Abstract
In this paper, we consider the following logarithmic Choquard equation
where \(n\ge 3, \lambda >0, 2<p<\frac{2n}{n-2}, a \in L^\infty ({\mathbb {R}}^n).\) For \(n=2, p>2\), this equation has been studied extensively. In this paper, we prove the existence of a mountain-pass solution and a ground state solution for \(n\ge 3\) by using the variational method.
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1 Introduction
This paper is devoted to the following logarithmic Choquard equation
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
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
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 \(I:X\rightarrow {\mathbb {R}},\)
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
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
- (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, \(c_{mp}>0.\)
- (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
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
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
We now define the functionals
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
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
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
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
-
(i)
The space X is compactly embedded in \(L^{s}({\mathbb {R}}^n)\) for all \(s\in [2,2^{*})\).
-
(ii)
The functionals \(V_{0},V_{1},V_{2}\) and I are of class \(C^{1}\) on X.
-
(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
and
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
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
Proposition 3.1
There exists a sequence \(({u_m})_{m}\) in X such that
Then after passing to a subsequence, there exist points \({x_m}\in {\mathbb {Z}}^n, m\in {\mathbb {N}}\) such that
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 \(m\rightarrow \infty \),
Now we divide into two cases.
Case 1: \(3\le p<2^{*}.\)
By using (3.2) and \(({\mathcal {A}})\), we have
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
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
According to the Gagliardo–Nirenberg inequality, there exists a constant \(C>0, \) such that
By using (3.2), (3.5) and (3.6), we have
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.
- (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).
- (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.
- (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
Thus, (3.3) gives
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
According to Lemma 2.5 and the definition of the set \({\mathcal {N}}\), we know that the sequence \({u_m}\) satisfies
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
Then, \(u\in {\mathcal {N}}\) and
which helps to end the proof. \(\square \)
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The first author is partially supported by the National Natural Science Foundation of China (Grant No. 11571268).
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Communicated by Syakila Ahmad.
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Guo, Q., Wu, J. Existence of Solutions to the Logarithmic Choquard Equations in High Dimensions. Bull. Malays. Math. Sci. Soc. 43, 1545–1553 (2020). https://doi.org/10.1007/s40840-019-00756-2
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DOI: https://doi.org/10.1007/s40840-019-00756-2