1 Introduction

This paper is devoted to the following logarithmic Choquard equation

$$\begin{aligned} -\, \Delta u + a(x)u + \lambda (\log |\cdot |*|u|{^2})u = b|u|^{p-2}u, ~~ \text{ in }~ {\mathbb {R}}^n, \end{aligned}$$
(1.1)

where \(\lambda>0, p>2, a \in L^\infty ({\mathbb {R}}^n).\)

For \(n=2\), Eq. (1.1) comes from the planar Schrödinger–Poisson system. In [8], Cingolani and Weth proved the existence of high-energy solutions for \(p\ge 4\). The case \(2<p<4\) was considered by Du and Weth [10]. For \(a>0\) and \(b\equiv 0\), the asymptotic decay of the unique positive, radially symmetric solution to (1.1), was also studied in [4]. For earlier related results, please see [18]. On the other hand, in the three-dimensional case, the following Choquard equation

$$\begin{aligned} -\, \Delta u + a(x)u=\lambda (|\cdot |^{2-n}*|u|{^2})u+b|u|^{p-2}u, ~~ \text{ in }~{\mathbb {R}}^3 \end{aligned}$$

arises in physics, see [12, 13, 15]. This three-dimensional Choquard equation has been studied extensively, see, for example, [1, 3, 5, 9, 11, 14, 17, 19,20,21,22] and the references therein.

In this paper, we are interested in the logarithmic Choquard equation (1.1) in the high-dimensional case \(n\ge 3\). To the best of our knowledge, the only result related to (1.1) for \(n\ge 3\) can be seen in [2], where the existence of a ground state solution is obtained as \(b=0\). In this paper, we will consider the case \(b\ge 0\) and prove the existence of a ground state solution and a mountain-pass solution. We will follow the line of [8] and omit the proofs of some lemmas to make the paper concise. Notice that in this case Eq. (1.1) is also related to the logarithmic Hardy–Littlewood–Sobolev inequality [7].

We assume the following throughout this paper.

\(({\mathcal {A}})\). a is \({\mathbb {Z}}^n-\)periodic, \(a\in L^{\infty }({\mathbb {R}}^n)\cap C^{1}({\mathbb {R}}^n), \ \inf \nolimits _{x\in {{\mathbb {R}}^n}}a(x)>0,\) and there exists a constant \( c_0>0\) such that \(a(x)\pm \frac{1}{n}(x\cdot \nabla a(x))\ge c_0.\)

Define for the measurable function \(u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) that

$$\begin{aligned}|u|^{2}_{*} := \int _{{{\mathbb {R}}^n}}\log (1+|x|)u^{2}(x)\hbox {d}x.\end{aligned}$$

Define the Hilbert space \(X:=\{u\in H^{1}({\mathbb {R}}^n):|u|_{*}^{2}<\infty \}\) with the norm given by \(\Vert u\Vert _{X}^{2}:=|u|_{*}^{2}+\Vert u\Vert ^{2},\) where \(\Vert u\Vert =(\int _{{\mathbb {R}}^n}|\nabla u|^2+a(x)|u|^2 \hbox {d}x)^{\frac{1}{2}}\) is the norm in \(H^1({\mathbb {R}}^n).\)

Without loss of generality, we assume \(\lambda =1\). The energy functional corresponding to (1.1) is defined by \(I:X\rightarrow {\mathbb {R}},\)

$$\begin{aligned} I(u):= & {} \frac{1}{2}\int _{{\mathbb {R}}^n}(|\nabla u|^{2}+a(x)u^{2}(x))\hbox {d}x+\frac{1}{4}\int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^n}\log (|x-y|)u^{2}(x)u^{2}(y)\hbox {d}x\hbox {d}y\\&-\,\frac{b}{p}\int _{{\mathbb {R}}^n}|u|^{p}\hbox {d}x. \end{aligned}$$

By using the assumption \(({\mathcal {A}})\), we know that I is invariant under translations with respect to \({\mathbb {Z}}^n\).

To continue, we first define the mountain-pass value

$$\begin{aligned} c_{mp}=\mathop {\inf }\limits _{\gamma \in \Gamma }\mathop {\sup }\limits _{t\in [0,1]} {I(\gamma (t))},~~~~\Gamma =\{\gamma \in C([0,1],X):\gamma (0)=0,I(\gamma (1))<0\}. \end{aligned}$$

Theorem 1.1

Let \(2<p<2^*:=\frac{2n}{n-2}, n\ge 3, b\ge 0,\lambda =1\). Assume that \(({\mathcal {A}})\) holds. Then

  1. (i)

    Equation (1.1) admits a mountain-pass solution, i.e., a solution \(u\in X{\setminus }{\{0\}}\), such that \(I(u)=c_{mp}\). Moreover, \(c_{mp}>0.\)

  2. (ii)

    Equation (1.1) admits a ground state solution, i.e., a solution \(u\in X{\setminus }{\{0\}}\), such that \(I(u)=c_{g}\), where \(c_{g}:=\inf \{I(v):v\in X{\setminus }{\{0\}},I'(v)=0\}\) is the ground state energy of Eq. (1.1).

2 Preliminaries

We introduce the scalar product

$$\begin{aligned} \langle u,v\rangle = \int _{{{\mathbb {R}}^n}}(\nabla u\nabla v+a(x)uv)\hbox {d}x,\ u,v\in H^{1}({\mathbb {R}}^n). \end{aligned}$$

For any measurable functional \(u,v:{\mathbb {R}}^n\rightarrow {\mathbb {R}},\) as in [8], we define the symmetric bilinear form \(B_{0}(u,v)=B_{1}(u,v)-B_{2}(u,v)\), where

$$\begin{aligned}&B_{1}(u,v)=\int _{{{\mathbb {R}}^n}}\int _{{{\mathbb {R}}^n}}\log (1+|x-y|)u(x)v(y)\hbox {d}x\hbox {d}y,\\&B_{2}(u,v)=\int _{{{\mathbb {R}}^n}}\int _{{{\mathbb {R}}^n}}\log \left( 1+\frac{1}{|x-y|}\right) u(x)v(y)\hbox {d}x\hbox {d}y. \end{aligned}$$

By using the Hardy–Littlewood–Sobolev inequality and the fact that \(0<\log (1+r)<r\) for \(r>0\), for \(u,v\in L^{\frac{2n}{2n-1}}({\mathbb {R}}^n)\), there is a constant  \(C_{0}\) such that

$$\begin{aligned} |B_{2}(u,v)|\le \int _{{{\mathbb {R}}^n}}\int _{{{\mathbb {R}}^n}}\frac{1}{|x-y|}|u(x)v(y)|\hbox {d}x\hbox {d}y\le C_{0}|u|_{\frac{2n}{2n-1}}|v|_{\frac{2n}{2n-1}}. \end{aligned}$$
(2.1)

We now define the functionals

$$\begin{aligned}&V_{1}:H^{1}({\mathbb {R}}^n)\mapsto [0,\infty ], V_{1}(u)=B_{1}(u^{2},u^{2})\\&\quad =\int _{{{\mathbb {R}}^n}}\int _{{{\mathbb {R}}^n}}\log (1+|x-y|)u^{2}(x)u^{2}(y)\hbox {d}x\hbox {d}y,\\&V_{2}:L^{\frac{4n}{2n-1}}({\mathbb {R}}^n)\mapsto [0,\infty ), V_{2}(u)=B_{2}(u^{2},u^{2})\\&\quad =\int _{{{\mathbb {R}}^n}}\int _{{{\mathbb {R}}^n}}\log \left( 1+\frac{1}{|x-y|}\right) u^{2}(x)u^{2}(y)\hbox {d}x\hbox {d}y,\\&V_{0}:H^{1}({\mathbb {R}}^n)\mapsto {\mathbb {R}}\cup \{{\infty }\}, V_{0}(u)\\&\quad =B_{0}(u^{2},u^{2})=\int _{{{\mathbb {R}}^n}}\int _{{{\mathbb {R}}^n}}\log (|x-y|)u^{2}(x)u^{2}(y)\hbox {d}x\hbox {d}y. \end{aligned}$$

By using (2.1), we have \(|V_{2}(u)|\le C_{0}|u|^{4}_{\frac{4n}{2n-1}}.\) Similar to that in [8], we have

$$\begin{aligned} | B_{1}(uv,wz)|\le |u|_{*}|v|_{*}|w|_{2}|z|_{2}+|u|_{2}|v|_{2}|w|_{*}|z|_{*},~~~\text{ for }~u,v,w,z\in L^{2}({\mathbb {R}}^n) \end{aligned}$$
(2.2)

with the conventions \(\infty \cdot 0=0,\infty \cdot s=\infty ~~(s>0).\) The following is an important lemma.

Lemma 2.1

Let  \((u_{m})_{m}\) be a sequence in  \(L^{2}({\mathbb {R}}^n)\) such that  \(u_{m}\rightarrow u\in L^{2}({\mathbb {R}}^n){\setminus }\{0\}\), and pointwise a.e. in \({\mathbb {R}}^n\). Moreover, let \((v_{m})_{m}\) be a bounded sequence in \(L^{2}({\mathbb {R}}^n)\) such that \(\mathop {\sup }\nolimits _{m\in {\mathbb {N}}} {B_{1}(u_{m}^{2},v_{m}^{2})}<\infty .\) Then, there exist  \(m_{0}\in {\mathbb {N}}~\) and \(C>0\) such that \(|v_{m}|_{*}<C \) for \(m\ge m_{0}.\) If, moreover,  \(B_{1}(u_{m}^{2},v_{m}^{2})\rightarrow 0\) and \(|v_{m}|_{2}\rightarrow 0,\) as \(m\rightarrow \infty \), then

$$\begin{aligned} |v_{m}|_{*}\rightarrow 0 ~~~~~~~~~as~m\rightarrow \infty . \end{aligned}$$

Proof

We omit the proof here since it is similar to the proof of Lemma 2.1 in [8]. \(\square \)

Here and hereafter, we always assume \(({\mathcal {A}})\) holds and \(b\ge 0,2<p<2^{*}\). We consider the functional

$$\begin{aligned} I:H^{1}({\mathbb {R}}^n)\mapsto {\mathbb {R}}\cup \{{\infty }\},~~~~~I(u) =\frac{1}{2}\Vert u\Vert ^2+\frac{1}{4}V_{0}(u)-\frac{b}{p}|u|_p^p . \end{aligned}$$

Note that the restriction of I to X only takes finite values in \({\mathbb {R}}\) by using (2.2).

The proofs of the following Lemmas 2.2, 2.3 and 2.4 are similar to that in [8], even here we consider the high-dimensional case. Thus, we omit all of them here to make the paper concise.

Lemma 2.2

  1. (i)

    The space X is compactly embedded in \(L^{s}({\mathbb {R}}^n)\) for all \(s\in [2,2^{*})\).

  2. (ii)

    The functionals \(V_{0},V_{1},V_{2}\) and  I are of class \(C^{1}\) on  X.

  3. (iii)

     \(V_{2}\) is continuously differentiable on \(L^{\frac{4n}{2n-1}}({\mathbb {R}}^n)\); \(V_{1}\) is weakly lower semicontinuous on \(H^{1}({\mathbb {R}}^n)\); I is weakly lower semicontinuous on X and lower semicontinuous on \(H^{1}({\mathbb {R}}^n)\).

Next, we introduce some geometric results about the functional I.

Lemma 2.3

There exists \(\alpha >0,\) such that

$$\begin{aligned} m_{\beta }:=\inf \left\{ {I\left( u \right) :u \in X:\left\| u \right\| = \beta } \right\} > 0,~\hbox {for}~ 0<\beta \le \alpha , \end{aligned}$$

and

$$\begin{aligned} n_{\beta }:=\inf \left\{ {I'\left( u\right) u:u \in X:\left\| u \right\| = \beta } \right\} > 0,~\hbox {for}~ 0<\beta \le \alpha . \end{aligned}$$

Lemma 2.4

Suppose that \((u_{m})_{m},(v_{m})_{m},(w_{m})_{m}\) are bounded sequences in X, and \(u_{m}\rightharpoonup u\) weakly in X as \(m\rightarrow \infty \). Then,  \(B_{1}({v_m}{w_m},z({u_m}-u))\rightarrow 0~~as~m\rightarrow \infty \) for all \(z\in X\).

The following Pohozaev-type identity of Eq. (1.1) can be seen in [6, 16].

Lemma 2.5

(Pohozaev-type identity) Assume that \(u\in X\) is a weak solution to (1.1). Then

$$\begin{aligned} P(u)&:=\frac{n}{2}\int _{{\mathbb {R}}^n}a(x)u^{2}(x)\hbox {d}x+\frac{n-2}{2}\int _{{\mathbb {R}}^n}|\nabla u|^{2}\hbox {d}x\\&\quad +\,\frac{n}{2}\int _{{\mathbb {R}}^n}\int _{{\mathbb {R}}^n}\log (|x-y|)u^{2}(x)u^{2}(y)\hbox {d}x\hbox {d}y\nonumber \\&\quad +\,\frac{1}{4}\left( \int _{{\mathbb {R}}^n}u^{2}(x)\hbox {d}x \right) ^{2}+\frac{1}{2}\int _{{\mathbb {R}}^n}(x\cdot \nabla a(x))u^{2}(x)\hbox {d}x-\frac{bn}{p}|u(x)|^{p}=0. \end{aligned}$$

3 Proof of Theorem 1.1

In order to prove Theorem 1.1, we need the following compactness result. Firstly, for a function \(u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) and \(x\in {\mathbb {R}}^n\), we define

$$\begin{aligned} x*u:{\mathbb {R}}^n\rightarrow {\mathbb {R}},~~~~~[x*u](y)=u(y-x), ~\text{ for }~y\in {\mathbb {R}}^n. \end{aligned}$$

Proposition 3.1

There exists a sequence \(({u_m})_{m}\) in  X such that

$$\begin{aligned} I({u_m})\rightarrow d=c_{mp}>0,~~~~\Vert I'({u_m})\Vert _{X'}(1+\Vert {u_m}\Vert _{X})\rightarrow 0,~~~~G({u_m})\rightarrow 0,\ \ m\rightarrow \infty . \end{aligned}$$
(3.1)

Then after passing to a subsequence, there exist points  \({x_m}\in {\mathbb {Z}}^n, m\in {\mathbb {N}}\) such that

$$\begin{aligned} {x_m}*{u_m}\rightarrow u \ \text{ in }\ X \ \text{ as }\ m\rightarrow \infty , \end{aligned}$$

and \(u\in X\) is a nonzero critical point of I.

To prove Proposition 3.1, the following lemma is important.

Lemma 3.2

Assume that the sequence  \(({u_m})_{m}\subset X\) satisfies  (3.1). Then \(({u_m})_{m}\) is bounded in  \(H^{1}({\mathbb {R}}^n)\).

Proof

By using (3.1), as \(m\rightarrow \infty \),

$$\begin{aligned} d+o(1)&= I({u_m})-\frac{1}{2n}G({u_m})\nonumber \\&=\frac{n-2}{4n}\int _{{\mathbb {R}}^n}|\nabla {u_m}|^{2}\hbox {d}x+\frac{1}{4}\int _{{\mathbb {R}}^n}a(x)u_m^{2}(x)\hbox {d}x\nonumber \\&\quad +\,\frac{(p-3)b}{2p}|{u_m}|_{p}^{p}+\frac{1}{8n}|{u_m}|_{2}^{4} +\frac{1}{4n}\int _{{\mathbb {R}}^n}(x\cdot \nabla a(x))u_m^{2}(x)\hbox {d}x. \end{aligned}$$
(3.2)

Now we divide into two cases.

Case 1: \(3\le p<2^{*}.\)

By using (3.2) and \(({\mathcal {A}})\), we have

$$\begin{aligned} d+o(1)\ge \frac{n-2}{4n}\Vert {u_m}\Vert ^{2}+\frac{1}{2n}C_{0}|{u_m}|_2^{2}+\frac{(p-3)b}{2p}|{u_m}|_{p}^{p}+\frac{1}{8n}|{u_m}|_{2}^{4} \end{aligned}$$
(3.3)

where \(C_{0}>0\). Therefore, \(({u_m})_{m}\) is bounded in  \(H^{1}({\mathbb {R}}^n)\).

Case 2: \(2<p<3\). Firstly, we claim that

$$\begin{aligned} |\nabla {u_m}|_{2}\le C, \ \ \ m=1,2,\ldots . \end{aligned}$$
(3.4)

Assume by contradiction that \(|\nabla {u_m}|_{2}\rightarrow \infty \) as \(m\rightarrow \infty .\) Then by letting \(t_{m}:=|\nabla {u_m}|_{2}^{-\frac{2}{n+2}}\rightarrow 0~\) as \(m\rightarrow \infty ,~\) and  \(v_{m}:=t_{m}^{n}u_{m}(t_{m}x)\in X,\) we have

$$\begin{aligned} |\nabla v_{m}|_{2}=1,|v_{m}|_{q}^{q}=t_{m}^{n(q-1)}|u_{m}|_{q}^{q}, \ \forall m \in {\mathbb {N}},\ 1\le q<\infty . \end{aligned}$$
(3.5)

According to the Gagliardo–Nirenberg inequality, there exists a constant  \(C>0, \) such that

$$\begin{aligned} |v_{m}|_{p}^{p}\le C|v_{m}|_{2}^{2}|\nabla v_{m}|_{2}^{p-2}=C|v_{m}|_{2}^{2},\ m=1,2,\ldots . \end{aligned}$$
(3.6)

By using (3.2), (3.5) and  (3.6), we have

$$\begin{aligned} dt_{m}^{2n}+o(t_{m}^{2n})\ge&\frac{n-2}{4n}t_{m}^{n-2}-\frac{(3-p)b}{2p}t_{m}^{n(3-p)}|{v_m}|_{p}^{p}+\frac{1}{8n}|{v_m}|_{2}^{4}\\\nonumber \ge&\frac{n-2}{4n}t_{m}^{n-2}-C\frac{(3-p)b}{2p}t_{m}^{n(3-p)}|{v_m}|_2^2+\frac{1}{8n}|{v_m}|_{2}^{4}. \end{aligned}$$
(3.7)

If \(\liminf \nolimits _{m\rightarrow \infty } |v_{m}|_{2}^{2}\ge c_0> 0,\) then by using (3.7) we know that \(dt_{m}^{2n}+o(t_{m}^{2n})\ge \frac{n-2}{4n}t_{m}^{n-2}+\frac{1}{16n}|{v_m}|_{2}^{4}\) as m large, which implies a contradiction.

If \( |v_{m}|_{2}^{2}\rightarrow 0,\) up to a subsequence, we consider the following three subcases.

  1. (i)

    For \(2<p\le \frac{2n+2}{n}~(i.e., n-2\le n(3-p)),\) a contradiction is obtained again by using (3.7).

  2. (ii)

    For \(\frac{2n+2}{n}<p<\frac{5n+2}{2n} (i.e., n(3-p)<n-2<2n(3-p)),\) by using (3.7) we know that

    $$\begin{aligned} \frac{1}{8n}|{v_m}|_{2}^{4}-C_1t_{m}^{n(3-p)}|{v_m}|_{2}^{2}+\frac{n-2}{8n}t_{m}^{n-2}\le 0 \end{aligned}$$

    as m large enough, where \(C_1:=C\frac{(3-p)b}{2p}\). But it is impossible since \(\Delta :=C_1^2 t_{m}^{2n(3-p)}-\frac{n-2}{16n^2}t_{m}^{n-2}<0\) as m large enough.

  3. (iii)

    For \(\frac{5n+2}{2n}\le p<3~( i.e., 2n(3-p)\le n-2),\) we calculate

    $$\begin{aligned} I(u_{m})-\frac{1}{4}I'(u_{m})u_{m}&=\frac{1}{4}\int _{{\mathbb {R}}^n}(|\nabla u_{m}|^{2}+a(x){u_m}^{2}(x))\hbox {d}x+b\left( \frac{1}{4}-\frac{1}{p}\right) |u_{m}|_{p}^{p}\\&=\frac{1}{4}t_{m}^{-(n+2)}+\frac{1}{4}t_{m}^{-n}\int _{{\mathbb {R}}^n}a(t_m y){v_m}^{2}(y)\hbox {d}y\\&\quad +b\left( \frac{1}{4}-\frac{1}{p}\right) t_{m}^{-n(p-1)}|v_{m}|_{p}^{p}\\&=d+o(1). \end{aligned}$$

    Then, \(|v_{m}|_{p}^{p}=\frac{p}{(4-p)b}t_{m}^{n(p-2)-2}(1+o(1))\). Moreover, we have

    $$\begin{aligned} dt_{m}^{2n}+o(t_{m}^{2n})\ge \left( \frac{n-2}{4n}-\frac{3-p}{2(4-p)}\right) t_{m}^{n-2}(1+o(1))+\frac{1}{8n}|{v_m}|_{2}^{4}, \end{aligned}$$

    which again implies a contradiction.

In conclusion, we know that  (3.4) holds for any  \(2<p\le 3\). Then by using the Gagliardo–Nirenberg inequality, we have

$$\begin{aligned} |u_{m}|_{p}^{p}\le C|u_{m}|_{2}^{2}|\nabla u_{m}|_{2}^{p-2}\le C|u_{m}|_{2}^{2},\ m=1,2,\ldots . \end{aligned}$$

Thus, (3.3) gives

$$\begin{aligned} d+o(1)\ge \frac{n-2}{4n}\Vert {u_m}\Vert ^{2}+\frac{1}{2n}C_{0}|{u_m}|_2^{2}+C\frac{(p-3)b}{2p}|{u_m}|_{2}^{2}+\frac{1}{8n}|{u_m}|_{2}^{4}, \end{aligned}$$

which implies that \(|{u_m}|_{2}\le C\) uniformly. Therefore, \(({u_m})_{m}\) is bounded in  \(H^{1}({\mathbb {R}}^n)\). \(\square \)

Proof of Proposition 3.1

By using Lemmas 2.1, 2.2, 2.4 and 3.2, we can prove it similar as that in [8, 10]. So we omit it here. \(\square \)

Proof of Theorem 1.1

By using Proposition  3.1, it is easy to see that there exists a critical point  \(u\in X{\setminus }{\{0\}}\) of  I such that \(I(u)=c_{mp},\) which helps to end the proof of (i).

To prove (ii), we first notice that the set  \({\mathcal {N}}:=\{u\in X{\setminus }{\{0\}}: I'(u)=0\}\) is not empty. Let the sequence  \((u_{m})_{m}\subset {{\mathcal {N}}}\) satisfy

$$\begin{aligned} I({u_m})\rightarrow c_{g}\le c_{mp}. \end{aligned}$$

According to Lemma 2.5 and the definition of the set  \({\mathcal {N}}\), we know that the sequence \({u_m}\) satisfies

$$\begin{aligned} \Vert I'({u_m})\Vert _{X'}(1+\Vert {u_m}\Vert _{X})=0,~~~~G({u_m})\rightarrow 0~~as~~m\rightarrow \infty . \end{aligned}$$

Then by using Lemma  2.3, there is a constant  \(\alpha >0,\) such that \(\mathop {\liminf }\nolimits _{m\rightarrow \infty }\Vert {u_m}\Vert \ge \alpha .\) Hence by using Proposition 3.1, there exist points \({x_m}\in {\mathbb {Z}}^n, m=1,2,\ldots \), such that there exists a nonzero critical point  \(u\in X\) of  I in X, with

$$\begin{aligned}{x_m}*{u_m}\rightarrow u \ \text{ in }\ X, (m\rightarrow \infty ).\end{aligned}$$

Then,  \(u\in {\mathcal {N}}\) and

$$\begin{aligned} I(u)=\mathop {\lim }\limits _{m\rightarrow \infty }I({x_m}*{u_m})=\mathop {\lim }\limits _{m\rightarrow \infty }I({u_m})=c_{g}, \end{aligned}$$

which helps to end the proof. \(\square \)