Local and global behavior of solutions to 2 D -elliptic equation with exponentially-dominated nonlinearities

Abstract

We study the family of blowup solutions to semilinear elliptic equations in two-space dimensions with exponentially-dominated nonnegative nonlinearities. Such a family admits an exclusion of the boundary blowup, finiteness of blowup points, and pattern formation. Then, Hamiltonian control of the location of blowup points, residual vanishing, and mass quantization arise under the estimate from below of the nonlinearity. Finally, if the principal growth rate of nonlinearity is exactly exponential and the residual part has a gap relative to this term, there is a locally uniform estimate of the solution which ensures its asymptotic non-degeneracy.

Keywords

Semilinear elliptic equation blowup analysis point vortex Hamiltonian recursive hierarchy Onsager’s theory

1. Introduction

Our purpose is to study the family of blowup solutions to exponentially-dominated elliptic boundary problems in two-space dimensions, particularly, formation of quantized bubbles and local uniform estimate of Y.Y. Li type [19]. The result is applicable to Neri’s limit equation [25] on many point vortices with one-sided stochastic intensities studied by [10], arising in Onsager’s theory of statistical mechanics [31].

Usually the limit equation of point vortices is represented by the form of mean field equation with non-local term, but here, we use the Gel’fand equation [13], $\begin{matrix} (1) & - Δ v = λ f (v), v > 0 in Ω, v = 0 on \partial Ω, \end{matrix}$ where $Ω \subset R^{2}$ is a bounded domain with smooth boundary $\partial Ω$ , and $λ > 0$ is a constant. Our result covers a wide class of $f = f (v) \in C [0, + \infty)$ ; the fundamental assumption is its non-negativity and exponentially-dominatedness: $\begin{matrix} (2) & 0 ⩽ f (v) ⩽ e^{v} . \end{matrix}$ This assumption is under the normalization, and if $\begin{matrix} 0 ⩽ f (v) ⩽ C e^{α v} \end{matrix}$ holds with positive constants C and α, then we take v and λ for $α v$ and $C α λ$ , respectively, and obtain comparable results in below. Here and henceforth, C denotes a generic positive constant.

We are concerned with a sequence of solutions $(λ, v) = (λ_{k}, v_{k})$ , $k = 1, 2, \dots$ , to (1), satisfying $\begin{matrix} (3) & ‖ v_{k} ‖_{\infty} \to + \infty, \int_{Ω} λ_{k} f (v_{k}) ⩽ C, λ_{k} ⩽ C . \end{matrix}$ The second condition of (3) describes the boundedness of (negative) inverse statistical temperatures in the context of statistical mechanics. If this condition does not hold, there is an entire blowup of the solution up to a subsequence. Thus we obtain $\begin{matrix} (4) & v_{k} \to + \infty locally uniformly in Ω \end{matrix}$ if $\begin{matrix} (5) & lim_{k \to \infty} \int_{Ω} λ_{k} f (v_{k}) = + \infty . \end{matrix}$ See [24] for the proof of this fact. The third condition of (3), on the other hand, is derived from the convexity and super-linearity of the nonlinearity, $\begin{matrix} (6) & v \in [0, + \infty) \mapsto f (v) convex, lim_{v ↑ + \infty} \frac{f (v)}{v} = + \infty \end{matrix}$ by a standard argument of Kaplan [8,12]. This convexity of $f = f (v)$ , however, is not necessary for the argument below under the cost of $λ_{k} ⩽ C$ in (3).

Remark 1.
The last two conditions of (3) imply $‖ v_{k} ‖_{W^{1, q} (Ω)} ⩽ C$ for $1 ⩽ q < 2$ by Stampacchia’s estimate [3,37], and hence $‖ v_{k} ‖_{p} ⩽ C$ for $1 ⩽ p < \infty$ by Sobolev’s embedding theorem. Hence (3) does not arise if the growth rate of $f (v)$ is up to the order of polynomials.

The first result of this paper is sometimes called a rough estimate. It assures an exclusion of the boundary blowup, finiteness of blowup points, and formation of blowup patterns of ${v_{k}}$ under the assumption of (2)–(3).

To state this result, let $\begin{matrix} S = {x_{0} \in \overline{Ω} ∣ \exists x_{k} \to x_{0}, v_{k} (x_{k}) \to + \infty} \end{matrix}$ be the blowup set of ${v_{k}}$ , and $G = G (x, x^{'})$ be the Green’s function $\begin{matrix} (7) & - Δ_{x} G (\cdot, x^{'}) = δ_{x^{'}} (d x), G (\cdot, x^{'}) |_{\partial Ω} = 0 \end{matrix}$ defined for $(x, x^{'}) \in \overline{Ω} \times Ω$ . Let, furthermore, $M (\overline{Ω}) = C {(\overline{Ω})}^{'}$ denote the set of measures on $\overline{Ω}$ .
Theorem 1.
Let ${(λ_{k}, v_{k})}$ be a sequence of solutions to ( 1 ) satisfying ( 3 ). Assume ( 2 ). Then it holds that $\begin{matrix} (8) & S \subset Ω, ♯ S < + \infty . \end{matrix}$ Passing to a subsequence, there arises $\begin{matrix} (9) & λ_{k} f (v_{k}) d x ⇀ \sum_{x_{0} \in S} m (x_{0}) δ_{x_{0}} (d x) + r (x) d x in M (\overline{Ω}) \end{matrix}$ for $\begin{matrix} (10) & m (x_{0}) ⩾ 4 π, 0 ⩽ r = r (x) \in L^{1} (Ω), \end{matrix}$ which implies $\begin{matrix} (11) & v_{k} \to v locally uniformly in \overline{Ω} ∖ S \end{matrix}$ for $\begin{matrix} (12) & v (x) = \sum_{x_{0} \in S} m (x_{0}) G (x, x_{0}) + \int_{Ω} G (x, x^{'}) r (x^{'}) d x . \end{matrix}$

Several features of the solution for the extremal nonlinearity, $f (v) = e^{v}$ , are described in [38,39] with their physical and geometrical backgrounds. There is, actually, mass quantization indicated by $\begin{matrix} (13) & m (x_{0}) = 8 π, \forall x_{0} \in S \end{matrix}$ in (10). Residual vanishing also arises for $f (v) = e^{v}$ , as $\begin{matrix} (14) & r (x) d x = 0 \end{matrix}$ in (10), which is a consequence of $\begin{matrix} (15) & λ_{k} \to 0 . \end{matrix}$

So far, Hamiltonian control of the location of blowup points is thought to be a striking feature of this extremal nonlinearity; recall that the Hamiltonian of the system of m-point vortices is given by $\begin{matrix} (16) & H (x) = \frac{1}{2} \sum_{j = 1}^{m} R (x_{j}) + \sum_{1 ⩽ i < j ⩽ m} G (x_{i}, x_{j}), x = (x_{1}, \dots, x_{m}), \end{matrix}$ and takes a fundamental role in Onsager’s theory [31]; a formation of the ordered structure in the negative inverse statistical temperature. Here in (16), $G = G (x, x^{'})$ is the Green’s function defined by (7) and $\begin{matrix} R (x) = {[G (x, x^{'}) + \frac{1}{2 π} log | x - x^{'} |]}_{x^{'} = x} \end{matrix}$ stands for the Robin function.
Remark 2.
After the formulation of [31], equation (1) for $f (v) = e^{v}$ has been derived by several methods, as a limit equation of the stream function when the number of point vortices tends to infinite. See [4,11] and the references therein.

The second and the third results of this paper assure the above residual vanishing and Hamiltonian control of the location of blowup points under a suitable bound from below for $f (v)$ . Let $\begin{matrix} (17) & F_{0} (v) = \int_{0}^{v} f (v) d v ⩾ 0 . \end{matrix}$
Theorem 2.
Assume, besides ( 2 ) and ( 3 ) in ( 1 ), that $\begin{matrix} (18) & 0 ⩽ β f (v) ⩽ F_{1} (v) \equiv F_{0} (v) + C \end{matrix}$ and $\begin{matrix} (19) & \underset{v ↑ + \infty}{lim inf} e^{- α v} f (v) > 0 \end{matrix}$ for $α, β > 0$ in $α β > \frac{1}{2}$ . Then ( 15 ) and in particular ( 14 ) in ( 9 ) hold.
Remark 3.
Since $f (v)$ satisfies $\begin{matrix} (20) & \underset{v ↑ + \infty}{lim inf} f (v) > 0 \end{matrix}$ by (19), inequality (18) is equivalent to $\begin{matrix} (21) & \underset{v ↑ + \infty}{lim sup} {(log F_{0})}^{'} (v) ⩽ β^{- 1} . \end{matrix}$ To apply Theorem 2 for $f (v) = e^{β^{- 1} v}$ with $β ⩾ 1$ , therefore, we take $α > 0$ in $\frac{1}{2 β} < α < \frac{1}{β}$ .
Theorem 3.
Assume ( 2 ), ( 3 ), ( 18 ), ( 19 ), and $\begin{matrix} (22) & 0 ⩽ F_{0} (v) ⩽ γ f (v) + C \end{matrix}$ in ( 1 ) for $α, β, γ > 0$ in $α β > \frac{1}{2}$ . Let, furthermore, $S = {x_{1}^{}, \dots, x_{m}^{}}$ . Then, $x_{} = (x_{1}^{}, \dots, x_{m}^{})$ is a critical point of* $H = H (x)$ defined by ( 16 ): $\begin{matrix} (23) & \nabla_{x_{i}} H (x_{1}^{}, \dots, x_{m}^{}) = 0, 1 ⩽ i ⩽ m . \end{matrix}$
Remark 4.
By (20), inequality (22) is equivalent to $\begin{matrix} (24) & \underset{v ↑ + \infty}{lim inf} {(log F_{0})}^{'} (v) ⩾ γ^{- 1} . \end{matrix}$

We have, furthermore, mass quantization (13) under more strict bounds of the nonlinearity.
Theorem 4.
Mass quantization ( 13 ) occurs, if, besides the assumption of Theorem 1 , any $ε > 0$ admits C such that $\begin{matrix} (25) & | F_{0} (v) - f (v) | ⩽ ε f (v) + C . \end{matrix}$

Theorems 1, 2, 3, and 4 are applicable to Neri’s equation studied in [10,34,35]. This equation is associated with the stochastic distribution of intensities of vortices, and the nonlinearity takes the form $\begin{matrix} (26) & f (v) = \int_{[0, 1]} α e^{α v} P (d α), \end{matrix}$ where $P (d α)$ is a Borel measure on $[0, 1]$ with $sup {α ∣ α \in supp P (d α)} = 1$ .

In fact, we have $\begin{matrix} F_{0} (v) = \int_{[0, 1]} e^{α v} P (d α) - 1 \end{matrix}$ and hence $\begin{array}{l} f (v) & = \int_{[0, 1]} α e^{α v} P (d α) \\ ⩽ F_{0} (v) + 1 = \int_{[0, 1]} e^{α v} P (d α) \\ = \int_{[1 - ε^{'}, 1]} e^{α v} P (d α) + \int_{[0, 1 - ε^{'})} e^{α v} P (d α) \\ ⩽ \frac{1}{1 - ε^{'}} \int_{[1 - ε^{'}, 1]} α e^{α v} P (d α) + e^{(1 - ε^{'}) v} \\ ⩽ \frac{1}{1 - ε^{'}} f (v) + e^{(1 - ε^{'}) v}, \end{array}$ which implies $\begin{matrix} - 1 ⩽ F_{0} (v) - f (v) ⩽ \frac{ε^{'}}{1 - ε^{'}} f (v) + e^{(1 - ε^{'}) v} ⩽ 2 ε^{'} f (v) + e^{(1 - ε^{'}) v} \end{matrix}$ for $0 < ε^{'} < \frac{1}{2}$ . It holds also that $\begin{array}{l} f (v) & ⩾ \int_{[1 - ε^{'} / 2, 1]} α e^{α v} P (d α) \\ ⩾ P [1 - ε^{'} / 2, 1] (1 - ε^{'} / 2) e^{(1 - ε^{'} / 2) v} \end{array}$ with $P [1 - ε^{'} / 2, 1] > 0$ . Therefore, any $0 < ε^{'} ≪ 1$ admits $p > 1$ and $C > 0$ such that $\begin{matrix} | F_{0} (v) - f (v) | ⩽ 2 ε^{'} f (v) + C (f {(v)}^{1 / p} + 1), \end{matrix}$ which implies (25) for $ε = 3 ε^{'}$ .
Remark 5.
This paper is concerned with the non-negative nonlinearity, and the conclusions of Theorems 1, 2, 3, and 4 are valid to Neri’s equation associated with the non-negative intensities (26). For the case of sign-changing $P (d α)$ , the exclusion of boundary blowup, finiteness of blowup points, residual vanishing, and a mass identity in place of the mass quantization are proven by [32].
Remark 6.
The extremal nonlinearity, $f (v) = e^{v}$ , has been known to be a threshold for the formation of a blowup pattern as in (11)–(12), or the finiteness of the blowup points (8) in two-space dimensions. In [26], a family of solutions ${(λ_{k}, v_{k})}$ to (1) for $Ω = B (0, 1) \subset R^{2}$ satisfying $\begin{matrix} (27) & λ_{k} \to 0, ‖ v_{k} ‖_{\infty} \to + \infty, \end{matrix}$ is studied. In this case it holds that $\begin{matrix} (28) & lim_{k \to \infty} v_{k} = 0 locally uniformly in \overline{B} ∖ {0} \end{matrix}$ and $\begin{matrix} (29) & lim_{k \to \infty} v_{k} = + \infty locally uniformly in B \end{matrix}$ if $\begin{matrix} (30) & lim_{v ↑ + \infty} {(log f)}^{'} (v) = + \infty \end{matrix}$ and $\begin{matrix} (31) & lim_{v ↑ + \infty} {(log f)}^{'} (v) = 0, \end{matrix}$ respectively. By Theorem 1, in particular, a family of radially symmetric solutions ${(λ_{k}, v_{k})}$ does not realize $\begin{matrix} (32) & ‖ v_{k} ‖_{\infty} \to + \infty, \int_{Ω} λ_{k} f (v_{k}) ⩽ C, λ_{k} \to 0, \end{matrix}$ if the growth rate of $f (v)$ is weaker than the exponential as in (31). A counter part of (30) or (31) is (21) or (24) because $\begin{matrix} lim_{v ↑ + \infty} {(log F_{0})}^{'} (v) = lim_{v ↑ + \infty} {(log f)}^{'} (v) \end{matrix}$ holds if the limit of the right-hand side exists. In Theorems 2–3, actually, equalities (30) and (31) are excluded by (21) and (19), respectively.

The fifth result of this paper is some estimates on $m (x_{0})$ . It is associated with the realization of (32). In contrast with the result in Remark 6 on radially symmetric solutions, here the family of solutions ${(λ_{k}, v_{k})}$ is required to satisfy the second condition of (3) additionally: $\begin{matrix} \int_{Ω} λ_{k} f (v_{k}) ⩽ C . \end{matrix}$
Theorem 5.
Assume ( 2 ) and let ${(λ_{k}, v_{k})}$ be a family of solutions to ( 1 ) satisfying ( 3 ). Then it holds that $m (x_{0}) ⩾ 8 π β$ and $m (x_{0}) ⩽ 8 π γ$ for any $x_{0} \in S$ , under the assumptions of ( 18 ) and ( 22 ), respectively.
Remark 7.
From the first inequality of $m (x_{0}) ⩾ 8 π β$ , there does not arise (3) if $β > 0$ can be arbitrarily large. Under (20), this property is equivalent to $β = + \infty$ in (21): $\begin{matrix} \underset{v ↑ + \infty}{lim sup} {(log F_{0})}^{'} (v) ⩽ 0 . \end{matrix}$ If $f (v)$ is a polynomial, it holds, more strongly, that $\begin{matrix} {(log F_{0})}^{'} (v) = O (\frac{1}{v}), v ↑ + \infty . \end{matrix}$ In this case the exclusion of (3) is proven directly by an a priori estimate of the solution as in Remark 1. Note, next, that the inequality $m (x_{0}) ⩾ 4 π$ in (10) is a consequence of the normalization (2). Then the second inequality of $m (x_{0}) ⩽ 8 π γ$ implies $γ ⩾ \frac{1}{2}$ . Actually, we have a more stronger requirement, $γ ⩾ 1$ , for (24) to be consistent with (2).
Remark 8.
Theorems 1, 2, 3, and 4 have been known for the case that $f (v)$ is a perturbation of $e^{v}$ , that is, $\begin{matrix} (33) & f (v) = e^{v} + g (v) . \end{matrix}$ with $\begin{matrix} (34) & g (v) = o (e^{v}), v ↑ + \infty . \end{matrix}$ If this condition holds then the assumptions of Theorems 1, 2, 3, and 4 are achieved. Hence we have the exclusion of the boundary blowup, finiteness of blowup points, residual vanishing, control of the Hamiltonian on the location of blowup points, and mass quantization. All of these profiles of the family of solutions, furthermore, are valid to (26), although this nonlinearity is not reduced to the form of (33)–(34) unless $1 \in supp P (d α)$ . See Remark 11 below.
Remark 9.
In [24], the nonlinearity is not required to be non-negatively definite, but a stronger assumption than (34), that is, $\begin{matrix} (35) & G (v) + | G^{'} (v) | = O (e^{β v}), v ↑ + \infty, β < \frac{1}{4} \end{matrix}$ for $G (v)$ satisfying $\begin{matrix} (36) & | g (v) - g^{'} (v) | ⩽ G (v), \end{matrix}$ is imposed. In [41] this condition of $g (v)$ is so relaxed as (34) under the presense of $f (v) ⩾ 0$ . Note that (35) with (36) implies $\begin{matrix} (37) & g (v) = O (e^{β v}), v ↑ + \infty \end{matrix}$ for $β < \frac{1}{4}$ .
Remark 10.
The above papers [24,41] commonly use complex variables, but the key observation of [41] is the transformation $\begin{matrix} (38) & u_{k} = v_{k} + log λ_{k}, V_{k} (x) = e^{- v_{k}} f (v_{k}), \end{matrix}$ which induces $\begin{matrix} (39) & - Δ u_{k} = V_{k} (x) e^{u_{k}}, 0 ⩽ V_{k} (x) ⩽ b in Ω, \int_{Ω} e^{u_{k}} ⩽ C \end{matrix}$ for $b = 1$ . Then the use of later result [2] makes it possible to avoid a technical a priori estimate of the solution needed in [24].
Remark 11.
In the non-degenerate case, $P ({1}) = τ > 0$ of (26), this nonlinearity takes the form $\begin{matrix} f (v) = τ e^{v} + g (v), g (v) = o (e^{v}), v ↑ + \infty . \end{matrix}$ Then we can assume (33) with (34) by replacing λ to $τ λ$ . In the degenerate case of $P ({1}) = 0$ , however, the transformation (38) does not imply (39). More precisely, the third condition of (39) does not follow from the second condition of (3) unless one also assumes that $\begin{matrix} (40) & \underset{v ↑ + \infty}{lim inf} e^{- v} f (v) > 0 . \end{matrix}$ We thus have to argue rather differently from the previous work [41], for the proof of Theorems 1, 2, 3, and 4.

The feature of the solution to (1) for $f (v) = e^{v}$ has been known in much more details. Among others are the degree calculation [5,6] and the deformation theory [23], which ensures a multiple existence of the solution in accordance with the topology of Ω. A locally uniform estimate of the solution due to [19] has been fundamental in these studies. This estimate holds if there is a gap of the growth rate in (34) in our case, which is the sixth result of this paper, Theorem 6 below.

This theorem thus treats the case of (37) with $β < 1$ . Then (8), (23) for $S = {x_{1}^{}, \dots, x_{m}^{}}$ , and (9) with $m (x_{0}) = 8 π$ and $r (x) d x = 0$ hold, that is in particular: $\begin{matrix} (41) & λ_{k} f (v_{k}) d x ⇀ \sum_{x_{0} \in S} 8 π δ_{x_{0}} (d x) . \end{matrix}$ Given $x_{0} \in S$ and $0 < r ≪ 1$ , thus define $x_{k} \in B_{2 r} (x_{0})$ by $\begin{matrix} (42) & max_{B_{2 r} (x_{0})} v_{k} = v_{k} (x_{k}) . \end{matrix}$ Then it holds that $\begin{matrix} x_{k} \to x_{0} \end{matrix}$ because (11) holds for $\begin{matrix} v (x) = \sum_{x_{0} \in S} 8 π G (x, x_{0}), x \in \overline{Ω} ∖ S \end{matrix}$ by Theorems 1, 2, and 4.
Theorem 6.
Assume ( 33 ) and ( 37 ) for some $β < 1$ in ( 1 ), and let ${(λ_{k}, v_{k})}$ be a sequence of solutions satisfying ( 3 ). Take $x_{0} \in S$ and define $x_{k} \to x_{0}$ by ( 42 ) for $0 < r ≪ 1$ . Then it holds that $\begin{matrix} (43) & | v_{k} (x) - v_{k} (x_{k}) - 2 log \frac{1}{1 + \frac{λ_{k}}{8} f (v_{k} (x_{k})) | x - x_{k} |^{2}} | ⩽ C \end{matrix}$ for any $x \in B_{r} (x_{k})$ and $k = 1, 2, \dots$ .

We use a scaling argument for the proof of Theorem 6. The formula (43) arises in accordance with the classification of the solution to the Liouville equation on the plane due to [7]. Thus we have $\begin{matrix} \tilde{v} (x) = 2 log \frac{1}{1 + \frac{| x |^{2}}{8}} \end{matrix}$ for $\tilde{v} = \tilde{v} (x)$ satisfying $\begin{matrix} (44) & - Δ \tilde{v} = e^{\tilde{v}}, \tilde{v} ⩽ \tilde{v} (0) = 0 in R^{2}, \int_{R^{2}} e^{\tilde{v}} < + \infty . \end{matrix}$

This paper is composed of six sections. We prove Theorem 1 in Section 2. Then Theorems 2, 3, 4, and 5 are proven in Section 3. Section 4 is devoted to the proof of Theorem 6. The last two sections are the remarks on the asymptotic non-degeneracy of the solution and spacially inhomogeneous nonlinearity.
2. Proof of Theorem 1

Exclusion of the boundary blowup $S \subset Ω$ is proven similarly to [24]. First, if Ω is convex, there is a uniformly decreasing property toward the boundary, independent of the nonlinearity. Since $f (v) ⩾ 0$ , furthermore, this property is valid without the convexity of $Ω \subset R^{2}$ by the Kelvin transformation. These results are established in [14].

Remark 12.
We extend this decreasing property of the solution for spatially inhomogeneous nonlinearities in the last section.

Having this property, we use the second condition of (3) to confirm $S \subset Ω$ . In fact, we have $\begin{matrix} (45) & ‖ v_{k} ‖_{W^{1, q}} = O (1), 1 ⩽ \forall q < 2 = \frac{n}{n - 1}, n = 2 \end{matrix}$ by (3) and the elliptic estimate of [3]. These properties of the solution, uniform decreasing toward the boundary and uniform local $L^{1}$ estimate of the solution near the boundary assured by (45), are sufficient to apply the argument of [9], and consequently, any m admits open set $ω_{m}$ containing $\partial Ω$ such that $\begin{matrix} {‖ D^{α} v_{k} ‖}_{L^{\infty} (ω_{m})} ⩽ C, | α | ⩽ m . \end{matrix}$ Thus we have $S \subset Ω$ in particular.
Remark 13.
The second condition of (3) is necessary here; otherwise there arises $S = \overline{Ω}$ by (4).
Remark 14.
For reader’s convenience, here we illustrate the argument [41] of deriving (41) under (33)–(34). Below subsequences of ${(λ_{k}, v_{k})}$ are not distinguished for simplicity of statements. In the first part we use only (40), which ensures (39) for (38). Then Theorem 3 of [2] is applicable to ${u_{k}}$ . Since $v_{k} ⩾ 0$ and $‖ v_{k} ‖_{\infty} \to + \infty$ , any alternatives of this theorem does not hold if $λ_{k} ⩾ δ > 0$ . Hence there arises that $λ_{k} \to 0$ . Next, by this convergence, the blowup set of ${u_{k}}$ , denoted by $S_{0}$ , is contained in $S$ , the blowup set of ${v_{k}}$ . If $x_{0} \notin S_{0}$ , conversely, there is $0 < r ≪ 1$ such that $λ_{k} f (v_{k}) ⩽ C$ on $B_{2 r} (x_{0})$ by (38) and (2). Then we obtain $v_{k} ⩽ C$ on $B_{r} (x_{0})$ by the elliptic estimate, recalling (45), which means $x_{0} \notin S$ . It thus holds that the third alternative of Theorem 3 of [2] to ${u_{k}}$ with $S = S_{0} \neq \emptyset$ , which ensures $\begin{matrix} (46) & \underset{k \to \infty}{lim inf} \int_{B_{r} (x_{0})} λ_{k} f (v_{k}) ⩾ 4 π, \forall x_{0} \in S, 0 < \forall r ≪ 1, \end{matrix}$ by $\begin{matrix} V_{k} (x) e^{u_{k}} = λ_{k} f (v_{k}) . \end{matrix}$ As in Remark 15 below, this (46) implies (9)–(10) with $♯ S < + \infty$ by the second condition of (3), and at the same time, $\begin{matrix} (47) & max_{B} u_{k} \to + \infty \end{matrix}$ and $\begin{matrix} (48) & u_{k} \to - \infty locally uniformly in B ∖ {x_{0}} \end{matrix}$ for $B = B_{r} (x_{0})$ with $0 < r ≪ 1$ . The second part is a refinement of the above rough estimate, based on the refined asymptotics of (40), that is, (33)–(34), which ensures (13), (14), and (23). This part is done by a modification of the argument [24], using the complex variable $z = x_{1} + \sqrt{- 1} x_{2}$ for $x = (x_{1}, x_{2}) \in Ω$ . Here we use the fact that (1) implies $\begin{matrix} {\tilde{H}}_{\overline{z}} = 0, \tilde{H} = H + \frac{α}{2} \end{matrix}$ to execute a residue analysis at each blowup point, where $\begin{matrix} H = v_{z}^{2}, α = W * T, T = - \frac{1}{π z^{2}}, \end{matrix}$ and $\begin{matrix} W = (λ e^{v} + P (v)) χ_{Ω}, P (u) = \int_{0}^{v} p (v) d v, f (v) = e^{v} + p (v) . \end{matrix}$ See [41] for more details.

The discrepancy between (2) and (40) is thus the first obstruction in our arguments; we have to derive (46) directly from (2) without (40). For this purpose we use the third condition of (3).

Given $x_{0} \in S$ , let $\overline{B} \subset Ω$ for $B = B_{r} (x_{0})$ and $0 < r ≪ 1$ . We take $φ \in C_{0}^{\infty} (B)$ satisfying $0 ⩽ φ ⩽ 1$ and $φ = 1$ on $B / 2 \equiv B_{r / 2} (x_{0})$ . Put $\begin{matrix} ξ_{k} = φ v_{k}, g_{k} = 2 \nabla v_{k} \cdot \nabla φ + v_{k} Δ φ, \end{matrix}$ to obtain $\begin{matrix} (49) & - Δ ξ_{k} = λ_{k} φ f (v_{k}) - g_{k} in B, ξ_{k} = 0 on \partial B . \end{matrix}$

We define $η_{k}$ by $\begin{matrix} - Δ η_{k} = g_{k} in B, η_{k} = 0 on \partial B . \end{matrix}$ It holds that $‖ g_{k} ‖_{q} ⩽ C$ for $1 ⩽ q < 2$ by (45) ane hence $‖ η_{k} ‖_{W^{2, q}} ⩽ C$ by the elliptic estimate, which implies $\begin{matrix} (50) & ‖ η_{k} ‖_{L^{\infty} (B)} ⩽ C \end{matrix}$ by Sobolev’s embedding theorem. Then $\begin{matrix} ζ_{k} = ξ_{k} + η_{k} \end{matrix}$ satisfies $\begin{matrix} - Δ ζ_{k} = λ_{k} φ f (v_{k}) in B, ζ_{k} = 0 on \partial B . \end{matrix}$

If $\begin{matrix} \underset{k \to \infty}{lim inf} \int_{B} λ_{k} f (v_{k}) < 4 π, \end{matrix}$ Theorem 1 of [2] implies $\begin{matrix} {‖ e^{γ ζ_{k}} ‖}_{L^{1} (B)} ⩽ C, γ = \frac{4 π}{4 π - δ} > 1 \end{matrix}$ and hence $\begin{matrix} {‖ e^{γ ξ_{k}} ‖}_{L^{1} (B)} ⩽ C \end{matrix}$ by (50). We thus obtain $\begin{matrix} {‖ e^{γ v_{k}} ‖}_{L^{1} (B / 2)} ⩽ C, \end{matrix}$ and therefore, $\begin{matrix} {‖ λ_{k} f (v_{k}) ‖}_{L^{γ} (B / 2)} ⩽ C \end{matrix}$ by (2) and the third relation of (3). Then $\begin{matrix} ‖ v_{k} ‖_{L^{\infty} (B / 4)} ⩽ C \end{matrix}$ follows from a local elliptic estimate of (1) with (45) and Sobolev’s embedding theorem, which implies $x_{0} \notin S$ . Hence (46) holds just by (2) and (3).
Remark 15.
Inequality (46) implies (9)–(10) with $♯ S < + \infty$ by a standard argument of measure theory. For reader’s convenience, we recall it here. First, the second condition of (3), $\begin{matrix} (51) & \int_{Ω} λ_{k} f (v_{k}) d x ⩽ C \end{matrix}$ ensures a subsequence, denoted by the same symbol, such that $λ_{k} f (v_{k})$ converges to a measure, denoted by $μ (d x)$ : $\begin{matrix} λ_{k} f (v_{k}) d x ⇀ μ (d x) in M (\overline{Ω}) . \end{matrix}$ Then (46) implies $\begin{matrix} (52) & μ ({x_{0}}) ⩾ 4 π, \forall x_{0} \in S \end{matrix}$ and hence $♯ S < + \infty$ , because $μ (\overline{Ω}) < + \infty$ is derived from (51). Second, by the elliptic estimate applied to (1) locally, the convergence (11) arises up to a subsequence with some $v = v (x)$ smooth in $\overline{Ω} ∖ S$ . Hence the support of the singular part $μ^{s} (d x)$ of $μ (d x)$ lies on the finite set $S$ , and therefore, is a sum of delta functions: $\begin{matrix} μ^{s} (d x) = \sum_{x_{0} \in S} m (x_{0}) δ_{x_{0}} (d x) . \end{matrix}$ Here, $m (x_{0}) ⩾ 4 π$ follows from (52), while $0 ⩽ r = r (x) \in L^{1} (Ω)$ in (9) arises as the Radon–Nikodym density of the regular part of $μ (d x)$ .

We finally show that $v = v (x)$ in the above Remark 15 is actually given by (12). For this purpose let $v = v (x)$ be as in (12), and put $\begin{matrix} w_{k} = v_{k} - v . \end{matrix}$ It holds that $v = v (x) \in W^{1, q} (Ω)$ , $1 ⩽ q < 2$ , by $S \subset Ω$ and the local profile of Green’s function, $\begin{matrix} (53) & G (x, x^{'}) = Γ (x - x^{'}) + K (x, x^{'}), K \in C^{\infty} (Ω \times \overline{Ω}) \end{matrix}$ for $\begin{matrix} Γ (x) = \frac{1}{2 π} log \frac{1}{| x |} . \end{matrix}$ We note that (53) is derived from (7) and the elliptic regularity.

Then we obtain $\begin{matrix} w_{k} \in W^{1, q} (Ω), w_{k} |_{\partial Ω} = 0 \end{matrix}$ and $\begin{matrix} Δ w_{k} \to 0 in M (\overline{Ω}) \end{matrix}$ by (9). By (45), on the other hand, we have $\begin{matrix} w_{k} ⇀ \exists w in W^{1, q} (Ω) \end{matrix}$ up to a subsequence, There arises that $Δ w = 0$ in Ω and $w = 0$ on $\partial Ω$ , in the sense of distributions and that of traces, respectively. We thus have $w = 0$ and hence $\begin{matrix} v_{k} ⇀ v in W^{1, q} (Ω), \end{matrix}$ which implies (11)–(12) by the elliptic regularity.
3. Proof of Theorems 2, 3, 4, and 5

So far, we have confirmed the exclusion of the boundary blowup, finiteness of blowup points, and pattern formation of the solution, as in (8), (9)–(10), and (11)–(12), respectively, under the assumption of (2) and (3). Using additional assumptions on the nonlinearity, now we show the residual vanishing (14)–(15), Hamiltonian control of the location of blowup points (23), and mass quantization (13) or some estimates of $m (x_{0})$ in (9).

Remark 16.
The first target below is the residual vanishing. Again, we review a standard method to confirm the essential difficulty in our case. The most common argument to deduce residual vanishing is to show $\begin{matrix} (54) & lim_{k \to \infty} \int_{Ω} f (v_{k}) = + \infty . \end{matrix}$ In fact, equality (54) implies (15) by the third condition of (3), and then (14) follows in (9). Once (40) holds, then (54) is a direct consequence of (10) with $S \neq \emptyset$ , (11)–(12), and (53) below with $\begin{matrix} (55) & \int_{B} | x - x_{0} |^{- 2} d x = + \infty \end{matrix}$ for $B = B (x_{0}, R)$ . This argument is adopted in the proof of Theorem 1 of [2]. Because of this structure, to derive (54) for the degenerate case $P ({1}) = 0$ of (26), for example, a slightly stronger condition $m (x_{0}) > 4 π$ is required in (10). This property is actually the second obstruction in the proof of Theorem 2 for the general case without (40). Then deriving mass quantization (13) before showing residual vanishing is proposed in [28–30] to overcome this difficulty by the method of duality described below.

Given the family ${(λ_{k}, v_{k})}$ in Theorem 1, thus we put $\begin{matrix} u_{k} = λ_{k} f (v_{k}), \end{matrix}$ to obtain $\begin{matrix} (56) & v_{k} (x) = \int_{Ω} G (x, x^{'}) u_{k} (x^{'}) d x \end{matrix}$ and $\begin{matrix} (57) & u_{k} (x) d x ⇀ \sum_{x_{0} \in S} m (x_{0}) δ_{x_{0}} (d x) + r (x) d x in M (\overline{Ω}) \end{matrix}$ with (10) by (2)–(3).

Let $F (v)$ stand for an integral of $f (v)$ : $\begin{matrix} F^{'} (v) = f (v) . \end{matrix}$ Hence it can be any of $F_{0} (v)$ , $F_{1} (v)$ , and $F_{2} (v)$ .

First, take $ψ \in C^{1} (\overline{Ω}, R^{2})$ satisfying $\begin{matrix} ψ \cdot ν = 0 on \partial Ω, \end{matrix}$ where ν denotes the outer unit normal vector. Since $\begin{matrix} u_{k} \nabla v_{k} = λ_{k} f (v_{k}) \nabla v_{k} = λ_{k} \nabla F (v_{k}) \end{matrix}$ we have $\begin{array}{l} - \int_{Ω} ψ \cdot u_{k} \nabla v_{k} & = - \iint_{Ω \times Ω} [ψ (x) \cdot \nabla_{x} G (x, x^{'})] u_{k} (x) u_{k} (x^{'}) d x d x^{'} \\ = - \int_{Ω} ψ \cdot λ_{k} \nabla F (v_{k}) \\ (58) & = \int_{Ω} (\nabla \cdot ψ) λ_{k} F (v_{k}) . \end{array}$ Next, given $x_{0} \in S$ , we take $0 < R ≪ 1$ such that $\begin{matrix} B_{3 R} (x_{0}) \subset Ω, B_{3 R} (x_{0}) \cap S = {x_{0}} . \end{matrix}$ Let $φ = φ_{x_{0}, R} (x) \in C_{0}^{\infty} (B (x_{0}, R))$ be the function satisfying $\begin{matrix} 0 ⩽ φ ⩽ 1, φ = 1 on B_{R / 2} (x_{0}) . \end{matrix}$ We put $\begin{matrix} ψ_{a, R} (x) = (x - a) φ_{x_{0}, R} \end{matrix}$ for $a \in R^{2}$ arbitrary.

The following lemma describes a fundamental identity in the method of duality introduced by [28].
Lemma 1.
Under the assumption of Theorem 1 it holds that $\begin{array}{l} lim_{k \to \infty} \int_{Ω} (\nabla \cdot ψ_{a, R}) λ_{k} F (v_{k}) \\ = \frac{m {(x_{0})}^{2}}{4 π} - m {(x_{0})}^{2} (x_{0} - a) \cdot (\nabla_{x} K (x_{0}, x_{0}) + \sum_{x_{0}^{'} \in S ∖ {x_{0}}} \nabla_{x} G (x_{0}, x_{0}^{'})) \\ (59) & + A_{1} (R, a) \end{array}$ with $A_{1} (R, a)$ satisfying $\begin{matrix} (60) & lim_{R ↓ 0} A_{1} (R, a) = 0 . \end{matrix}$
Proof.
Fix $a \in R^{2}$ and $0 < R ≪ 1$ , and write $ψ = ψ_{a, R}$ . We treat the left-hand side of (58) as follows, using $\hat{φ} = φ_{x_{0}, 2 R}$ : $\begin{array}{l} \int_{Ω} ψ \cdot u_{k} \nabla v_{k} & = \iint_{Ω \times Ω} ψ (x) \cdot \nabla_{x} G (x, x^{'}) u_{k} (x) u_{k} (x^{'}) d x d x^{'} \\ = \iint_{Ω \times Ω} ψ (x) \cdot \nabla_{x} G (x, x^{'}) u_{k} (x) \hat{φ} (x) u_{k} (x^{'}) \hat{φ} (x^{'}) d x d x^{'} \\ (61) & + \iint_{Ω \times Ω} ψ (x) \cdot \nabla_{x} G (x, x^{'}) u_{k} (x) \hat{φ} (x) u_{k} (x^{'}) (1 - \hat{φ} (x^{'})) d x d x^{'} . \end{array}$ The second term of the right-hand side on (61) is treated as $\begin{array}{l} lim_{k \to \infty} \iint_{Ω \times Ω} ψ (x) \cdot \nabla_{x} G (x, x^{'}) u_{k} (x) \hat{φ} (x) u_{k} (x^{'}) (1 - \hat{φ} (x^{'})) d x d x^{'} \\ (62) & = m {(x_{0})}^{2} (x_{0} - a) \cdot \sum_{x_{0}^{'} \in S ∖ {x_{0}}} \nabla_{x} G (x_{0}, x_{0}^{'}) + I (R) \end{array}$ with $\begin{matrix} (63) & lim_{R ↓ 0} I (R) = 0 \end{matrix}$ by (57) with (10), and (53).

For the first term of the right-hand side on (61) we use (53) again to get $\begin{array}{l} \iint_{Ω \times Ω} ψ (x) \cdot \nabla_{x} G (x, x^{'}) u_{k} (x) \hat{φ} (x) u_{k} (x^{'}) \hat{φ} (x^{'}) d x d x^{'} \\ = \frac{1}{2} \iint_{Ω \times Ω} ρ_{ψ}^{0} (x, x^{'}) u_{k} (x) \hat{φ} (x) u_{k} (x^{'}) \hat{φ} (x^{'}) d x d x^{'} \\ (64) & + \iint_{Ω \times Ω} ψ (x) \cdot \nabla_{x} K (x, x^{'}) u_{k} (x) \hat{φ} (x) u_{k} (x^{'}) \hat{φ} (x^{'}) d x d x^{'} \end{array}$ by the method of symmetrization, where $\begin{matrix} ρ_{ψ}^{0} (x, x^{'}) = - \frac{1}{2 π} \frac{x - x^{'}}{| x - x^{'} |^{2}} \cdot (ψ (x) - ψ (x^{'})) . \end{matrix}$ For the second term of the right-hand side on (64) it follows that $\begin{array}{l} lim_{k \to \infty} \iint_{Ω \times Ω} ψ (x) \cdot \nabla_{x} K (x, x^{'}) u_{k} (x) \hat{φ} (x) u_{k} (x^{'}) \hat{φ} (x^{'}) d x d x^{'} \\ (65) & = m {(x_{0})}^{2} (x_{0} - a) \cdot \nabla_{x} K (x_{0}, x_{0}) + II (R) \end{array}$ with $\begin{matrix} lim_{R ↓ 0} II (R) = 0 \end{matrix}$ from (57) with (10) again. For the first term of the right-hand side on (64), finally, we use $\begin{matrix} | ρ_{ψ}^{0} (x, x^{'}) | ⩽ C \end{matrix}$ and $\begin{array}{l} ρ_{ψ}^{0} (x, x^{'}) = & ρ_{ψ}^{0} (x, x^{'}) \tilde{φ} (x) \tilde{φ} (x^{'}) + ρ_{ψ}^{0} (x, x^{'}) (1 - \tilde{φ} (x)) \tilde{φ} (x^{'}) \\ + ρ_{ψ}^{0} (x, x^{'}) \tilde{φ} (x) (1 - \tilde{φ} (x)) \end{array}$ with $\begin{matrix} ρ_{ψ}^{0} (x, x^{'}) \tilde{φ} (x) \tilde{φ} (x^{'}) = - \frac{1}{2 π} \end{matrix}$ for $\tilde{φ} = φ_{x_{0}, R / 2}$ . We thus obtain $\begin{array}{l} lim_{k \to \infty} \frac{1}{2} \iint_{Ω \times Ω} ρ_{ψ}^{0} (x, x^{'}) u_{k} (x) \hat{φ} (x) u_{k} (x^{'}) \hat{φ} (x^{'}) d x d x^{'} \\ (66) & = - \frac{1}{4 π} m {(x_{0})}^{2} + III (R) \end{array}$ with $\begin{matrix} lim_{R ↓ 0} III (R) = 0 \end{matrix}$ similarly. We now end up with (59), and the proof is complete. □

Theorems 2, 3, 4, and 5 are the consequence the following lemma. Recall (53).
Lemma 2.
Under the assumption of Theorem 1 it holds that $\begin{array}{l} lim_{R ↓ 0} lim_{k \to \infty} \int_{Ω} λ_{k} F (v_{k}) \nabla φ_{x_{0}, R} \\ (67) & = - m {(x_{0})}^{2} (\nabla_{x} K (x_{0}, x_{0}) + \sum_{x_{0}^{'} \in S ∖ {x_{0}}} \nabla_{x} G (x_{0}, x_{0}^{'})) \end{array}$ and $\begin{matrix} (68) & lim_{R ↓ 0} lim_{k \to \infty} \int_{Ω} λ_{k} F (v_{k}) (2 φ_{x_{0}, R} + (x - x_{0}) \cdot \nabla φ_{x_{0}, R}) = \frac{m {(x_{0})}^{2}}{4 π} . \end{matrix}$
Proof.
First, since $\begin{matrix} \nabla \cdot ψ_{a, R} = 2 φ_{x_{0}, R} + (x - a) \cdot \nabla φ_{x_{0}, R} \end{matrix}$ it holds that $\begin{array}{l} \int_{Ω} λ_{k} F (v_{k}) (2 φ_{x_{0}, R} + (x - a) \cdot \nabla φ_{x_{0}, R}) \\ = \frac{m {(x_{0})}^{2}}{4 π} - m {(x_{0})}^{2} (x_{0} - a) \cdot (\nabla_{x} K (x_{0}, x_{0}) + \sum_{x_{0}^{'} \in S ∖ {x_{0}}} \nabla_{x} G (x_{0}, x_{0}^{'})) \\ (69) & + A_{1} (R, a) + o (1) \end{array}$ by (59) and (60). Putting $a = x_{0}$ , we obtain $\begin{matrix} \int_{Ω} λ_{k} F (v_{k}) (2 φ_{x_{0}, R} + (x - x_{0}) \cdot \nabla φ_{x_{0}, R}) = \frac{m {(x_{0})}^{2}}{4 π} + A_{1} (R, x_{0}) + o (1) \end{matrix}$ as $k \to \infty$ , and hence (68).

Subtracting these equalities (69) and (68), second, we get $\begin{array}{l} (x_{0} - a) \cdot \int_{Ω} λ_{k} F (v_{k}) \nabla φ_{x_{0}, R} \\ + m {(x_{0})}^{2} (x_{0} - a) \cdot (\nabla_{x} K (x_{0}, x_{0}) + \sum_{x_{0}^{'} \in S ∖ {x_{0}}} \nabla_{x} G (x_{0}, x_{0}^{'})) \\ = A_{2} (R, a) + o (1) \end{array}$ as $k \to \infty$ with $\begin{matrix} lim_{R ↓ 0} A_{2} (R, a) = 0, \end{matrix}$ more precisely, $\begin{array}{l} (x_{0} - a) \cdot {lim_{R ↓ 0} lim_{k \to \infty} \int_{Ω} λ_{k} F (v_{k}) \nabla φ_{x_{0}, R} \\ + m {(x_{0})}^{2} (\nabla_{x} K (x_{0}, x_{0}) + \sum_{x_{0}^{'} \in S ∖ {x_{0}}} \nabla_{x} G (x_{0}, x_{0}^{'}))} = 0 \end{array}$ for any $a \in R^{2}$ . Hence it holds that (67). □

Now we show the following proof. Proof of Theorem 5.
Up to a subsequence it holds that $\begin{matrix} λ_{k} f (v_{k}) d x ⇀ \sum_{x_{0} \in S} m (x_{0}) δ_{x_{0}} (d x) + r (x) d x in M (\overline{Ω}) \end{matrix}$ with $0 ⩽ r = r (x) \in L^{1} (Ω)$ and $m (x_{0}) ⩾ 4 π$ , which implies $\begin{matrix} (70) & lim_{R ↓ 0} lim_{k \to \infty} \int_{Ω} λ_{k} f (v_{k}) (2 φ_{x_{0}, R} + (x - x_{0}) \cdot \nabla φ_{x_{0}, R}) = 2 m (x_{0}) \end{matrix}$ as in the proof of the above lemmas. Then inequality (18) implies $\begin{matrix} \frac{m {(x_{0})}^{2}}{4 π} ⩾ 2 β m (x_{0}), \end{matrix}$ from (68) applied to $F (v) = F_{0} (v) + γ_{1}$ , which implies $m (x_{0}) ⩾ 8 π β$ . Similarly, we obtain $m (x_{0}) ⩽ 8 π γ$ from (22). □
Proof of Theorem 2.
We have readily shown that $\begin{matrix} m (x_{0}) ⩾ 8 π β \end{matrix}$ follows from (18). Since there arises $\begin{matrix} v (x) ⩾ m (x_{0}) G (x, x_{0}) ⩾ 4 β log \frac{1}{| x - x_{0} |} - C, x \in B = B_{r} (x_{0}) \end{matrix}$ for $0 < r ≪ 1$ from (12) and (53), it holds that $\begin{matrix} \int_{B} e^{α v (x)} d x = + \infty \end{matrix}$ by $α β > \frac{1}{2}$ , which implies (54) by (19) and Fatou’s lemma as in [2]. Then we obtain (15). □
Proof of Theorem 3.
We have (15) by Theorem 2. From (22), therefore, it follows that $\begin{matrix} (71) & λ_{k} F_{2} (x) d x ⇀ μ_{2} (d x) in M (\overline{Ω}) \end{matrix}$ with $\begin{matrix} (72) & 0 ⩽ μ_{2} (d x) = \sum_{x_{0} \in S} m_{2} (x_{0}) δ_{x_{0}} (d x), 0 ⩽ m_{2} (x) ⩽ γ m (x_{0}), \end{matrix}$ up to a subsequence. We obtain $\begin{matrix} lim_{k \to \infty} \int_{Ω} λ_{k} F_{2} (v_{k}) \nabla φ_{x_{0}, R} = 0 \end{matrix}$ for $0 < R ≪ 1$ by (71)–(72), and then, the equality $\begin{matrix} \nabla_{x} (K (x_{0}, x_{0}) + \sum_{x_{0}^{'} \in S ∖ {x_{0}}} G (x_{0}, x_{0}^{'})) = 0, \forall x_{0} \in S \end{matrix}$ follows from (67) for $F = F_{2}$ , which is equivalent to (23). □
Remark 17.
We expect that the requirement of the residual vanishing is not necessary in Theorem 3.
Proof of Theorem 4.
By (25) it holds that $\begin{matrix} (73) & λ_{k} F_{0} (v_{k}) d x ⇀ \sum_{x_{0} \in S} m (x_{0}) δ_{x_{0}} (d x) + R (x) d x in M (\overline{Ω}) \end{matrix}$ with $0 ⩽ R = R (x) \in L^{1} (Ω)$ besides (9)–(10). Then we obtain $\begin{matrix} lim_{R ↓ 0} lim_{k \to \infty} \int_{Ω} λ_{k} F_{0} (v_{k}) (2 φ_{x_{0}, R} + (x - x_{0}) \cdot \nabla φ_{x_{0}, R}) = 2 m (x_{0}) \end{matrix}$ similarly, and hence $\begin{matrix} 2 m (x_{0}) = \frac{m {(x_{0})}^{2}}{4 π} \end{matrix}$ by (68). Hence $m (x_{0}) = 8 π$ follows. □

4. Proof of Theorem 6

This section is devoted to the proof of Theorem 6. We assume (33) and (37) for the nonlinearity. Given the sequence of solutions ${(λ_{k}, v_{k})}$ to (1) satisfying (3), define $(u_{k}, V_{k} (x))$ by (38): $\begin{matrix} (74) & u_{k} = v_{k} + log λ_{k}, V_{k} (x) = e^{- v_{k}} f (v_{k}) . \end{matrix}$ Since (33) and (37) imply (40), it holds that (39): $\begin{matrix} (75) & - Δ u_{k} = V_{k} (x) e^{u_{k}}, 0 ⩽ V_{k} (x) ⩽ b in Ω, \int_{Ω} e^{u_{k}} ⩽ C \end{matrix}$ for $b = 1$ . We have also (47) and (48), $\begin{matrix} (76) & max_{B} u_{k} \to + \infty, u_{k} \to - \infty locally uniformly in B ∖ {x_{0}}, \end{matrix}$ by [2], or, by (11)–(12) in the result of Theorem 2. We recall the following result.

Theorem 7 ([19]).

Let $B = B_{R} (x_{0}) \subset R^{2}$ be a ball, and let ${(u_{k}, V_{k} (x))}$ be a sequence of solutions to $\begin{matrix} (77) & - Δ u_{k} = V_{k} (x) e^{u_{k}}, 0 ⩽ V_{k} (x) ⩽ b in B, \int_{B} e^{u_{k}} ⩽ C, \end{matrix}$ satisfying $\begin{matrix} (78) & ‖ \nabla V_{k} ‖_{\infty} ⩽ C, \end{matrix}$ where $b > 0$ is a constant. Assume, furthermore, ( 76 ) and $\begin{matrix} (79) & max_{\partial B} u_{k} - min_{\partial B} u_{k} ⩽ C . \end{matrix}$

Then, passing to a subsequence there arises $\begin{matrix} (80) & V_{k} (x) e^{u_{k}} d x ⇀ 8 π δ_{x_{0}} (d x) in M (B) . \end{matrix}$ There is, furthermore, $0 < r_{0} ≪ 1$ , such that $\begin{matrix} (81) & | u_{k} (x) - u_{k} (x_{k}) - 2 log \frac{1}{1 + \frac{V_{k} (x_{k})}{8} e^{u_{k} (x_{k})} | x - x_{k} |^{2}} | ⩽ C \end{matrix}$ for any $x \in B_{r_{0}} (x_{k})$ and $k = 1, 2, \dots$ , where $x_{k} \to x_{0}$ is taken as ${max}_{B} u_{k} = u_{k} (x_{k})$ .

If $\begin{matrix} (82) & V_{k} \to V uniformly in B \end{matrix}$ in (75) we have $\begin{matrix} V_{k} (x) e^{u_{k}} d x ⇀ 8 π m δ_{x_{0}} (d x), m \in N, in M (B) \end{matrix}$ by [20]. Theorem 7 says that the simplicity $m = 1$ and uniform estimate (81) follow under the cost of (78) and (79). Here, we notice that $V > 0$ in B may be assumed, because $V (0) > 0$ follows from the property of scaling limit. See [20] for the proof of this fact.

Inequality (78) assumed in [40], actually holds for $(u_{k}, V_{k}) = ({\tilde{u}}_{k}, {\tilde{V}}_{k} (x))$ defined below, if $β < \frac{1}{2}$ is valid in (37). In fact, under (37) we obtain $\begin{matrix} λ_{k} {‖ g (v_{k}) ‖}_{p} ⩽ C λ_{k} {‖ e^{v_{k}} ‖}_{p}^{1 / p} ⩽ C λ_{k}^{1 - \frac{1}{p}} = o (1), p = \frac{1}{β} \end{matrix}$ by (3) and (40), which implies $\begin{matrix} (83) & ‖ z_{k} ‖_{W^{2, p}} ⩽ C \end{matrix}$ for $z_{k} = z_{k} (x)$ defined by $\begin{matrix} (84) & - Δ z_{k} = λ_{k} g (v_{k}) in Ω, z_{k} = 0 in \partial Ω . \end{matrix}$ if $p = \frac{1}{β} > 2$ it follows that $\begin{matrix} (85) & ‖ z_{k} ‖_{C^{1, θ}} ⩽ C, 0 < θ < 1 \end{matrix}$ from (83).

Then we put $\begin{matrix} {\tilde{u}}_{k} = v_{k} + log λ_{k} - z_{k}, \end{matrix}$ and deduce (75) for $(u_{k}, V_{k} (x)) = ({\tilde{u}}_{k}, {\tilde{V}}_{k} (x))$ defined by $\begin{matrix} (86) & {\tilde{V}}_{k} (x) = e^{z_{k}}, \end{matrix}$ that is, $\begin{matrix} (87) & - Δ {\tilde{u}}_{k} = {\tilde{V}}_{k} (x) e^{{\tilde{u}}_{k}}, 0 ⩽ {\tilde{V}}_{k} (x) ⩽ \tilde{b} in Ω, \int_{Ω} e^{{\tilde{u}}_{k}} ⩽ C . \end{matrix}$ This ${\tilde{V}}_{k} = {\tilde{V}}_{k} (x)$ satisfies (78) by (85): $\begin{matrix} ‖ \nabla {\tilde{V}}_{k} ‖_{\infty} ⩽ C . \end{matrix}$ We may assume, furthermore, $B_{2 R} (x_{0}) \cap S = {x_{0}}$ , recalling that $S$ denotes the blowup set of ${v_{k}}$ . Then, (79) follows from (11)–(12) and (85): $\begin{matrix} max_{\partial B} {\tilde{u}}_{k} - min_{\partial B} {\tilde{u}}_{k} ⩽ C . \end{matrix}$

We thus obtain (81) by Theorem 7, which means $\begin{matrix} (88) & | v_{k} (x) - v_{k} (x_{k}) - 2 log \frac{1}{1 + \frac{λ_{k}}{8} e^{v_{k} (x_{k})} | x - x_{k} |^{2}} | ⩽ C \end{matrix}$ for $x \in B (x_{k}, R_{1})$ with $0 < R_{1} ≪ 1$ and $k = 1, 2, \dots$ , because of $\begin{matrix} {\tilde{V}}_{k} e^{{\tilde{u}}_{k}} = λ_{k} e^{v_{k}} . \end{matrix}$ Inequality (88), finally, implies (43) by $\begin{array}{l} | \frac{1 + \frac{λ_{k}}{8} f (v_{k} (x_{k})) | x - x_{k} |^{2}}{1 + \frac{λ_{k}}{8} e^{v_{k} (x_{k})} | x - x_{k} |^{2}} - 1 | & = \frac{\frac{λ_{k}}{8} | g (v_{k} (x_{k})) | | x - x_{k} |^{2}}{1 + \frac{λ_{k}}{8} e^{v_{k} (x_{k})} | x - x_{k} |^{2}} \\ ⩽ e^{- v_{k} (x_{k})} | g (v_{k} (x_{k})) | = o (1) . \end{array}$ Theorem 7, therefore, implies Theorem 6 under the assumption of (33) and (37) for $β < \frac{1}{2}$ .

The main event of the proof of Theorem 6 is, thus, to improve the growth rate of the residual nonlinearity $g (v)$ in (33) up to near optimal, $β < 1$ as in (37), to assure (43) for (1). Here, we use the alternative proof of Theorem 7 by [21], different from the original one of [19] based on moving planes. The idea of this proof goes back to [1] for more difficult problem with singular source. We thus use the method of [7], and control of tails of the rescaled solution is achieved by the coincidence of the local mass $8 π$ of bubbles in (80) and that of entire solution arising in the scaling limit, that is, $\begin{matrix} \int_{R^{2}} e^{\tilde{v}} = 8 π \end{matrix}$ for the solution $\tilde{v} = \tilde{v} (x)$ to (44).

We obtain, consequently, a refinement of Theorem 7 in the following form. The convergence (82) assumed below is just used to assure uniform continuity at $x = x_{0}$ of $V_{k} (x)$ , that is, there is $c ⩾ 0$ such that any $ε > 0$ admits $δ > 0$ such that $\begin{matrix} | V_{k} (x) - c | < ε \end{matrix}$ for any $k = 1, 2, \dots$ , provided that $| x - x_{0} | < δ$ .

Theorem 8 ([21]).

Let $B = B_{r} (x_{0}) \subset R^{2}$ be a ball and let ${(u_{k}, V_{k} (x))}$ be a sequence of solutions to ( 77 ). Let ( 76 ), ( 79 ), and ( 80 ) be satisfied, and asssume ( 82 ). Then, it holds that ( 81 ) for any $x \in B (x_{k}, R_{0})$ with $0 < R_{0} ≪ 1$ , and $k = 1, 2, \dots$ , provided that $V_{k} \in C^{1} (\overline{B})$ and $\begin{matrix} (89) & \int_{B_{k}} [(x - x_{k}) \cdot \nabla V_{k} (x)] e^{u_{k}} ⩽ C {(log \frac{1}{δ_{k}})}^{- 1} \end{matrix}$ for $δ_{k}, r_{k} > 0$ defined by $\begin{matrix} (90) & B_{k} = B (x_{k}, r_{k}), r_{k} = δ_{k} log \frac{1}{δ_{k}}, δ_{k} = e^{- u_{k} (x_{k}) / 2} . \end{matrix}$

Turning to the proof of Theorem 6, we note that inequality (85) is reduced to $\begin{matrix} (91) & ‖ z_{k} ‖_{C^{θ}} ⩽ C, 0 < θ < 1 \end{matrix}$ if $p = \frac{1}{β} > 1$ in (83), where $z_{k} = z_{k} (x)$ is defined by (84). This (91), however, still guarantees (82) for ${\tilde{V}}_{k} = {\tilde{V}}_{k} (x)$ in (86). Thus there is $V = \tilde{V} (x)$ such that $\begin{matrix} {\tilde{V}}_{k} \to \tilde{V} uniformly in B, \end{matrix}$ passing to a subsequence. Inequality (89) for this ${\tilde{V}}_{k} = {\tilde{V}}_{k} (x)$ is left to complete the proof.

To confirm this inequality, we note that its left-hand side is equal to $\begin{matrix} \int_{{\tilde{B}}_{k}} [(x - x_{k}) \cdot \nabla {\tilde{V}}_{k} (x)] e^{{\tilde{u}}_{k}} = \int_{{\tilde{B}}_{k}} [(x - x_{k}) \cdot \nabla z_{k}] e^{z_{k}} e^{{\tilde{u}}_{k}} \end{matrix}$ for ${\tilde{B}}_{k} = B (x_{k}, {\tilde{r}}_{k})$ defined by $\begin{matrix} {\tilde{r}}_{k} = {\tilde{δ}}_{k} log \frac{1}{{\tilde{δ}}_{k}}, {\tilde{δ}}_{k} = e^{- {\tilde{u}}_{k} (x_{k}) / 2} . \end{matrix}$ Since (83) for $p > 1$ implies $\begin{matrix} ‖ \nabla z_{k} ‖_{q} = O (1), 1 ⩽ \forall q < \infty \end{matrix}$ and ${max}_{B} u_{k} = u_{k} (x_{k})$ , we obtain $\begin{array}{l} | \int_{{\tilde{B}}_{k}} [(x - x_{k}) \cdot \nabla {\tilde{V}}_{k} (x)] e^{{\tilde{u}}_{k}} | & ⩽ | \int_{{\tilde{B}}_{k}} [(x - x_{k}) \cdot e^{z_{k}} \nabla z_{k}] e^{{\tilde{u}}_{k}} | \\ ⩽ {\tilde{δ}}_{k}^{- 2} {‖ e^{z_{k}} ‖}_{\infty} ‖ \nabla z_{k} ‖_{q} ‖ x - x_{k} ‖_{L^{q^{'}} ({\tilde{B}}_{k})} \\ ⩽ C {\tilde{δ}}_{k}^{- 2} {\tilde{r}}_{k}^{(q^{'} + 2) / q^{'}} \\ = C {\tilde{δ}}_{k}^{- 1 + \frac{2}{q^{'}}} {(log \frac{1}{{\tilde{δ}}_{k}})}^{1 + \frac{2}{q^{'}}} \end{array}$ for $\frac{1}{q} + \frac{1}{q^{'}} = 1$ . We have $- 1 + \frac{2}{q^{'}} > 0$ for $q > 2$ , and therefore, inequality (89) follows for this $(u_{k}, V_{k} (x)) = ({\tilde{u}}_{k}, {\tilde{V}}_{k} (x))$ . We thus conclude the proof of Theorem 6.

5. Asymptotic non-degneracy

The last two sections are appendix. First observation is that the local uniform estimate (43) induces the asymptotic non-degeneracy of the solution in the extremal case of $f (v) = e^{v}$ . Thus, if $S = {x_{1}^{*}, \dots, x_{m}^{*}}$ is the blowup set of ${v_{k}}$ and $x_{*} = (x_{1}^{*}, \dots, x_{m}^{*})$ is a non-degenerate critical point of $H = H (x)$ , $x = (x_{1}, \dots, x_{m})$ , defined by (16), then the linearized operator $L_{k} = - Δ - λ_{k} f^{'} (v_{k})$ provided with the Dirichlet boundary condition is non-degenerate for k sufficiently large. The property is proven by [18] and related results are done by [15,27].

Here, we just note the key identity used there, $\begin{array}{l} \int_{B_{r} (x_{k})} [(x - x_{k}) \cdot \nabla v_{k}] Δ w_{k} + Δ v_{k} [(x - x_{k}) \cdot \nabla w_{k}] d x \\ = \int_{B_{r} (x_{k})} (x - x_{k}) \cdot \nabla (λ_{k} f (v_{k}) w_{k}), \end{array}$ is still valid to the general case of (1), where $w_{k} = w_{k} (x)$ is a solution to $\begin{matrix} - Δ w_{k} = λ_{k} f^{'} (v_{k}) w_{k} in Ω, w_{k} = 0 on \partial Ω . \end{matrix}$ Equivalence of Morse indices is also expected as in [16,17,36], which implies the above described asymptotic non-degeneracy. Some careful analysis, however, is left in future for this generalization.

6. Spatially inhomogeneous nonlinearity

Several results are extended for spatially inhomogeneous nonlinearity in (1): $\begin{matrix} (92) & - Δ v = λ f (x, v), v > 0 in Ω, v = 0 on \partial Ω, \end{matrix}$ for $f = f (x, v) \in C^{1, 0} (\overline{Ω} \times [0, + \infty))$ satisfyig $\begin{matrix} (93) & 0 ⩽ f (x, v) ⩽ e^{v} . \end{matrix}$ Given the sequence of the solution ${(λ_{k}, v_{k})}$ satisfying $\begin{matrix} (94) & ‖ v_{k} ‖_{\infty} \to + \infty, \int_{Ω} λ_{k} f (x, v_{k}) ⩽ C, λ_{k} ⩽ C, \end{matrix}$ first, we obtain the same result as in Theorem 1, provided that $\begin{matrix} (95) & f (x, v) > 0, | \nabla_{x} log f (x, v) | ⩽ C, x \in ω \cap Ω, v ⩾ 0, \end{matrix}$ where $\partial Ω \subset ω$ is an open set. In fact, we have $\partial Ω \cap S = \emptyset$ in this case by the method of [14,22].

More precisely, given $x_{0} \in \partial Ω$ , we take $B = B_{R} (x_{1}) \subset Ω^{c}$ satisfying $\overline{B} \cap \overline{Ω} = {x_{0}}$ . We apply the Kelvin transformation $\begin{matrix} y = R^{2} \frac{x - x_{1}}{| x - x_{1} |^{2}}, w (y) = v (x) \end{matrix}$ in (92), to reach $\begin{matrix} (96) & - Δ w = g (y, w) in Ω^{'}, w = 0 on \partial Ω^{'}, \end{matrix}$ where $Ω^{'} \subset B_{R} (0)$ is the image of $x \in Ω \mapsto y \in Ω^{'}$ and $\begin{matrix} g (y, w) = \frac{R^{4}}{| y |^{4}} f (x_{1} + R^{2} \frac{y}{| y |^{2}}, w) . \end{matrix}$

It holds that $\begin{array}{l} \frac{\partial g}{\partial y_{i}} = & {(\frac{R}{| y |})}^{4} f (x_{1} + R^{2} \frac{y}{| y |^{2}}, w) \frac{y_{i}}{| y |^{2}} \\ \cdot {- 4 + \frac{R^{2}}{| y |} \frac{\partial}{\partial x_{i}} log f (x_{1} + R^{2} \frac{y}{| y |^{2}}, w)} . \end{array}$ Since $\begin{matrix} \frac{R^{2}}{| y |} = | x - x_{1} |, \end{matrix}$ furthermore, we have $\begin{matrix} | x_{1} + R^{2} \frac{y}{| y |^{2}} - x_{0} | ⩽ | x_{1} - x_{0} | + \frac{R^{2}}{| y |} = R + | x - x_{1} | \end{matrix}$ and $\begin{array}{l} ξ \cdot \nabla g = & {(\frac{R}{| y |})}^{4} f (x_{1} + R^{2} \frac{y}{| y |^{2}}, w) \cdot \frac{1}{| y |} \\ \cdot (ξ \cdot \frac{y}{| y |}) \cdot {- 4 + O (| x - x_{1} |)} \end{array}$ for $ξ \in R^{2}$ in $| ξ | = 1$ . We have then $0 < R_{0} ≪ 1$ and a neighbourhood $ω_{x_{0}}$ of $x_{0}$ such that $\begin{matrix} ξ \cdot \nabla g (y, w) ⩽ 0 in (ω {(x_{0})}^{'} \cap Ω^{'}) \times [0, + \infty) \end{matrix}$ for $0 < R ⩽ R_{0}$ , provided that $\begin{matrix} ξ \cdot \frac{y}{| y |} ⩾ 0, \end{matrix}$ where $ω {(x_{0})}^{'}$ denotes the image of $x \in ω (x_{0}) \mapsto y \in ω {(x_{0})}^{'}$ .

At this stage we apply the method of moving plane [14] to assure the existence of $0 < r, δ ≪ 1$ independent of the solution $v = v (x)$ to (92) such that $\begin{matrix} \frac{d}{d t} v (x_{0} + t ξ) < 0, - r < t < 0 \end{matrix}$ for any $x_{0} \in \partial Ω$ and $ξ \in R^{2}$ with $| ξ | = 1$ satisfying $ξ \cdot ν_{x_{0}} ⩾ δ$ , where $ν_{x_{0}}$ denotes the outer normal unit vector at $x_{0} \in \partial Ω$ . Based on this fact, we apply argument of [9] as in Section 2 to obtain $S \subset Ω$ , because (45) follows from the second condition of (94).

Second, the duality method used for the proof of Theorems 2, 3, 4, and 5 is valid if $f (x, v)$ takes the form of separation of variables $\begin{matrix} (97) & f (x, v) = W (x) h (v), 0 < W = W (x) \in C^{1} (\overline{Ω}), \end{matrix}$ with $h = h (v) ⩾ 0$ satisfying the assumptions there required for $f = f (v)$ .

In fact, given the sequence of solutions to (92) satisfying (94), we put $\begin{matrix} u_{k} = λ_{k} f (x, v_{k}) . \end{matrix}$ It holds that $\begin{matrix} u_{k} \nabla v_{k} = λ_{k} W (x) \nabla F (v_{k}) \end{matrix}$ if $f = f (x, v)$ takes the form of (97), where $F^{'} (v) = h (v)$ . The fundamental equality (69) is now replaced by $\begin{array}{l} lim_{R ↓ 0} lim_{k \to \infty} \int_{Ω} λ_{k} F (v_{k}) (2 φ_{x_{0}, R} + (x - a) \cdot \nabla φ_{x_{0}, R} + (x - a) φ_{x_{0}, R} \cdot \nabla W) \\ = \frac{m {(x_{0})}^{2}}{4 π} - m {(x_{0})}^{2} (x_{0} - a) \cdot (\nabla_{x} K (x_{0}, x_{0}) + \sum_{x_{0}^{'} \in S ∖ {x_{0}}} \nabla_{x} G (x_{0}, x_{0}^{'})), \end{array}$ which implies $\begin{array}{l} lim_{R ↓ 0} lim_{k \to \infty} \int_{Ω} λ_{k} F (v_{k}) (2 φ_{x_{0}, R} + (x - x_{0}) \cdot \nabla φ_{x_{0}, R} + (x - x_{0}) φ_{x_{0}, R} \cdot \nabla W) \\ = \frac{m {(x_{0})}^{2}}{4 π} \end{array}$ and $\begin{array}{l} lim_{R ↓ 0} lim_{k \to \infty} \int_{Ω} λ_{k} F (v_{k}) (\nabla φ_{x_{0}, R} + φ_{x_{0}, R} \nabla W) \\ (98) & = - m {(x_{0})}^{2} (\nabla_{x} K (x_{0}, x_{0}) + \sum_{x_{0}^{'} \in S ∖ {x_{0}}} \nabla_{x} G (x_{0}, x_{0}^{'})), \end{array}$ instead of (68) and (67), respectively.

Then we obtain the residual vanishing (15) if (18) holds for $α, β > 0$ in $α β > \frac{1}{2}$ , where $f (v) = h (v)$ and $\begin{matrix} F_{0} (v) = \int_{0}^{v} h (v) d v . \end{matrix}$ It thus follows that $\begin{matrix} λ_{k} h (v_{k}) W (x) d x ⇀ \sum_{x_{0} \in S} m (x_{0}) δ_{x_{0}} (d x) in M (\overline{Ω}) \end{matrix}$ in this case, where $m (x_{0}) ⩾ 4 π$ .

If we have (22) for this $F_{0} (v)$ and $f (v) = h (v)$ , furthermore, there arises $\begin{matrix} λ_{k} F_{0} (v_{k}) W (x) d x ⇀ \sum_{x_{0} \in S} m_{0} (x_{0}) δ_{x_{0}} (d x) in M (\overline{Ω}) \end{matrix}$ with $\begin{matrix} (99) & 2 m_{0} (x_{0}) = \frac{m {(x_{0})}^{2}}{4 π} . \end{matrix}$ Then it follows that $\begin{matrix} lim_{k \to \infty} \int_{Ω} λ_{k} F (v_{k}) (\nabla φ_{x_{0}, R} + φ_{x_{0}, R} \nabla W) = m_{0} (x_{0}) \nabla log W (x_{0}) \end{matrix}$ for $0 < R ≪ 1$ from $\begin{matrix} λ_{k} F (v_{k}) (\nabla φ_{x_{0}, R} + φ_{x_{0}, R} \nabla W) = λ_{k} F (v_{k}) W (\frac{\nabla φ_{x_{0}, R}}{W} + φ_{x_{0}, R} \nabla log W), \end{matrix}$ which implies $\begin{matrix} \nabla_{x} K (x_{0}, x_{0}) + \sum_{x_{0}^{'} \in S ∖ {x_{0}}} \nabla_{x} G (x_{0}, x_{0}^{'}) + \frac{1}{8 π} \nabla log W (x_{0}) = 0 \end{matrix}$ by (98)–(99). The conclusion of Theorem 3 thus arises in this case of (97), if $h (v)$ satisfies the condition there required for $f (v)$ , with $H = H (x)$ , $x = (x_{1}, \dots, x_{m})$ , replaced by $\begin{matrix} (100) & H (x) = \frac{1}{2} \sum_{j = 1}^{m} R (x_{j}) + \sum_{1 ⩽ i < j ⩽ m} G (x_{i}, x_{j}) + \frac{1}{8 π} \sum_{j = 1}^{m} log W (x_{j}) . \end{matrix}$

Mass quantization (13) also arises under the assumption of Theorem 3 for the above $F_{0} (v)$ and $f (v) = h (v)$ . We thus have a generalization of the result in [22] for the extremal case of $f (x, v) = W (x) e^{v}$ .

A typical examples are $h (v) = e^{v} + g (v)$ with $g (v) = o (e^{v})$ as $v ↑ + \infty$ and $\begin{matrix} h (v) = \int_{[0, 1]} α e^{α v} P (d α), \end{matrix}$ where $P (d α)$ is a Borel measure on $[0, 1]$ . Then the blowup set of the sequence of solutions ${(λ_{k}, v_{k})}$ to (92) for (97) satisfying (94) is composed of a finite number of interior points denoted by $S = {x_{1}^{*}, \dots, x_{m}^{*}}$ , and it holds that $\begin{matrix} λ_{k} h (v_{k}) W (x) d x ⇀ 8 π \sum_{x_{0} \in S} δ_{x_{0}} (d x) in M (\overline{Ω}) \end{matrix}$ and $\begin{matrix} v_{k} \to 8 π \sum_{x_{0} \in S} G (\cdot, x_{0}) locally uniformly in \overline{Ω} ∖ S . \end{matrix}$ We have also that $x_{*} = (x_{1}^{*}, \dots, x_{m}^{*})$ is a critical point of $H (x)$ for $x = (x_{1}, \dots, x_{m})$ defined by (100).

Analogous result to Theorems 6 is expected similarly. If we have (97) with $\begin{matrix} (101) & h (v) = e^{v} + g (v), g (v) = O (e^{β v}), v ↑ + \infty \end{matrix}$ for $β < 1$ , it holds that $\begin{matrix} | v_{k} (x) - v_{k} (x_{k}) - 2 log \frac{1}{1 + \frac{λ_{k}}{8} W (x_{k}) h (v_{k} (x_{k})) | x - x_{k} |^{2}} | ⩽ C \end{matrix}$ for $x \in B_{r} (x_{k})$ with $0 < r ≪ 1$ , where $x_{k} \to x_{0}$ satisfies ${max}_{B_{2 r} (x_{0})} v_{k} = v_{k} (x_{k})$ for $x_{0} \in S$ .

Footnotes

Acknowledgement

This work is supported by JSPS Grand-in-Aid for Scientific Reserach 19H 01799.

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