \subset E$ satisfying (3.2) and ${u_{n}}$ is bounded in E. We may thus assume, passing to a subsequence again if necessary, that $u_{n} ⇀ \bar{u}$ in E, $u_{n} \to \bar{u}$ in $L^{s} (R^{2})$ , $s \in [2, \infty)$ and $u_{n} \to \bar{u}$ a.e. on $R^{2}$ . Hence it follows from (2.4) that $\begin{matrix} (3.30) & A_{2} (u_{n}^{2}, u_{n} (u_{n} - \bar{u})) = o (1), A_{2} ({\bar{u}}^{2}, u_{n} (u_{n} - \bar{u})) = o (1) . \end{matrix}$ As in the proof of (3.26), we can deduce that $\begin{matrix} (3.31) & \int_{R^{2}} {| f (u_{n}) (u_{n} - \bar{u}) |}^{4 / (2 + α)} d x = o (1) . \end{matrix}$ It follows from (2.13), (2.15), (3.31), Lemma 2.5 ii) and Hölder inequality that $\begin{array}{l} \int_{R^{2}} [I_{α} F (u_{n})] f (u_{n}) (u_{n} - \bar{u}) d x \\ ⩽ {(\int_{R^{2}} [I_{α} * F (u_{n})] F (u_{n}) d x)}^{\frac{1}{2}} {(\int_{R^{2}} [I_{α} * f (u_{n}) (u_{n} - u_{n})] f (u_{n}) (u_{n} - \bar{u}) d x)}^{\frac{1}{2}} \\ ⩽ C_{9} {‖ f (u_{n}) (u_{n} - \bar{u}) ‖}_{4 / (2 + α)} \\ (3.32) & = o (1) . \end{array}$ Since ${‖ u_{n} ‖_{}}$ and ${‖ u_{n} ‖_{2}}$ are bounded, it follows that $\begin{array}{l} \int_{R^{2}} ln (2 + | y |) | \bar{u} (y) | | u_{n} (y) - \bar{u} (y) | d y \\ ⩽ ln (2 + | R |) ‖ \bar{u} ‖_{2} ‖ u_{n} - \bar{u} ‖_{2} + ‖ u_{n} - \bar{u} ‖_{} {[\int_{R^{2} ∖ B_{R} (0)} ln (2 + | y |) {\bar{u}}^{2} (y) d y]}^{1 / 2} \\ = o_{n} (1) + o_{R} (1), as n \to \infty, R \to \infty, \end{array}$ which implies $\begin{matrix} (3.33) & \int_{R^{2}} ln (2 + | y |) | \bar{u} (y) | | u_{n} (y) - \bar{u} (y) | d y = o (1) . \end{matrix}$ By (2.8), (3.33) and the fact that $‖ u_{n} - \bar{u} ‖_{2} \to 0$ , we have $\begin{array}{l} | A_{1} (u_{n}^{2}, \bar{u} (u_{n} - \bar{u})) | \\ ⩽ \int_{R^{2}} \int_{R^{2}} [ln (2 + | x |) + ln (2 + | y |)] u_{n}^{2} (x) | \bar{u} (y) | | u_{n} (y) - \bar{u} (y) | d x d y \\ ⩽ ‖ u_{n} ‖_{}^{2} ‖ \bar{u} ‖_{2} ‖ u_{n} - \bar{u} ‖_{2} + ‖ u_{n} ‖_{2}^{2} \int_{R^{2}} ln (2 + | y |) | \bar{u} (y) | | u_{n} (y) - \bar{u} (y) | d y \\ (3.34) & = o (1) . \end{array}$ From (2.4), we also have $\begin{matrix} (3.35) & A_{2} (u_{n}^{2}, \bar{u} (u_{n} - \bar{u})) = o (1), A_{2} (u_{n}^{2} - {\bar{u}}^{2}, {\bar{u}}^{2}) = o (1) . \end{matrix}$ From (2.19), (2.10), (3.2), (3.30), (3.32), (3.34) and (3.35), one has $\begin{array}{l} o (1) & = ⟨ Φ^{'} (u_{n}), u_{n} - \bar{u} ⟩ \\ = ‖ \nabla u_{n} ‖_{2}^{2} - ‖ \nabla \bar{u} ‖_{2}^{2} + A_{1} (u_{n}^{2}, {(u_{n} - \bar{u})}^{2}) + A_{1} (u_{n}^{2}, \bar{u} (u_{n} - \bar{u})) \\ - A_{1} ({\bar{u}}^{2}, \bar{u} (u_{n} - \bar{u})) - A_{2} (u_{n}^{2}, u_{n} (u_{n} - \bar{u})) + A_{2} ({\bar{u}}^{2}, \bar{u} (u_{n} - \bar{u})) \\ - \int_{R^{2}} [I_{α} F (u_{n})] f (u_{n}) (u_{n} - \bar{u}) d x \\ = ‖ u_{n} - \bar{u} ‖^{2} + A_{1} (u_{n}^{2}, {(u_{n} - \bar{u})}^{2}) + o (1) \\ ⩾ ‖ \nabla u_{n} ‖_{2}^{2} - ‖ \nabla \bar{u} ‖_{2}^{2} + \frac{1}{8 π} ‖ u_{n} ‖_{2}^{2} ‖ u_{n} - \bar{u} ‖_{*}^{2} + o (1), \end{array}$ which, together with $δ_{0} = {lim sup}_{n \to \infty} ‖ u_{n} ‖_{2} > 0$ , implies that $u_{n} \to \bar{u}$ in E. Hence, $0 < c = {lim}_{n \to \infty} Φ (u_{n}) = Φ (\bar{u})$ and $Φ^{'} (\bar{u}) = 0$ . □

4. Ground state solutions

First, we establish some key inequalities.

Lemma 4.1.

Assume that (F1) and (F5) hold. Then for all $s > 0$ and $t \in R$ , $\begin{matrix} (4.1) & g (s, t) : = \frac{4}{s^{(2 + α) / 2}} F (s^{2} t) + (1 - s^{2}) [4 f (t) t - (2 + α) F (t)] - 4 F (t) ⩾ 0 . \end{matrix}$

Proof.

It is evident that $g (s, 0) ⩾ 0$ for $s > 0$ . For $t \neq 0$ , it follows from (F4) that $\begin{array}{rcl} \frac{d}{d s} g (s, t) & = & \frac{2 s | t |^{(6 + α) / 4}}{2} [\frac{4 f (s^{2} t) s^{2} t - (2 + α) F (s^{2} t)}{| s^{2} t |^{(6 + α) / 4}} - \frac{4 f (t) t - (2 + α) F (t)}{| t |^{(6 + α) / 4}}] \\ \{\begin{matrix} ⩾ 0, & s ⩾ 1, \\ ⩽ 0, & 0 < s < 1, \end{matrix} \end{array}$ which implies that $g (s, t) ⩾ g (1, t) = 0$ for all $s > 0$ and $t \in (- \infty, 0) \cup (0, + \infty)$ . □

Lemma 4.2.

Assume that (F1) and (F5) hold. Then $\begin{matrix} (4.2) & \frac{F (t)}{t | t |^{(2 + α) / 4}} is nondecreasing on both (- \infty, 0) and (0, + \infty) . \end{matrix}$

Proof.

By (F1) and (4.1), one has $\begin{matrix} (4.3) & lim_{s \to 0} g (s, t) = 4 f (t) t - (6 + α) F (t) ⩾ 0, \forall t \in R . \end{matrix}$ Note that $\begin{matrix} (4.4) & \frac{d}{d t} (\frac{F (t)}{t | t |^{(2 + α) / 4}}) = \frac{1}{4 | t |^{(10 + α) / 4}} [4 f (t) t - (6 + α) F (t)] . \end{matrix}$ Thus, the conclusion follows from (4.3) and (4.4). □

Lemma 4.3.

Assume that (F1), (F2) and (F5) hold. Then $\begin{array}{rcl} h (t, u) & : = & \int_{R^{2}} {\frac{1}{t^{2 + α}} (I_{α} * F (t^{2} u)) F (t^{2} u) - (I_{α} * F (u)) F (u) \\ + \frac{1 - t^{4}}{4} (I_{α} * F (u)) [4 f (u) u - (2 + α) F (u)]} d x \\ (4.5) & ⩾ & 0, \forall t > 0, u \in H^{1} (R^{2}) . \end{array}$

Proof.

Note that (F1) and (4.2) imply $\begin{matrix} (4.6) & I_{α} * (\frac{F (t^{2} u)}{| t |^{(6 + α) / 2}}) - I_{α} * F (u) \{\begin{matrix} ⩾ 0, & t ⩾ 1, \\ ⩽ 0, & 0 < t < 1 . \end{matrix} \end{matrix}$ By (F1), (F5) and (4.6), we have $\begin{array}{rcl} \frac{d}{d t} h (t, u) & = & \int_{R^{2}} {\frac{1}{t^{3 + α}} (I_{α} * F (t^{2} u)) [4 f (t^{2} u) t^{2} u - (α + 2) F (t^{2} u)] \\ - t^{3} (I_{α} * F (u)) [4 f (u) u - (2 + α) F (u)]} \\ = & t^{3} \int_{R^{2}} | u |^{(6 + α) / 4} {(I_{α} * \frac{F (t^{2} u)}{t^{(6 + α) / 2}}) \frac{4 f (t^{2} u) t^{2} u - (2 + α) F (t^{2} u)}{| t^{2} u |^{(6 + α) / 4}} \\ - (I_{α} * F (u)) \frac{4 f (u) u - (2 + α) F (u)}{| u |^{(6 + α) / 4}}} d x \\ (4.7) & \{\begin{matrix} ⩾ 0, & t ⩾ 1, \\ ⩽ 0, & 0 < t < 1, \end{matrix} \end{array}$ which implies $h (t, u) ⩾ h (1, u) = 0$ for $t > 0$ and $u \in H^{1} (R^{2})$ . This shows that (4.5) holds. □

Lemma 4.4.

Assume that (F1), (F2) and (F5) hold. Then $\begin{matrix} (4.8) & Φ (u) ⩾ Φ (t^{2} u_{t}) + \frac{1 - t^{4}}{4} J (u) + \frac{1 - t^{4} + 4 t^{4} ln t}{32 π} ‖ u ‖_{2}^{4}, \forall u \in E, t > 0 . \end{matrix}$

Proof.

Thus, by (2.18), (3.14), (3.1) and (4.5), one has $\begin{array}{l} Φ (u) - Φ (t^{2} u_{t}) \\ = \frac{1 - t^{4}}{2} ‖ \nabla u ‖_{2}^{2} + \frac{1 - t^{4}}{4} I_{0} (u) + \frac{t^{4} ln t}{8 π} ‖ u ‖_{2}^{4} \\ + \frac{1}{2} \int_{R^{2}} [\frac{1}{t^{2 + α}} (I_{α} * F (t^{2} u)) F (t^{2} u) - (I_{α} * F (u)) F (u)] d x \\ = \frac{1 - t^{4}}{4} J (u) + \frac{1 - t^{4} + 4 t^{4} ln t}{32 π} ‖ u ‖_{2}^{4} \\ + \frac{1}{2} \int_{R^{2}} {\frac{1}{t^{2 + α}} (I_{α} * F (t^{2} u)) F (t^{2} u) - (I_{α} * F (u)) F (u) \\ + \frac{1 - t^{4}}{4} (I_{α} * F (u)) [4 f (u) u - (2 + α) F (u)]} d x \\ ⩾ \frac{1 - t^{4}}{4} J (u) + \frac{1 - t^{4} + 4 t^{4} ln t}{32 π} ‖ u ‖_{2}^{4}, \forall u \in E, t > 0 . \end{array}$ This shows that (4.8) holds. □

By Lemma 4.4, we can get the following corollary.

Corollary 4.5.

Assume that (F1), (F2) and (F5) hold. Then $\begin{matrix} (4.9) & Φ (u) = max_{t > 0} Φ (t^{2} u_{t}), \forall u \in M . \end{matrix}$

Next, we show $M \neq \emptyset$ by seeking for the saddle point structure of Φ with respect to the fibres ${t^{2} u_{t} : t > 0} \subset E$ , $u \in E ∖ {0}$ .

Lemma 4.6.

Assume that (F1), (F2) and (F5) hold. Then for any $u \in E ∖ {0}$ , there exists a constant $t_{u} > 0$ such that $t_{u}^{2} u_{t_{u}} \in M$ .

Proof.

Let $u \in E ∖ {0}$ be fixed and define a function $ζ (t) : = Φ (t^{2} u_{t})$ on $(0, \infty)$ . Clearly, by (3.14) and (3.1), we have $\begin{array}{rcl} ζ^{'} (t) = 0 & \Leftrightarrow & 2 t^{3} ‖ \nabla u ‖_{2}^{2} + t^{3} I_{0} (u) - \frac{4 t^{3} ln t + t^{3}}{8 π} ‖ u ‖_{2}^{4} \\ - \frac{1}{2 t^{α + 3}} \int_{R^{2}} (I_{α} * F (t^{2} u)) [4 f (t^{2} u) t^{2} u - (α + 2) F (t^{2} u)] d x = 0 \\ \Leftrightarrow & J (t^{2} u_{t}) = 0 \Leftrightarrow t^{2} u_{t} \in M, \forall t > 0 . \end{array}$ Then it follows from (F5) and (4.2) that $\begin{array}{l} \frac{ζ^{'} (t)}{t^{3}} & ⩾ 2 ‖ \nabla u ‖_{2}^{2} + I_{0} (u) - \frac{4 ln t + 1}{8 π} ‖ u ‖_{2}^{4} \\ (4.10) & - \frac{1}{2} \int_{R^{2}} (I_{α} * F (u)) [4 f (u) u - (α + 2) F (u)] d x, \forall 0 < t < 1 \end{array}$ and $\begin{array}{l} \frac{ζ^{'} (t)}{t^{3}} & ⩽ 2 ‖ \nabla u ‖_{2}^{2} + I_{0} (u) - \frac{4 ln t + 1}{8 π} ‖ u ‖_{2}^{4} \\ (4.11) & - \frac{1}{2} \int_{R^{2}} (I_{α} * F (u)) [4 f (u) u - (α + 2) F (u)] d x, \forall t > 1 . \end{array}$ Plainly, (4.10) and (4.11) imply that $ζ^{'} (t) > 0$ for $0 < t < 1$ small and $ζ^{'} (t) < 0$ for $t > 1$ large. Therefore there exists $t_{u} > 0$ such that $ζ^{'} (t_{u}) = 0$ and $t_{u}^{2} u_{t_{u}} \in M$ . □

Combining Corollary 4.5 and Lemma 4.6, we have the following lemma.

Lemma 4.7.

Assume that (F1), (F2) and (F5) hold. Then $\begin{matrix} (4.12) & inf_{u \in M} Φ (u) : = m = inf_{u \in E ∖ {0}} max_{t > 0} Φ (t^{2} u_{t}) . \end{matrix}$

Lemma 4.8.

Assume that (F1), (F2) and (F5) hold. Then

there exists $ϵ > 0$ such that $‖ \nabla u ‖_{2}^{2} + ‖ u ‖_{2}^{4} ⩾ ϵ$ , $\forall u \in M$ ;

$m = {inf}_{u \in M} Φ (u) > 0$ .

Proof.

(i) Arguing as in (3.7)–(3.10), we can prove that $\begin{array}{l} \int_{R^{2}} (I_{α} * F (u)) [4 f (u) u - (α + 2) F (u)] d x \\ ⩽ \frac{ln 2}{8 π} (‖ u ‖_{2}^{4} + ‖ \nabla u ‖_{2}^{2}) - C_{1} (‖ u ‖_{2}^{9 / 2} + ‖ \nabla u ‖_{2}^{3}) - C_{2} (‖ u ‖_{2}^{6} + ‖ \nabla u ‖_{2}^{6}), \\ (4.13) & \forall u \in E, ‖ u ‖ ⩽ \sqrt{α π / 2 β} . \end{array}$ Since $J (u) = 0$ for $u \in M$ , if $ρ (u) = ‖ \nabla u ‖_{2}^{2} + ‖ u ‖_{2}^{4} ⩽ α π / 2 β$ , then it follows from (2.5), (2.11), (3.1), (3.4), (3.5), (3.6), (4.13) and the Sobolev embedding theorem that $\begin{array}{l} 2 ‖ \nabla u ‖_{2}^{2} + \frac{ln 2}{8 π} ‖ u ‖_{2}^{4} \\ ⩽ 2 ‖ \nabla u ‖_{2}^{2} + I_{1} (u) \\ = I_{2} (u) + \frac{1}{8 π} ‖ u ‖_{2}^{4} + \frac{1}{2} \int_{R^{2}} (I_{α} * F (u)) [4 f (u) u - (α + 2) F (u)] d x \\ ⩽ C_{3} (\frac{2}{3} ‖ u ‖_{2}^{9 / 2} + \frac{1}{3} ‖ \nabla u ‖_{2}^{3}) + \frac{1}{8 π} ‖ u ‖_{2}^{4} + \frac{ln 2}{16 π} (‖ u ‖_{2}^{4} + ‖ \nabla u ‖_{2}^{2}) \\ + C_{1} (‖ u ‖_{2}^{9 / 2} + ‖ \nabla u ‖_{2}^{3}) + C_{2} (‖ u ‖_{2}^{6} + ‖ \nabla u ‖_{2}^{6}), \forall u \in M with ρ (u) ⩽ α π / 2 β, \end{array}$ which implies that there exists $\bar{ρ} > 0$ such that $\begin{matrix} (4.14) & ρ (u) ⩾ \bar{ρ}, \forall u \in M with ρ (u) ⩽ α π / 2 β . \end{matrix}$ This shows $\begin{matrix} inf_{u \in M} (‖ \nabla u ‖_{2}^{2} + ‖ u ‖_{2}^{4}) ⩾ min {\bar{ρ}, α π / 2 β} : = ϵ . \end{matrix}$

(ii) Let ${u_{n}} \subset M$ be such that $Φ (u_{n}) \to m$ . We have the following two cases:

Case 1) ${inf}_{n \in N} ‖ u_{n} ‖_{2} : = ϱ_{1} > 0$ . Then (4.8) implies $\begin{matrix} m + o (1) = Φ (u_{n}) ⩾ \frac{1}{32 π} ‖ u_{n} ‖_{2}^{4} ⩾ \frac{1}{32 π} ϱ_{1}^{4} . \end{matrix}$

Case 2) ${inf}_{n \in N} ‖ u_{n} ‖_{2} = 0$ . Passing to a subsequence, we can assume that $‖ u_{n} ‖_{2} \to 0$ and $‖ \nabla u_{n} ‖_{2}^{2} ⩾ \frac{ϵ}{2}$ for all $n \in N$ . Let $t_{n} = ‖ \nabla u_{n} ‖_{2}^{- 1 / 2}$ . Then $t_{n} ⩽ {(2 / ϵ)}^{1 / 4}$ . By (2.4) and (3.3), we have $\begin{matrix} (4.15) & t_{n}^{4} I_{2} (u_{n}) ⩽ K_{0} t_{n}^{4} ‖ u_{n} ‖_{8 / 3}^{4} ⩽ C_{4} t_{n}^{4} ‖ u_{n} ‖_{2}^{3} ‖ \nabla u_{n} ‖_{2} = C_{4} t_{n}^{2} ‖ u_{n} ‖_{2}^{3} = o (1) . \end{matrix}$ Noting that ${t_{n}^{2} u_{n}}$ is bounded in $H^{1} (R^{2})$ , arguing as in the proof of (3.25), we then deduce that $\begin{array}{l} (4.16) & \int_{R^{2}} {| F (t_{n}^{2} u_{n}) |}^{4 / (2 + α)} d x ⩽ 2 [{‖ t_{n}^{2} u_{n} ‖}_{(α + 6) / (α + 2)}^{(α + 6) / (α + 2)} + C_{4} {‖ t_{n}^{2} u_{n} ‖}_{6}^{12 / (2 + α)}] . \end{array}$ Thus (2.13), (3.3) and (4.16) yield $\begin{array}{l} \frac{1}{2 t_{n}^{2 + α}} \int_{R^{2}} [I_{α} * F (t_{n}^{2} u_{n})] F (t_{n}^{2} u_{n}) d x \\ ⩽ \frac{C_{0}}{2 t_{n}^{2 + α}} {(\int_{R^{2}} {| F (t_{n}^{2} u_{n}) |}^{4 / (2 + α)} d x)}^{\frac{2 + α}{2}} \\ ⩽ \frac{C_{5}}{t_{n}^{2 + α}} [{‖ t_{n}^{2} u_{n} ‖}_{(α + 6) / (α + 2)}^{(α + 6) / 2} + {‖ t_{n}^{2} u_{n} ‖}_{6}^{6}] \\ ⩽ \frac{C_{5}}{t_{n}^{2 + α}} [C_{(α + 6) / (α + 2)}^{(α + 2) / 2} t_{n}^{α + 6} ‖ u_{n} ‖_{2}^{α + 2} ‖ \nabla u_{n} ‖_{2}^{(2 - α) / 2} + C_{6} t_{n}^{12} ‖ u_{n} ‖_{2}^{2} ‖ \nabla u_{n} ‖_{2}^{4}] \\ = C_{6} (t_{n}^{α + 2} ‖ u_{n} ‖_{2}^{α + 2} + t_{n}^{2 - α} ‖ u_{n} ‖_{2}^{2}) \\ (4.17) & = o (1) . \end{array}$ Since $J (u_{n}) = 0$ , it follows from (3.14), (4.15) (4.17) and Corollary 4.5 that $\begin{array}{l} m + o (1) & = Φ (u_{n}) ⩾ Φ (t_{n}^{2} {(u_{n})}_{t_{n}}) \\ = \frac{t_{n}^{4}}{2} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{t_{n}^{4}}{4} [I_{1} (u_{n}) - I_{2} (u_{n})] - \frac{t_{n}^{4} ln t_{n}}{8 π} ‖ u_{n} ‖_{2}^{4} \\ - \frac{1}{2 t_{n}^{2 + α}} \int_{R^{2}} [I_{α} * F (t_{n}^{2} u_{n})] F (t_{n}^{2} u_{n}) d x \\ (4.18) & ⩾ \frac{t_{n}^{4}}{2} ‖ \nabla u_{n} ‖_{2}^{2} + o (1) = \frac{1}{2} + o (1) . \end{array}$ Cases 1) and 2) show that $m = {inf}_{u \in M} Φ (u) > 0$ . □

In the following, we will show that the Cerami sequence ${u_{n}}$ obtained in Lemma 2.4 is a minimizing sequence for Φ. This idea goes back to Tang [30,32].

Lemma 4.9.

Assume that (F1)–(F3) and (F6) hold. Then there exists a sequence ${u_{n}} \subset E$ satisfying $\begin{matrix} (4.19) & Φ (u_{n}) \to c \in (0, m], {‖ Φ^{'} (u_{n}) ‖}_{E^{*}} (1 + ‖ u_{n} ‖_{E}) \to 0, J (u_{n}) \to 0 . \end{matrix}$

Proof.

In view of Lemmas 4.7 and 4.8, we choose $v_{k} \in M$ such that $\begin{matrix} (4.20) & 0 < m ⩽ Φ (v_{k}) < m + \frac{1}{k}, k \in N . \end{matrix}$ Applying Lemma 3.1, there exists a sequence ${u_{n}}_{n \in N} \subset E$ satisfying (3.2). Now we choose $T_{k} > 0$ such that $Φ (T_{k}^{2} {(v_{k})}_{T_{k}}) < 0$ . Let $γ_{k} (t) = {(t T_{k})}^{2} {(v_{k})}_{t T_{k}}$ for $t \in [0, 1]$ . Then $γ_{k} \in Γ$ . Using (3.12), it is easy to see that $c \in [κ_{0}, {sup}_{t > 0} Φ (t^{2} {(v_{k})}_{t})]$ . Moreover, Corollary 4.5 yields that $Φ (v_{k}) = {sup}_{t > 0} Φ (t^{2} {(v_{k})}_{t})$ . Hence, by (4.20), one has $\begin{matrix} (4.21) & c ⩽ sup_{t > 0} Φ (t^{2} {(v_{k})}_{t}) < m + \frac{1}{k}, k \in N . \end{matrix}$ Let $k \to \infty$ . Then we deduce the conclusion by Lemma 3.1. □

Proof of Theorem 1.2.

In view of Lemma 4.9, there exists a sequence ${u_{n}} \subset E$ satisfying (4.19). Then, it follows from (4.2) and (4.19) that $\begin{matrix} (4.22) & c + o (1) = Φ (u_{n}) - \frac{1}{4} J (u_{n}) ⩾ \frac{1}{32 π} ‖ u_{n} ‖_{2}^{4} . \end{matrix}$ This shows that ${‖ u_{n} ‖_{2}}$ is bounded. Now, we prove that ${‖ \nabla u_{n} ‖_{2}}$ is also bounded. Arguing by contradiction, suppose that $‖ \nabla u_{n} ‖_{2} \to \infty$ . Let ${\hat{t}}_{n} = {(2 \sqrt{m} / ‖ \nabla u_{n} ‖_{2})}^{1 / 2}$ . Then $t_{n} \to 0$ . Similarly as in the proofs of (4.15) and (4.17), we can get $\begin{matrix} (4.23) & {\hat{t}}_{n}^{4} I_{2} (u_{n}) = o (1) \end{matrix}$ and $\begin{matrix} (4.24) & \frac{1}{2 {\hat{t}}_{n}^{2 + α}} \int_{R^{2}} [I_{α} * F ({\hat{t}}_{n}^{2} u_{n})] F ({\hat{t}}_{n}^{2} u_{n}) d x = o (1) . \end{matrix}$ Thus, it follows from (2.5), (3.14), (4.3), (4.19), (4.23) and (4.24) that $\begin{array}{l} m + o (1) & ⩾ c + o (1) = Φ (u_{n}) \\ ⩾ Φ ({\hat{t}}_{n}^{2} {(u_{n})}_{{\hat{t}}_{n}}) + \frac{1 - {\hat{t}}_{n}^{4}}{4} J (u_{n}) \\ = \frac{{\hat{t}}_{n}^{4}}{2} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{{\hat{t}}_{n}^{2}}{2} ‖ u_{n} ‖_{2}^{2} + \frac{{\hat{t}}_{n}^{4}}{4} [I_{1} (u_{n}) - I_{2} (u_{n})] - \frac{{\hat{t}}_{n}^{4} ln {\hat{t}}_{n}}{8 π} ‖ u_{n} ‖_{2}^{4} \\ - \frac{1}{2 {\hat{t}}_{n}^{2 + α}} \int_{R^{2}} [I_{α} * F ({\hat{t}}_{n}^{2} u_{n})] F ({\hat{t}}_{n}^{2} u_{n}) d x \\ ⩾ \frac{{\hat{t}}_{n}^{4}}{2} ‖ \nabla u_{n} ‖_{2}^{2} + o (1) = 2 m + o (1) . \end{array}$ This contradiction implies that ${‖ \nabla u_{n} ‖_{2}}$ is also bounded, and so ${u_{n}}$ is bounded in $H^{1} (R^{2})$ . By the same argument as in the first part of the proof of Theorem 1.1, we conclude that there exists $\bar{u} \in E ∖ {0}$ such that $Φ^{'} (\bar{u}) = 0$ and $Φ (\bar{u}) = c \in (0, m]$ . Moreover, since $\bar{u} \in M$ , we have $Φ (\bar{u}) ⩾ m$ . This shows that $\bar{u} \in E$ is a ground state solution for (1.1) with $Φ (\bar{u}) = m = {inf}_{M} Φ > 0$ . □

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Schrödinger–Poisson system with zero mass and convolution nonlinearity in R 2

Abstract

Keywords

1. Introduction

Lemma 2.1 ([11]).

Corollary 2.2. The following inequality holds (2.11) I 1 ( u ) ⩾ 1 8 π ‖ u ‖ 2 2 ‖ u ‖ ∗ 2 , ∀ u ∈ E . To give the variational framework, we now present the following three lemmas. Lemma 2.3 (Hardy–Littlewood–Sobolev inequality [22]).

Lemma 2.4 (Cauchy–Schwarz type inequality [1] and [24, Section 5]).

Lemma 2.5 (Trudinger–Moser inequality [2,5,6]).

Lemma 2.6. Assume that (F1) and (F2) hold. If u is a critical point of Φ restricted to E, then u is a critical point of Φ on X. 3. Nontrivial solutions

References

Lemma 2.6.
Assume that (F1) and (F2) hold. If u is a critical point of Φ restricted to E, then u is a critical point of Φ on X.

3. Nontrivial solutions