Local uniqueness and non-degeneracy of blow up solutions of mean field equations with singular data

TL;DR

This paper proves local uniqueness and non-degeneracy of bubbling solutions in singular mean field equations on bounded domains, using sharp estimates and Pohozaev identities.

Contribution

It establishes the conditions for local uniqueness and non-degeneracy of blow-up solutions in singular mean field equations, advancing understanding of their behavior.

Findings

Proved local uniqueness of bubbling solutions.

Established non-degeneracy under certain conditions.

Developed sharp estimates and Pohozaev identities for analysis.

Abstract

We are concerned with the mean field equation with singular data on bounded domains. Under suitable non-degeneracy conditions we prove local uniqueness and non-degeneracy of bubbling solutions blowing up at singular points. The proof is based on sharp estimates for bubbling solutions of singular mean field equations and suitably defined Pohozaev-type identities.

Equations

$$\{\begin{array}{ll}-\Delta u_n={\rho }_n\frac{h{e}^{u_n}}{{\int }_{\Omega }h{e}^{u_n}}& \mathrm{in}\Omega ,\\ u_n=0& \mathrm{on}\partial \Omega ,\end{array}$$

$$\{\begin{array}{ll}-\Delta G(x,p)={\delta }_p& \mathrm{in}\Omega ,\\ G(x,p)=0& \mathrm{on}\partial \Omega .\end{array}$$

$$\frac{h{e}^{u_n}}{{\int }_{\Omega }h{e}^{u_n}dx}\rightharpoonup 8\pi \sum _{j=1}^m{\delta }_{q_j},$$

$$\frac{h{e}^{u_n}}{{\int }_{\Omega }h{e}^{u_n}dx}\rightharpoonup 8\pi \sum _{j=1}^m(1+{\alpha }_j){\delta }_{p_j},$$

$${\ell }_{\text{mbox}reg}(\mathbf{q})=\sum _{j=1}^m[\Delta \log h(q_j)]h(q_j)e^{G_j^{\ast }(q_j)}.$$

$$\displaystyle\mathcal{H}_{m}(x_{1},x_{2},\cdots,x_{m})=\sum_{j=1}^{m}\big{[}\log(h(x_{j}))+4\pi R(x_{j},x_{j})\big{]}+4\pi\sum_{l\neq j}^{1,\cdots,m}G(x_{l},x_{j}).$$

$$\{\begin{array}{ll}\Delta \varphi +{\rho }_n\frac{h{e}^{u_n}}{{\int }_{\Omega }h{e}^{u_n}dx}\left(\varphi -\frac{{\int }_{\Omega }h{e}^{u_n}\varphi dx}{{\int }_{\Omega }h{e}^{u_n}dx}\right)=0& \text{mbox}in\Omega ,\\ \varphi =0& \text{mbox}on\partial \Omega ,\end{array}$$

$$\Vert u_n{\Vert }_{L^{\infty }(\Omega )}\to +\infty \text{mbox}asn\to +\infty .$$

$$\ell (p_i)=\frac{2{\pi }^2}{(1+{\alpha }_i)\sin \left(\frac{\pi }{1+{\alpha }_i}\right)}{\left(\frac{(1+{\alpha }_i)}{\pi {\overline{h}}_1(p_i)}\right)}^{\frac{1}{1+{\alpha }_i}}\Delta \log h_{\ast }(p_i),$$

$$\displaystyle\mathcal{H}_{p_{i}}(x)=8\pi(1+\alpha_{i})\bigr{(}R(x,p_{i})-R(p_{i},p_{i})\bigr{)}+\bigr{(}\log\overline{h}_{1}(x)-\log\overline{h}_{1}(p_{i})\bigr{)}.$$

$${\xi }_n=\frac{u_n^{(1)}-u_n^{(2)}}{\Vert u_n^{(1)}-u_n^{(2)}{\Vert }_{L^{\infty }(\Omega )}}.$$

$$\Delta v+|x{|}^{2{\alpha }_i}{e}^v=0\text{in}{\mathbb{R}}^2.$$

$$v\left(z\right)=v_{\mu }(z)=\log \frac{8(1+{\alpha }_i{)}^2{e}^{\mu }}{(1+e^{\mu }|z{|}^{2(1+{\alpha }_i)}{)}^2},\mu \in \mathbb{R}.$$

$$L\varphi :=\Delta \varphi +\frac{8(1+{\alpha }_i{)}^2|z{|}^{2{\alpha }_i}}{(1+|z{|}^{2(1+{\alpha }_i)}{)}^2}\varphi \text{in}{\mathbb{R}}^2.$$

$$Y_{0}(z)=\frac{1-|z|^{2(1+\alpha_{i})}}{1+|z|^{2(1+\alpha_{i})}}=\frac{\partial v_{\mu}}{\partial\mu}\Big{|}_{\mu=0}.$$

$${\Xi }_n=\frac{{\varphi }_n-\frac{{\int }_{\Omega }h{e}^{u_n}{\varphi }_{n}dx}{{\int }_{\Omega }h{e}^{u_n}dx}}{{\Vert {\varphi }_n-\frac{{\int }_{\Omega }h{e}^{u_n}{\varphi }_{n}dx}{{\int }_{\Omega }h{e}^{u_n}dx}\Vert }_{L^{\infty }(\Omega )}},$$

$${\tilde{u}}_n=u_n-\log \left({\int }_{\Omega }h{e}^{u_n}dx\right),{\lambda }_n=\underset{\Omega }{\max }{\tilde{u}}_n,{\sigma }_n^{2(1+\alpha )}=e^{-{\lambda }_n},$$

$$U_n(x)={\lambda }_n-2\log (1+{\gamma }_n{e}^{{\lambda }_n}|x-p{|}^{2+2\alpha }),{\gamma }_n=\frac{{\rho }_n{\overline{h}}_1(p)}{8(1+\alpha {)}^2},$$

$${\overline{h}}_1(x)=h_{\ast }\exp (-4\pi {\overline{G}}_1(x)){\overline{G}}_1(x)=\sum _{i=2}^N{\alpha }_{i}G(x,p_i)+R(x,p),$$

$$h(x)={\overline{h}}_1(x)|x-p{|}^{2\alpha },$$

$$|{\tilde{u}}_n(x)-U_n(x)|\le C,\forall x\in B_r(p).$$

$${\rho }_n-8\pi (1+\alpha )=\ell (p)e^{-\frac{{\lambda }_n}{1+\alpha }}+O(e^{-{\lambda }_n\frac{1+{ϵ}_0}{1+\alpha }})\mathrm{as}n\to +\infty ,$$

$${\rho }_{n,1}-8\pi (1+\alpha )=\ell (p)e^{-\frac{{\lambda }_n}{1+\alpha }}+O(e^{-{\lambda }_n\frac{1+{ϵ}_0}{1+\alpha }})\mathrm{as}n\to +\infty ,$$

$${\rho }_{n,1}={\int }_{B(p,r_0)}h{e}^{{\tilde{u}}_n},\ell (p)=\frac{2{\pi }^2}{(1+\alpha )\sin \left(\frac{\pi }{1+\alpha }\right)}{\left(\frac{(1+\alpha )}{\pi {\overline{h}}_1(p)}\right)}^{\frac{1}{1+\alpha }}\Delta \log h_{\ast }(p),$$

$${ϵ}_0=2-2(1-\alpha {)}^{+}=\{\begin{array}{ll}2,& \mathrm{if}\alpha \ge 1,\\ & \\ 2\alpha ,& \mathrm{if}\alpha \in (0,1).\end{array}$$

$$R_{n,1}={\rho }_{n,1}R(x,p),$$

$$v_n={\tilde{u}}_n-(R_{n,1}(x)-R_{n,1}(p)),x\in B(p,r_0),$$

$$\{\begin{array}{ll}\Delta {\psi }_n=0,& \mathrm{in}{B}_{(}p,4r_0),\\ & \\ {\psi }_n=v_n-\frac{1}{8\pi r_0}{\int }_{|x-p|=4r_0}{v}_{n}ds& \mathrm{on}\partial B(p,4r_0).\end{array}$$

$$v_n-U_n-{\psi }_n(x)={\sigma }_n{\psi }_{n,1}\left(\frac{x-p}{{\sigma }_n}\right)+{\sigma }_n^2{\psi }_{n,2}\left(\frac{x-p}{{\sigma }_n}\right)+O({\sigma }_n^2)\mathrm{in}B(p,4r_0),$$

$${\psi }_{n,1}(y)=-\frac{2(1+\alpha )a_{n,1}}{\alpha }\frac{y_1}{1+{\gamma }_n|y{|}^{2(1+\alpha )}},y=(y_1,y_2)\in {\mathbb{R}}^2,$$

Full text

2010 Mathematics Subject classification: 35B32, 35J25, 35J61, 35J99, 82D15.

D. Bartolucci is partially supported by FIRB project ”Analysis and Beyond”, by PRIN project 2012, ERC PE1_11, ”Variational and perturbative aspects in nonlinear differential problems”, and by the Consolidate the Foundations project 2015 (sponsored by Univ. of Rome ”Tor Vergata”), ERC PE1_11, ”Nonlinear Differential Problems and their Applications”. Y. Lee is partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2018R1C1B6003403). W. Yang is partially supported by NSFC No.11801550.

Keywords: Mean field equations, uniqueness, non-degeneracy, blow up solutions, singular data.

1. Introduction

We are concerned with a sequence of solutions of the following mean field equation with singular data

[TABLE]

where $\Omega\subset\mathbb{R}^{2}$ is a smooth bounded domain, $h=h_{*}\exp(-4\pi\sum_{i=1}^{N}\alpha_{i}G(x,p_{i}))$, $p_{i}$ are distinct points in $\Omega$, $\alpha_{i}\in(0,\infty)\setminus\mathbb{N}$, $h_{*}\in C^{\infty}(\overline{\Omega})$, and $G$ is the Green function satisfying

[TABLE]

The mean field equation ${\textbf{(}\mathbf{P}_{\rho_{n}}\textbf{)}}$ (and its counterpart on compact surfaces) have been widely discussed in the last decades because of their several applications in Mathematics and Physics, such as Electroweak and Chern-Simons self-dual vortices [47, 49, 53], conformal metrics on surfaces with [50] or without conical singularities [35], statistical mechanics of two-dimensional turbulence [20] and of self-gravitating systems [52] and cosmic strings [45], and the theory of hyperelliptic curves [22] and of the Painlevé equations [24]. There are by now many results concerning existence [1, 3, 4, 5, 15, 21, 26, 28, 30, 31, 32, 36, 42], multiplicity [5, 29], uniqueness [6, 7, 10, 11, 12, 13, 14, 23, 33, 34, 40, 41, 48] and blow up analysis [2, 9, 16, 18, 17, 19, 25, 27, 37, 38, 39, 51, 54].

Our goal is to show that bubbling solutions of ${\textbf{(}\mathbf{P}_{\rho_{n}}\textbf{)}}$ blowing up at singular points $p_{i}$ are unique and non-degenerate for $n$ large enough.

Definition 1.1**.**

Let $u_{n}$ be a sequence of solutions of ${\textbf{(}\mathbf{P}_{\rho_{n}}\textbf{)}}$. We say that $u_{n}$ is a regular $m$-bubbling solution blowing up at the points $q_{j}\notin\{p_{1},\cdots,p_{N}\}$, $j=1,\cdots,m$, if,

[TABLE]

weakly in the sense of measures in $\Omega$.

We say that $u_{n}$ is a singular $m$-bubbling solution blowing up at the points $p_{j}\in\{p_{1},\cdots,p_{N}\}$, $j=1,\cdots,m$, $m\leq N$ if,

[TABLE]

weakly in the sense of measures in $\Omega$.

To state the main result and to compare it with the existing literature we introduce some notation. Let $R(x,y)=\frac{1}{2\pi}\log|x-y|+G(x,y)$ be the regular part of $G(x,y)$. For what concerns regular bubbling solutions, for $\mathbf{q}=(q_{1},\cdots,q_{m})\in\overline{\Omega}\times\cdots\times\overline{\Omega}$, we let $G_{j}^{*}(x)=8\pi R(x,q_{j})+8\pi\sum^{1,\cdots,m}_{l\neq j}G(x,q_{l})$ and

[TABLE]

For $(x_{1},\cdots,x_{m})\in\overline{\Omega}\times\cdots\overline{\Omega}$, we also define the $m$-vortex Hamiltonian,

[TABLE]

Then, by assuming suitable non-degeneracy conditions the authors in [8, 9] proved that regular $m$-bubbling solutions are unique and non-degenerate (see also [10] for an analogous result for the Gelfand equation).

Theorem A ([8, 9]). Let $u_{n}^{(1)}$ and $u_{n}^{(2)}$ be two regular $m$-bubbling solutions of ${\textbf{(}\mathbf{P}_{\rho_{n}}\textbf{)}}$, with $\rho^{(1)}_{n}=\rho_{n}=\rho^{(2)}_{n}$, blowing up at the points $q_{j}\notin\{p_{1},\cdots,p_{N}\}$, $j=1,\cdots,m$, where $\mathbf{q}=(q_{1},\cdots,q_{m})$ is a critical point of $\mathcal{H}_{m}$. Assume that,

1. (1)

$\textrm{det}(D^{2}\mathcal{H}_{m}(\mathbf{q}))\neq 0$, 2. (2)

$\ell_{\mbox{\footnotesize\emph{reg}}}(\mathbf{q})\neq 0$.

Then there exists $n_{0}\geq 1$ such that $u_{n}^{(1)}=u_{n}^{(2)}$ for all $n\geq n_{0}$. Moreover, the linearized problem at a $m$-bubbling solution $u_{n}$

[TABLE]

admits only the trivial solution $\phi\equiv 0$ for any $n\geq n_{0}$.

The above condition (2) can be relaxed by assuming $\ell_{\mbox{\footnotesize reg}}(\mathbf{q})=0$ and $D(\mathbf{q})\neq 0$, where $D(\mathbf{q})$ is a geometric quantity. Our aim is to extend the latter result to singular bubbling solutions. Even though the argument works out for more general situations we focus here on singular $1$-bubbling solution blowing up at $p_{i}$ for some $i\in\{1,\cdots,N\}$, see also Remark 1.3. More precisely, we assume without loss of generality that $\alpha_{i}\neq\alpha_{j}$ for $i\neq j$ and we study the case $\rho_{n}\to 8\pi(1+\alpha_{i})$ for some fixed $i\in\{1,\cdots,N\}$ and

[TABLE]

We define

[TABLE]

where $h(x)=\overline{h}_{1}(x)|x-p_{i}|^{2\alpha_{i}}$. Moreover, we define the ’desingularized’ $1$-vortex Hamiltonian to be

[TABLE]

Our main results are the following.

Theorem 1.1**.**

Let $u_{n}^{(1)}$ and $u_{n}^{(2)}$ be two singular $1$-bubbling solutions of ${\textbf{(}\mathbf{P}_{\rho_{n}}\textbf{)}}$, with $\rho^{(1)}_{n}=\rho_{n}=\rho^{(2)}_{n}$, blowing up at the point $p_{i}$ for some $i\in\{1,\cdots,N\}$, $\alpha_{i}\in(0,\infty)\setminus\mathbb{N}$. Assume that,

1. (1)

$p_{i}$* is a critical point of $\mathcal{H}_{p_{i}}$,* 2. (2)

$\ell(p_{i})\neq 0$.

Then there exists $n_{0}\geq 1$ such that $u_{n}^{(1)}=u_{n}^{(2)}$ for all $n\geq n_{0}$.

Theorem 1.2**.**

Let $u_{n}$ be a singular $1$-bubbling solution of ${\textbf{(}\mathbf{P}_{\rho_{n}}\textbf{)}}$, blowing up at the point $p_{i}$ for some $i\in\{1,\cdots,N\}$, $\alpha_{i}\in(0,\infty)\setminus\mathbb{N}$. Assume that the conditions (1)-(2) of Theorem 1.1 hold true. Then there exists $n_{0}\geq 1$ such that, for any $n\geq n_{0}$, (1.2) admits only the trivial solution $\phi\equiv 0$.

Observe that we do not need the non-degeneracy of the Hamiltonian as in condition (1) of Theorem A. This is essentially due to the difference of the linearized problem, see (1.8) and the discussion later on. On the other hand, we do need to assume $p_{i}$ to be a critical point of $\mathcal{H}_{p_{i}}$. For the regular blow up this is always the case since it is well-known [44] that for a regular $m$-bubbling solution blowing up at the points $q_{j}\notin\{p_{1},\cdots,p_{N}\}$, then $\mathbf{q}=(q_{1},\cdots,q_{m})$ has to be a critical point of $\mathcal{H}_{m}$.

Remark 1.3**.**

The argument yielding Theorems 1.1 and 1.2 works out for more general situations and can be carried out to prove local uniqueness of singular $m$-bubbling and even for mixed scenarios of singular $m$-bubbling and regular $m^{\prime}$-bubbling solutions. The decision to focus on singular $1$-bubbling is twofold: on one side the latter case is very subtle since in general the singular blow up point does not need be a critical point of the Hamiltonian $\mathcal{H}_{p_{i}}$ and furthermore we are not assuming any non-degeneracy of $\mathcal{H}_{p_{i}}$, and on the other side we want to highlight the differences with respect to the regular case. We postpone the general situation to a future paper. The case $\alpha\in(-1,0)$ will be treated in a separate paper since we first need to derive suitable sharp estimates for bubbling solutions, which are still missing in this case. Finally, the case $\alpha\in\mathbb{N}$ is by now out of reach due to the presence of non-simple (and non-radial) blow up [18, 37].

To prove Theorem 1.1 we argue by contradiction and we analyze the asymptotic behavior of the (normalized) difference of two distinct solutions for ${\textbf{(}\mathbf{P}_{\rho_{n}}\textbf{)}}$,

[TABLE]

Near the blow up point $p_{i}$, and after a suitable scaling, $\xi_{n}$ converges to an entire solution of the linearized problem of the Liouville equation

[TABLE]

Solutions of (1.6) with finite mass are completely classified [46] and for $\alpha_{i}\in(0,\infty)\setminus\mathbb{N}$ take the form,

[TABLE]

The freedom in the choice of $\mu$ is due to the invariance of equation (1.6) under dilations. The linearized operator $L$ relative to $v_{0}$ is defined by,

[TABLE]

It follows from [27, Corollary 2.2] that the $L^{\infty}$-bounded kernel of $L$ has one eigenfunction $Y_{0}$, where,

[TABLE]

The main part of the proof of Theorem 1.1 is to show that, after scaling and for large $n$, $\xi_{n}$ is orthogonal to $Y_{0}$. This is done by a delicate analysis of a suitably defined Pohozaev-type identity first introduced in [43] and then exploited in [8, 10].

The proof of Theorem 1.2 follows the same strategy by analyzing the asymptotic behavior of

[TABLE]

for a non-trivial solution $\phi_{n}$ of (1.2), which plays the role of (1.5).

The paper is organized as follows. In section 2 we introduce some preliminary results, in section 3 we estimate the $L^{\infty}$-norm of the difference of two solutions to ${\textbf{(}\mathbf{P}_{\rho_{n}}\textbf{)}}$ and in section 4 we then deduce the first estimates of $\xi_{n}$, the normalized difference of two solutions, away from the blow up point. In section 5 we introduce a Pohozaev-type identity to get refined estimates on $\xi_{n}$ and prove Theorem 1.1. Finally, in section 6 we give the sketch of the proof of Theorem 1.2.

2. Preliminary estimates about the blow up Phenomenon at the singular point

In this section we collect some preliminary results which will be used in the sequel. Let us assume that $i=1$ and set $p=p_{1}$, $0\neq\alpha=\alpha_{1}\in(0,+\infty)\setminus\mathbb{N}$. We define

[TABLE]

and

[TABLE]

where

[TABLE]

and $R(x,y)=G(x,y)+\frac{1}{2\pi}\log|x-y|$ is the regular part of the Green function. Therefore, we have

[TABLE]

and in any small enough ball centered at $p$ it holds that $\overline{h}_{1}>0$. It has been shown in [2] (for $\alpha\in(0,+\infty)\setminus\mathbb{N}$) and [17] (for $\alpha\in(-1,0)$) that

[TABLE]

Actually the proofs in [18, 2] show that this estimate holds locally near $p$, but then the global estimate follows by looking at the Definition 1 and the Green representation formula.

More recently, it has been proved in [27] that if $\alpha\in(0,+\infty)\setminus\mathbb{N}$, then

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

Next, we set

[TABLE]

let $r_{0}>0$ be a small positive number and set

[TABLE]

and as in [54], we denote $\psi_{n}$ as the solution of

[TABLE]

By the Mean value Therorem, we have $\psi(0)=0$. It has been proved in [54] that

[TABLE]

where

[TABLE]

and

[TABLE]

for suitable $R_{0}\geq 1$. Here $a_{n,0}$ is a uniformly bounded sequence,

[TABLE]

and, composing with suitable rotations, we can assume that

[TABLE]

Moreover, it has been shown in [27, Lemma 3.2] that

[TABLE]

Since $\psi_{n}$ is harmonic, then we also have

[TABLE]

We also have, see [27, Lemma 3.1],

[TABLE]

Also, we will need the fllowing improved estimate obtained by matching (2.6) and (2.12).

Lemma 2.1**.**

It holds,

[TABLE]

Proof.

Putting $c_{n}=\log\left(\int_{\Omega}he^{u_{n}}\right)$ and picking any $|x-p|=2r_{0}$ in (2.6) and (2.12), we conclude that

[TABLE]

Clearly we have

[TABLE]

and we find that

[TABLE]

and then the desired conclusion easily follows from (2.3) and (2.4). ∎

Finally, similar arguments used in the estimate (2.12), yield

[TABLE]

3. Estimate of the $L^{\infty}$-norm

The proof of Theorem 1.1 is obtained by contradiction and we assume that two distinct solutions $u_{n}^{(i)},~{}i=1,2$, exist for ${\textbf{(}\mathbf{P}_{\rho_{n}}\textbf{)}}$, whence in particular with the same $\rho_{n}$, which satisfy

[TABLE]

where $\alpha=\alpha_{1}$. We also assume without loss of generality that

[TABLE]

Then we define

[TABLE]

and in particular $v_{n}^{(i)}$ defined as in (2.5). Also we set

[TABLE]

There is no loss of generality in assuming that

[TABLE]

To simplify the notation, we set

[TABLE]

Then we have

Lemma 3.1**.**

• (i)

$|\lambda_{n}^{(1)}-\lambda_{n}^{(2)}|=O(\sum_{i=1}^{2}e^{-\frac{\epsilon_{0}}{\alpha+1}\lambda_{n}^{(i)}})=O(e^{-\frac{\epsilon_{0}}{\alpha+1}\lambda_{n}^{(1)}})=O(\sigma_{n}^{2\epsilon_{0}}).$**

• (ii)

$\|\tilde{u}_{n}^{(1)}-\tilde{u}_{n}^{(2)}\|_{L^{\infty}(B(0,r_{0}))}\leq|\lambda_{n}^{(2)}-\lambda_{n}^{(1)}|+O(\lambda_{n}^{(1)}\sigma_{n}^{2})$.

• (iii)

$\|\tilde{u}_{n}^{(1)}-\tilde{u}_{n}^{(2)}\|_{L^{\infty}(\Omega\setminus B(0,r_{0}))}\leq O(\sigma_{n})$.

Proof.

(i) In view of (2.3), we find that

[TABLE]

which immediately implies, since $\ell(p)\neq 0$,

[TABLE]

as claimed.

(ii) By using $\lambda_{n}^{(1)}\leq\lambda_{n}^{(2)}$, it is not difficult to see that

[TABLE]

uniformly in $B(0,r)$ for any $r>0$. Also, in view of (2.10), and since the $\psi_{n}^{(i)}$’s are harmonic, we find that

[TABLE]

uniformly in $B(0,3r_{0})$, we use this gradient estimate to evaluate the difference,

[TABLE]

which implies that

[TABLE]

uniformly in $B(0,r_{0})$. Also it is easy to see that

[TABLE]

Therefore, in view of (2.6) and Lemma 3.1, we finally conclude that,

[TABLE]

which is (ii).

(iii) Next we obtain the estimate in $\Omega\setminus B(0,r_{0})$, by using the Green’s representation formula,

[TABLE]

In view of (2.2) and since $\rho_{n}$ is the same for the two solutions, then we have

[TABLE]

Then, by using (2.2) once more, for $x\in\Omega\setminus B(0,r_{0})$ we have,

[TABLE]

uniformly in $x\in\Omega\setminus B(0,r_{0})$. Therefore we conclude that

[TABLE]

as claimed. ∎

4. Estimate of the difference away from the blow up point

Let

[TABLE]

Clearly $\xi_{n}$ satisfies

[TABLE]

for some constant $d_{n}$ satisfying $|d_{n}|\leq 1$ and

[TABLE]

To simplify the notations, we set

[TABLE]

Then by defining

[TABLE]

we prove the following

Lemma 4.1**.**

There exists a constant $b_{0}\in\mathbb{R}$, such that $\hat{\xi}_{n}(z)\to b_{0}\hat{\xi}_{0}(z)$ in $C_{\mathrm{loc}}^{0}(\mathbb{R}^{2})$, where

[TABLE]

where $\gamma=\frac{\pi\overline{h}_{1}(0)}{1+\alpha}.$

Proof.

By Lemma 3.1, we see that

[TABLE]

and then by (2.6), (2.10) and (2.11)

[TABLE]

where $\gamma=\frac{\pi\overline{h}_{1}(0)}{1+\alpha}.$

We define

[TABLE]

By using (4.2), we have

[TABLE]

and since $|\hat{\xi}_{n}|\leq 1$, then we conclude that $\hat{\xi}_{n}\to\hat{\xi}$ in $C_{\mathrm{loc}}^{0}(\mathbb{R}^{2})$, where $\hat{\xi}$ is a solution of

[TABLE]

It follows from [27, Corollary 2.2] that $\hat{\xi}(z)=b_{0}\xi_{0}(z)$, for some constant $b_{0}$, as claimed. ∎

Next, we have

Lemma 4.2**.**

For any $r_{0}$ small enough we have

[TABLE]

where $b_{0}$ is defined by Lemma 4.1.

Proof.

It follows from (2.2) that

[TABLE]

Since $\|\xi_{n}\|_{L^{\infty}(\Omega)}\leq 1$, then (4.2) implies that

[TABLE]

where

[TABLE]

As a consequence, $\xi_{0}$ is smooth in $\Omega\setminus\{0\}$ and in particular

[TABLE]

Therefore $\xi_{0}=-b$ in $\Omega$ for some constant $b$ and

[TABLE]

In particular $-\xi_{n}(x)=d_{n}\to b$ for $x\in\partial\Omega.$ Let $\phi_{n}=\frac{1-\gamma_{n}e^{\lambda_{n}}|x|^{2+2\alpha}}{1+\gamma_{n}e^{\lambda_{n}}|x|^{2+2\alpha}}$ and let us fix $d\in(0,r_{0})$. Then, by using (2.6), (2.10) and (2.11), we find that

[TABLE]

Therefore, by the scaling $x=\sigma_{n}z$, we see that,

[TABLE]

In view of Lemma 3.1 we obtain

[TABLE]

Let $\zeta_{n}=\int_{0}^{2\pi}\xi_{n}(r,\theta)d\theta$, where $r=|x|$. Then, for any fixed $R>0$, (4.4) yields

[TABLE]

Also for any $R>0$ large enough, and for any $r\in(R\sigma_{n},r_{0}]$, we also obtain that

[TABLE]

and so we conclude that

[TABLE]

Integrating (4.5) we obtain that

[TABLE]

In view of Lemma 4.1, we also have

[TABLE]

where $\lim_{R\to+\infty}o_{R}(1)=0$ and $\lim_{n\to+\infty}o_{n}(1)=0$. Then by (4.6) we have

[TABLE]

In view of (4.3), we see that

[TABLE]

which implies that $b=b_{0}$. Hence, we finish the proof. ∎

Next, we need a refined estimate about $\xi_{n}$ which will be needed in next section.

Lemma 4.3**.**

[TABLE]

where

[TABLE]

Moreover, there is a constant $C>0$, which does not depend on $R>0$, such that

[TABLE]

Proof.

By the Green representation formula we find that,

[TABLE]

while, by Lemma 3.1, we also find that

[TABLE]

Thus, for $x\in\Omega\setminus B(0,2r_{0})$, we see from (2.2), (2.6) that

[TABLE]

By using (2.2), (2.6) and Lemma 3.1, after scaling we see that for $x\in\Omega\setminus B(0,2r_{0})$, it holds

[TABLE]

Therefore, in view of Lemma 4.1, for $x\in\Omega\setminus B(0,2r_{0})$ we find that,

[TABLE]

From (4.10)-(4.13), we see that the estimate (4.8) holds in $C^{0}(\Omega\setminus B(0,r_{0}))$. The proof of the fact that (4.8) holds in $C^{1}(\Omega\setminus B(0,r_{0}))$ is similar and we skip it here to avoid repetitions.

From (4.11), (2.6) and suitable scaling, we see that there exists $C>0$, which is independent of $R>0$ such that for $x\in B(0,2r_{0})\setminus B(0,\sigma_{n}R)$, it holds that

[TABLE]

By (4.10), (4.11) and (2.6), we also see that for $x\in\Omega\setminus B(0,2r_{0})$, it holds that

[TABLE]

By (4.14) and (4.15) we obtain (4.9), which concludes the proof of Lemma 4.3. ∎

5. Estimates via Pohozaev identities

From now on, for a given function $f(y,x)$, we shall use $\partial$ and $D$ to denote the partial derivatives with respect to $y$ and $x$ respectively. With a small abuse of notation, for a function $f(x)$ we will use both $\nabla$ and $D$ to denote its gradient.

We define

[TABLE]

and

[TABLE]

Recall the definition of $\xi_{n}$ which satisfies (4.2). Our aim is to show that the projection of $\xi_{n}$ on the radial part kernel is zero, i.e., $b_{0}=0$. We shall accomplish it by exploiting the following Pohozaev identity to derive a more accurate estimate on $\xi_{n}.$

Lemma 5.1**.**

([43])*

For any fixed $r\in(0,r_{0})$, it holds*

[TABLE]

Proof.

See [8] for a proof of this identity. ∎

Let

[TABLE]

Recall the definition of $A_{n}$ given in Lemma 4.3. Then we have

Lemma 5.2**.**

[TABLE]

Proof.

Let

[TABLE]

so that

[TABLE]

In view of (2.14), we have

[TABLE]

for any fixed small $r_{0}>0.$ As a consequence, for fixed $r>r_{0}$, we find that

[TABLE]

where we used (2.3). Therefore, as a consequence of Lemma 4.3, we conclude that

[TABLE]

In this particular case, we have $A_{n}=0.$

To estimate the right hand side of (5.8), we need a refined estimate about $\xi_{n}$ on $\partial B(0,r)$. So, by the Green representation formula with $x\in\partial B(0,r)$, we find that

[TABLE]

where

[TABLE]

and

[TABLE]

At this point, let us fix $\overline{\theta}\in(0,\frac{r}{2})$. By Lemma 3.1 and Lemma 4.2, we find that,

[TABLE]

for any $y\in\partial\Omega\setminus B(0,\overline{\theta})$. By (2.4), (2.12), (2.13) and (5.10), we conclude that

[TABLE]

where

[TABLE]

On the other hand, by (2.6), we have for $y\in B(0,\overline{\theta})$,

[TABLE]

Next, by (5.10), for $y\in B(0,\overline{\theta})$ and $x\in\partial B(0,r)$, we get

[TABLE]

Let us define

[TABLE]

so that, by (5.11)-(5.13), we conclude that for $x\in\partial B(0,r)$, it holds

[TABLE]

where

[TABLE]

Let us set

[TABLE]

and then subsititute (5.15) into (5.8), to derive that

[TABLE]

To estimate the right hand side of (5.17), we notice that for any pair of (smooth enough) functions $u$ and $v$, it holds

[TABLE]

In view of (5.14), we also see that, for any $\underline{\theta}\in(0,r)$,

[TABLE]

and moreover, by using (5.5) and (5.1), we have

[TABLE]

By using (5.18)-(5.20) and (5.6), we conclude that

[TABLE]

and thus,

[TABLE]

At this point, let us denote by $o_{\underline{\theta}}(1)$ any quantity which converges to [math] as $\underline{\theta}\to 0^{+}$, and then observe that,

[TABLE]

Since, $D_{i}D_{h}\log|x|=\frac{\delta_{ih}|x|^{2}-2x_{i}x_{h}}{|x|^{4}}$, then we find that,

[TABLE]

We observe that, if $h=k$ then $D_{i}\log|x|=\frac{x_{i}}{|x|^{2}}$,

[TABLE]

and thus,

[TABLE]

If $h\neq k$, then

[TABLE]

which implies that

[TABLE]

By (5.21)-(5.25), we conclude that

[TABLE]

Next we estimate the other terms in (5.17), that is $4(1+\alpha)\int_{\partial B(0,r)}\left\langle\nu,D_{x}\zeta_{n}^{*}(x)\right\rangle d\sigma$, where $\zeta_{n}^{*}$ is defined in (5.16). Clearly we have

[TABLE]

If $y\in\Omega\setminus B(0,\overline{\theta})$ and $x\in\partial B(0,\theta)$ with $\theta\ll(\overline{\theta})^{2}$, then we find that

[TABLE]

which implies

[TABLE]

Thus (5.23)-(5.25) and (5.27) imply that

[TABLE]

We observe that

[TABLE]

and let us choose $u(x)=\Psi_{n}(y,x)$ and $v(x)=G_{n}(x)-\varphi_{n}(x)$ in (5.18). Then we consider the following two cases:

• (i)

If $y\in\partial B(0,r)\setminus B(0,\overline{\theta})$, then from (5.18) and (5.28), we obtain that

[TABLE]

• (ii)

If $y\in\Omega\setminus B(0,r)$, then we see from (5.18) and (5.28) that

[TABLE]

and by (5.16), and (5.29)-(5.30), we finally conclude that

[TABLE]

Obviously from (5.17), (5.26) and (5.31) we get the conclusion of Lemma 5.2 ∎

To estimate the right hand side of (5.3) of Lemma 5.1, we recall, see for example (5.10), that

[TABLE]

Recall also the definitions of $\Phi(x,0)$ and $\mathcal{H}_{p}=\mathcal{H}_{0}$ in (5.4) and (1.4), respectively and the definition of $\ell(p)$ after (2.4). A crucial point in our proof is the following estimate.

Lemma 5.3**.**

• (i)

[TABLE]

• (ii)

[TABLE]

• (iii)

[TABLE]

where $O(m_{n,1}(\alpha))$ is defined after (5.36) and $O(1)$ is used to denote any quantity uniformly bounded with respect to $r$, $R$ and $n.$

Proof.

(i) We first observe that (5.11) implies that

[TABLE]

Clearly we have

[TABLE]

By (5.32) and (5.33), we obtain,

[TABLE]

which proves (i).

(ii) We notice that $A_{n}=\int_{\Omega}f_{n}^{*}=0$, and thus

[TABLE]

By (5.11) we see that

[TABLE]

which proves (ii).

(iii) By (2.3) and (2.6), we see that

[TABLE]

where

[TABLE]

see (2.7), (2.8) and (2.10). Thus, we set

[TABLE]

and use Lemma 3.1 and (2.4), we deduce that

[TABLE]

Set

[TABLE]

[TABLE]

Using (5.36) together with (2.5), (2.7) and Lemma 3.1, we conclude that

[TABLE]

where

[TABLE]

and

[TABLE]

In view of (2.9), (2.10), (2.11) and (2.4), we have

[TABLE]

and then, putting $a_{1}=\partial_{x_{1}}\mathcal{H}_{0}$ and $\Lambda(z)=\rho_{n}\overline{h}_{1}(0)|z|^{2\alpha}$, we conclude that

[TABLE]

where we used the properties of $\Gamma(x)$, and thus

[TABLE]

On the other hand, in view of Lemma 4.1, for any fixed $R\geq 1$ large, we have

[TABLE]

Finally we have

[TABLE]

On the other side, in view of (4.9), we also see that if $R\leq|z|\leq r/\sigma_{n}$, then it holds

[TABLE]

and thus

[TABLE]

As a consequence, by Lemma 3.1, we find that

[TABLE]

Collecting the above estimates we conclude that

[TABLE]

∎

Recall that $p=0$. Using the assumptions $\ell(p)\neq 0$ and $\nabla\mathcal{H}_{0}(0)=0$ we can now prove that $b_{0}=0$.

Lemma 5.4**.**

$b_{0}=0.$**

Proof.

By (5.3) and Lemmas 5.2-5.3, we have for any $r\in(0,1)$ and $R>1$,

[TABLE]

Recall $A_{n}=0$. Since $\nabla\mathcal{H}_{0}(0)=0$ by assumption, after some manipulations, for $r\in(0,r_{0})$ and any $R>1$, we find that

[TABLE]

which implies

[TABLE]

provided $\ell(p)\neq 0$. Hence we finish the proof. ∎

Proof of Theorem 1.1.

Let $x_{n}^{*}$ be a maximum point of $\xi_{n}$, then we have,

[TABLE]

By Lemma 4.2 and Lemma 5.4 we have that $x_{n}^{*}\to p$. By Lemma 5.4, it holds that

[TABLE]

Setting $\bar{\xi}_{n}(x)=\xi(s_{n}x+p)$, then we have $\bar{\xi}_{n}$ satisfies

[TABLE]

On the other hand, by (5.39), we also have

[TABLE]

In view of (5.40) and $|\bar{\xi}_{n}|\leq 1$ we see that $\bar{\xi}_{n}\to\bar{\xi}_{0}$ on any compact subset of $\mathbb{R}^{2}\setminus\{0\}$, where $\bar{\xi}_{0}$ satisfies $\Delta\bar{\xi}_{0}=0$ in $\mathbb{R}^{2}\setminus\{0\}$. Since $|\bar{\xi}_{0}|\leq 1$, we have $\Delta\bar{\xi}_{0}=0$ in $\mathbb{R}^{2}$, which implies $\bar{\xi}_{0}$ is a constant. At this point, since $\frac{|x_{n}^{*}-p|}{s_{n}}=1$ and in view of (5.41), we find $\bar{\xi}_{0}=1$ or $\bar{\xi}_{0}=-1$. From which we have $|\bar{\xi}_{n}(x)|\geq\frac{1}{2}$ when $s_{n}\leq|x-p|\leq\frac{1}{2}s_{n}$, which contradicts to (4.6)-(4.8) since $e^{-\frac{\lambda_{n}^{(1)}}{2(1+\alpha)}}\ll s_{n}$ and $\lim_{n\to+\infty}s_{n}=0$ and $b_{0}=0$. This fact concludes the proof of Theorem 1.1. ∎

6. The proof of Theorem 1.2

In this section we give the proof of the non-degeneracy result stated in Theorem 1.2. Since the argument is similar to the one yielding local uniqueness of bubbling solutions we will be sketchy to avoid repetitions, referring to [9] for full details.

Suppose by contradiction the linearized problem (1.2) admits a non-trivial solution $\phi_{n}$, where $u_{n}$ is a singular $1$-bubbling solution of ${\textbf{(}\mathbf{P}_{\rho_{n}}\textbf{)}}$ blowing up at the point $p_{i}$ for some $i\in\{1,\cdots,N\}$. We suppose with no loss of generality that $p_{i}=0\in\Omega$, set $\alpha_{i}=\alpha$ and

[TABLE]

Define

[TABLE]

which plays the role of the difference of two bubbling solutions, see (4.1) in the proof of Theorem 1.1. Then, $\Xi_{n}$ satisfies

[TABLE]

for some constant $d_{n}$ satisfying $|d_{n}|\leq 1$ and $c_{n}(x)=e^{\tilde{u}_{n}(x)}$.

Step 1. We start by considering the asymptotic behavior of $\Xi_{n}$ near the blow up point $p_{i}$. After a suitable scaling, $\Xi_{n}$ converges in $C_{\mathrm{loc}}^{0}(\mathbb{R}^{2})$ to a solution $\hat{\xi}$ of the linearized problem

[TABLE]

where $\gamma=\frac{\pi\overline{h}_{1}(0)}{1+\alpha}$, see for example Lemma 4.1. It follows from [27, Corollary 2.2] that there exists a constant $b_{0}\in\mathbb{R}$ such that

[TABLE]

Step 2. We next consider the global behavior of $\Xi_{n}$ away from the blow up point $p_{i}$. It follows from (2.2) that

[TABLE]

Using then $\|\Xi_{n}\|_{L^{\infty}(\Omega)}\leq 1$ and (6.1) it is not difficult to see that

[TABLE]

Therefore $\xi_{0}=-b$ in $\Omega$ for some constant $b$ and

[TABLE]

Finally, by an O.D.E. argument as in Lemma 4.2 one can show $b=b_{0}$.

Step 3. We then study the asymptotic in the Pohozaev-type identity given by Lemma 5.1 (with suitable minor modifications, see for example [9]). Using the assumption $\nabla\mathcal{H}_{p_{i}}(p_{i})=0$ it is possible to prove that

[TABLE]

see section 5. Since by assumption $\ell(p_{i})\neq 0$ we deduce $b_{0}=0$.

Step 4. The contradiction is then obtained by a blow up argument using $b=b_{0}=0$ jointly with (6.2) and (6.3) exactly as in the proof of Theorem 1.1, see the end of section 5. The proof of Theorem 1.2 is completed.