Abstract
In this paper we are interested in the following critical Hartree equation
where \(N\ge 4\), \(0<\mu \le 4\), \(\varepsilon >0\) is a small parameter, \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), and \(2_{\mu }^*=\frac{2N-\mu }{N-2}\) is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. By establishing various versions of local Pohozaev identities and applying blow-up analysis, we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation. Next we prove the local uniqueness of the blow-up solutions that concentrates at the non-degenerate critical point of the Robin function for \(\varepsilon \) small.
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1 Introduction and main results
In a celebrated paper [4], Brezis and Nirenberg introduced the following Sobolev critical problem
where \(2^{*}=\frac{2N}{N-2}\) with \(N\ge 3\), \(\varepsilon >0\) is a real positive parameter, \(\Omega \) is a smooth bounded domain be \(\mathbb {R}^N\). The existence of a positive solution \(u_{\varepsilon }\) to (1.1), i.e., a solution which achieves the infimum
has been proved by Brezis and Nirenberg in [4] provided \(\varepsilon \in (0,\lambda _1)\) in dimension \(N\ge 4\) and when \(\varepsilon \in (\lambda _{*},\lambda _1)\) in dimension \(N=3\), where \(\lambda _1\) is the first eigenvalue of \(-\Delta \) with Dirichlet boundary condition and \(\lambda _{*}\in (0,\lambda _1)\) depends on the domain \(\Omega \). On the other hand, when \(\varepsilon =0\), problem (1.1) becomes much more delicate. Pohozaev first proved in [33] that (1.1) does not have any solutions in the case where \(\Omega \) is a star-shaped domain. Bahri and Coron [2] proved that (1.1) has a solution when \(\Omega \) has a nontrivial topology and \(\varepsilon =0\).
As \(\varepsilon \rightarrow 0\), Rey [34] proved that if a solution \(u_{\varepsilon }\) of (1.1) satisfies
where \(\delta _x\) denotes the Dirac mass at x and S the best Sobolev constant defined by
Then \(x_{0}\in \Omega \) is a critical point of Robin function \(\mathcal {R}(x)\) (see (1.4)). Conversely, if \(N\ge 5\) and \(x_0\) is a nondegenerate critical point of \(\mathcal {R}(x)\), then for \(\varepsilon \) sufficiently small (1.1) has a family of solutions \(u_\varepsilon \) satisfying (1.2). Let \(\Omega \) be a smooth bounded domain in \(\mathbb {R}^N\) and \(N\ge 4\), Rey [35] (independently by Han [27]) considered
and studied the localtion of blow-up point for solutions to (1.3) and blowing up rate, namely,
where \(\omega _N\) is a measure of the unit sphere of \(\mathbb {R}^N\), \(\rho _N=\int _{0}^{\infty }\frac{r^{N-1}}{(1+r^2)^{N-2}}dr\) and
is called the Robin function of \(\Omega \) at point x. The Green’s function of the Dirichlet problem for the Laplacian is then defined by
and it satisfies
Musso and Pistoia in [31] and Bahri, Li and Rey in [3] studied existence of solutions which blow-up at \(k\ge 1\) different points of \(\Omega \).
To investigate the uniqueness of the blow-up solutions, Grossi in [24] proved the uniqueness of the solutions to (1.1) under suitable assumptions on the the domain \(\Omega \), see also [25]. If \(N\ge 5\) and for \(\varepsilon \) small enough, Cerqueti in [12] proved that if the domain \(\Omega \) is symmetric with respect to the coordinate hyperplanes \({x_k=0}\) and convex in the \(x_k\)-directions, there exists a unique solution \(u_{\varepsilon }\) of (1.3) with the property that
and this solution is nondegenerate. Later inspired by Li in [30], Cerqueti and Grossi in [6] follow closely the line of [30] for the blow-up analysis which be used to prove the uniqueness result for the solutions of (1.3), and they proved that all solutions of (1.3) satisfy the property (1.6) under the same hypothesis on the domain \(\Omega \). In [22], Glangetas considered the problem (1.3) and it is shown that if \(N\ge 5\), the uniqueness of solutions \(u_\varepsilon \) of (1.3) with the property that (1.2) for \(\varepsilon \) small enough, where \(x_0\) is a nondegenerate critical point of Robin function \(\mathcal {R}(x)\). Recently, considering the uniqueness result of Glangetas in [22], Cao, Luo and Peng [9] proved that if \(\varepsilon \) is small, problem (1.1) has a unique solution provided the domain \(\Omega \) is convex and \(N\ge 6\). For other related results, we refer the readers to [7, 8, 10, 16, 26] and their references for the existence and uniqueness of solutions for nonlinear elliptic equations.
There is wide literature about the study of the asymptotic behavior of the solutions for the almost critical problem
Atkinson and Peletier [1] studied the asymptotic behavior of subcritical solutions \(u_\varepsilon \) to (1.7).
Brezis and Peletier [5] used the method of PDE to obtain the same results as that in [1] for the spherical domains. Wei in [38] further locate the blow-up point \(x_0\) and to give a precise asymptotic expansion of the least energy solutions for problem (1.7). Rey in [36] and Musso and Pistoia in [32] proved, for \(\varepsilon >0\) small enough, a positive solutions with two positive blow-up points provided the domain \(\Omega \) have a small hole. For \(\varepsilon <0\), Del Pino, Felmer and Musso in [14] established a positive solutions which blows-up at two positive points when the domain \(\Omega \) have a hole and for \(\varepsilon \) small enough. Del Pino, Felmer and Musso in [15] found solutions with three or more positive blow-up points under suitable assumptions on the domain \(\Omega \). Towers of positive bubbles for problem (1.7) were constructed by Del Pino, Dolbeault and Musso in [13] under suitable assumptions on the nondegeneracy of Robin’s function \(\mathcal {R}(x)\) and Green’s function.
In this paper we are interested in the following critical Hartree equation
where \(N\ge 4\), \(0<\mu \le 4\), \(\varepsilon >0\) is a small parameter, \(\Omega \) is a smooth and bounded domain in \(\mathbb {R}^N\) and the exponent \(2_{\mu }^*:=\frac{2N-\mu }{N-2}\) is critical in the sense of the Hardy–Littlewood–Sobolev inequality. To under the critical growth of the nonlocal problem, we need to recall the famous Hardy–Littlewood–Sobolev inequality.
Proposition 1.1
( [29]) Let \(\theta ,r>1\) and \(0<\mu <N\) with \(\frac{1}{\theta }+\frac{1}{r}=2-\frac{\mu }{N}\). Let \(f\in L^\theta (\mathbb {R}^N)\) and \(g\in L^r(\mathbb {R}^N)\), there exists a sharp constant \(C(\theta ,r,\mu ,N)\) independent of f, g, such that
If \(\theta =r=\frac{2N}{2N-\mu }\), then
There is equality in (1.9) if and only if \(f\equiv (const.)g\) and
for some \(A\in \mathbb {C}\), \(\lambda \in \mathbb {R}\setminus \{0\}\) and \(z\in \mathbb {R}^N\).
According to Proposition 1.1, the functional
is well defined in \(H^1(\mathbb {R}^N)\times H^1(\mathbb {R}^N)\) if \(\frac{2N-\mu }{N}\le p\le \frac{2N-\mu }{N-2}\). Here, it is quite natural to call \(\frac{2N-\mu }{N}\) the lower Hardy–Littlewood–Sobolev critical exponent and \(2_{\mu }^{*}:=\frac{2N-\mu }{N-2}\) the upper Hardy–Littlewood–Sobolev critical exponent. In the following, we use \(S_{H,L}\) to denote best constant defined by
In this way, we know that (1.13) is closely related to the nonlocal Euler-Lagrange equation
For the critical nonlocal equation (1.11), Du and Yang in [17] and Guo, Hu, Peng and Shuai in [23] studied equation (1.11) with critical exponent \(\frac{2N-\mu }{N-2}\) by analyzing the corresponding integral system. They also classified the uniqueness of the positive solutions and concluded that every positive solution of (1.11) must assume the form (see [17, 20])
where (see [37]),
is the unique family of positive solutions of
In a recent paper [20], Gao and Yang considered the Hartree type Brezis-Nirenberg problem (1.13). They proved a Brezis-Nirenberg type result saying that: if \(N\ge 4\), (1.13) has a nontrivial solution for \(\varepsilon >0\); if \(N=3\), then there exists \(\lambda _{*}\) such that (1.8) has a nontrivial solution for \(\varepsilon >\lambda _{*}\), where \(\varepsilon \) is not an eigenvalue of \(-\Delta \) with homogeneous Dirichlet boundary data; if \(N\ge 3\) and \(\varepsilon \le 0\), (1.8) admits no solutions when \(\Omega \) is star-shaped. More recently, Yang and Zhao in [40] proved that the solution \(u_\varepsilon \) of (1.8) blows up exactly at a critical point of the Robin function that cannot be on the boundary of \(\Omega \) via the Lyapunov-Schmit reduction method. Existence of bubbling solutions for equation (1.8) were constructed by Yang, Ye and Zhao in [41] under suitable assumptions on the nondegeneracy of Robin’s function \(\mathcal {R}(x)\).
Naturally, one would like to know whether the local uniqueness results of the blow-up solutions hold true for the Hartree equation and if it is possible to prove the location of blow-up point for the critical problem via local Pohozaev identities. For \(N\ge 4\) and \(\varepsilon >0\) is small, one of the main purposes of this paper is to locate the blow-up point of single bubbling solutions for the following critical Hartree equation by local Pohozaev identities and blow-up analysis,
and study the local uniqueness of the blow-up solutions for problem (1.13) provided \(N\ge 6\) and \(\varepsilon \) small enough.
Before stating the main results, it is useful to introduce some notations. We denote by
We know that \(U_{z,\lambda }(x)\) is the solution of
We denote by \(PU_{z,\lambda }\) the projection of a function \(U_{z,\lambda }\) onto \(H_{0}^{1}(\Omega )\), namely,
Let us set
We remark that \(\psi _{z,\lambda }\) is a harmonic function such that
A first result that we obtain is the following.
Theorem 1.1
Let \(N\ge 4\) and \(\mu \in (0, N)\). Assume that \(u_{\varepsilon }\) is a sequence of solutions of \(H_{0}^1(\Omega )\) satisfying
with its maximum at \(x_{\varepsilon }\) and \(\lambda _{\varepsilon }^{\frac{N-2}{2}}=\max _{x\in \Omega }u_{\varepsilon }(x)=u_{\varepsilon }(x_{\varepsilon })\). Then there exists \(x_0\in \Omega \) such that as \(\varepsilon \rightarrow 0\), \(x_{\varepsilon }\rightarrow x_0\), and \(x_0\) is a critical point of Robin function \(\mathcal {R}\), i.e., \(\nabla \mathcal {R}(x_{0})=0\).
Remark 1.1
The above results have been be proved by Yang and Zhao [40] by using reduction arguments under different conditions, in this paper we will prove this theorem via the local Pohozaev identity (2.2).
In [41], authors constructed the existence of single bubbling solutions for (1.13) via the Lyapunov-Schmit reduction method. Along with this interesting results, we will obtain a type of local uniqueness results of these. More precisely, we can prove the following result.
Theorem 1.2
Let \(N\ge 6\) and \(\mu \in (0,4)\). Assume that \(\{u_{\varepsilon }^{(j)}\}(j=1,2)\) are two families of functions of \(H_{0}^1(\Omega )\) such that \(u_{\varepsilon }^{(j)}\) is a solution of (1.13) and satisfies condition (1.16). If \(x_0\in \Omega \) is an isolated non-degenerate critical point of the Robin function \(\mathcal {R}(x)\), then there exists \(\varepsilon _0^{\prime }>0\) such that for any \(\varepsilon \in (0,\varepsilon _0^{\prime }]\), such type of solutions
are unique, that is, \(u_{\varepsilon }^{(1)}=u_{\varepsilon }^{(2)}\), \(x_{\varepsilon }^{(1)}=x_{\varepsilon }^{(2)}\), \(\lambda _{\varepsilon }^{(1)}=\lambda _{\varepsilon }^{(2)}\) and \(w_{\varepsilon }^{(1)}=w_{\varepsilon }^{(2)}\).
Remark 1.2
In a first version of the main resutls were obtained under the assumption \(\mu \) is close N or 0 and that \(N=6\) and \(\mu =4\). The same results was improved to the present version due to the work [28] by Li, Liu, Tang and Xu, since the nondegeneracy property of the limit critical Hartree equation was generalized to a wider range of the parameters.
Remark 1.3
We remark that there are some restriction on the dimension N and parameter \(\mu \), since some estimates do not work well in applying the local Pohozaev identities and applying blow-up analysis. For example, we note that in the case that \(N=6\) and \(\mu =4\), it is difficult to prove that \(c_0=0\) in Lemma 3.9 by (4.25) and (4.26) (see proof of Lemma 3.9 below). Moreover, for \(N=5\), if \(x_0\) is a nondegerate critical point of Robin function \(\mathcal {R}\), from Lemma 3.2, we have
However, by (1.17), we can not derive the estimates of (3.23) and (3.24) in Lemma 3.23.
The proof of the main results is mainly inspired by [16, 26], let \(u_{\varepsilon }^{(1)}\) and \(u_{\varepsilon }^{(2)}\) be two different positive solutions of (1.1). Set
then for any fixed \(\theta \in (0,1)\) and small \(\varepsilon \), we want to prove \(|\eta _{\varepsilon }(x)|<\theta \) for all \(x\in \Omega \), which is incompatible with the fact \(\Vert \eta _{\varepsilon }\Vert _{L^{\infty }}=1\). Compared with the local Brezis-Nirenberg problem, the appearance of nonlocal critical term in problem (1.13) brings new difficulties. For example, the corresponding local Pohozaev identities will have various new terms involving volume integral, which causes new difficulties in estimates of the order of each terms in the local Pohozaev identities precisely. To apply the blow-up analysis, we need to use some nondegeneracy results. Du and Yang in [17] showed that if \(\mu \) is close to N with \(N=3\) or 4, \(\bar{U}_{z,\lambda }\) as in (1.11) is nondegenerate in the sense that solutions of the linearized equation. Recently, Gao et al. in [19] proved that a nondegeneracy result at \(\bar{U}_{z,\lambda }\) for (1.11) when \(N=6\) and \(\mu =4\), and also proposed that the problem is an open within the remaining range of \(N,\mu \). Later, X. Li, C. Liu, X. Tang and G. Xu. in [28] gave a affirmative answers and it also completely solves the interval of the remaining exponents N and \(\mu \).
Lemma 1.1
Assume that \(N\ge 3\), \(0<\mu <N\) with \(0<\mu \le 4\), and \(\bar{U}_{z,\lambda }\) be the corresponding family of unique family of positive solutions of (1.11). Then the linearized operator of (1.11) at \(\bar{U}_{z,\lambda }\) defined by
only admits solutions in \(D^{1,2}(\mathbb {R}^N)\) of the form
where \(\bar{a}\in \mathbb {R}\), \(\textbf{b}\in \mathbb {R}^N\).
Notation. In what follows we let
as the standard norm in the Sobolev space \(H_{0}^1(\Omega )\) and \(L^q(\Omega )\)-standard norm for any \(q\in [1,+\infty )\), respectively. Moreover, \(A=o(\bar{\alpha })\) means \(A/\bar{\alpha }\rightarrow 0\) as \(\varepsilon \rightarrow 0\) and \(A=O(\bar{\alpha })\) means that \(|A/\bar{\alpha }|\le C\).
This paper is organized as follows: in Sect. 2, we first construct the local Pohozaev type identities for critical Hartree equation and give the proof of Theorem 1.1. In Sect. 3, we give the proofs of some crucial estimates for blow-up solutions and Green’s function, and completed the proof of Theorem 1.2. The proof of Theorem 1.2 requires some technical computations which are given in Sect. 5 and Appendixs A-D.
2 Local Pohozaev identities and blow-up points
The first goal of this section is to establish the local Pohozaev type identities for the critical Hartree equation. As an application of the local Pohozaev type identities, we are able to locate the blow-up points in Theorem 1.1.
2.1 Local Pohozaev type identities
Lemma 2.1
Suppose that \(u_{\varepsilon }\) be a solution of the equation (1.13). Then, for any bounded domain \(\Omega ^{\prime }\subset \Omega \), one has the following identity holds:
and
for \(j=1,\ldots ,N\), where \(\nu =\nu (x)\) denotes the unit outward normal to the boundary \(\partial \Omega ^{\prime }\).
Proof
By elliptic regularity theory, we know that the solution \(u_{\varepsilon }\) of (1.13) is of \( C^2\). Without loss of generality, we may suppose that \(x_\varepsilon =0\). Since \(u_\varepsilon \) satisfies
Then we multiply the equation (2.3) by \(\langle x,\nabla u_{\varepsilon }\rangle \) and integrating on \(\Omega ^{\prime }\), we obtain
Notice that
We calculate the first term on the right-hand side to obtain
Similarly, we can deduce
Thus we can prove that
For the second term, integration by parts, we have
On the other hand, we have
and
In view of Green’s formulas, we have
Hence by (2.4), (2.5), (2.6), (2.7), (2.8) and (2.9) imply that (2.1).
To prove (2.2), We multiply (2.3) by \(\frac{\partial u_{\varepsilon }}{\partial x_j}\) and integrating on \(\Omega ^{\prime }\), we have
Similar to the above argument, we have
Then, we can deduce
Similarly, we also have
Hence we can get
Now similar to the calculations of (2.11), we know
So, by (2.10), (2.12) and (2.13), we can prove (2.2). This finishes the proof. \(\square \)
2.2 Location of the blow-up point
We first prove the following lemma.
Lemma 2.2
Assume that \(N\ge 4\) and \(u_{\varepsilon }\) is a sequence of solutions of problem (1.13) satisfying the assumptions of Theorem 1.1. Then there holds \(\lambda _{\varepsilon }d_{\varepsilon }\rightarrow +\infty \) for \(\varepsilon \) small enough.
Proof
Assume that \(\lambda _{\varepsilon }d_{\varepsilon }\rightarrow \tilde{c}<+\infty \) as \(\varepsilon \rightarrow 0\) and \(u_{\varepsilon }\) is a solution of (1.13) with \(\lambda _{\varepsilon }^{\frac{N-2}{2}}=\max \limits _{x\in \Omega }u_{\varepsilon }(x)=u_{\varepsilon }(x_{\varepsilon })\rightarrow +\infty \) as \(\varepsilon \rightarrow 0\). Set \(v_{\varepsilon }=\lambda _{\varepsilon }^{-\frac{N-2}{2}}u_{\varepsilon }(\lambda _{\varepsilon }^{-1}x+x_{\varepsilon })\). Then \(v_{\varepsilon }(x)\) satisfies
As \(\varepsilon \rightarrow 0\), by the elliptic regularity, we have \(v_{\varepsilon }\rightarrow v\) in \(C^2_{loc}(\mathbb {R}^N_{+})\) and v satisfies
It follows from the Pohozaev identity that \(v\equiv 0\), which contradicts with \(v(0)=1\). \(\square \)
We are ready to give the estimate of \(u_{\varepsilon }\) away from \(x_{\varepsilon }\).
Lemma 2.3
Assume that \(N\ge 4\) and \(u_{\varepsilon }\) is a sequence of solutions of problem (1.13) satisfying the assumptions of Theorem 1.1 and \(x\in \Omega {\setminus } B_{R\lambda _{\varepsilon }^{-1}}(x_\varepsilon )\) for \(R>0\) is any fixed large constant. Then
and
Here \(d=|x_{\varepsilon }-x|\) and \(A_{N,\mu }=\displaystyle {\int _{ B_{\frac{1}{2}d\lambda _{\varepsilon }}(0)}\int _{B_{\frac{1}{2}d\lambda _{\varepsilon }}(0)}\frac{v_{\varepsilon }^{2_{\mu }^*}(\xi )v_{\varepsilon }^{2_{\mu }^*-1}(x)}{|x-\xi |^{\mu }}d\xi dx}\).
Proof
By the potential theory and (1.13), we have
First we remark that, as a consequence of the moving sphere method, the Talenti bubbles satisfy
(see [21][Proof Theorem 1.2] for example). Combining (1.16), (2.17) and \(G(x,z)=O\big (\frac{1}{|z-x|^{N-2}}\big )\), we know
where \(d=|x_{\varepsilon }-x|\). Similar to the above estimates, we can also obtain
Furthermore, we have
where since
On the other hand, by (1.16), \(G(x,y)=O\big (\frac{1}{|y-x|^{N-2}}\big )\) and the definition of \(U_{x_{\varepsilon },\lambda _{\varepsilon }}\), we can deduce
It follows from (2.16)–(2.21) that the inequality (2.14).
To prove (2.15), we know
Similar to estimate of \(u_{\varepsilon }(x)\), we can also obtain the inequality (2.15). Hence we finish the proof of Lemma 2.3. \(\square \)
We are going to prove Theorem 1.1 by applying Lemmas 2.2, 2.3 and the local Pohozaev identity (2.1).
Proof of Theorem 1.1
We will prove the theorem by excluding the case \(x_{0}\in \partial \Omega \). In fact, takeing \(d_{\varepsilon }=\frac{1}{2}d(x_{\varepsilon },\partial \Omega )\), and by Lemma 2.2, we have
Then, by repeating the similar calculations of (A.3) in Lemma A.5, we know
By the Hardy–Littlewood–Sobolev inequality, we have
Also, we have
In view of Lemma 2.3, we know the estimates (2.14) and (2.15) hold on \(\partial B_{d_{\varepsilon }}(x_{\varepsilon })\). By(2.24) and (2.25), taking \(\Omega ^{\prime }=B_{d_{\varepsilon }}(x_\varepsilon )\) in the local Pohozaev identity (2.2) in Lemma 2.1, we have
Since we have the identity (see [11])
then we know
However, recall the following estimate established in [11, 34]
where \(\tilde{x}\in \partial \Omega \) is the unique point, satisfying \(d(x_{\varepsilon },\partial \Omega )=|x_{\varepsilon }-\tilde{x}|\). The estimates in (2.27) and (2.28), lead to a contradiction as \(\varepsilon \rightarrow 0\) immediatelly.
From the above arguments, we know there must hold \(x_0\in \Omega \). We have the following estimate that its proof has been postponed to Lemma A.5 in the Appendix,
By Lemma 2.3 and (2.29), we get by taking \(\Omega ^{\prime }=B_{\delta }(x_\varepsilon )\) in the local Pohozaev identity (2.2) in Lemma 2.1,
On the other hand, by Hardy–Littlewood–Sobolev inequality, we can also find
It follows from (2.2) and (4.14) that
Hence (2.31) and (2.26) imply that
This means that \(\nabla \mathcal {R}(x_{0})=0\) as \(\varepsilon \rightarrow 0\). Thus the conclusion follows. \(\square \)
3 Local uniqueness of the blow-up solutions
3.1 Estimates for blow-up solutions and Green’s function
Before we prove that local uniqueness of such type of solutions, we need some preparations. The following lemma plays a crucial role.
Lemma 3.1
Assume that \(N\ge 6\), \(\mu \in (0,4)\) and \(u_{\varepsilon }\) is a sequence of solutions of problem (1.13) in \(H_{0}^1(\Omega )\). Then we have
and
where \(A_{N,\mu }\) from Lemma 2.3 and \(d=|x_{\varepsilon }-x|\).
Proof
We know that the solution of (1.13) can be rewritten as:
From the estimate in (2.18), we know
and
Since
then we know
where
by \(G(x,x_{\varepsilon })=G(x_{\varepsilon },x)\). Similarly, we can calculate that
Finally, similar to the calculation of (2.22), from \(\lambda _{\varepsilon }\sim \varepsilon ^{-\frac{1}{N-4}}\) from [41, subsection 2.2], we obtain
Then (3.3), (3.4), (3.5), (3.6) and (3.7) imply that (3.1). Finally, we get (3.2) from the fact that
Hence the proof is finished. \(\square \)
Lemma 3.2
Assume that \(N\ge 6\), \(\mu \in (0,4)\) and \(u_{\varepsilon }\) is a solution of (1.13). Then we have
and
where \(A_0\) is a strictly positive constant.
Proof
Similar to the arguments of (2.32), by applying the Pohozaev identity in Lemma 2.1 and Lemma 3.1, we can obtain (3.9) by taking \(\Omega ^{\prime }=B_{\tau }(x_\varepsilon )\). Next we shall prove (3.10). By Lemma 2.3, we find
and
By (3.11) and (3.12), taking \(\Omega ^{\prime }=B_{\tau }(x_\varepsilon )\) in the local Pohozaev identity (2.1), we obtain
Similarly, we can also calculate that
Inserting (3.11) and (3.12) into the Pohozaev identity (2.1), we know
By applying the following identity (see [9])
we get from (3.13) that
On the one hand, by Lemma A.5, we know
On the other hand, by \(PU_{x_{\varepsilon },\lambda _{\varepsilon }}\le U_{x_{\varepsilon },\lambda _{\varepsilon }}\), we have
Therefore, together with (3.14), (3.15) and (3.16), we can deduce
which implies that (3.10) is true. \(\square \)
3.2 The local uniqueness result
The purpose of this subsection is devoted to complete the proof of Theorem 1.2. There are some prelimilaries to be done before we go into the proof. First of all, we let \(u_{\varepsilon }^{(1)}\) and \(u_{\varepsilon }^{(2)}\) be two different solutions of (1.1). We will use \(x_{\varepsilon }^{(j)}\) and \(\lambda _{\varepsilon }^{(j)}\) to denote the center and the height of the bubbles appearing in \(u_{\varepsilon }^{(j)}(j=1,2)\), respectively.
Let
then \(\eta _{\varepsilon }(x)\) satisfies \(\Vert \eta _{\varepsilon }\Vert _{L^{\infty }}\)=1 and
where
with
Lemma 3.3
For \(N\ge 6\), \(\mu \in (0,4]\), it holds that
Proof
First we remark that
Combining (3.9) and \(x_0\) is a nondegerate critical point of Robin function \(\mathcal {R}\), we see that for \(N\ge 6\)
A direct computations, we get (3.21) from (3.10) and (3.22). \(\square \)
Lemma 3.4
For \(N\ge 6\), \(\mu \in (0,4)\), it holds that
and
Proof
In view of Lemma 3.3, we first note that
which implies that
Then (3.26) can deduce that (3.23) and (3.24). \(\square \)
From [39], we have the following estimate:
Lemma 3.5
For any constant \(0<\sigma \le N-2\), there is a constant \(C>0\) such that
Lemma 3.6
For any constant \(\sigma \ge N-2-\frac{\mu }{2}\) and \(\mu \in (0,4]\), there is a constant \(C>0\) such that
Proof
We just need to obtain the estimate for \(|y|\ge 2\), the other is similar. Let \(d=\frac{1}{2}|y|\). Then we have
for any \(\sigma >N-2-\frac{\mu }{2}\). And we have
Assume that \(x\in \mathbb {R}^N\setminus \big (B_d(0)\cup B_{d}(y)\big )\). Then we know
Hence by a direct computation, we have
for any \(\sigma \ge N-2-\frac{\mu }{2}\). This finishes the proof. \(\square \)
Lemma 3.7
For \(\eta _{\varepsilon }(x)\) defined by (3.18), we have
where \(\delta >0\) is a any small fixed constant.
Proof
By the potential theory, we know
where
Firstly, we can deduce
Combining \(|\eta _{\varepsilon }|\le 1\), (1.16), (2.17) and Lemma 3.5, then we get
Next repeating the above process, we know
Then we can proceed as in the above argument for finite number of times to prove
Next, we find
Hence by Lemma 3.6 and Hardy–Littlewood–Sobolev inequality, we can calculate that for \(d(\Omega ):=diam(\Omega )\)
Finally, we have
Therefore, together with the estimates of \(\eta _{\varepsilon ,1}\) and \(\eta _{\varepsilon ,2}\), we can deduce
Hence (3.27) can be deduced by (3.32). \(\square \)
According to the above nondegeneracy result in Lemma 1.1, we have the following crucial lemma.
Lemma 3.8
Suppose that the exponents N, \(\mu \) satisfy the assumptions of Theorem 1.2. Let \(\tilde{\eta }_{\varepsilon }(x)=\eta _{\varepsilon }\big (\frac{x}{\lambda _{\varepsilon }^{(1)}}+x_{\varepsilon }^{(1)}\big )\). Then we have that
where \(c_k\), \(k=1,\ldots ,N\) are some constants and
Proof
Since \(|\tilde{\eta }_{\varepsilon }|\le 1\), by the regularity theorem [18], we know that
for any fixed large \(\tilde{R}\) and \(\alpha \in (0,1)\). Hence we may assume that \(\tilde{\eta }_{\varepsilon }\rightarrow \tilde{\eta }_0\) in \(C^1(B_{\tilde{R}}(0))\) for any large \(\tilde{R}>0\). Now by a direct calculation, we have
where
Then for any \(\varphi (x)\in C_{0}^{\infty }(\mathbb {R}^N)\) with \(supp~\varphi (x)\subset B_{\lambda _{\varepsilon }^{(1)}\delta }(x_{\varepsilon }^{(1)})\) for a small fixed \(\delta \), we have
In view of the elementary inequality (A.2) in Appendix A, we know
where
By the definition of \(U_{x_{\varepsilon }^{(1)},\lambda _{\varepsilon }^{(1)}}\), Lemma 3.4 and Lemma A.5, we have
And we have the following rest other estimates for which the proof is left in Lemma A.6 in Appendix A:
Now similar to the calculations of (3.35), by inequality (A.2), we can deduce
On the other hand, from \(\Vert \eta _{\varepsilon }\Vert =1\), we have
Consequently, in view of (3.34)-(3.39), we obtain
Taking \(\varepsilon \rightarrow 0\) in (3.40), we find that \(\tilde{\eta }_0\) satisfies
From the non-degeneracy results of Lemma 1.1, which gives \(\tilde{\eta }_0=\sum \nolimits _{k=0}^{N}c_k\phi _k\). Hence the conclusion (3.33) follows by (3.40) and (3.41). \(\square \)
The proof of the following lemma is postponed to Sect. 4.
Lemma 3.9
Suppose that the exponents N, \(\mu \) satisfy the assumptions of Theorem 1.2, there holds
where \(c_k\) are the constants in Lemma 3.8.
We are going to prove Theorem 1.2 by using Lemmas 3.8 and 3.9.
Proof of Theorem 1.2
From (3.32), we find that
which means that for any fixed \(\theta \in (0,1)\) and small \(\varepsilon \), there exists \(\tilde{R}>0\) such that
Also for above fixed \(\tilde{R}\), in view of Lemma 3.9, we know
Then for any fixed \(\theta \in (0,1)\) and small \(\varepsilon \), we can deduce that \(|\eta _{\varepsilon }(x)|<\theta \) for all \(x\in \Omega \). This is a contradiction to \(\Vert \eta _{\varepsilon }\Vert _{L^{\infty }}=1\). So \(u_{\varepsilon }^{(1)}=u_{\varepsilon }^{(2)}\) for small \(\varepsilon \). This finishes the proof of Theorem 1.2. \(\square \)
4 Proof of Lemma 3.9
This section is devoted to the proof of Lemma 3.9.
Lemma 4.1
For \(N\ge 6\) and \(\mu \in (0,4)\), let \(\eta _{\varepsilon }(x)\) be the function defined by (3.18). Then we have the following estimate:
where \(\delta >0\) is any small fixed constant, \(\partial _{k}G(z,x)=\frac{G(z,x)}{\partial z_k}\),
Proof
By the potential theory and (3.19), we have
According to Lemma 3.7, for any \(z\in \Omega {\setminus } B_{2\delta }(a_{\varepsilon }^{(1)})\), we obtain the estimate of the third term in (4.6) as
Decomposing the first term of (4.6) by
We are going to estimate \(G_1\), \(G_2\) and \(G_3\), respectively. By using Hardy–Littlewood–Sobolev inequality, Lemma 3.7 and (3.28), then we have
where \(A_{\varepsilon }^{(1)}\) and \(B_{\varepsilon ,k}^{(1)}\) are defined in (4.2) and (4.4). Moreover, we can also find
Analogously, we also have
For the second term in (4.6), we decompose it by
Similar to the estimate for \(G_1\), by Hardy–Littlewood–Sobolev inequality and the fact \(|\eta _{\varepsilon }(x)|\le 1\), (3.28) and (3.31), a direct calculation shows that
where \(A_{\varepsilon }^{(2)}\) and \(B_{\varepsilon ,k}^{(2)}\) are defined in (4.3)and (4.5). By Lemma 3.7, we can calculate that
And similar to the estimate of (4.8), we can also find
where we using \(\frac{2N(N-\mu +2)}{2N-\mu }>N\) and \(\frac{2N(N-2)}{2N-\mu }<N\). Combining (4.6)-(4.7) and estimates of \(G_1\), \(G_2\), \(G_3\), \(L_1\), \(L_2\), \(L_3\), then we get
in the last step we have used \(\varepsilon =O\big (\frac{1}{(\lambda _{\varepsilon }^{(1)})^{N-4}}\big )=O\big (\frac{1}{(\lambda _{\varepsilon }^{(1)})^{2}}\big )\).
On the other hand, from (4.6), we obtain
Similar to the above estimates of \(\eta _{\varepsilon }(x)\), we know for \(N\ge 6\),
By Hardy–Littlewood–Sobolev inequality and the fact that \(|\eta _{\varepsilon }(x)|\le 1\), Lemma 3.7, (3.28) and (3.31), then we can get
and
Therefore, we deduce
According to the above argument of \(\eta _{\varepsilon }(x)\) and \(\frac{\partial \eta _{\varepsilon }(x)}{\partial x_i}\). Then we can finish the proof of Lemma 4.1. \(\square \)
Lemma 4.2
Assume that \(N\ge 6\), \(\mu \in (0,4)\) and \(u_{\varepsilon }^{(j)}\) with \(j=1,2\) be the solutions of (1.1). Then we have
where \(A_{N,\mu }\) is from Lemma 2.3.
Proof
Firstly, in view of Lemma 3.1, we know that (4.9) holds for \(j=1\) and
By a direct calculate shows that
Since \(B_{\delta }(x_\varepsilon ^{(1)})\subset B_{2\delta }(x_\varepsilon ^{(1)})\) for small \(\varepsilon \), we deduce that (4.9) for \(j=2\) from Lemma 3.3 and (4.10). \(\square \)
Lemma 4.3
For \(\eta _{\varepsilon }(x)\) defined by (3.18), we have the following pohozaev identities:
and
where \(\Omega ^{\prime }\subset \Omega \) is a smooth domain, \(\nu =\nu (x)\) denotes the unit outward normal to the boundary \(\partial \Omega ^{\prime }\) and
Proof
In view of Lemma 2.1, taking \(u_{\varepsilon }=u_{\varepsilon }^{(j)}\) with \(j=1,2\) in (2.2), and then making a difference between those respectively. By a direct calculations, we can obtain (4.11). Similarly, taking \(u_{\varepsilon }=u_{\varepsilon }^{(j)}\) with \(j=1,2\) in (2.1), and then making a difference between those respectively, we can also derive that (4.12). \(\square \)
Now we are ready to prove Lemma 3.9 by using the local Pohozaev identities.
Proof of Lemma 3.9
We divide the argument into two steps: Step 1. We prove that \(c_k=0,k=1,\ldots ,N\). We define the following quadratic form
For \(N\ge 6\), taking \(\Omega ^{\prime }=B_{\tau }(x_{\varepsilon }^{(1)})\) in (4.11), by (4.1) and (4.9), we know
We next estimate \(A_{\varepsilon }^{(1)}\) and \(A_{\varepsilon }^{(2)}\), respectively. In fact,
since
Also, we have
by \(\frac{2N(N-\mu +2)}{2N-\mu }>N\). On the other hand, from [11], we know
Hence it follow from (3.9) that
Next we are going to estimate each term of the right hand side of (4.11) with \(\Omega ^{\prime }=B_{\tau }(x_{\varepsilon }^{(1)})\). We define
Firstly, we can deduce
Then we have
Together with (2.17), (3.25) and (3.27), we obtain
and
Similar to the above estimates, we can also prove
Moreover, we can find
This means that \(P_4=P_6=0\). Furthermore, note that \(\varepsilon =O\big (\frac{1}{(\lambda _{\varepsilon }^{(1)})^{N-4}}\big )=O\big (\frac{1}{(\lambda _{\varepsilon }^{(1)})^{2}}\big )\) if \(N\ge 6\), so we have
Hence we know that
Then it follows from (4.13) that
Using the estimate (see [11])
and \(x_0\) is a nondegenerate critical point of Robin function \(\mathcal {R}(x)\), we see that
On the other hand, we consider that the estimates of \(B_{\varepsilon ,l}^{(1)}\) and \(B_{\varepsilon ,l}^{(2)}\) in (4.4)-(4.5). Using the elementary inequality (A.2) in Appendix A, then we know that
where
Then by Lemmas B.1, B.2 and B.3 in Appendix B, we get
Noting that
where
Then by Lemma C.1 in Appendix C, we get
Thus by (4.18), (4.21) and (4.24) imply \(c_k=0\), \(k=1,2,\ldots ,N\).
Step 2. We prove that \(c_0=0\). First we define the following quadratic form
Taking \(\Omega ^{\prime }=B_{\tau }(x_{\varepsilon }^{(1)})\) in (4.12), from (4.1) and (4.9), we have
Since we have the estimate (see [11])
which implies that
Note that by (3.15), we know
On the other hand, from Lemma D.1 in Appendix D, we can find
A direct calculation, we can also find
Therefore, together with the above estimates, we can deduce
From Lemma D.2 in Appendix D, we know
As a result,
Notice that, from the proof of Lemma 3.9 in Sect. 3, we can find the basic estimate
Then (4.25) and (4.26) imply that \(c_0=0\). This finishes the proof of Lemma. \(\square \)
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Acknowledgements
The authors would like to thank the anonymous referee for his/her useful comments and suggestions which help to improve the presentation of the paper greatly.
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Appendices
Appendix A. Estimates of \(A_{N,\mu }\) and \(\mathcal {F}_2\), \(\mathcal {F}_3\), \(\mathcal {F}_{4}\) in (3.35)
In this section, we give that have been used in the previous sections. Let recall that
Some basic estimates as follow:
Lemma A.1
where where \(d=dist(x, \partial \Omega )\) is the distance between x and the boundary of \(\Omega \).
Proof
This follows from the definition of \(U_{z,\lambda }\), \(PU_{z,\lambda }\), \(\psi _{z,\lambda }\) and direct computations. See also [34]. \(\square \)
Lemma A.2
It holds
where \(\delta >0\) is any small fixed constant.
Proof
For a proof of this lemma, we refer to [34]. \(\square \)
Lemma A.3
It holds
Proof
See Lemma 4.1 in [40]. \(\square \)
Lemma A.4
For any \(a>0\), \(b>0\), one has
Proof
This follows from a direct calculation. \(\square \)
Lemma A.5
For \(N\ge 4\), \(\mu \in (0,4]\), it holds
where \(A_{N,\mu }\) and \(A_{H,L}\) from (1.14) and Lemma 2.3, respectively.
Proof
We have
where
Combining (A.2) and a direct calculation shows that
where the estimate of (A.5) follows by the following some computations. Firstly we remark that
since
and
Secondly, we have
Moreover, we deduce
and
Similar to the calculation of (A.7), (A.8) and (A.9), we find
Combining (A.6) and (A.7)-(A.9), the estimate (A.5) is reached.
Using \(|u_{\varepsilon }(x)|\le CU_{x_{\varepsilon },\lambda _{\varepsilon }}\), we compute
And similar to the estimate of (A.10), we can also obtain
Then (A.4), (A.5) and (A.10)-(A.11) imply that (A.3). \(\square \)
Lemma A.6
For any fixed small \(\delta >0\), it holds
Proof
Notice that
Let us write \(\mathcal {F}_2=\mathcal {F}_{2,1}+\mathcal {F}_{2,2}\). Now by Lemma 3.4 and (A.1), we can calculate that
Next, similar to the calculations of (A.12), by Lemma A.2, we can also get
Hence we prove that \(\frac{1}{(\lambda _{\varepsilon }^{(1)})^{N-\mu +2}}\mathcal {F}_{2}=o\Big (\frac{1}{\lambda _{\varepsilon }^{(1)}}\Big )\). Analogously, we have
This finishes the proof. \(\square \)
Appendix B. Estimates of \(\mathcal {G}_{1}, \mathcal {G}_{2}, \mathcal {G}_{3}\) and \(\mathcal {G}_{4}\) in (4.20)
Lemma B.1
For any \(N\ge 6\) and \(\mu \in (0,4)\), it holds that
Proof
In view of \(PU_{z,\lambda }=U_{z,\lambda }-\psi _{z,\lambda }\), we know
Then \(\mathcal {G}_1\) can be written as follows:
where
Combining (2.17), (3.33), (3.23) and oddness of the function, we can prove that as \(\varepsilon \rightarrow 0\)
Together with (3.23), (A.1), Lemma A.1, oddness of the function, Hardy–Littlewood–Sobolev, Hölder and Sobolev inequalities, we can prove
by
where we have used \(\frac{2N(N-\mu +2)}{2N-\mu }>N\) and \(\frac{2N(N-\mu +4)}{2N-\mu }>N+\frac{4N}{2N-\mu }\).
and
And analogously, from \(0\le \psi _{x_{\varepsilon }^{(1)},\lambda _{\varepsilon }^{(1)}}\le U_{x_{\varepsilon }^{(1)},\lambda _{\varepsilon }^{(1)}}\), we have
since \(\frac{2N(N-\mu +2)}{2N-\mu }>N\) and \(\frac{2N(N-\mu +4)}{2N-\mu }>N+\frac{4N}{2N-\mu }\). Then (B.2), (B.3), (B.4) and (B.5) imply (B.1). \(\square \)
Lemma B.2
For any \(N\ge 6\) and \(\mu \in (0,4)\), it holds that
Proof
Firstly, \(\mathcal {G}_2\) can be written as follows:
where
Now by (3.23), and Lemma A.1, we have
by
and
Next similar to the calculations of \(\mathcal {A}_{2,1}\), we know
Then (B.7), (B.8) and (B.9) imply (B.6). \(\square \)
Similar to the proof of Lemmas B.1 and B.2, we can find following two estimates.
Lemma B.3
For any \(N\ge 6\) and \(\mu \in (0,4)\), it holds that
Appendix C. Estimates of \(\mathcal {H}_{1}, \mathcal {H}_{2}\) and \(\mathcal {H}_{3}\) in (4.23)
Lemma C.1
For any \(N\ge 6\) and \(\mu \in (0,4)\), it holds that
Proof
Firstly, let us write \(\mathcal {B}_1\) \( \mathcal {H}_1=\mathcal {H}_{1,1}+O\big (\mathcal {H}_{1,2}\big ), \) where
Now, let us write
Note that, we have
which imply
On the other hand, we divide our argument into three cases: \(\mathbf {(1).}\) For \(0<\mu <2\), and \(\frac{2N(N-\mu +2)}{2N-\mu }>\frac{2N}{2N-\mu }+N\),
Using Hardy–Littlewood–Sobolev, Hölder inequality and (A.1), then we have
\(\mathbf {(2).}\) For \(\mu =2\), and \(\frac{2N(N-\mu +2)}{2N-\mu }=\frac{2N}{2N-\mu }+N\),
Using the definition of \(\mathcal {H}_{1,1,3}\) and (A.1), we can also obtain
\(\mathbf {(3).}\) For \(2<\mu <4\), and \(\frac{2N(N-\mu +2)}{2N-\mu }<\frac{2N}{2N-\mu }+N\),
Using the definition of \(\mathcal {H}_{1,1,3}\) and (A.1), we can also get
Thus, from (C.4), (C.5) and (C.6) imply \(\mathcal {H}_{1,1}=o\big (\frac{1}{(\lambda _{\varepsilon }^{(1)})^{N-1}}\big ).\) Similarly, we can also prove
Similar to the above argument of \(\mathcal {H}_{1}\), we can also get
Then the conclusion follows by the above estimates. \(\square \)
Appendix D. Estimates of \(A_{\varepsilon }^{(1)}\) and \(A_{\varepsilon }^{(2)}\) in (4.2)-(4.3) and \(\text {RHS of}\) (4.12) when \(\Omega ^{\prime }=B_{\tau }(x_{\varepsilon }^{(1)})\)
Lemma D.1
For any \(N\ge 6\) and \(\mu \in (0,4)\), it holds that
Proof
The proof is similar to that of Lemmas B.1, B.2 and B.3. Then we can estimate (D.1) by (2.17), (3.23), (3.24), (3.31), (3.27), (3.33), (4.2), (4.3) and (A.3). \(\square \)
Lemma D.2
For any \(N\ge 6\) and \(\mu \in (0,4)\), it holds that
Proof
Taking \(\Omega ^{\prime }=B_{\tau }(x_{\varepsilon }^{(1)})\) in (4.12), \(\text {RHS of}\) (4.12) can be written as follows:
where
Using Lemmas 3.27, 3.8, 4.1, (3.31) and Hardy–Littlewood–Sobolev inequality, we can calculate that
and analogously
Using Hardy–Littlewood–Sobolev inequality, we obtain
Similarly, we can also obtain
Moreover, we know
Combining (3.33), then we get
The conclusion can be reached by the above estimates \(\mathcal {J}_1\), \(\mathcal {J}_2\), \(\mathcal {J}_3\), \(\mathcal {J}_4\), \(\mathcal {J}_5\), \(\mathcal {J}_6\), \(\mathcal {J}_7\), \(\mathcal {J}_8\), and \(\mathcal {J}_9\). \(\square \)
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Squassina, M., Yang, M. & Zhao, S. Local uniqueness of blow-up solutions for critical Hartree equations in bounded domain. Calc. Var. 62, 217 (2023). https://doi.org/10.1007/s00526-023-02551-1
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DOI: https://doi.org/10.1007/s00526-023-02551-1