Abstract
In this paper, we first prove that each positive solution of
is radially symmetric, monotone decreasing about some point and has the form
where \(0<\alpha <N\) if \(N=3\) or 4, and \(N-4\le \alpha <N\) if \(N\ge 5\), \({2_{\alpha }^{*}}:=\frac{N+\alpha }{N-2}\) is the upper Hardy–Littlewood–Sobolev critical exponent, \(t>0\) is a constant and \(c_\alpha >0\) depends only on \(\alpha \) and N. Based on this uniqueness result, we then study the following nonlinear Choquard equation
By using Lions’ Concentration-Compactness Principle, we obtain a global compactness result, i.e. we give a complete description for the Palais–Smale sequences of the corresponding energy functional. Adopting this description, we are succeed in proving the existence of at least one positive solution if \(\Vert V(x)\Vert _{L^\frac{N}{2}}\) is suitable small. This result generalizes the result for semilinear Schrödinger equation by Benci and Cerami (J Funct Anal 88:90–117, 1990) to Choquard equation.
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1 Introduction and main results
Recently, the following nonlinear Choquard problem
has been investigated by many authors, where \(I_{\alpha }:{\mathbb {R}}^N{\setminus } \{0\}\rightarrow {\mathbb {R}}\) is the Riesz potential defined by
and \(\Gamma \) is the Gamma function, see [31, 34].
Equation (1.1) is usually called the nonlinear Choquard or Choquard–Pekar equation. It has several physical motivations. In the physical case \(N = 3\), \(p = 2\) and \(\alpha = 2\), the problem
appeared as early as in 1954, in a work by Pekar describing the quantum mechanics of a polaron at rest [33]. See also [24, 30] for more physical background of Eqs. (1.1)–(1.2). In particular, Lieb [21] proved that the ground state solution of Eq. (1.2) is radial and unique up to translations (see also [25]). Later, Wei and Winter [37] showed that the ground state solution is nondegenerate.
Problem (1.1) has a variational structure, setting \(V(x)\equiv 1\) for example, the corresponding energy functional is defined by
It follows by the Hardy–Littlewood–Sobolev inequality that the functional \(E_{\alpha ,p}(u)\) is well defined and belongs to \(\mathcal {C}^1(H^1({\mathbb {R}}^N),{\mathbb {R}})\) if \(p\in [\frac{N+\alpha }{N},\,\frac{N+\alpha }{N-2}]\). Moreover, the critical points of \(E_{\alpha ,p}\) are weak solutions of Eq. (1.1).
Theorem A
(See [22, 23], Hardy–Littlewood–Sobolev inequality) Suppose \(\alpha \in (0,N)\), and p, \(r>1\) with \( \frac{1}{p}+\frac{1}{r}=1+\frac{\alpha }{ N}\). Let \(f\in L^{p}({\mathbb {R}}^N)\), \(g\in L^{r}({\mathbb {R}}^N)\), then there exists a sharp constant \(C(p,r,\alpha ,N)\), independent of f and g, such that
where \(\Vert \cdot \Vert _{L^p}=\left( \int _{{\mathbb {R}}^N}|u|^{p}dx\right) ^{\frac{1}{p}}\). If \(p=r=\frac{2N}{N+\alpha }\), then
In this case, the equality in (1.4) is achieved if and only if \(f\equiv \text {(const.)}g\) and
for some \(A\in \mathbb {C}\), \(\widetilde{a}\in {\mathbb {R}}^N\) and \(0\ne \widetilde{\gamma }\in {\mathbb {R}}\).
For \(N\ge 3\), \(0<\alpha <N\), let \(2^\alpha _*=\frac{N+\alpha }{N}\) and \(2^*_\alpha =\frac{N+\alpha }{N-2}\). By the Sobolev embedding theorem, \(W^{1,2}({\mathbb {R}}^N)\subset L^{\frac{2Np}{N+\alpha }}({\mathbb {R}}^N)\) if and only if \(p\in [2^\alpha _*, 2^*_\alpha ]\). In [31], Moroz and Van Schaftingen proved that \(E_{\alpha , p}(u)\) has no nontrivial critical points when \(p\not \in \left[ 2^\alpha _*, 2^*_\alpha \right] \). Hence, \(2^\alpha _*\) and \(2^*_\alpha \) are critical exponents for existence and nonexistence of solutions to Eq. (1.1). In the past few years, there is plenty of work dealt with Eq. (1.1) with \(p\in (2^\alpha _*, 2^*_\alpha )\) by variational methods, see for example [2, 28,29,30,31,32, 37]. When \(p=2^\alpha _*\), Moroz and Van Schaftingen [32] proved the existence of one nontrivial solution to Eq. (1.1) if V(x) satisfies
As for the upper Hardy–Littlewood–Sobolev exponent, Gao and Yang [12] considered the following Brézis–Nirenberg type problem on bounded domains
In this paper, we first consider
By using Theorem A, one can verify that, up to translations and scalings, the ground state solution of Eq. (1.6) is unique and has the form
where \(t>0\), \(x_0\in {\mathbb {R}}^N\) and
here S is the best Sobolev constant for the embedding \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\hookrightarrow L^{2^{*}}({\mathbb {R}}^N)\).
A natural question is whether positive solution of Eq. (1.6) is unique and has the form of (1.7). Our result on this aspect can be stated as follows.
Theorem 1.1
Suppose \(0<\alpha <N\) if \(N=3\) or 4, and \(N-4\le \alpha <N\) if \(N\ge 5\), let u(x) be a positive solution of Eq. (1.6), then u(x) is radially symmetric and monotone decreasing about some point \(x_0\in {\mathbb {R}}^N\). Moreover, u(x) has the form of (1.7).
Remark 1.1
-
(i)
If \(p<\frac{N+\alpha }{N-2}\), by Pohozaev type identity, the following equation
$$\begin{aligned} -\Delta u=\big (I_{\alpha }*|u|^p\big )|u|^{p-2}u, \quad x\in {\mathbb {R}}^{N} \end{aligned}$$(1.9)has no nontrivial solution \(u\in W^{1,2}({\mathbb {R}}^N)\cap L^{\frac{2Np}{N+\alpha }}({\mathbb {R}}^N)\) with \(\nabla u\in W_{loc}^{1,2}({\mathbb {R}}^N)\cap L_{loc}^{\frac{2Np}{N+\alpha }}({\mathbb {R}}^N)\).
-
(ii)
We prove Theorem 1.1 by a moving plane method, which was invented by Alexanderov in [1]. Later, it was further developed by Serrin [35], Gidas et al. [14], Caffarelli et al. [5] when classifying the solutions of semilinear elliptic equation
$$\begin{aligned} -\Delta u=u^{\frac{N+2}{N-2}}, \quad x\in {\mathbb {R}}^N. \end{aligned}$$Subsequently, Chen and Li [8] and Li [17] simplified the proof, Wei and Xu [38] and Chen et al. [11] generalized the classification result to the solutions of higher order conformally invariant equations
$$\begin{aligned} (-\Delta )^{s} u=u^{\frac{N+s}{N-s}}, \quad x\in {\mathbb {R}}^N,\ 0<s<N. \end{aligned}$$Li [18] used the method of moving spheres to obtain the same classification result as that in [11]. For other applications, we refer the readers to [7, 9, 10, 16, 28].
Based on the uniqueness result, we can investigate the following Choquard equation
where the potential function \(V(x)\in L^{\frac{N}{2}}({\mathbb {R}}^N)\cap \mathcal {C}^{\gamma }({\mathbb {R}}^N)\) is nonnegative for some \(\gamma \in (0,1)\). Define the energy functionals I, \(I_{\infty }\) corresponding to Eqs. (1.10), (1.6) respectively by
and
The Nehari manifolds corresponding to I and \(I_{\infty }\) denoted by \(\mathcal {N}\) and \(\mathcal {N}_{\infty }\) respectively are
Moreover, we define
and
Obviously, m is the mountain pass level of the functional I and
Our main result on Eq. (1.10) can be stated as follows.
Theorem 1.2
Let \(0<\alpha <N\) if \(N=3\) or 4, and \(N-4\le \alpha <N\) if \(N\ge 5\), and suppose that \(V(x)\in L^{\frac{N}{2}}({\mathbb {R}}^N)\cap \mathcal {C}^{\gamma }({\mathbb {R}}^N)\) is nonnegative for some \(\gamma \in (0,1)\), then \(m=m_{\infty }\) holds and m is not achieved. If V(x) in addition satisfies
then Eq. (1.10) possesses at least one positive solution.
We prove Theorem 1.2 by following the variational approach developed by Benci and Cerami [3], in which a similar result was proved for the following Schrödinger equation
However, we cannot apply this approach directly, several difficulties arise because of the nonlocal nonlinearity with critical exponent. The main obstacle is lack of compactness, even if we get a \((PS)_c\) sequence with \(c\in (m_\infty , 2m_\infty )\), we still cannot obtain the strongly convergence of \((PS)_c\) sequence, because the nodal solutions of Eq. (1.6) doesn’t possess the double energy property (see [39]), i.e. there may exist nodal solutions of Eq. (1.6) with energy between \(m_\infty \) and \(2m_\infty \) (see Theorem 3, [13]), but the double energy property is crucial for proving the main result in [3]. We solve this difficulty by using Linking Theorem to seek a nonnegative \((PS)_c\) sequence with \(c\in (m_\infty , 2m_\infty )\) and analysing carefully the nonlocal nonlinearity. To this end, a nonlocal version of the Concentration-Compactness Principle (see Lemma 2.1, [27]) is used, which is totally different from the usual local case.
The following splitting result for Palais–Smale sequences is crucial for proving Theorem 1.2, while the local case on bounded domain has been established by Struwe [36].
Theorem 1.3
Suppose \(V(x)\ge 0\) and \(V(x)\in L^{\frac{N}{2}}({\mathbb {R}}^N)\), let \(\{u_n\}\) be a Palais–Smale sequence of I at level c. Then \(\{u_n\}\) has a subsequence which converges strongly in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\), or otherwise, replacing \(\{u_n\}\) if necessary by a subsequence, there exists a function \(\bar{u}\in \mathcal {D}^{1,2}({\mathbb {R}}^N)\) satisfying \(u_n\rightharpoonup \bar{u}\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\). Moreover, there exists a number \(k\in {\mathbb N}\), k functions \(u^1, \ldots , u^k\in \mathcal {D}^{1,2}({\mathbb {R}}^N)\); k sequences of points \(\{y_{n}^{i}\}\subset {\mathbb {R}}^N\), \(1\le i \le k\) and k sequences of positive numbers \(\{\sigma _{n}^{i}\}\), \(1\le i\le k\), such that
where \(\bar{u}\) is a nontrivial solution of Eq. (1.10) and \(u^{i}\), \(1 \le i \le k\), are the nontrivial solutions of Eq. (1.6). Moreover, as \(n\rightarrow +\infty \), we have
and
where \(\Vert u\Vert ^2=\int _{{\mathbb {R}}^N}|\nabla u|^2dx\) for \(u\in \mathcal {D}^{1,2}({\mathbb {R}}^N)\).
The paper is organized as follows. In Sect. 2, via the moving plane method, we prove that, up to translations and scalings, the positive solution of Eq. (1.6) is unique. In Sect. 3, by studying the behavior of Pslais–Smale sequences, we obtain a global compactness result, which provides a complete description of Palais–Smale sequences. In Sect. 4, we first show that the mountain pass value is not achieved. Then, combining Linking Theorem with Theorem 1.3, we prove the existence of at least one positive solution for Eq. (1.10).
2 Uniqueness of positive solution
In this section, we set \(A_\alpha \equiv 1\) for convenience. We will use the moving planes method to show the uniqueness of the positive solution of Eq. (1.6). To do this, we first show the invariance of (1.6) under Kelvin transform. Denote \(K_u\) the Kelvin transform of u, that is,
Lemma 2.1
Let u(x) be a solution of Eq. (1.6), then, \(U=K_u\) is still a solution of Eq. (1.6).
Proof
Note that
On the other hand,
where we use the identity
in the third step. Therefore, we have
This shows that \(U=K_u\) is also a solution of Eq. (1.6), which implies that Eq. (1.6) is invariant under Kelvin transform. \(\square \)
Now, we transform Eq. (1.6) to an equivalent integral system. Let \(v(x)=|x|^{-N+\alpha }*|u|^{2^*_\alpha }\). Then, up to a normalization constant, Eq. (1.6) is equivalent to
By \(u\in L^{\frac{2N}{N-2}}({\mathbb {R}}^N)\) and Hardy–Littlewood–Sobolev inequality, we know that \(v\in L^{\frac{2N}{N-\alpha }}({\mathbb {R}}^N)\). Making use of the moving plane method in integral forms, we show that each positive solution (u, v) of system (2.1) in \(L^{\frac{2N}{N-2}}({\mathbb {R}}^N)\times L^{\frac{2N}{N-\alpha }}({\mathbb {R}}^N)\) is radially symmetric and monotone decreasing about some point \(x_0\in {\mathbb {R}}^N\).
For this purpose, we first introduce some notation. For \(x=(x_1,x_2,\dots ,x_N)\in {\mathbb {R}}^N\), \(\lambda \in {\mathbb {R}}\), we define \(x^{\lambda }=(2\lambda -x_1, x_2,\dots ,x_N)\) and
Let \(\Sigma _{\lambda }=\{x=(x_1,x_2,\dots ,x_N)\in {\mathbb {R}}^N:x_1\ge \lambda \}\). We set
Moreover, we denote the complement of \(\Sigma _\lambda \) in \({\mathbb {R}}^N\) by \(\Sigma _{\lambda }^{c}\), and the reflection of \(\Sigma _{\lambda }^{u}\) about the plane \(x_1=\lambda \) by \(\big (\Sigma _{\lambda }^{u}\big )^*\).
We decompose \(u_{\lambda }(x)\), u(x) in \(\Sigma _{\lambda }\) and \(v_{\lambda }(x)\), v(x) in \(\Sigma _{\lambda }\) as follows.
Lemma 2.2
For each positive solution (u, v) of system (2.1), we have
and
Proof
By (2.1) and the fact that \(|x-y^{\lambda }|=|x^{\lambda }-y|\), we then obtain
which leads to
From (2.4) and (2.5), we then get (2.2). By a similar argument, we can also prove (2.3). \(\square \)
Using the above preliminaries, we then prove the following proposition.
Proposition 2.3
Suppose \(0<\alpha <N\) if \(N=3\) or 4 and \(N-4\le \alpha <N\) if \(N\ge 5\), and let (u, v) be a positive solution of system (2.1) in \(L^{\frac{2N}{N-2}}({\mathbb {R}}^N)\times L^{\frac{2N}{N-\alpha }}({\mathbb {R}}^N)\). Then u and v are both radially symmetric and decreasing about some point \(x_0\in {\mathbb {R}}^N\).
Proof
The proof consists of three steps.
Step 1 There exists \(l_0>0\) such that for any \(\lambda <-l_0\), we have
For the sufficiently negative value of \(\lambda \), we show that both \(\Sigma _{\lambda }^{u}\) and \(\Sigma _{\lambda }^{v}\) must be empty.
In fact, for any \(x\in \Sigma ^u_{\lambda }\), we have
Hence, if \(2^*_\alpha \ge 2\), we then get
By Lemma 2.2 and Hölder’s inequality, we obtain
and
Hence, substituting (2.9) into (2.8), we then obtain
Recalling that \(u\in L^{\frac{2N}{N-2}}({\mathbb {R}}^N)\), \(v\in L^{\frac{2N}{N-\alpha }}({\mathbb {R}}^N)\), by the dominated convergence theorem that we can choose \(l_0\) sufficiently large such that \(\lambda <-l_0\) and
Thus, it follows by (2.10) and (2.11) that
This implies that \(\Sigma _{\lambda }^{u}\) must be a set with zero measure, hence must be empty up to a set with zero measure. By (2.9), \(\Sigma _{\lambda }^{v}\) must be empty.
Step 2 We move the plane continuously from \(\lambda <-l_0\) to the right as long as (2.6) holds. We show that if the procedure stops at \(x_1=\lambda _0\) for some \(\lambda _0\), then u(x) and v(x) must be symmetric and monotone about the plane \(x_1=\lambda _0\). Otherwise, we can move the plane all the way to the right.
Moving the plane \(x_1=\lambda \) to the right as long as (2.6) holds. Suppose that at some \(\lambda _0\), we have
but
In the following, we show that the plane can be moved further to the right. More precisely, there exists \(\delta =\delta (N, u, v)\) such that \(u(x)\ge u_{\lambda }(x)\) and \(v(x)\ge v_{\lambda }(x)\) on \(\Sigma _{\lambda }\) for all \(\lambda \in [\lambda _0,\lambda _0+\delta )\).
Assume that
By (2.2), we have \(u(x)> u_{\lambda _0}(x)\) in the interior of \(\Sigma _{\lambda _0}\). Note that
where \(meas(\overline{\Sigma _{\lambda _0}^{u}})\) denotes the Lebesgue measure of \(\overline{\Sigma _{\lambda _0}^{u}}\). Since \(u\in L^{\frac{2N}{N-2}}({\mathbb {R}}^N)\), \(v\in L^{\frac{2N}{N-\alpha }}({\mathbb {R}}^N)\) and \(meas(\overline{\Sigma _{\lambda _0}^{u}})=0\), then using the dominated convergence theorem, we can choose \(\delta >0\) sufficiently small, such that for all \(\lambda \in [\lambda _0,\lambda _0+\delta )\), we have
It follows from (2.10) that
Hence, \(\Sigma _{\lambda }^{u}\) must be empty for all \(\lambda \in [\lambda _0, \lambda _0+\delta )\), which also implies that \(\Sigma _{\lambda }^{v}\) is empty for all \(\lambda \in [\lambda _0,\lambda _0+\delta )\).
Assume that
By (2.3), we see \(v(x)> v_{\lambda _0}(x)\) in the interior of \(\Sigma _{\lambda _0}\). By the above analysis, we know that \(\Sigma _{\lambda }^{u}\) and \(\Sigma _{\lambda }^{v}\) must also be empty for all \(\lambda \in [\lambda _0,\lambda _0+\delta )\). This completes the proof.
Step 3 By step 1, we know that the plane cannot keep moving all the way to the right in Step 2. That is, the plane will eventually stop at some point. In fact, with the similar analysis as that in Step 1 and Step 2, we then assert that there exists a large \(\bar{l}\), such that for \(\lambda >\bar{l}\),
Now we can move the plane continuously from \(\lambda >\bar{l}\) to the left as long as the above fact holds. The planes moved from the left and the right will eventually meet at some point. Finally, since the direction of \(x_1\) can be chosen arbitrarily, we deduce that u(x) and v(x) must be radially symmetric and decreasing about some point. \(\square \)
Now we use the elliptic regularity theory to show the following proposition.
Proposition 2.4
Assume that u is a positive solution of (1.6). Then u is uniformly bounded in \({\mathbb {R}}^N\). Furthermore, u is \(\mathcal {C}^{\infty }({\mathbb {R}}^N)\) and
for some positive constant \(u_{\infty }\).
Proof
Step 1 We first show that u is uniformly bounded and smooth. For \(A>0\), we define
Hence
Since u is a solution of Eq. (1.6), we have
which implies that for any \(x\in \Omega \),
Next we divide our argument into three cases.
Case 1 \(0<\alpha \le 2\). For any \(r\ge \frac{2N}{N-2}\), by Hardy–Littlewood–Sobolev inequality, we see
where \(q=\frac{2N+4r}{r(2+\alpha )}\) and \(1/p+ 1/q=1\). One can easily check that \(q>1\) for every \(r\ge \frac{2N}{N-2}\) and \(0<\alpha \le 2\). Thus, using the Hardy–Littlewood–Sobolev inequality again, one finds
On one hand, by \(u\in L^{\frac{2N}{N-2}}({\mathbb {R}}^N)\), we can choose A large enough, such that
On the other hand, by \(u\in L^{\frac{2N}{N-2}}({\mathbb {R}}^N)\) and (2.14), we verify that
Substituting (2.16) and (2.17) into (2.15), we then assert that, for any \(r\ge \frac{2N}{N-2}\)
which implies that \(u_A \in L^{r}({\mathbb {R}}^N)\) for any \(r\ge \frac{2N}{N-2}\). Therefore, we have \(u\in L^{r}({\mathbb {R}}^N)\) for any \(r\ge \frac{2N}{N-2}\). Using Hardy–Littlewood–Sobolev inequality again, we get
Using the \(L^{p}\)-theory and Sobolev embedding theorem (see Theorem 9.9, [15]), we know that u is uniformly bounded and belongs to \(\mathcal {C}^{0,s}({\mathbb {R}}^N)\) for all \(0<s<1\). In fact, we also conclude \(u\in \mathcal {C}^{\infty }({\mathbb {R}}^N)\) from Theorem 4.4.8 in [6].
Case 2 \(2<\alpha <N-4\). Let \(p=\frac{N+\alpha }{N-2}<2\). First, we claim that \(u_A\in L^{s}\) for every \(2^*=\frac{2N}{N-2} \le s\le \frac{Np}{\alpha }\). Set \(s_0=2^*\), we assume that \(u_A\in L^s\) for every \(s\in [2^*, s_n]\) and \(s_n<\frac{Np}{\alpha }\). We will prove that \(u_A\in L^r\) if \(r\ge s_n\) satisfies
Moreover, we compare \(r_0=(\frac{p-1}{s_n}-\frac{2}{N})^{-1}\) with \(\frac{N p}{\alpha }\). If \(r_0\ge \frac{Np}{\alpha }\), then the claim is proved. If \(r_0<\frac{Np}{\alpha }\), set \(s_{n+1}=r_0\) and proceed again. Since
our argument must terminate at a finite number of steps. We should note that if \(s_n<\frac{N}{\alpha }p\),
Then using the Hardy–Littlewood–Sobolev inequality and the condition (2.19)–(2.21), we find
and
Setting \(t={\frac{s_n Nrp}{s_n (N+2r)-(p-1)Nr+s_nr\alpha }}\), we know that \(s_n<t<r\). Hence \(t=(1-\theta )s_n+\theta r\) where \(\theta =\frac{t-s_n}{r-s_n}\). It yields that
Similarly to (2.18), we have
Then we choose \(A>0\) sufficiently large such that
Note that \(\theta p<1\). To see this, we only need to prove
which is equivalent to
Since \(s\ge \frac{2N}{N-2}\), we compute that
From this, by (2.22) we know
It follows that \(u_A\in L^{r}({\mathbb {R}}^N)\) for \(A>0\) sufficiently large. Thus \(u\in L^{\frac{N p}{\alpha }}({\mathbb {R}}^N)\) and \(I_\alpha *|u|^p\in L^\infty ({\mathbb {R}}^N)\).
Finally, since u satisfies
Then, by standard elliptic regularity theory, \(u\in \mathcal {C}^\infty ({\mathbb {R}}^N)\).
Case 3 \(N-4\le \alpha < N\). In this case, \(2_\alpha ^*=\frac{N+\alpha }{N-2}\ge 2\). Then \(a(x):=(I_\alpha *|u|^{2_\alpha ^*})u^{2_\alpha ^*-2}\in L^{N/2}({\mathbb {R}}^N)\). The Brézis–Kato theorem [4] implies that \(u\in L^t_{loc}({\mathbb {R}}^N)\) for all \(1\le t<\infty \). Thus, \(u\in W^{2,t}({\mathbb {R}}^N)\) for all \(1\le t<\infty \). By elliptic regularity theory, \(u\in \mathcal {C}^\infty ({\mathbb {R}}^N)\).
Step 2 We want to prove the asymptotic behavior at infinity of u. We prove it by contradiction. Consider the Kelvin transform:
Applying Proposition 2.3 to U(x), we conclude that U(x) must be radially symmetric about some point and continuous. Hence
which completes the proof of Proposition 2.4. \(\square \)
Lemma 2.5
Let u be a solution of Eq. (1.6), then there exist \(\lambda >0\) and \(x\in {\mathbb {R}}^N\) such that
Proof
Let u be a solution of Eq. (1.6). By Proposition 2.3, we can assume that u(x) is symmetric about the origin, and we prove this lemma with \(x=0\). Moreover, without loss of generality we assume that \(\lambda =1\). Otherwise, we just need to make a translation or a scaling.
By Proposition 2.4, suppose that \(\lim \limits _{|x|\rightarrow +\infty }|x|^{N-2}u(x)=u_\infty =u(0)\). Let e be any unit vector in \({\mathbb {R}}^N\). We define
Obviously, w(y) is the Kelvin transform of \(u(y-e)\). By Lemma 2.1, w satisfies the Eq. (1.6) and hence should be radially symmetric about some point \(z_0\in {\mathbb {R}}^N\). Note that
Thus, w must be symmetric about the plane \(\Pi =\{x: (x-\frac{e}{2})\cdot e=0\}\). Now, choosing \(y=\frac{e}{2}-he\) for any \(h>0\), similarly to the proof of Lemma 3.1 in [11], we can prove that
Taking \(y=\frac{e}{2}+he\), \(h>0\), we have
Combining (2.25) with (2.26) and noticing the radial symmetry of u, we find
Let \(t=(\frac{1}{2}-h)/(\frac{1}{2}+h)\), then
Replacing |t|, e by 1 / |y|, y / |y|, respectively, we obtain
Furthermore, we can take a translation transform to obtain (2.24). \(\square \)
To prove Theorem 1.1, we also need the following proposition from Li and Zhang [19]. Earlier version with stronger assumptions was first proved by Li and Zhu [20].
Proposition 2.6
[19] Let \(f\in \mathcal {C}^1({\mathbb {R}}^N,{\mathbb {R}})\), \(\lambda >0\) and \(\mu >0\). Suppose that for every \(x\in {\mathbb {R}}^N\), there exists \(\lambda (x)>0\) such that
Then,
for some \(a\ge 0\), \(d>0\) and \(\bar{x}\in {\mathbb {R}}^N\).
Proof of Theorem 1.1
Using Lemma 2.5 and Proposition 2.6, we obtain that the solution of Eq. (1.6) must be of form (1.7). \(\square \)
3 A global compactness result
In this section, we study the behavior of Palais–Smale sequences of the energy functional I and then prove Theorem 1.3. The following result is a Brézis–Lieb’s type lemma for problem (1.10), and the proof is similar as Lemma 2.4 in [31].
Lemma 3.1
Let \(N\ge 3\) and \(\alpha \in (0,N)\). If \(\{u_n\}\) is a bounded sequence in \(L^{\frac{2N}{N-2}}({\mathbb {R}}^N)\) such that \(u_n\rightarrow u\) almost everywhere in \({\mathbb {R}}^N\) as \(n\rightarrow +\infty \), then
and
where \(\big (\mathcal {D}^{1,2}({\mathbb {R}}^N)\big )'\) is the dual space of \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\).
In order to prove Theorem 1.3, we need the following concentration principle for Riesz potential.
Lemma 3.2
Let \(\{u_n\} \subset \mathcal {D}^{1,2}({\mathbb {R}}^N)\) be a sequence of functions such that
Assume that there exist a bounded open set \(Q\subset {\mathbb {R}}^N\) and a positive constant \(\varrho >0\) such that
and
Moreover, suppose that
where \(\chi _n \in \big (\mathcal {D}^{1,2}({\mathbb {R}}^N)\big )'\) and
with \(\Omega \) being an open neighborhood of Q and \(\{\varepsilon _n\}\) being a sequence of positive numbers converging to 0. Then there exist a sequence of positive numbers \(\{\sigma _n\}\) and a sequence of points \(\{y_n\}\subset \bar{Q}\) such that
converges weakly in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) to v, which is a nontrivial solution of Eq. (1.6).
Proof
Since \(u_n \rightharpoonup 0\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\), then by Concentration Compactness Principle II (see Lemma I.1, [26]), we obtain an at most countable index set \(\Gamma \), a sequence of \(\{ x_i\}_{i\in \Gamma }\subset {\mathbb {R}}^N \) and a family of \(\{\nu _i\}_{i\in \Gamma }\subset (0,+\infty )\) such that
where \(\phi _{\Omega }(x)\) is a cut-off function with \(\phi _{\Omega }(x)=1\) in Q; \(\phi _{\Omega }(x)=0\) in \({\mathbb {R}}^N{\setminus } \Omega \) and \(0\le \phi _{\Omega }(x)\le 1\).
For the readers’ convenience, we will prove this lemma through three claims.
Claim 1
There is at least one \(i_0\in \Gamma \) such that \(x_{i_0}\in \bar{Q}\) with \(\nu _{i_0}>0\).
Proof
Otherwise, then \(u_n\rightarrow 0\) in \(L^{2^*}(Q)\), which together with Hardy–Littlewood–Sobolev inequality implies that
This is a contradiction to the assumption (3.4) and the claim is proved. \(\square \)
Now, we define the concentration function
For a given small \(\tau \in \big (0, \big [\frac{S}{A_{\alpha } C(N,\alpha )}\big ]^{\frac{N}{\alpha +2}}\big )\), we choose \(\sigma _n=\sigma _n(\tau )>0\), \(y_n \in \bar{Q}\) such that
Let \(v_n(x):=\sigma _{n}^{\frac{N-2}{2}}u_{n}(\sigma _n x+ y_n)\), then
where \(\bar{Q}_n:=\{x\in {\mathbb {R}}^N: \sigma _n x+y_n \in \bar{Q}\}\). It follows by (3.7) and (3.8) that
Claim 2
There exists some \(\tau \in \big (0, \big [\frac{S}{A_{\alpha } C(N,\alpha )}\big ]^{\frac{N}{\alpha +2}}\big )\) such that \(\sigma _{n}(\tau )\rightarrow 0\) as \(n\rightarrow +\infty \).
Proof
Assume by contradiction, for any \(\varepsilon >0\), that there exists \(r_0>0\) such that \(\sigma _n(\varepsilon )\ge r_0\). Then a direct calculation shows that
In particular
where \(o_{n}(1)\rightarrow 0\) as \(n\rightarrow +\infty \). Then, it follows by (3.11) that we have \(\nu _{i_0}\le 0\), which contradicts Claim 1.
By the definition of \(v_n\), we have \(\int _{{\mathbb {R}}^N}|\nabla v_n|^2dx=\int _{{\mathbb {R}}^N}|\nabla u_n|^2dx\), which together with the boundness of \(\{u_n\}\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) implies that \(\{v_n\}\) is bounded in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\). Without loss of generality, we may assume that there exists some \(v\in \mathcal {D}^{1,2}({\mathbb {R}}^N)\) such that \(v_n\rightharpoonup v\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) up to a subsequence.
Claim 3
v is a nontrivial solution of Eq. (1.6).
Proof
In fact, for any \(\varphi \in C_{0}^{\infty }({\mathbb {R}}^N)\), we define
Since \(\sigma _n\rightarrow 0\) and \(y_n \in \bar{Q}\), then we assert that \(\widetilde{\varphi }_{n}(x)\in C_{0}^{\infty }(\bar{\Omega })\) for n large enough. In virtue of (3.5) and (3.6), we obtain that
Thus, v is a weak solution of Eq. (1.6). Before concluding the proof, we still need to prove \(v\ne 0\). To this end, it is sufficient to prove that, up to a subsequence,
Since \(v_n\rightharpoonup v\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\), by Concentration Compactness Principle (see Lemma I.1 in [26] and Lemma 2.1 in [27]), we may assume that there exist three bounded nonnegative measures \(\widetilde{\mu }\), \(\widetilde{\nu }\), \(\widetilde{\omega }\), such that \(|\nabla v_n|^2 \rightharpoonup \widetilde{\mu }\), \(|v_n|^{2^*}\rightharpoonup \widetilde{\nu }\) and \(\big |I_{\alpha }* |v_n|^{2_{\alpha }^{*}}\big |^{\frac{2N}{N-\alpha }}\rightharpoonup \widetilde{\omega }\) weakly in finite measure space \(\mathcal {M}({\mathbb {R}}^N)\) (see Page 26 in [40]). Moreover,
and
where \(\widetilde{\Gamma }\) is an at most countable index set. In order to prove (3.14), we only need to prove
If not, we suppose that there exists \(x_{j_0}\in \overline{B_{1}(0)}\) for some \(j_0 \in \widetilde{\Gamma }\) and define \(\phi _{\rho }(x):=\phi \big (\frac{x-x_{j_0}}{\rho }\big )\), \(\phi \) is a cut-off function which satisfies \(\phi =1\) on \(B_{1}(0)\), \(supp \phi \subset B_{2}(0)\) and \(0\le \phi \le 1\). Denote by \(\widetilde{\phi }_{\rho ,n}(x)=\phi _{\rho }\big (\frac{x-y_n}{\sigma _n}\big )\), by the facts that \(y_n\in \bar{Q}\), \(x_{j_0}\in \overline{B_{1}(0)}\) and \(\sigma _n\rightarrow 0\), we then observe that \(supp \widetilde{\phi }_{\rho ,n}(x) \subset B_{2\sigma _n\rho }(y_n+\sigma _n x_{j_0})\subset \Omega \), which implies \(\widetilde{\phi }_{\rho ,n}(x) u_n \in \mathcal {D}_{0}^{1,2}(\Omega )\). A direct calculation yields that
Hence, \(\{ \widetilde{\phi }_{\rho ,n} u_n\}\) is bounded in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) and the bound is independent of \(\rho \). Combining (3.5), (3.6) with the fact that \(C_{0}^{\infty }({\mathbb {R}}^N)\) is dense in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\), we then get
Let \(\rho \rightarrow 0\), then
Moreover,
and
It follows from (3.18)–(3.20) that
then
Combining the inequality above and (3.9), then we get
which contradicts the assumption \(\tau \in \big (0, \left[ \frac{S}{A_{\alpha } C(N,\alpha )}\big ]^{\frac{N}{\alpha +2}}\right) \). Therefore, (3.14) is proved. Combining (3.9) and (3.14), we have
which implies that \(v\ne 0\). Thus, combining Claims 1–3, we can complete the proof. \(\square \)
Lemma 3.3
Let \(\{u_n\}\) be a Palais–Smale sequence for \(I_\infty \), such that \(u_n \in C_{0}^{\infty }({\mathbb {R}}^N)\) and
Then there exist a sequence of points \(\{y_n\}\subset {\mathbb {R}}^N\), a sequence of positive numbers \(\{\sigma _n\}\) such that
converges weakly in \(\mathcal {\mathcal {D}}^{1,2}({\mathbb {R}}^N)\) to v, which is a nontrivial solution of Eq. (1.6). Moreover,
Proof
Since \(u_n\rightharpoonup 0\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\), then \(\{u_n\}\) is bounded in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\). Furthermore, as \(\{u_n\}\) is a Palais–Smale sequence for \(I_\infty \), we then know that
where \(\chi _n \in \big (\mathcal {D}^{1,2}({\mathbb {R}}^N)\big )'\) satisfies
Multiplying by \(u_n\) on both sides of (3.24) and integrating on \({\mathbb {R}}^N\), we then have
Let us decompose \({\mathbb {R}}^N\) in N-dimensional hypercubes \(Q_i\) with unitary sides and vertices with integer coordinates. Next, we assert that for any \(n\in {\mathbb N}\), there exists some \(\widetilde{\varrho }>0\) satisfying
If not, then we have \(d_n \rightarrow 0\) as \(n\rightarrow +\infty \). A direct calculation shows that
Combining (3.26) with (3.27) and letting \(d_n\rightarrow 0\) as \(n\rightarrow +\infty \), we observe that \(\Vert {u}_n\Vert \rightarrow 0\), which leads to a contradiction.
In the following, let \(\widetilde{y}_n\) be the center of a hypercube \(Q_i\) such that
Set \(w_n=u_n(x+\widetilde{y}_{n})\), then
where Q denote a hypercube of unitary side centered at the origin. Using the Hardy–Littlewood–Sobolev inequality and the boundedness of \(\{u_n\}\) in \(\mathcal D^{1,2}({\mathbb {R}}^N)\) again, we get
Hence we can deduce that there exists \(\bar{\varrho }>0\) such that
At this point, we have verified the conditions (3.3)–(3.5) in Lemma 3.2 for \(\{w_n\}\). The first part of Lemma 3.3 follows from Lemma 3.2. Obviously,
Then we can prove (3.23). Similarly, (3.22) follows from (3.1). \(\square \)
It follows from Theorems A and 1.1 that
Now, we are ready to prove Theorem 1.3.
Proof of Theorem 1.3
Since \(\{u_n\}\) is a Palais–Smale sequence for I at level c, then it is easy to prove that \(\{u_n\}\) is bounded in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) and consequently bounded in \(L^{2^*}({\mathbb {R}}^N)\). Without loss of generality, we may assume that \(u_n\rightharpoonup \bar{u}\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) and \(L^{2^*}({\mathbb {R}}^N)\) as \(n\rightarrow +\infty \). Moreover, \(\bar{u}\) is a weak solution of Eq. (1.10). In fact, for any \(\varphi _1 \in C_{0}^{\infty }({\mathbb {R}}^N)\), we have
By Lemma 3.1, we know
Moreover, by Lemma 2.13 [40], we have
and
Thus, it follows by (3.31)–(3.33) that
which leads to \(I'(\bar{u})=0\), \(I(\bar{u})=I(u_n)-I_\infty (u_n-\bar{u})+o_n(1)\).
Let \(z_{n}^{1}:=u_n-\bar{u}\), then \(z_{n}^{1}\rightharpoonup 0\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) and \(\{z_{n}^{1}\}\) is a Palais–Smale sequence for \(I_\infty \). In fact, for any \(\varphi _{2}\in C_{0}^{\infty }({\mathbb {R}}^N)\), we have
where (3.33) and (3.2) are used. Hence \(\{z_{n}^{1}\}\) is a Palais–Smale sequence of \(I_\infty \).
For any \(n\in {\mathbb N}^{+}\), there exists a sequence \(\{u_{n}^{1}\}\subset C_{0}^{\infty }({\mathbb {R}}^N)\) such that
It is not difficult to verify that
Furthermore, one has
and
If \(u_n^1\rightarrow 0\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\), then we have done. Now we suppose that \(u_{n}^{1}\nrightarrow 0\) strongly in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\). From (3.35) that we know that \(\{u_n^1\}\) is a Palais–Smale sequence of \(I_\infty \) and \(\{u_{n}^{1}\}\subset C_{0}^{\infty }({\mathbb {R}}^N)\) satisfies
Applying Lemma 3.3 to \(\{u_n^1\}\), we assert that there exist a sequence of points \(\{x_n^{1}\}\subset {\mathbb {R}}^N\), a sequence of positive numbers \(\{\eta _{n}^{1}\}\subset {\mathbb {R}}\) such that
converges weakly in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) to a nontrivial solution \(u^{1}\) of Eq. (1.6). Moreover,
Combining (3.38) with (3.35), we obtain that
and
Let \(z_{n}^{j}=v_{n}^{j-1}-u^{j-1}\) and repeat the above procedure. If \(z_{n}^{j}\rightarrow 0\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\), we have done. If \(z_{n}^{j}\nrightarrow 0\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\), the analogously \(\{z_{n}^{j}\}\) is a Palais–Smale sequence of \(I_\infty \), then we can find \(\{u_n^{j}\}\subset C_{0}^{\infty }({\mathbb {R}}^N)\) such that
and there exist a sequence of positive numbers \(\{\eta _{n}^{j}\} \subset {\mathbb {R}}\) and a sequence of points \(\{x_{n}^{j}\} \subset {\mathbb {R}}^N\) such that
converges weakly in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) to a nontrivial solution \(u^j\) of Eq. (1.6). Moreover, the following properties hold:
Furthermore, we deduce that
and
Since \(u^{j}\) is a nontrivial weak solution of Eq. (1.6), then \(\Vert u^j\Vert ^2\ge S_{\alpha }^{\frac{N+\alpha }{\alpha +2}}\), which together with (3.44) and the fact that \(u_n\) is bounded in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) tells us that the iteration procedure must terminate after finitely-many steps. Therefore, we complete the proof of Theorem 1.3. \(\square \)
4 Existence of positive bound state solution
In this section, we prove the existence of bound state solutions to Eq. (1.10). Firstly, we show that, providing \(V(x)\ge 0\) and \(V(x) \in L^{\frac{N}{2}}({\mathbb {R}}^N)\), then there is no minimizer for functional I restrict on the Nehari manifold \(\mathcal {N}\).
Proposition 4.1
Assume that \(V(x)\ge 0\) and \(V(x) \in L^{\frac{N}{2}}({\mathbb {R}}^N)\), then \(m=m_{\infty }\) holds and m is not attained.
Proof
Obviously, for \(u\in \mathcal {D}^{1,2}({\mathbb {R}}^N){\setminus } \{0\}\), there exist unique \(t_u\), \(s_u>0\) such that \(t_uu\in \mathcal {N}\) \(s_uu\in \mathcal {N}_{\infty }\), moreover \(I(t_uu)=\max _{t>0}I(tu)\) and \(I_{\infty }(s_uu)=\max _{s>0}I_{\infty }(su)\). Especially, if \(u\in \mathcal {N}\) and \(s_uu \in \mathcal {N}_{\infty }\), then we have \(s_u\in (0,1]\). Therefore, for \(u\in \mathcal {N}\),
which implies that \(m_{\infty } \le m\).
Next, we prove \(m\le m_\infty \). In fact, we consider a sequence \(\{u_n:=t_nw_n \} \subset \mathcal {N}\), where \(w_n(\cdot )=w(\cdot -z_n)\) with w being a positive solution centered at zero to Eq. (1.6), \(\{z_n\}\subset {\mathbb {R}}^N\) satisfying \(|z_n|\rightarrow +\infty \) as \(n\rightarrow +\infty \) and \(t_n:=t_{w_n}\). It follows by the definition of \(w_n\) that
and
Furthermore, by Lemma 2.13 [40], we know that
Thus, in virtue of (4.2)–(4.4), we can prove easily that
Since \(w_n \in \mathcal {N}_{\infty }\) and \(t_nw_n\in \mathcal {N}\), then
and
Combining (4.6) and (4.7), we then have
From (4.2), (4.3) and (4.8), then \(\{t_n\}\) is bounded and \(t_n\rightarrow 1\) as \(n\rightarrow +\infty \). Therefore, we have \(I(u_n)\rightarrow m_{\infty }\) as \(n\rightarrow +\infty \) which implies that \(m\le m_{\infty }\). Thus \(m = m_{\infty }\).
In the following, we prove that m cannot be attained. If not, we suppose that there exists \(u_0\in \mathcal {N}\) such that \(I(u_0)=m\) and \(s_{u_0}u_0\in \mathcal {N}_{\infty }\) with \(s_{u_0}\in (0,1]\). With a direct calculation, we get
which leads to
Thus, \(u_0\in \mathcal {N}_{\infty }\) and \(I_{\infty }(u_0)=m_{\infty }\). Recalling that \(u_0\) must be of form (1.7) and \(u_0>0\), then
which contradicts to (4.10). Thus, m is not achieved. \(\square \)
The following corollaries can be regarded as a direct consequence of Theorem 1.3 and Proposition 4.1.
Corollary 4.2
Let \(\{u_n\}\subset \mathcal {D}^{1,2}({\mathbb {R}}^N)\) be a nonnegative Palais–Smale sequence satisfying the assumptions of Theorem 1.3 with \(c\in (m,2m)\), then up to a subsequence, \(\{u_n\}\) converges to a nonnegative nontrivial solution \(\bar{u}\) of Eq. (1.10) strongly in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\).
Proof
Obviously, \(u_n\rightharpoonup \bar{u}\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) and \(\bar{u}\) is nonnegative. Since \(c\in (m,2m)\), we conclude \(k\le 1\) in (1.12). If \(\bar{u}\ne 0\) and \(k=1\), then \(c\ge 2m\) by (1.14). If \(\bar{u}= 0\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) and \(k=1\), then \(u^1\) is a nonnegative solution of Eq. (1.6). By using the property of super harmonic function, we deduce that \(u^1\) is positive and \(c=m\). This is a contradiction, since \(c\in (m,2m)\).
\(\square \)
Corollary 4.3
If \(\{u_n\}\) is a minimizing sequence for I on \(\mathcal {N}\), then there exist a sequence of points \(\{y_n\} \subset {\mathbb {R}}^N\), a sequence of positive numbers \(\{\delta _n\}\subset {\mathbb {R}}^{+}\) and \(\{w_n\}\subset \mathcal {D}^{1,2}({\mathbb {R}}^N)\) such that
where
and \(w_n \rightarrow 0\) strongly in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\).
Now, we prove the existence of positive solutions of Eq. (1.10) via classical Linking Theorem. A direct calculation shows that
In order to build a suitable min–max sequence for our problem, we introduce a barycenter type function and define \(\mathcal {G} : \mathcal {D}^{1,2}({\mathbb {R}}^N)\rightarrow {\mathbb {R}}^N \times {\mathbb {R}}^{+}\) by
where \(\zeta (x)\) is a cut-off function such that
Moreover,
and
Lemma 4.4
If \(|y|\ge \frac{1}{2}\), then
Proof
A direct calculation shows that
Then, for each \(\varepsilon >0\), there exists \(\delta _0:=\delta _0(\varepsilon )\) such that for any \(\delta \in (0, \delta _0]\),
Furthermore
Let \(\varepsilon \) be small enough such that for \(|y|\ge \frac{1}{2}\), the following property holds
Then, by (4.15), we have
Therefore, it follows by (4.16) and (4.17) that we can easily deduce that
and then we complete the proof of lemma. \(\square \)
In the sequel, we denote by
a subset of Nehari manifold \(\mathcal {N}\) and define \(c_{\mathcal {M}}:=\displaystyle \inf _{u\in \mathcal {M}}I(u)\).
Lemma 4.5
Let \(V(x)\ge 0\) and \(V(x) \in L^{\frac{N}{2}}({\mathbb {R}}^N)\). Then \(c_{\mathcal {M}}>m\).
Proof
Obviously \(c_{\mathcal {M}}\ge m\). In order to show the identity cannot hold, we shall argue by contradiction and then assume that there exists a sequence of \(\{u_n\} \subset \mathcal {M}\) such that
Moreover, for any \(n\in {\mathbb N}\),
By Corollary 4.3, we deduce that there exist a sequence of points \(\{y_n\}\subset {\mathbb {R}}^N\), a sequence of positive numbers \(\{\delta _n\}\subset {\mathbb {R}}^N\) and a sequence of functions \(\{w_n\}\subset \mathcal {D}^{1,2}({\mathbb {R}}^N)\) with \(w_n\rightarrow 0\) in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) such that \(u_n(x)=w_{n}(x)+\psi _{\delta _n,y_n}\). By the definition of \(\mathcal {G}\), we get
Therefore, by (4.18), we deduce that
There exists a subsequence \((\delta _n, y_n)\) such that one of the following cases may happen
-
(1)
\(\delta _{n}\rightarrow +\infty \) as \(n\rightarrow \infty \);
-
(2)
\(\delta _{n}\rightarrow \bar{\delta }\ne 0\) as \(n\rightarrow \infty \);
-
(3)
\(\delta _{n}\rightarrow 0 \) and \(y_n \rightarrow \bar{y}\), \(|\bar{y}|<\frac{1}{2}\) as \(n\rightarrow \infty \);
-
(4)
\(\delta _{n}\rightarrow 0 \) as \(n\rightarrow \infty \) and \(|y_n|\ge \frac{1}{2}\) for n large.
Now we prove that none of the possibilities listed above can be true. Obviously, by Lemma 4.4 and (4.19), case (4) can not happen. If (1) holds, then
which contradicts to (4.19). If (2) happens, we first assert that \(|y_n|\rightarrow +\infty \). If not, up to a subsequence, we notice that \(\psi _{\delta _n, y_n}\) would converge strongly in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\), so \(u_n\) converges strongly in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\), which is impossible by Proposition 4.1. Thus, for \(n\rightarrow +\infty \), we have
which is absurd in the sense of (4.19). If (3) is true, then for n large,
which is also impossible. Then the proof is completed. \(\square \)
In the following, we define a mapping \(\theta : \mathcal {D}^{1,2}({\mathbb {R}}^N){\setminus } \{0\} \rightarrow \mathcal {N}\) by
where \(t_u\) is the unique positive number such that \(t_u u \in \mathcal {N}\). Also we define the operator \(T:{\mathbb {R}}^N \times (0,+\infty )\rightarrow \mathcal {D}^{1,2}({\mathbb {R}}^N)\) by
Then we have the following lemma.
Lemma 4.6
Assume that \(V(x)\ge 0\) and \(V(x)\in L^{\frac{N}{2}}({\mathbb {R}}^N)\). Then for any \(\varepsilon >0\), there exists \(\delta _1=\delta _1(\varepsilon )\) and \(\delta _2=\delta _2(\varepsilon )\) (without loss of generality, we assume that \(\delta _1\le \delta _2\)) such that
for any \(\delta \in (0, \delta _1]\cup [\delta _2, +\infty )\) and \(y\in {\mathbb {R}}^N\).
Proof
Since \(V(x)\in L^{\frac{N}{2}}({\mathbb {R}}^N)\), then for any \(\varepsilon >0\), there exists \(r>0\) small enough such that
A direct calculation shows that
Thus, there exists \(\delta _1=\delta _{1}(\varepsilon )\) small enough, such that for any \(\delta \in (0, \delta _1)\),
From (4.20) and (4.22), we obtain
Using \(V(x)\in L^{\frac{N}{2}}({\mathbb {R}}^N)\) again, we assume that for any \(\varepsilon >0\), there exists \(R>0\) big enough such that
Recalling that \(\displaystyle \lim _{\delta \rightarrow +\infty }\sup _{x\in {\mathbb {R}}^N}|\psi _{\delta ,y}|=0\), then we obtain that
which implies that there exists \(\delta _2:=\delta _2(\varepsilon )>0\), such that for any \(\delta \ge \delta _2\),
In virtue of (4.24) and (4.26), we can prove that that for any \( \delta \ge \delta _2\),
Thus, combining (4.23) and (4.27) that we can conclude that
for any \(y\in {\mathbb {R}}^N \) and \(\delta \in (0,\delta _1]\cup [\delta _2, \infty )\).
For any \(\psi _{\delta ,y}\), there exists \(t_{\psi }:=t(\psi _{\delta ,y})\ge 1\) such that \(t_{\psi }\psi _{\delta ,y}\in \mathcal {N}\). With a similar argument to the proof in (4.6)–(4.8), we prove that for uniformly \(y\in {\mathbb {R}}^N\), \( t_{\psi }\rightarrow 1\) as \(\delta \rightarrow 0\) or \(\delta \rightarrow +\infty \). Thus, inspired by (4.28), for any \(\delta \in (0,\delta _1]\cup [\delta _2,+\infty )\),
\(\square \)
Lemma 4.7
Assume that \(V(x) \ge 0\) and \(V(x) \in L^{\frac{N}{2}}({\mathbb {R}}^N)\). Then for any fixed \(\delta >0\),
Proof
First, we claim that for any fixed \(\delta >0\),
Indeed, for a given \(\varepsilon >0\), we can choose some \(R>0\) large enough such that
and
Taking y with \(|y|>2R\), we see
With a similar argument as the proof in (4.6)–(4.8) again, we can also prove that \(t_{\psi }\rightarrow 1\) as \(|y|\rightarrow +\infty \), where \(t_{\psi }\) satisfies \(t_{\psi } \psi _{\delta ,y}\in \mathcal {N}\). Thus, as \(|y|\rightarrow +\infty \), by (4.30),
Thus \( \displaystyle \lim _{|y|\rightarrow +\infty } I(\theta \circ T(y,\delta ))=m\). \(\square \)
From Lemma 4.5, we can deduce that there exists some \(\sigma >0\) such that \(m+\sigma <c_{\mathcal {M}}\). In the following, we give some estimates.
Lemma 4.8
There exists \(\delta _1\in (0,\frac{1}{2})\) such that for any \(0<\delta \le \delta _1\), the following properties hold.
- (a):
-
\(I(\theta \circ T(y,\delta ))< m+\sigma ,\) for any \(y \in {\mathbb {R}}^N\);
- (b):
-
\( \big | \beta (\theta \circ T(y,\delta ))-\frac{y}{|y|}\big |<\frac{1}{4},\) for any \(y\in {\mathbb {R}}^N\) with \(|y|\ge \frac{1}{2}\);
- (c):
-
\(\vartheta (\theta \circ T(y,\delta ))<\frac{1}{2}\), for any \(y\in {\mathbb {R}}^N\) with \(|y|<\frac{1}{2}\).
Proof
(a) and (b) are easy to prove. In fact, (a) can be seen as a direct consequence of Lemma 4.6. In Lemma 4.6, we have proved that \(t_{\psi }\rightarrow 1\) as \(\delta \rightarrow 0\), which together with Lemma 4.4 yields (b). Now we only need to prove (c). A direct calculation shows that
where in the last equality we have used the fact \(\int _{{\mathbb {R}}^N{\setminus } B_1(y)}|\nabla \psi _{\delta ,0}|^2dx \rightarrow 0\) for \(|y|<\frac{1}{2}\) as \(\delta \rightarrow 0\). \(\square \)
Lemma 4.9
There exist \(\delta _2\in (\frac{1}{2}, +\infty )\) such that for any \(\delta \ge \delta _2\), the following properties hold.
- (a):
-
\(I(\theta \circ T(y,\delta ))< m+\sigma \), for any \(y \in {\mathbb {R}}^N\);
- (b):
-
\(\vartheta (\theta \circ T(y,\delta ))>\frac{1}{2}\), for any \(y\in {\mathbb {R}}^N\).
Proof
By Lemma 4.6, (a) is true. Since
and \(t_{\psi }\rightarrow 1\) as \(\delta \rightarrow +\infty \), we obtain
Hence (b) holds. \(\square \)
Lemma 4.10
There exists some \(R>0\) such that for any \(\delta \in [\delta _1,\delta _2]\), the following properties hold.
- (a):
-
\(I(\theta \circ T(y,\delta ))< m+\sigma ,\) for any \(y \in {\mathbb {R}}^N\) with \(|y|\ge R\);
- (b):
-
\(\langle \beta (\theta \circ T(y,\delta )),y\rangle >0,\) for any \(y \in {\mathbb {R}}^N\) with \(|y|\ge R\).
Proof
For any fixed \(\delta \), let \(|y|\rightarrow +\infty \) and repeating the argument in the proof of (4.6)–(4.8) again, we know \(t_{\psi }=t(\psi _{\delta ,y})\rightarrow 1\), where \(t_{\psi }\) satisfies \(t_{\psi }\psi _{\delta ,y}\in \mathcal {N}\). Using Lemma 4.7 and the compactness of \([\delta _1,\delta _2]\), we deduce that there exists some \(R_1>0\) such that
Let \(({\mathbb {R}}^N)_{y}^{+}=\{x\in {\mathbb {R}}^N:\langle x, y\rangle >0\}\) and \(({\mathbb {R}}^N)_{y}^{-}={\mathbb {R}}^N {\setminus } ({\mathbb {R}}^N)_{y}^{+}\). Since \(\delta \in [\delta _1, \delta _2]\), we assert that there exists \(R_2>0\) large enough and \(r\in (0,\frac{1}{4})\) such that the following properties holds: for any y with \(|y|\ge R_2\),
with \(|\widetilde{y}-y |=\frac{1}{2}\) and for any \(x\in B_{r}(\widetilde{y})\),
where \(K_1\) only depend on N and \(\alpha \), \(H_1\) is a positive constant. Moreover, for each \(x\in ({\mathbb {R}}^N)_{y}^{-} \),
Thus, for any y satisfying \(|y|\ge R_2\), we have
where \(H_{3}\) is a positive constant. Taking \(R=\max \{R_1,R_2\} \), we then complete the proof. \(\square \)
In the sequel, we define a bounded domain \(D\subset {\mathbb {R}}^N \times {\mathbb {R}}\) by
where \(\delta _1\), \(\delta _2\) and R are given in Lemmas 4.8–4.10.
Lemma 4.11
Define a mapping \(\Upsilon : D\rightarrow {\mathbb {R}}^N \times {\mathbb {R}}^+\) by
Then
Proof
Consider the following homotopy
Since \(deg\big (id, D, \left( 0,\frac{1}{2})\right) =1\), then by the homotopy invariance of topological degree, we can complete the proof. In order to use the homotopy invariance of the topological degree, we must prove
For the readers’ convenience, we divide the proof into several cases and discuss them respectively.
Case 1 If \(|y|<\frac{1}{2}\) and \(\delta =\delta _1\), by Lemma 4.8(c), we know
for any \(s\in [0,1]\).
Case 2 If \(\frac{1}{2}\le |y|\le R\) and \(\delta =\delta _1\), then it follows by Lemma 4.8(b) that
Thus
Case 3 If \(|y|\le R\) and \(\delta =\delta _2\), from Lemma 4.9(b), we know that
for any \(s\in [0,1]\).
Case 4 If \(|y|=R\) and \(\delta \in [\delta _1,\delta _2]\), by Lemma 4.10(b),
for \(s\in [0,1]\). \(\square \)
Proof of Theorem 1.2
Obviously, the first part of Theorem 1.2 follows from Proposition 4.1. In order to apply the classical Linking Theorem (see [40]), we define
and
We claim that \(\mathcal {M}\) and \(\partial \mathcal {H}\) is a link, that is
- (a):
-
\(\partial \mathcal {H}\cap \mathcal {M}=\emptyset \);
- (b):
-
\(h(\mathcal {H})\cap \mathcal {M} \ne \emptyset \) for any \(h\in \Lambda =\{h\in \mathcal {C}(\mathcal {H}, \mathcal {N}): h(\partial \mathcal {H})=id\}\).
In fact, if \(u\in \theta \circ T(\partial D)\), then it follows from Lemmas 4.8(a), 4.9(a) and 4.10(a) that
which implies \(u\notin \mathcal {M}\) and we prove (a).
Next, we prove (b). In fact, for any \(h\in \Lambda \), we define a continuous mapping \(\widetilde{\eta }: D\rightarrow {\mathbb {R}}^N \times {\mathbb {R}}^{+}\) by
If \((y,\delta )\in \partial D\), then \(\theta \circ T(y,\delta )\in \partial \mathcal {H}\), hence \(h\circ \theta \circ T(y,\delta )=\theta \circ T(y,\delta )\). Therefore
By the homotopy invariance of the topological degree and Lemma 4.11, we have
which implies that there exists \((y',\delta ')\in D\) such that \(h\circ \theta \circ T(y',\delta ')\in \mathcal {M}\). Hence (b) holds.
Since \(\mathcal {N}\) is a natural constraint for I, with classical minimal arguments we obtain a Palais–Smale sequence for I at level d with
From (b) and Lemma 4.5, we have
Moreover, by definition of d and \(\mathcal {H}\), we get
As \(t_{\psi }\psi _{\delta ,y}\in \mathcal {N}\), we know that
On the other hand,
Recall that
By (4.35), we obtain that
which implies that \(t_{\psi }^{2\cdot 2_{\alpha }^{*}}\le \left( 1+\frac{\Vert V(x)\Vert _{L^{\frac{N}{2}}}}{S}\right) ^{\frac{N+\alpha }{\alpha +2}}\). Since \(\Vert V(x)\Vert _{L^{\frac{N}{2}}}<(2^{\frac{\alpha +2}{N+\alpha }}-1)S\), we have
which combining together with (4.34) and the fact that
yields \(m<d<2m\).
We claim that there exists a nonnegative \((PS)_d\) sequence of I with \(d\in (m,2m)\). In fact, we can modify the energy functional I into
Suppose \(\{u_n\}\) is a \((PS)_d\) sequence of \(\widetilde{I}\) with \(d\in (m,2m)\), then \(\{u_n\}\) is bounded in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) and
It follows that
Thus, \(\{u_n^+\}\) is a nonnegative \((PS)_d\) sequence of \(\widetilde{I}\) with \(d\in (m,2m)\).
As a direct consequence of Corollary 4.2, up to a subsequence, we may suppose that \(u_n^+\rightarrow u\) strongly in \(\mathcal {D}^{1,2}({\mathbb {R}}^N)\) , and u is a nonnegative of (1.10). Since \(V(x)\in L^{\frac{N}{2}}({\mathbb {R}}^N)\cap \mathcal {C}^{\gamma }({\mathbb {R}}^N)\) is nonnegative for some \(\gamma \in (0,1)\), by a similar argument as the proof of Proposition 2.4, one can deduce that \(u\in \mathcal {C}^{2,\iota }({\mathbb {R}}^N)\) for some \(0<\iota <\gamma \). Then, the positivity of u follows from the strong maximum principle. Thus we complete the proof of the Theorem 1.2. \(\square \)
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Acknowledgements
The authors would like to thank the anonymous referees for carefully reading this paper and making valuable comments and suggestions. This research was partially supported by the NSFC (Nos. 11831009, 11701203), the program for Changjiang Scholars and Innovative Research Team in University No. IRT_17R46 and CCNU18CXTD04. Hu is also supported by the Project funded by China Postdoctoral Science Foundation (No. 2018M643389). Shuai is also supported by the NSF of Hubei Province (No. 2018CFB268).
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Guo, L., Hu, T., Peng, S. et al. Existence and uniqueness of solutions for Choquard equation involving Hardy–Littlewood–Sobolev critical exponent. Calc. Var. 58, 128 (2019). https://doi.org/10.1007/s00526-019-1585-1
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DOI: https://doi.org/10.1007/s00526-019-1585-1