A note on a sinh-Poisson type equation with variable intensities on pierced domains

Abstract

We consider a sinh-Poisson type equation with variable intensities and Dirichlet boundary condition on a pierced domain $\begin{matrix} \{\begin{matrix} Δ u + ρ (V_{1} (x) e^{u} - V_{2} (x) e^{- τ u}) = 0 & in Ω_{ϵ} : = Ω ∖ ⋃_{i = 1}^{m} \overline{B (ξ_{i}, ϵ_{i})} \\ u = 0 & on \partial Ω_{ϵ}, \end{matrix} \end{matrix}$ where $ρ > 0$ , $V_{1}, V_{2} > 0$ are smooth potentials in Ω, $τ > 0$ , Ω is a smooth bounded domain in $R^{2}$ and $B (ξ_{i}, ϵ_{i})$ is a ball centered at $ξ_{i} \in Ω$ with radius $ϵ_{i} > 0$ , $i = 1, \dots, m$ . When $ρ > 0$ is small enough and $m_{1} \in {1, \dots, m - 1}$ , there exist radii $ϵ = (ϵ_{1}, \dots, ϵ_{m})$ small enough such that the problem has a solution which blows-up positively at the points $ξ_{1}, \dots, ξ_{m_{1}}$ and negatively at the points $ξ_{m_{1} + 1}, \dots, ξ_{m}$ as $ρ \to 0$ . The result remains true in cases $m_{1} = 0$ with $V_{1} \equiv 0$ and $m_{1} = m$ with $V_{2} \equiv 0$ , which are Liouville type equations.

Keywords

sinh-Poisson equation pierced domain blowing-up solutions

1. Introduction

Let $Ω \subset R^{2}$ be a smooth bounded domain. Given $ϵ : = (ϵ_{1}, \dots, ϵ_{m})$ and m different points $ξ_{1}, \dots, ξ_{m} \in Ω$ , define $Ω_{ϵ} : = Ω ∖ ⋃_{i = 1}^{m} \overline{B (ξ_{i}, ϵ_{i})}$ , a pierced domain, where $B (ξ_{i}, ϵ_{i})$ is a ball centered at $ξ_{i}$ with radius $ϵ_{i} > 0$ . Inspired by recent results in [14], we are interested in this paper in building solutions to a sinh-Poisson type equation with variable intensities and Dirichlet boundary condition on pierced domains: $\begin{matrix} (1.1) & \{\begin{matrix} Δ u + ρ (V_{1} (x) e^{u} - ν V_{2} (x) e^{- τ u}) = 0 & in Ω_{ϵ} \\ u = 0 & on \partial Ω_{ϵ}, \end{matrix} \end{matrix}$ where $ρ > 0$ is small, $V_{1}, V_{2} > 0$ are smooth potentials in Ω, $ν ⩾ 0$ and $τ > 0$ . This equation and its variants have attracted a lot of attention in recent years due to its relevance in the statistical mechanics description of 2D-turbulence, as initiated by Onsager [27]. Precisely, in this context Caglioti, Lions, Marchioro, Pulvirenti [5] and Sawada, Suzuki [37] derive the following equation: $\begin{matrix} (1.2) & \{\begin{matrix} - Δ u = λ \int_{[- 1, 1]} \frac{α e^{α u}}{\int_{Ω} e^{α u} d x} d P (α) & in Ω \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where Ω is a bounded domain in $R^{2}$ , u is the stream function of the flow, $λ > 0$ is a constant related to the inverse temperature and $P$ is a Borel probability measure in $[- 1, 1]$ describing the point-vortex intensities distribution. We observe that (1.2) is obtained under a deterministic assumption on the distribution of the vortex circulations.

On the other hand, on a bounded domain $Ω \subset R^{2}$ a similar mean field equation to (1.2) is derived by Neri in [25]: $\begin{matrix} (1.3) & \{\begin{matrix} - Δ u = λ \int_{[- 1, 1]} \frac{α e^{α u} d P (α)}{\int \int_{Ω \times [- 1, 1]} e^{α u} d P (α) d x} & in Ω \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ under the stochastic assumption that the point vortex circulations are independent identically distributed random variables with probability distribution $P$ . For Neri’s model, several blow-up and existence results have been obtained by Ricciardi and Zecca in [35]. Moreover, the same authors show some common properties between such deterministic and stochastic models in [34].

Equation (1.2) (and also (1.3)) includes several well-known problems depending on a suitable choice of $P$ . For instance, if $P = δ_{1}$ is concentrated at 1, then (1.2) corresponds to the classical mean field equation $\begin{matrix} (1.4) & \{\begin{matrix} - Δ u = λ \frac{e^{u}}{\int_{Ω} e^{u} d x} & in Ω \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ which has been widely studied in the last decades as shown in [23]. When $P = σ δ_{1} + (1 - σ) δ_{- τ}$ with $τ \in [- 1, 1]$ and $σ \in [0, 1]$ , equation (1.2) becomes $\begin{matrix} (1.5) & \{\begin{matrix} - Δ u = λ (σ \frac{e^{u}}{\int_{Ω} e^{u} d x} - (1 - σ) τ \frac{e^{- τ u}}{\int_{Ω} e^{- τ u} d x}) & in Ω \\ u = 0 & on \partial Ω . \end{matrix} \end{matrix}$ Notice that solutions of (1.4) are critical points of the functional $\begin{matrix} J_{λ} (u) = \frac{1}{2} \int_{Ω} | \nabla u |^{2} - λ log (\int_{Ω} e^{u}), u \in H_{0}^{1} (Ω), \end{matrix}$ which can be found as minimizers of $J_{λ}$ if $λ < 8 π$ , by using Moser-Trudinger’s inequality. In the supercritical regime $λ ⩾ 8 π$ , the situation becomes subtler since the existence of solutions could depend on the topology and the geometry of the domain. In [7,8], Chen and Lin proved that (1.4) has a solution when $λ \notin 8 π N$ and Ω is not simply connected using a degree argument. On Riemann surfaces the degree argument in [7,8] is still available and has received a variational counterpart in [12,24] by means of improved forms of the Moser-Trudinger inequality. When $λ = 8 π$ problem (1.4) is solvable on a long and thin rectangle, as showed by Caglioti et al. [6], but not on a ball. Bartolucci and Lin [1] proved that (1.4) has a solution for $λ = 8 π$ when the Robin function of Ω has more than one maximum point.

Setting $λ_{1} = λ σ$ , $λ_{2} = λ (1 - σ)$ and $V_{1} = V_{2} = 1$ problem (1.5) can be rewritten as $\begin{matrix} (1.6) & \{\begin{matrix} - Δ u = λ_{1} \frac{V_{1} e^{u}}{\int_{Ω} V_{1} e^{u} d x} - λ_{2} τ \frac{V_{2} e^{- τ u}}{\int_{Ω} V_{2} e^{- τ u} d x} & in Ω \\ u = 0 & on \partial Ω . \end{matrix} \end{matrix}$ If $τ = 1$ and $V_{1} = V_{2} \equiv 1$ problem (1.6) reduces to mean field equation of the equilibrium turbulence, see [4,18,21,26,31] or its related sinh-Poisson version, see [2,3,17,20,22], which have received a considerable interest in recent years.

To the extent of our knowledge, there are by now just few results in a more general situation. Pistoia and Ricciardi built in [29] sequences of blowing-up solutions to (1.6) when $τ > 0$ and $λ_{1}$ , $λ_{2} τ^{2}$ are close to $8 π$ , while in [30] the same authors built an arbitrary large number of sign-changing blowing-up solutions to (1.6) when $τ > 0$ and $λ_{1}$ , $λ_{2} τ^{2}$ are close to suitable (not necessarily integer) multiples of $8 π$ . Ricciardi and Takahashi in [32] provided a complete blow-up picture for solution sequences of (1.6) and successively in [33] Ricciardi et al. constructed min-max solutions when $λ_{1} \to 8 π^{+}$ and $λ_{2} \to 0$ on a multiply connected domain (in this case the nonlinearity $e^{- τ u}$ may be treated as a lower-order term with respect to the main term $e^{u}$ ). A blow-up analysis and some existence results are obtained when $τ > 0$ in a compact Riemann surface in [19,36].

A matter of interest to us is whether do there exist solutions to (1.1) for small values of ρ or (1.6) for general values of the parameters $λ_{1}, λ_{2} > 0$ on multiply connected domain Ω. Ould-Ahmedou and Pistoia in [28] proved that on a pierced domain $D_{ϵ} : = Ω ∖ \overline{B (ξ_{0}, ϵ)}$ , $ξ_{0} \in Ω$ , there exists a solution to the classical mean field equation (1.4) (in $D_{ϵ}$ instead of Ω) which blows-up at $ξ_{0}$ as $ϵ \to 0$ for any $λ > 8 π$ (extra symmetric conditions are required when $λ \in 8 π N$ ). Recently, in [14] the authors studied the mean field equation with variable intensities on pierced domains $\begin{matrix} (1.7) & \{\begin{matrix} - Δ u = λ_{1} \frac{V_{1} e^{u}}{\int_{Ω_{ϵ}} V_{1} e^{u} d x} - λ_{2} τ \frac{V_{2} e^{- τ u}}{\int_{Ω_{ϵ}} V_{2} e^{- τ u} d x} & in Ω_{ϵ} \\ u = 0 & on \partial Ω_{ϵ}, \end{matrix} \end{matrix}$ in the super-critical regime $λ_{1} > 8 π m_{1}$ and $λ_{2} τ^{2} > 8 π (m - m_{1})$ with $m_{1} \in {0, 1, \dots, m}$ . This equation is related, but not equivalent, to problem (1.1) by using the change $\begin{matrix} ρ = \frac{λ_{1}}{\int_{Ω_{ϵ}} V_{1} e^{u}} and ρ ν = \frac{λ_{2} τ}{\int_{Ω_{ϵ}} V_{2} e^{- τ u}} . \end{matrix}$ More precisely, the authors constructed solutions to (1.7) $u ϵ$ in $Ω_{ϵ}$ blowing-up positively and negatively at $ξ_{1}, \dots, ξ_{m_{1}}$ and $ξ_{m_{1} + 1}, \dots, ξ_{m}$ , respectively, as $ϵ_{1}, \dots, ϵ_{m} \to 0$ under the assumption $\begin{array}{l} λ_{1} = 4 π (α_{1} + \dots + α_{m_{1}}), λ_{2} τ^{2} = 4 π (α_{m_{1} + 1} + \dots + α_{m}), \\ m_{1} \in {0, 1, \dots, m}, α_{i} > 2, α_{i} \notin 2 N . \end{array}$ Nevertheless, the result in [14] may not tell us whether (1.1) has solutions with $m_{1}$ positive bubbles and $m - m_{1}$ negative bubbles for all small $ρ > 0$ . Therefore, we perform directly to problem (1.1) a similar procedure. Our main result reads as follows.

Theorem 1.1.
Let m be a positive integer and $m_{1} \in {0, \dots, m}$ . Then, for all $ρ > 0$ small enough there are radii $ϵ (ρ) = (ϵ_{1} (ρ), \dots, ϵ_{m} (ρ))$ small enough such that the problem ( 1.1 ) has a solution $u_{ρ}$ in $Ω_{ϵ}$ blowing-up positively at $ξ_{1}, \dots, ξ_{m_{1}}$ and negatively at $ξ_{m_{1} + 1}, \dots, ξ_{m}$ as ρ goes to zero.

In Theorem 1.1 we intend that $m_{1} = m$ if $ν = 0$ (or $V_{2} \equiv 0$ ) and $m_{1} = 0$ if $V_{1} \equiv 0$ . Thus, (1.1) becomes a Liouville type equation. Let us stress that $Ω_{ϵ}$ is an example of a non-simply connected domain. Without loss of generality, we shall assume in the rest of the paper that $ν = 1$ , since we can replace $ν V_{2}$ by $V_{2}$ . However, we need the presence of ν when we compare (1.1) with equation (1.7).

Finally, we point out some comments about the proof of the theorem. Following the main ideas presented in [14], we find a solution $u_{ρ}$ using a perturbative approach, precisely, we look for a solution of (1.1) as $\begin{matrix} (1.8) & u_{ρ} = U + ϕ, \end{matrix}$ where U is a suitable ansatz built using the projection operator $P_{ϵ}$ onto $H_{0}^{1} (Ω_{ϵ})$ (see (2.3)) and U is defined as follows $\begin{matrix} U = \sum_{k = 1}^{m_{1}} P_{ϵ} w_{k} - \frac{1}{τ} \sum_{k = m_{1} + 1}^{m} P_{ϵ} w_{k} with w_{i} (x) = log \frac{2 α_{i}^{2} δ_{i}^{α_{i}}}{{(δ_{i}^{α_{i}} + | x - ξ_{i} |^{α_{i}})}^{2}}, \end{matrix}$ where $δ_{i} > 0$ , $i = 1, \dots, m$ and $α_{i}$ ’s are real parameters satisfying $α_{i} > 2$ with $α_{i} \notin 2 N$ for all $i = 1, \dots, m$ ; and $ϕ \in H_{0}^{1} (Ω_{ϵ})$ is a small remainder term. A careful choice of the parameters $δ_{j}$ ’s and the radii $ϵ_{j}$ ’s is made in Section 2 (see (2.7)) in order to make U be a good approximated solution. Indeed, the error term R given by $\begin{matrix} (1.9) & R = Δ U + ρ (V_{1} e^{U} - V_{2} e^{- τ U}) \end{matrix}$ is small in $L^{p}$ -norm for $p > 1$ close to 1 (see Lemma 2.4). A linearization procedure around U leads us to re-formulate (1.1) in terms of a nonlinear problem for ϕ (see equation (3.1)). We will prove the existence of such a solution ϕ to (3.1) by using a fixed point argument, thanks to some estimates in Section 3 (see (3.6)). The corresponding solution $u_{ρ}$ in (1.8) blows-up at the points $ξ_{i}$ ’s thanks to the asymptotic properties of its main order term U (see (2.10) in Corollary 2.2). In Section 4 we will prove the invertibility of the linear operator naturally associated to the problem (see (3.2)) stated in Proposition 3.1.
2. The ansatz

Following the main ideas in [14], in this section we shall make a choice of the parameters $δ_{j}$ ’s in order to make U a good approximation. Let $G (x, y) = - \frac{1}{2 π} log | x - y | + H (x, y)$ be the Green function of $- Δ$ in Ω, where the regular part H is a harmonic function in Ω so that $H (x, y) = \frac{1}{2 π} log | x - y |$ on $\partial Ω$ . Let us introduce the coefficients $β_{i j}$ , $i, j = 1, \dots, m$ , as the solution of the linear system $\begin{array}{l} β_{i j} (\frac{1}{2 π} log ϵ_{j} - H (ξ_{j}, ξ_{j})) - \sum_{k \neq j} β_{i k} G (ξ_{j}, ξ_{k}) \\ (2.1) & = - 4 π α_{i} H (ξ_{i}, ξ_{j}) + \{\begin{matrix} 2 α_{i} log δ_{i} & if i = j \\ 2 α_{i} log | ξ_{i} - ξ_{j} | & if i \neq j . \end{matrix} \end{array}$ Notice that (2.1) can be re-written as the diagonally-dominant system $\begin{array}{l} β_{i j} log ϵ_{j} - 2 π [β_{i j} H (ξ_{j}, ξ_{j}) + \sum_{k \neq j} β_{i k} G (ξ_{j}, ξ_{k})] \\ = - 8 π^{2} α_{i} H (ξ_{i}, ξ_{j}) + \{\begin{matrix} 4 π α_{i} log δ_{i} & if i = j \\ 4 π α_{i} log | ξ_{i} - ξ_{j} | & if i \neq j \end{matrix} \end{array}$ for $ϵ_{j}$ small, which has a unique solution satisfying $\begin{matrix} (2.2) & β_{i j} = \frac{4 π α_{i} log δ_{i}}{log ϵ_{j}} δ_{i j} + O (| log ϵ_{j} |^{- 1}), \end{matrix}$ where $δ_{i j}$ is the Kronecker symbol. Introduce the projection $P_{ϵ} w$ as the unique solution of $\begin{matrix} (2.3) & \{\begin{matrix} Δ P_{ϵ} w = Δ w & in Ω_{ϵ} \\ P_{ϵ} w = 0 & on \partial Ω_{ϵ} . \end{matrix} \end{matrix}$ Notice that $w_{i}$ is a solution of the singular Liouville equation $\begin{matrix} \{\begin{matrix} Δ w + | x - ξ_{i} |^{α_{i} - 2} e^{w} = 0 in R^{2} \\ \int_{R^{2}} | x - ξ_{i} |^{α_{i} - 2} e^{w} d x < + \infty . \end{matrix} \end{matrix}$ By studying the harmonic function $\begin{matrix} ψ = P_{ϵ} w_{i} - w_{i} + log [2 α_{i}^{2} δ_{i}^{α_{i}}] - 4 π α_{i} H (x, ξ_{i}) + \sum_{k = 1}^{m} β_{i k} G (x, ξ_{k}) \end{matrix}$ and using the maximum principle the following asymptotic expansion of $P_{ϵ} w_{i}$ was proved in [14].

Lemma 2.1.
There hold $\begin{array}{rcl} P_{ϵ} w_{i} & = & w_{i} - log [2 α_{i}^{2} δ_{i}^{α_{i}}] + 4 π α_{i} H (x, ξ_{i}) - \sum_{k = 1}^{m} β_{i k} G (x, ξ_{k}) \\ (2.4) & + O (δ_{i}^{α_{i}} + (1 + \frac{log δ_{i}}{log ϵ_{i}}) \sum_{k = 1}^{m} ϵ_{k} + {(\frac{ϵ_{i}}{δ_{i}})}^{α_{i}}) \end{array}$ uniformly in $Ω_{ϵ}$ and $\begin{matrix} (2.5) & P_{ϵ} w_{i} = 4 π α_{i} G (x, ξ_{i}) - \sum_{k = 1}^{m} β_{i k} G (x, ξ_{k}) + O (δ_{i}^{α_{i}} + (1 + \frac{log δ_{i}}{log ϵ_{i}}) \sum_{k = 1}^{m} ϵ_{k} + {(\frac{ϵ_{i}}{δ_{i}})}^{α_{i}}) \end{matrix}$ locally uniformly in $\overline{Ω} ∖ {ξ_{1}, \dots, ξ_{m}}$ .

From the definition of U and using (2.4)–(2.5), we need to impose $\begin{matrix} (2.6) & \{\begin{matrix} \sum_{j = 1}^{m_{1}} β_{j i} - \frac{1}{τ} \sum_{j = m_{1} + 1}^{m} β_{j i} = 2 π (α_{i} - 2) & i = 1, \dots, m_{1} \\ - τ \sum_{j = 1}^{m_{1}} β_{j i} + \sum_{j = m_{1} + 1}^{m} β_{j i} = 2 π (α_{i} - 2) & i = m_{1} + 1, \dots, m, \end{matrix} \end{matrix}$ see Corollary 2.2. Taking into account (2.2), (2.6) requires at main order that $α_{i} log δ_{i} ≃ \frac{α_{i} - 2}{2} log ϵ_{i}$ , i.e. $δ_{i}^{α_{i}} \sim ϵ_{i}^{\frac{α_{i} - 2}{2}}$ . Moreover, due to the presence of $log [2 α_{i}^{2} δ_{i}^{α_{i}}]$ in (2.4)–(2.5) we need further to assume that the $δ_{i}^{α_{i}}$ ’s have the same rate, as it is well known in elliptic problems with exponential nonlinearity in dimension two, see [7–9,11,13,15,16].

Summarizing, for any $i = 1, \dots, m$ we choose $\begin{matrix} (2.7) & δ_{i}^{α_{i}} = d_{i} ρ, ϵ_{i}^{\frac{α_{i} - 2}{2}} = r_{i} ρ, \end{matrix}$ for the small parameter $ρ > 0$ , where $d_{i}$ , $r_{i}$ will be specified below, and introduce $\begin{matrix} ρ_{i} = \{\begin{matrix} (α_{i} + 2) H (ξ_{i}, ξ_{i}) + \sum_{\begin{array}{c} j = 1 \\ j \neq i \end{array}}^{m_{1}} (α_{j} + 2) G (ξ_{i}, ξ_{j}) \\ - \frac{1}{τ} \sum_{j = m_{1} + 1}^{m} (α_{j} + 2) G (ξ_{i}, ξ_{j}) & i = 1, \dots, m_{1} \\ (α_{i} + 2) H (ξ_{i}, ξ_{i}) - τ \sum_{j = 1}^{m_{1}} (α_{j} + 2) G (ξ_{i}, ξ_{j}) \\ + \sum_{\begin{array}{c} j = m_{1} + 1 \\ j \neq i \end{array}}^{m} (α_{j} + 2) G (ξ_{i}, ξ_{j}) & i = m_{1} + 1, \dots, m . \end{matrix} \end{matrix}$ Setting $A_{i} = \overline{B (ξ_{i}, η)} ∖ B (ξ_{i}, ϵ_{i})$ for $η < \frac{1}{2} min {| ξ_{i} - ξ_{j} | : i \neq j; dist (ξ_{i}, \partial Ω) : i = 1, \dots, m}$ , by Lemma 2.1 we deduce the following expansion.
Corollary 2.2.
Assume the validity of ( 2.6 ). There hold $\begin{matrix} (2.8) & U = w_{i} - log [2 α_{i}^{2} δ_{i}^{α_{i}}] + (α_{i} - 2) log | x - ξ_{i} | + 2 π ρ_{i} + O (ρ + \sum_{k = 1}^{m} ρ^{\frac{2}{α_{k} - 2}} + | x - ξ_{i} |) \end{matrix}$ uniformly in $A_{i}$ , $i = 1, \dots, m_{1}$ , $\begin{matrix} (2.9) & - τ U_{=} w_{i} - log [2 α_{i}^{2} δ_{i}^{α_{i}}] + (α_{i} - 2) log | x - ξ_{i} | + 2 π ρ_{i} + O (ρ + \sum_{k = 1}^{m} ρ^{\frac{2}{α_{k} - 2}} + | x - ξ_{i} |) \end{matrix}$ uniformly in $A_{i}$ , $i = m_{1} + 1, \dots, m$ , and $\begin{matrix} (2.10) & U = 2 π \sum_{i = 1}^{m_{1}} (α_{i} + 2) G (x, ξ_{i}) - \frac{2 π}{τ} \sum_{i = m_{1} + 1}^{m} (α_{i} + 2) G (x, ξ_{i}) + O (ρ + \sum_{k = 1}^{m} ρ^{\frac{2}{α_{k} - 2}}) \end{matrix}$ locally uniformly in $\overline{Ω} ∖ {ξ_{1}, \dots, ξ_{m}}$ .
Proof.
The following expansions hold uniformly in $A_{i}$ , $i = 1, \dots, m_{1}$ $\begin{array}{rcl} U & = & w_{i} - log [2 α_{i}^{2} δ_{i}^{α_{i}}] + 4 π α_{i} H (x, ξ_{i}) - (\sum_{j = 1}^{m_{1}} β_{j i} - \frac{1}{τ} \sum_{j = m_{1} + 1}^{m} β_{j i}) G (x, ξ_{i}) - \sum_{\begin{array}{c} k = 1 \\ k \neq i \end{array}}^{m} β_{i k} G (x, ξ_{k}) \\ + \sum_{\begin{array}{c} k = 1 \\ k \neq i \end{array}}^{m_{1}} [4 π α_{k} G (x, ξ_{k}) - \sum_{\begin{array}{c} l = 1 \\ l \neq i \end{array}}^{m} β_{k l} G (x, ξ_{l})] - \frac{1}{τ} \sum_{k = m_{1} + 1}^{m} [4 π α_{k} G (x, ξ_{k}) - \sum_{\begin{array}{c} l = 1 \\ l \neq i \end{array}}^{m} β_{k l} G (x, ξ_{l})] \\ + O (ρ + \sum_{k = 1}^{m} ρ^{\frac{2}{α_{k} - 2}}) . \end{array}$ Notice that $\begin{array}{l} - \sum_{\begin{array}{c} k = 1 \\ k \neq i \end{array}}^{m} β_{i k} G (x, ξ_{k}) + \sum_{\begin{array}{c} k = 1 \\ k \neq i \end{array}}^{m_{1}} [4 π α_{k} G (x, ξ_{k}) - \sum_{\begin{array}{c} l = 1 \\ l \neq i \end{array}}^{m} β_{k l} G (x, ξ_{l})] \\ - \frac{1}{τ} \sum_{k = m_{1}}^{m} [4 π α_{k} G (x, ξ_{k}) - \sum_{\begin{array}{c} l = 1 \\ l \neq i \end{array}}^{m} β_{k l} G (x, ξ_{l})] \\ = \sum_{j = 1, j \neq i}^{m_{1}} [4 π α_{j} - \sum_{k = 1}^{m_{1}} β_{k j} + \frac{1}{τ} \sum_{k = m_{1} + 1}^{m} β_{k j}] G (x, ξ_{j}) \\ - \frac{1}{τ} \sum_{j = m_{1} + 1}^{m} [4 π α_{j} + τ \sum_{k = 1}^{m_{1}} β_{k j} - \sum_{k = m_{1} + 1}^{m} β_{k j}] G (x, ξ_{j}) . \end{array}$ Hence, (2.8) follows by using (2.6) and the choice of $ρ_{i}$ . Similarly, we conclude (2.9) and (2.10). □

As in [14], we obtain the validity of (2.6) by a suitable choice of $r_{i}$ and $d_{i}$ . The following Lemma states a relation between $r_{i}$ and $d_{i}$ , see [14, Lemma 2.3] for a proof.
Lemma 2.3.
If $r_{i} = d_{i} e^{- π ρ_{i}}$ for all $i = 1, \dots, m$ , then ( 2.6 ) does hold.

Finally, we need that $- Δ P_{ϵ} w_{i} = | x - ξ_{i} |^{α_{i} - 2} e^{w_{i}}$ match with $ρ V_{1} e^{U}$ and $ρ τ V_{2} e^{- τ U}$ in $A_{i}$ for $i = 1, \dots, m_{1}$ and for $i = m_{1} + 1, \dots, m$ , respectively. As we will see below, this is achieved by requiring that $\begin{matrix} (2.11) & 2 α_{j}^{2} δ_{j}^{α_{j}} = ρ V_{1} (ξ_{j}) e^{2 π ρ_{j}} and 2 α_{j}^{2} δ_{j}^{α_{j}} = ρ τ V_{2} (ξ_{j}) e^{2 π ρ_{j}} \end{matrix}$ for $i = 1, \dots, m_{1}$ and for $i = m_{1} + 1, \dots, m$ , respectively. The choice $\begin{matrix} (2.12) & d_{i} = \{\begin{matrix} \frac{V_{1} (ξ_{i}) e^{2 π ρ_{i}}}{2 α_{i}^{2}} & i = 1, \dots, m_{1} \\ \frac{V_{2} (ξ_{i}) e^{2 π ρ_{i}}}{2 α_{i}^{2} τ} & i = m_{1} + 1, \dots, m, \end{matrix} r_{i} = \{\begin{matrix} \frac{V_{1} (ξ_{i}) e^{π ρ_{i}}}{2 α_{i}^{2}} & i = 1, \dots, m_{1} \\ \frac{V_{2} (ξ_{i}) e^{π ρ_{i}}}{2 α_{i}^{2} τ} & i = m_{1} + 1, \dots, m \end{matrix} \end{matrix}$ guarantees the validity of (2.6) and (2.11) in view of Lemma 2.3. We are now ready to estimate the precision of our ansatz U. Lemma 2.4.
There exists $ρ_{0} > 0$ , $p_{0} > 1$ and $C > 0$ such that for any $ρ \in (0, ρ_{0})$ and $p \in (1, p_{0})$ $\begin{matrix} (2.13) & ‖ R ‖_{p} ⩽ C ρ^{σ_{p}} \end{matrix}$ for some $σ_{p} > 0$ .
Proof.
First, note that $\begin{matrix} Δ U = \{\begin{matrix} - | x - ξ_{i} |^{α_{i} - 2} e^{w_{i}} + O (ρ) & in A_{i}, i = 1, \dots, m \\ \frac{1}{τ} | x - ξ_{i} |^{α_{i} - 2} e^{w_{i}} + O (ρ) & in A_{i}, i = m_{1} + 1, \dots, m \\ O (ρ) & in Ω_{ϵ} ∖ ⋃_{i = 1}^{m} A_{i} \end{matrix} \end{matrix}$ in view of (2.7). By (2.8) we have that $\begin{matrix} (2.14) & V_{1} e^{U} = \frac{V_{1} (ξ_{i}) e^{2 π ρ_{i}}}{2 α_{i}^{2} δ_{i}^{α_{i}}} | x - ξ_{i} |^{α_{i} - 2} e^{w_{i}} [1 + O (ρ + \sum_{k = 1}^{m} ρ^{\frac{2}{α_{k} - 2}} + | x - ξ_{i} |)] \end{matrix}$ uniformly in $A_{i}$ for any $i = 1, \dots, m_{1}$ and by (2.9) $\begin{matrix} (2.15) & V_{1} e^{U} = O ({[\frac{| x - ξ_{i} |^{α_{i} - 2}}{δ_{i}^{α_{i}}} e^{w_{i}}]}^{- \frac{1}{τ}}) uniformly in A_{i}, for i = m_{1} + 1, \dots, m . \end{matrix}$ Similarly, by (2.9) we have that $\begin{matrix} (2.16) & V_{2} e^{- τ U} = \frac{V_{2} (ξ_{i}) e^{2 π ρ_{i}}}{2 α_{i}^{2} δ_{i}^{α_{i}}} | x - ξ_{i} |^{α_{i} - 2} e^{w_{i}} [1 + O (ρ + \sum_{k = 1}^{m} ρ^{\frac{2}{α_{k} - 2}} + | x - ξ_{i} |)] \end{matrix}$ uniformly in $A_{i}$ for any $i = m_{1} + 1, \dots, m$ and by (2.8) $\begin{matrix} (2.17) & V_{2} e^{- τ U} = O ({[\frac{| x - ξ_{i} |^{α_{i} - 2}}{δ_{i}^{α_{i}}} e^{w_{i}}]}^{- τ}) uniformly in A_{i}, for i = 1, \dots, m_{1} . \end{matrix}$ Hence, by using (2.7), (2.11), (2.12) and (2.14)–(2.17) we can estimate the error term R as: $\begin{matrix} (2.18) & R = | x - ξ_{i} |^{α_{i} - 2} e^{w_{i}} O (ρ^{σ} + | x - ξ_{i} |) + O (ρ^{σ}) \end{matrix}$ in $A_{i}$ , $i = 1, \dots, m_{1}$ , and $\begin{matrix} (2.19) & R = - \frac{1}{τ} | x - ξ_{i} |^{α_{i} - 2} e^{w_{i}} O (ρ^{σ} + | x - ξ_{i} |) + O (ρ^{σ}) \end{matrix}$ in $A_{i}$ , $i = m_{1} + 1, \dots, m$ , where $σ = min {\frac{1}{α_{i}} : i = 1, \dots, m}$ , while $R = O (ρ)$ does hold in $Ω_{ϵ} ∖ ⋃_{i = 1}^{m} A_{i}$ . By (2.18)–(2.19) we finally get that there exist $ρ_{0} > 0$ small, $p_{0} > 1$ close to 1 so that $‖ R ‖_{p} = O (ρ^{σ_{p}})$ for all $0 < ρ ⩽ ρ_{0}$ and $1 < p ⩽ p_{0}$ , for some $σ_{p} > 0$ . □

3. The nonlinear problem and proof of main result

In this section we shall study the existence of a function $ϕ_{ρ} \in H_{0}^{1} (Ω_{ϵ})$ , a small remainder term which satisfies the following nonlinear problem: $\begin{matrix} (3.1) & \{\begin{matrix} L (ϕ) = - [R + N (ϕ)] & in Ω_{ϵ} \\ ϕ = 0, & on \partial Ω_{ϵ}, \end{matrix} \end{matrix}$ where the linear operator L is defined as $\begin{matrix} (3.2) & L (ϕ) = Δ ϕ + W (x) ϕ, W (x) : = ρ V_{1} (x) e^{U} + ρ τ V_{2} (x) e^{- τ U} \end{matrix}$ and the nonlinear term $N (ϕ)$ is given by $\begin{matrix} (3.3) & N (ϕ) = ρ V_{1} (x) e^{U} (e^{ϕ} - ϕ - 1) - ρ V_{2} (x) e^{- τ U} (e^{- τ ϕ} + τ ϕ - 1) . \end{matrix}$ It is readily checked that ϕ is a solution to (3.1) if and only if $u_{ϵ}$ given by (1.8) is a solution to (1.1). In Section 4 we will prove the following result.

Proposition 3.1.
For any $p > 1$ , there exists $ρ_{0} > 0$ and $C > 0$ such that for any $ρ \in (0, ρ_{0})$ and $h \in L^{p} (Ω_{ϵ})$ there exists a unique $ϕ \in H_{0}^{1} (Ω_{ϵ})$ solution of $\begin{matrix} (3.4) & L (ϕ) = h in Ω_{ϵ}, ϕ = 0 on \partial Ω_{ϵ}, \end{matrix}$ which satisfies $\begin{matrix} (3.5) & ‖ ϕ ‖ ⩽ C | log ρ | ‖ h ‖_{p} . \end{matrix}$

The latter proposition implies that the unique solution $ϕ = T (h)$ of (3.4) defines a continuous linear map from $L^{p} (Ω_{ϵ})$ into $H_{0}^{1} (Ω_{ϵ})$ , with norm bounded by $C | log ρ |$ . Concerning problem (3.1) we have the following fact.
Proposition 3.2.
There exist $p_{0} > 1$ and $ρ_{0} > 0$ so that for any $1 < p < p_{0}$ and all $0 < ρ ⩽ ρ_{0}$ , the problem ( 3.1 ) admits a unique solution $ϕ (ρ) \in H_{0}^{1} (Ω_{ϵ})$ , where R, L and N are given by ( 1.9 ), ( 3.2 ) and ( 3.3 ), respectively. Moreover, there is a constant $C > 0$ such that $\begin{matrix} ‖ ϕ ‖_{\infty} ⩽ C ρ^{σ_{p}} | log ρ |, \end{matrix}$ for some $σ_{p} > 0$ .

Here, $σ_{p}$ is the same as in (2.13). We shall use the following estimate.
Lemma 3.3.
There exist $p_{0} > 1$ and $ρ_{0} > 0$ so that for any $1 < p < p_{0}$ and all $0 < ρ ⩽ ρ_{0}$ it holds $\begin{matrix} (3.6) & {‖ N (ϕ_{1}) - N (ϕ_{2}) ‖}_{p} ⩽ C ρ^{σ_{p}^{'}} ‖ ϕ_{1} - ϕ_{2} ‖ \end{matrix}$ for all $ϕ_{i} \in H_{0}^{1} (Ω_{ϵ})$ with $‖ ϕ_{i} ‖ ⩽ M ρ^{σ_{p}} | log ρ |$ , $i = 1, 2$ , and for some $σ_{p}^{'} > 0$ . In particular, we have that $\begin{matrix} (3.7) & {‖ N (ϕ) ‖}_{p} ⩽ C ρ^{σ_{p}^{'}} ‖ ϕ ‖ \end{matrix}$ for all $ϕ \in H_{0}^{1} (Ω_{ϵ})$ with $‖ ϕ ‖ ⩽ M ρ^{σ_{p}} | log ρ |$ .
Proof.
We will argue in the same way as in [28, Lemma 5.1], see also [14, Lemma 3.4]. Assume that $‖ ϕ_{i} ‖ ⩽ M ρ^{σ_{p}} | log ρ |$ , $i = 1, 2$ , and for some $σ_{p}^{'} > 0$ . First, we point out that $\begin{matrix} N (ϕ) = \sum_{i = 1}^{2} ρ {f_{i} (ϕ) - f_{i} (0) - f_{i}^{'} (0) [ϕ]}, where f_{i} (ϕ) = V_{i} (x) e^{{(- τ)}^{i - 1} (U + ϕ)} . \end{matrix}$ Hence, by the mean value theorem we get that $\begin{matrix} (3.8) & \begin{matrix} N (ϕ_{1}) - N (ϕ_{2}) & = \sum_{i = 1}^{2} ρ {f_{i}^{'} (ϕ_{θ_{i}}) - f_{i}^{'} (0)} [ϕ_{1} - ϕ_{2}] = \sum_{i = 1}^{2} ρ f_{i}^{″} ({\tilde{ϕ}}_{μ_{i}}) [ϕ_{θ_{i}}, ϕ_{1} - ϕ_{2}], \end{matrix} \end{matrix}$ where $ϕ_{θ_{i}} = θ_{i} ϕ_{1} + (1 - θ_{i}) ϕ_{2}$ , ${\tilde{ϕ}}_{μ_{i}} = μ_{i} ϕ_{θ_{i}}$ for some $θ_{i}, μ_{i} \in [0, 1]$ , $i = 1, 2$ , and $\begin{matrix} f_{i}^{″} (ϕ) [ψ, v] = τ^{2 (i - 1)} V_{i} (x) e^{{(- τ)}^{i - 1} (U + ϕ)} ψ v . \end{matrix}$ Using Hölder’s inequalities we get that $\begin{matrix} (3.9) & {‖ f_{i}^{″} (ϕ) [ψ, v] ‖}_{p} ⩽ | τ |^{2 (i - 1)} {‖ V_{i} e^{{(- τ)}^{i - 1} (U + ϕ)} ‖}_{p r_{i}} ‖ ψ ‖_{p s_{i}} ‖ v ‖_{p t_{i}} \end{matrix}$ with $\frac{1}{r_{i}} + \frac{1}{s_{i}} + \frac{1}{t_{i}} = 1$ . We have used the Hölder’s inequality and $‖ u v w ‖_{q} ⩽ ‖ u ‖_{q r} ‖ v ‖_{q s} ‖ w ‖_{q t}$ with $\frac{1}{r} + \frac{1}{s} + \frac{1}{t} = 1$ . Now, let us estimate $‖ V_{i} e^{{(- τ)}^{i - 1} (U + ϕ)} ‖_{p r_{i}}$ with $ϕ = {\tilde{ϕ}}_{μ_{i}}$ , $i = 1, 2$ . By (2.8) and (2.14) and the change of variable $x = δ_{i} y + ξ_{i}$ let us estimate $\begin{array}{rcl} \int_{A_{i}} {| V_{1} e^{U} |}^{q} d x & = & O (δ_{i}^{2 - (α_{i} + 2) q} \int_{\frac{ϵ_{i}}{δ_{i}} ⩽ | y | ⩽ \frac{η}{δ_{i}}} {| \frac{| y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} |}^{q} {[1 + O (ρ + \sum_{k = 1}^{m} ρ^{\frac{2}{α_{k} - 2}} + δ_{i} | y |)]}^{q} d y) \\ (3.10) & = & O (δ_{i}^{2 - (α_{i} + 2) q}) \end{array}$ for any $i = 1, \dots, m_{1}$ and similarly, by (2.9) and (2.16) we get that $\begin{matrix} (3.11) & \int_{A_{i}} {| V_{2} e^{- τ U} |}^{q} d x = O (δ_{i}^{2 - (α_{i} + 2) q}) \end{matrix}$ for any $i = m_{1} + 1, \dots, m$ , in view of (2.7) and (2.12). By (2.7), (2.9) and (2.15) we get the estimate $\begin{array}{rcl} \int_{A_{i}} {| V_{1} e^{U} |}^{q} d x & = & O (δ_{i}^{\frac{α_{i} q}{τ}} \int_{A_{i}} {[| x - ξ_{i} |^{α_{i} - 2} e^{U_{i}}]}^{- \frac{q}{τ}} d x) \\ = & O (δ_{i}^{\frac{(α_{i} + 2) q}{τ} + 2} \int_{\frac{ϵ_{i}}{δ_{i}} ⩽ | y | ⩽ \frac{η}{δ_{i}}} {[\frac{| y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}}]}^{- \frac{q}{τ}} d y) \\ = & O (δ_{i}^{\frac{(α_{i} + 2) q}{τ} + 2} [\int_{\frac{ϵ_{i}}{δ_{i}}}^{1} s^{1 - \frac{(α_{i} - 2) q}{τ}} d s + \int_{1}^{\frac{η}{δ_{i}}} s^{1 + \frac{(α_{i} + 2) q}{τ}} d s]) \\ (3.12) & = & O (1) \end{array}$ for all $i = m_{1} + 1, \dots, m$ . Similarly, by (2.7)–(2.8) we deduce that $\begin{matrix} (3.13) & \int_{A_{i}} {| V_{2} e^{- τ U} |}^{q} d x = O (1) \end{matrix}$ for $i = 1, \dots, m_{1}$ , in view of (2.17). Therefore, by using (2.10), (3.10) and (3.12) we deduce that $\begin{matrix} {‖ V_{1} e^{U} ‖}_{q}^{q} = \sum_{i = 1}^{m_{1}} O (δ_{i}^{2 - (α_{i} + 2) q}) = \sum_{i = 1}^{m_{1}} O (ρ^{\frac{2 - (α_{i} + 2) q}{α_{i}}}) for any q ⩾ 1 \end{matrix}$ and, by using (2.10), (3.11) and (3.13) we obtain that $\begin{matrix} {‖ V_{2} e^{- τ U} ‖}_{q}^{q} = \sum_{i = m_{1} + 1}^{m} O (ρ^{\frac{2 - (α_{i} + 2) q}{α_{i}}}) for any q ⩾ 1 . \end{matrix}$

On the other hand, using the estimate $| e^{a} - 1 | ⩽ | a |$ uniformly for any a in compact subsets of $R$ we have that $\begin{matrix} {‖ V_{i} e^{{(- τ)}^{i - 1} (U + {\tilde{ϕ}}_{μ_{i}})} - V_{i} e^{{(- τ)}^{i - 1} U} ‖}_{q} & = {(\int_{Ω_{ϵ}} {| V_{i} e^{{(- τ)}^{i - 1} U} |}^{q} {| e^{{(- τ)}^{i - 1} {\tilde{ϕ}}_{μ_{i}}} - 1 |}^{q})}^{1 / q} \\ ⩽ {‖ V_{i} e^{{(- τ)}^{i - 1} U} ‖}_{q s_{i}^{'}} ‖ {\tilde{ϕ}}_{μ_{i}} ‖_{q t_{i}^{'}}, \end{matrix}$ with $\frac{1}{s_{i}^{'}} + \frac{1}{t_{i}^{'}} = 1$ , $i = 1, 2$ , in view of $‖ {\tilde{ϕ}}_{μ_{i}} ‖ ⩽ M ρ^{σ_{p}} | log ρ | ⩽ C$ , $i = 1, 2$ . Hence, it follows that $\begin{matrix} {‖ V_{i} e^{{(- τ)}^{i - 1} (U + {\tilde{ϕ}}_{μ_{i}})} - V_{i} e^{{(- τ)}^{i - 1} U} ‖}_{q} ⩽ C ρ^{σ_{0, q s_{i}^{'}} - 1} ‖ {\tilde{ϕ}}_{μ_{i}} ‖ ⩽ C ρ^{σ_{0, q s_{i}^{'}} - 1} ρ^{σ_{p}} | log ρ |, \end{matrix}$ where for $q > 1$ we denote $σ_{0, q} = min {\frac{2 - 2 q}{α_{j} q} : j = 1, \dots, m}$ , in view of $‖ {\tilde{ϕ}}_{μ_{i}} ‖ ⩽ M ρ^{σ_{p}} | log ρ |$ , $i = 1, 2$ . Note that $σ_{0, q} - 1 ⩽ \frac{2 - (α_{i} + 2) q}{α_{i} q}$ for any $i = 1, \dots, m$ and $σ_{0, q} < 0$ for any $q > 1$ . By the previous estimates we find that $\begin{matrix} (3.14) & {‖ V_{i} e^{{(- τ)}^{i - 1} (U + {\tilde{ϕ}}_{μ_{i}})} ‖}_{q} = O (ρ^{σ_{0, q s_{i}^{'}} - 1 + σ_{p}} | log ρ | + ρ^{σ_{0, q} - 1}) . \end{matrix}$ Also, choosing q and $s_{i}^{'}$ , $i = 1, 2$ , close enough to 1, we get that $0 < σ_{p} + σ_{0, q s_{i}^{'}}$ . Now, we can conclude the estimate by using (3.8)–(3.14) to get $\begin{array}{l} {‖ N (ϕ_{1}) - N (ϕ_{2}) ‖}_{p} & ⩽ \sum_{i = 1}^{2} ρ {‖ f_{i}^{″} ({\tilde{ϕ}}_{μ_{i}}) [ϕ_{θ_{i}}, ϕ_{1} - ϕ_{2}] ‖}_{p} \\ ⩽ C \sum_{i = 1}^{2} ρ {‖ V_{i} e^{{(- τ)}^{i - 1} (U + {\tilde{ϕ}}_{μ_{i}})} ‖}_{p r_{i}} ‖ ϕ_{θ_{i}} ‖ ‖ ϕ_{1} - ϕ_{2} ‖ \\ ⩽ C \sum_{i = 1}^{2} ρ^{σ_{p} + σ_{0, p r_{i}}} | log ρ | ‖ ϕ_{1} - ϕ_{2} ‖ ⩽ C ρ^{σ_{p}^{'}} ‖ ϕ_{1} - ϕ_{2} ‖, \end{array}$ where $σ_{p}^{'} = \frac{1}{2} min {σ_{p} + σ_{0, p r_{i}} : i = 1, 2} > 0$ choosing $r_{i}$ close to 1 so that $σ_{p} + σ_{0, p r_{i}} > 0$ for $i = 1, 2$ . Let us stress that $p > 1$ is chosen so that $σ_{p} > 0$ and $σ_{p}^{'} > 0$ . □

We are now in position to study the nonlinear problem (3.1) and to prove our main result Theorem 1.1. Proof of the Proposition 3.2.
Notice that from Proposition 3.1 problem (3.1) becomes $\begin{matrix} ϕ = - T (R + N (ϕ)) : = A (ϕ) . \end{matrix}$ For a given number $M > 0$ , let us consider $F_{M} = {ϕ \in H : ‖ ϕ ‖ ⩽ M ρ^{σ_{p}} | log ρ |}$ . From the Proposition 3.1, (2.13) and (3.7), we get for any $ϕ \in F_{M}$ , $\begin{matrix} ‖ A (ϕ) ‖ & ⩽ C | log ρ | [‖ R ‖_{p} + {‖ N (ϕ) ‖}_{p}] ⩽ C | log ρ | [ρ^{σ_{p}} + ρ^{σ_{p}^{'}} ‖ ϕ ‖] \\ ⩽ C ρ^{σ_{p}} | log ρ | [1 + M ρ^{σ_{p}^{'}} | log ρ |] . \end{matrix}$ Given any $ϕ_{1}, ϕ_{2} \in F_{M}$ , we have that $A (ϕ_{1}) - A (ϕ_{2}) = - T (N (ϕ_{1}) - N (ϕ_{2}))$ and $\begin{matrix} ‖ A (ϕ_{1}) - A (ϕ_{2}) ‖ ⩽ C | log ρ | {‖ N (ϕ_{1}) - N (ϕ_{2}) ‖}_{p} ⩽ C ρ^{σ_{p}^{'}} | log ρ | ‖ ϕ_{1} - ϕ_{2} ‖, \end{matrix}$ with C independent of M, by using Proposition 3.1 and (3.6). Therefore, for some $σ > 0$ we get that $‖ A (ϕ_{1}) - A (ϕ_{2}) ‖ ⩽ C ρ^{σ} | log ρ | ‖ ϕ_{1} - ϕ_{2} ‖$ . It follows that for all ρ sufficiently small $A$ is a contraction mapping of $F_{M}$ (for M large enough), and therefore a unique fixed point of $A$ exists in $F_{M}$ . □
Proof of the Theorem 1.1.
Taking into account (1.8) and the definition of U, the existence of a solution $\begin{matrix} u_{ρ} = \sum_{j = 1}^{m_{1}} P_{ϵ} w_{j} - \frac{1}{τ} \sum_{j = m_{1} + 1}^{m} P_{ϵ} w_{j} + ϕ \end{matrix}$ to equation (1.1) follows directly by Proposition 3.2. The asymptotic behavior of $u_{ρ}$ as $ρ \to 0^{+}$ follows from (2.10) in Lemma 2.1 and estimate for ϕ in Proposition 3.2. Precisely, we have that as $ρ \to 0^{+}$ $\begin{matrix} u_{ρ} = 2 π \sum_{i = 1}^{m_{1}} (α_{i} + 2) G (\cdot, ξ_{i}) - \frac{2 π}{τ} \sum_{i = m_{1} + 1}^{m} (α_{i} + 2) G (\cdot, ξ_{i}) + o (1) \end{matrix}$ locally uniformly in $\bar{Ω} ∖ {ξ_{1}, \dots, ξ_{m}}$ . Therefore, from the behavior of Green’s function we conclude that $u_{ρ}$ blows-up positively at $ξ_{1}, \dots, ξ_{m_{1}}$ and negatively at $ξ_{m_{1} + 1}, \dots, ξ_{m}$ as ρ goes to zero. □

4. The linear theory

In this section, following ideas presented in [14, Section 4] we present the invertibility of the linear operator L defined in (3.2). Roughly speaking in the scale annulus $δ_{i}^{- 1} (B_{i} - ξ_{i})$ the operator L apporaches to the following linear operator in $R^{2}$ $\begin{matrix} L_{i} (ϕ) = Δ ϕ + \frac{2 α_{i}^{2} | y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} ϕ, i = 1, \dots, m . \end{matrix}$ It is well known that the bounded solutions of $L_{i} (ϕ) = 0$ in $R^{2}$ are precisely linear combinations of the functions $\begin{matrix} Y_{1 i} (y) = \frac{| y |^{\frac{α_{i}}{2}}}{1 + | y |^{α_{i}}} cos (\frac{α_{i}}{2} θ), Y_{2 i} (y) = \frac{| y |^{\frac{α_{i}}{2}}}{1 + | y |^{α_{i}}} sin (\frac{α_{i}}{2} θ) and Y_{0 i} (y) = \frac{1 - | y |^{α_{i}}}{1 + | y |^{α_{i}}}, \end{matrix}$ which are written in polar coordinates for $i = 1, \dots, m$ . See [10] for a proof. In our case, we will consider solutions of $L_{i} (ϕ) = 0$ such that $\int_{R^{2}} | \nabla ϕ (y) |^{2} d y < + \infty$ . See [28, Theorem A.1] for a proof.

Let us introduce the following Banach spaces for $j = 1, 2$ $\begin{matrix} (4.1) & L_{α_{i}} (R^{2}) = {u \in W_{loc}^{1, 2} (R^{2}) : \int_{R^{2}} \frac{| y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} {| u (y) |}^{2} d y < + \infty} \end{matrix}$ and $\begin{matrix} (4.2) & H_{α_{i}} (R^{2}) = {u \in W_{loc}^{1, 2} (R^{2}) : \int_{R^{2}} {| \nabla u (y) |}^{2} d y + \int_{R^{2}} \frac{| y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} {| u (y) |}^{2} d y < + \infty} \end{matrix}$ endowed with the norms $\begin{matrix} ‖ u ‖_{L_{α_{i}}} : = {(\int_{R^{2}} \frac{| y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} {| u (y) |}^{2} d y)}^{1 / 2} \end{matrix}$ and $\begin{matrix} ‖ u ‖_{H_{α_{i}}} : = {(\int_{R^{2}} {| \nabla u (y) |}^{2} d y + \int_{R^{2}} \frac{| y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} {| u (y) |}^{2} d y)}^{1 / 2} . \end{matrix}$ It is important to point out the compactness of the embedding $i_{α_{i}} : H_{α_{i}} (R^{2}) \to L_{α_{i}} (R^{2})$ , (see for example [17]).

Proof of the Proposition 3.1.

The proof will be done in several steps. Let us assume the opposite, namely, the existence of $p > 1$ , sequences $ρ_{n} \to 0$ , $ε_{n} : = ϵ (ρ_{n}) \to 0$ , functions $h_{n} \in L^{p} (Ω_{ε_{n}})$ , $ϕ_{n} \in W^{2, 2} (Ω_{ε_{n}})$ such that $\begin{matrix} (4.3) & L (ϕ_{n}) = h_{n} in Ω_{ε_{n}}, ϕ_{n} = 0 on \partial Ω_{ε_{n}} \end{matrix}$ $‖ ϕ_{n} ‖ = 1$ and $| log ρ_{n} | ‖ h_{n} ‖_{p} = o (1)$ as $n \to + \infty$ . We will shall omit the subscript n in $δ_{i, n} = δ_{i}$ . Recall that $δ_{i}^{α_{i}} = d_{i, n} ρ_{n}$ and points $ξ_{1}, \dots, ξ_{m} \in Ω$ are fixed.

Now, define $Φ_{i, n} (y) : = ϕ_{n} (ξ_{i} + δ_{i} y)$ for $y \in Ω_{i, n} : = δ_{i}^{- 1} (Ω_{ε_{n}} - ξ_{i})$ , $i = 1, \dots, m$ . Thus, extending $ϕ_{n} = 0$ in $R^{2} ∖ Ω_{ε_{n}}$ we can prove the following fact. Claim 1.

There holds that the sequence $Φ_{i, n}$ converges (up to subsequence) to $Φ_{i}^{*} = a_{j} Y_{0 i}$ for $i = 1, \dots, m$ , weakly in $H_{α_{i}} (R^{2})$ and strongly in $L_{α_{i}} (R^{2})$ as $n \to + \infty$ for some constant $a_{i} \in R$ , $i = 1, \dots, m$ .

Proof.

First, we shall show that the sequence ${Φ_{i, n}}_{n}$ is bounded in $H_{α_{i}} (R^{2})$ . Notice that for $i = 1, \dots, m$ $\begin{matrix} ‖ Φ_{i, n} ‖_{H_{0}^{1} (Ω_{i, n})} = \int_{Ω_{i, n}} δ_{i}^{2} {| \nabla ϕ_{i, n} (ξ_{i} + δ_{i} y) |}^{2} d y = \int_{Ω_{ε_{n}}} {| \nabla ϕ_{n} (x) |}^{2} d x = 1 . \end{matrix}$ Thus, we want to prove that there is a constant $M > 0$ such for all n (up to a subsequence) $\begin{matrix} ‖ Φ_{i, n} ‖_{L_{α_{i}}}^{2} = \int_{Ω_{i, n}} \frac{| y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} Φ_{i, n}^{2} (y) d y ⩽ M . \end{matrix}$ Notice that for any $i \in {1, \dots, m}$ we find that in $Ω_{i, n}$ $\begin{matrix} (4.4) & \begin{matrix} Δ Φ_{i, n} & + δ_{i}^{2} W (ξ_{i} + δ_{i} y) Φ_{i, n} = δ_{i}^{2} h_{n} (ξ_{i} + δ_{i} y) . \end{matrix} \end{matrix}$ Furthermore, it follows that $Φ_{i, n} \to Φ_{i}^{*}$ weakly in $H_{0}^{1} (Ω_{i, n})$ and strongly in $L^{p} (K)$ for any K compact sets in $R^{2}$ . Now, we multiply (4.4) by $Φ_{i, n}$ for any $i \in {1, \dots, m}$ and we get $\begin{matrix} - \int_{Ω_{i, n}} | \nabla Φ_{i, n} |^{2} + \int_{Ω_{i, n}} δ_{i}^{2} W (ξ_{i} + δ_{i} y) Φ_{i, n}^{2} = \int_{Ω_{i, n}} δ_{i}^{2} h_{n} (ξ_{i} + δ_{i} y) Φ_{i, n} . \end{matrix}$ Using (2.7), (2.8), (2.9) and (2.12), we obtain that $\begin{matrix} (4.5) & δ_{i}^{2} W (ξ_{i} + δ_{i} y) = \frac{2 α_{i}^{2} | y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} + O (δ_{i}^{2} ρ) for all i = 1, \dots, m \end{matrix}$ uniformly for y on compact subsets of $R^{2}$ . Thus, we deduce that $\begin{matrix} (4.6) & \sum_{i = 1}^{m} 2 α_{i}^{2} ‖ Φ_{i, n} ‖_{L_{α_{i}}}^{2} = 1 + o (1) \end{matrix}$ in view of $\begin{matrix} \int_{Ω_{i, n}} δ_{i}^{2} W (ξ_{i} + δ_{i} y) Φ_{i, n}^{2} d y = \int_{Ω_{ϵ_{n}}} W ϕ_{n}^{2} d x = \sum_{i = 1}^{m} 2 α_{i}^{2} ‖ Φ_{i, n} ‖_{L_{α_{i}}}^{2} + o (1) \end{matrix}$ since $\begin{matrix} \int_{Ω_{i, n}} \frac{2 α_{i}^{2} | y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} Φ_{i, n}^{2} (y) d y = 2 α_{i}^{2} ‖ Φ_{i, n} ‖_{L_{α_{i}}}^{2}, for i = 1, \dots, m . \end{matrix}$ Therefore, the sequence ${Φ_{i, n}}_{n}$ is bounded in $H_{α_{i}} (R^{2})$ , so that there is a subsequence ${Φ_{i, n}}_{n}$ and functions $Φ_{i}^{*}$ , $i = 1, 2$ such that ${Φ_{i, n}}_{n}$ converges to $Φ_{i}^{*}$ weakly in $H_{α_{i}} (R^{2})$ and strongly in $L_{α_{i}} (R^{2})$ . Furthermore, we have that $\begin{matrix} \int_{Ω_{i, n}} {(δ_{i}^{2} | h_{n} (ξ_{i} + δ_{i} y) |)}^{p} d y = δ_{i}^{2 p - 2} \int_{Ω_{ϵ_{n}}} {| h_{n} (x) |}^{p} d x = δ_{i}^{2 p - 2} ‖ h_{n} ‖_{p}^{p} = o (1) . \end{matrix}$ Hence, taking into account (4.4)–(4.5) we deduce that $Φ_{i}^{*}$ a solution to $\begin{matrix} Δ Φ + \frac{2 α_{i}^{2} | y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} Φ = 0, i = 1, \dots, m, in R^{2} ∖ {0} . \end{matrix}$ It is standard that $Φ_{i}^{*}$ , $i = 1, \dots, m$ extend to a solution in the whole $R^{2}$ . Hence, by using symmetry assumptions if necessary, we get that $Φ_{i}^{*} = a_{i} Y_{0 i}$ for some constant $a_{i} \in R$ , $i = 1, \dots, m$ . □

For the next step we construct some suitable test functions. To this aim, introduce the coefficients $γ_{i j}$ ’s and ${\tilde{γ}}_{i j}$ ’s, $i, j = 1, \dots, m$ , as the solution of the linear systems $\begin{matrix} (4.7) & γ_{i j} [- \frac{1}{2 π} log ϵ_{i} + H (ξ_{i}, ξ_{i})] + \sum_{k = 1, k \neq i}^{m} γ_{k j} G (ξ_{k}, ξ_{i}) = 2 δ_{i j} \end{matrix}$ and $\begin{array}{l} {\tilde{γ}}_{i j} [- \frac{1}{2 π} log ϵ_{i} + H (ξ_{i}, ξ_{i})] + \sum_{k = 1, k \neq i}^{m} {\tilde{γ}}_{k j} G (ξ_{k}, ξ_{i}) \\ (4.8) & = \{\begin{matrix} \frac{4}{3} α_{j} log δ_{j} + \frac{8}{3} + \frac{8 π}{3} α_{j} H (ξ_{j}, ξ_{j}) & if i = j \\ \frac{8 π}{3} α_{j} G (ξ_{i}, ξ_{j}), & if i \neq j, \end{matrix} \end{array}$ respectively. Notice that both systems (4.7) and (4.8) are diagonally dominant, system (4.7) has solutions $\begin{matrix} γ_{i j} = - \frac{4 π}{log ϵ_{j}} δ_{i j} + O (\frac{1}{| log ρ |^{2}}) = - \frac{2 π (α_{j} - 2)}{log ρ} δ_{i j} + O (\frac{1}{| log ρ |^{2}}) \end{matrix}$ and for the system (4.8) we get $\begin{matrix} {\tilde{γ}}_{i j} = - \frac{8 π}{3} \frac{α_{j} log δ_{j}}{log ϵ_{j}} δ_{i j} + O (\frac{1}{| log ρ |}) = - \frac{4 π}{3} (α_{j} - 2) δ_{i j} + O (\frac{1}{| log ρ |}), \end{matrix}$ where $δ_{i j}$ is the Kronecker symbol. Here, we have used (2.7). Consider now for any $j \in {1, \dots, m}$ the functions $\begin{matrix} η_{0 j} (x) = - \frac{2 δ_{j}^{α_{j}}}{δ_{j}^{α_{j}} + | x - ξ_{j} |^{α_{j}}} and \end{matrix}$ $\begin{matrix} η_{j} (x) = \frac{4}{3} log (δ_{j}^{α_{j}} + | x - ξ_{j} |^{α_{j}}) \frac{δ_{j}^{α_{j}} - | x - ξ_{j} |^{α_{j}}}{δ_{j}^{α_{j}} + | x - ξ_{j} |^{α_{j}}} + \frac{8}{3} \frac{δ_{j}^{α_{j}}}{δ_{j}^{α_{j}} + | x - ξ_{j} |^{α_{j}}}, \end{matrix}$ so that $\begin{array}{l} Δ η_{0 j} + | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} η_{0 j} = - | x - ξ_{j} |^{α_{j}} e^{w_{j}} and \\ Δ η_{j} + | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} η_{j} = | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} Z_{0 j}, \end{array}$ where $\begin{matrix} Z_{0 j} (x) = Y_{0 j} (δ_{j}^{- 1} [x - ξ_{j}]) = \frac{δ_{j}^{α_{j}} - | x - ξ_{j} |^{α_{j}}}{δ_{j}^{α_{j}} + | x - ξ_{j} |^{α_{j}}} . \end{matrix}$ Notice that $η_{0 j} + 1 = - Z_{0 j}$ and, by similar arguments as to obtain expansion (2.4), by studying the harmonic functions $\begin{array}{l} f (x) = P_{ϵ} η_{0 j} (x) - η_{0 j} (x) - \sum_{i = 1}^{m} γ_{i j} G (x, ξ_{i}) and \\ \tilde{f} (x) = P_{ϵ} η_{j} (x) - η_{j} (x) - \frac{8 π}{3} α_{j} H (x, ξ_{j}) + \sum_{i = 1}^{m} {\tilde{γ}}_{i j} G (x, ξ_{i}), \end{array}$ we have that the following fact, as shown in [14]. Lemma 4.1.

There hold $\begin{array}{l} P_{ϵ} η_{0 j} = η_{0 j} + \sum_{i = 1}^{m} γ_{i j} G (x, ξ_{i}) + O (ρ^{\tilde{σ}}) and \\ P_{ϵ} η_{j} = η_{j} + \frac{8 π}{3} α_{j} H (x, ξ_{j}) - \sum_{i = 1}^{m} {\tilde{γ}}_{i j} G (x, ξ_{i}) + O (ρ^{\tilde{σ}}) \end{array}$ uniformly in $Ω_{ϵ}$ for some $\tilde{σ} > 0$ .

Claim 2.

There hold that $a_{j} = 0$ for all $j = 1, \dots, m$ .

Proof.

To this aim let us construct suitable tests functions and from the assumption on $h_{n}$ , $| log ρ_{n} | ‖ h_{n} ‖_{*} = o (1)$ , we get the additional relation $\begin{matrix} (4.9) & a_{j} \int_{R^{2}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} Y_{0 j}^{2} - \frac{α_{j} (α_{j} - 2)}{3} a_{j} \int_{R^{2}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} Y_{0 j} log | y | d y = 0, \end{matrix}$ which implies $a_{j} = 0$ as claimed, since $\begin{matrix} \int_{R^{2}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} Y_{0 j}^{2} = \int_{R^{2}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} {(\frac{1 - | y |^{α_{j}}}{1 + | y |^{α_{j}}})}^{2} d y = \frac{4 π}{3} α_{j} \end{matrix}$ and $\begin{matrix} \int_{R^{2}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} Y_{0 j} log | y | = \int_{R^{2}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} \frac{1 - | y |^{α_{j}}}{1 + | y |^{α_{j}}} log | y | d y = - 4 π . \end{matrix}$

Next, as in [14, Claim 3, Section 4], we define the following test function $P_{ϵ} Z_{j}$ , where $Z_{j} = η_{j} + γ_{j}^{*} η_{0 j}$ and $γ_{j}^{*}$ is given by $\begin{matrix} γ_{j}^{*} = \frac{\frac{{\tilde{γ}}_{j j}}{2 π} log δ_{j} + (\frac{8 π}{3} α_{j} - {\tilde{γ}}_{j j}) H (ξ_{j}, ξ_{j}) - \sum_{i = 1, i \neq j}^{m} {\tilde{γ}}_{i j} G (ξ_{i}, ξ_{j})}{1 - γ_{j j} H (ξ_{j}, ξ_{j}) - \sum_{i = 1, i \neq j}^{m} γ_{i j} G (ξ_{i}, ξ_{j}) + \frac{γ_{j j}}{2 π} log δ_{j}}, \end{matrix}$ so that, $\begin{array}{rcl} γ_{j}^{*} & = & (\frac{8 π}{3} α_{j} - {\tilde{γ}}_{j j} + γ_{j j} γ_{j}^{*}) H (ξ_{j}, ξ_{j}) - \sum_{i = 1, i \neq j}^{m} ({\tilde{γ}}_{i j} - γ_{j}^{*} γ_{i j}) G (ξ_{i}, ξ_{j}) \\ (4.10) & + \frac{1}{2 π} ({\tilde{γ}}_{j j} - γ_{j}^{*} γ_{j j}) log δ_{j} . \end{array}$ From the (2.7) and the expansions for $γ_{j j}$ and ${\tilde{γ}}_{j j}$ we obtain that $\begin{matrix} γ_{j}^{*} = \frac{\frac{{\tilde{γ}}_{j j}}{2 π} log δ_{j} + O (1)}{1 + \frac{γ_{j j}}{2 π} log δ_{j} + O (\frac{1}{| log ρ |})} = - \frac{α_{j} - 2}{3} log ρ + O (1) . \end{matrix}$ Notice that $P_{ϵ} Z_{j}$ expands as $\begin{array}{rcl} P_{ϵ} Z_{j} & = & Z_{j} + \frac{8 π}{3} α_{j} H (x, ξ_{j}) - {\tilde{γ}}_{j j} (- \frac{1}{2 π} log | x - ξ_{i} | + H (x, ξ_{j})) - \sum_{i = 1, i \neq j}^{m} {\tilde{γ}}_{i j} G (x, ξ_{i}) \\ + O (ρ^{\tilde{σ}}) + γ_{j}^{*} [γ_{j j} (- \frac{1}{2 π} log | x - ξ_{i} | + H (x, ξ_{j})) + \sum_{i = 1, i \neq j}^{m} γ_{i j} G (x, ξ_{i}) + O (ρ^{\tilde{σ}})] \\ = & Z_{j} + (\frac{8 π}{3} α_{j} - {\tilde{γ}}_{j j} + γ_{j j} γ_{j}^{*}) H (x, ξ_{j}) - \sum_{i = 1, i \neq j}^{m} ({\tilde{γ}}_{i j} - γ_{i j} γ_{j}^{*}) G (x, ξ_{i}) \\ (4.11) & + \frac{1}{2 π} ({\tilde{γ}}_{j j} - γ_{j}^{*} γ_{j j}) log | x - ξ_{j} | + O (ρ^{\tilde{σ}}) + γ_{j}^{*} O (ρ^{\tilde{σ}}) . \end{array}$ Hence, multiplying equation (4.3) by $P_{ϵ} Z_{j}$ and integrating by parts we obtain that $\begin{matrix} \int_{Ω_{ϵ}} h P Z_{j} = \int_{Ω_{ϵ}} Δ Z_{j} ϕ + \int_{Ω_{ϵ}} W ϕ P_{ϵ} Z_{j}, \end{matrix}$ in view of $P_{ϵ} Z_{j} = 0$ on $\partial Ω_{ϵ_{n}}$ and $\int_{Ω_{ϵ}} Δ ϕ P_{ϵ} Z_{j} = \int_{Ω_{ϵ}} ϕ Δ P_{ϵ} Z_{j}$ . Furthermore, we have that $\begin{array}{rcl} \int_{Ω_{ϵ}} h P Z_{j} & = & \int_{Ω_{ϵ}} ϕ [| x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} Z_{0 j} - | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} η_{j} \\ + γ_{j}^{*} (- | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} - | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} η_{0 j})] + \int_{Ω_{ϵ}} W ϕ P_{ϵ} Z_{j} \\ = & \int_{Ω_{ϵ}} ϕ | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} Z_{0 j} + \int_{Ω_{ϵ}} | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} (P Z_{j} - Z_{j} - γ_{j}^{*}) \\ + \int_{Ω_{ϵ}} (W - | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}}) P_{ϵ} Z_{j} ϕ . \end{array}$ Now, estimating every integral term we find that $\begin{matrix} \int_{Ω_{ϵ}} h P Z_{j} = O (| log ρ | ‖ h ‖_{p}) = o (1), \end{matrix}$ in view of $P Z_{j} = O (| log ρ |)$ and $G (x, ξ_{k}) = O (| log ϵ_{k} |)$ . Next, scaling we obtain that $\begin{matrix} \int_{Ω_{ϵ}} ϕ | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} Z_{0 j} = \int_{Ω_{j, n}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} Φ_{j, n} Y_{0 j} d y = a_{j} \int_{R^{2}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} Y_{0 j}^{2} d y + o (1) . \end{matrix}$ Also, by using (4.10) and (4.11) we get that $\begin{array}{l} \int_{Ω_{ϵ}} | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} (P Z_{j} - Z_{j} - γ_{j}^{*}) \\ = \int_{Ω_{ϵ}} | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}} O (ϵ^{\tilde{σ}}) + \int_{Ω_{j, n}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} Φ_{j, n} O (δ_{j} | y |) d y \\ + \frac{1}{2 π} ({\tilde{γ}}_{j j} - γ_{j}^{*} γ_{j j}) \int_{Ω_{j, n}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} Φ_{j, n} log | y | d y \\ = - \frac{α_{j} (α_{j} - 2)}{3} a_{j} \int_{R^{2}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} Y_{0 j} log | y | d y + o (1), \end{array}$ in view of $\frac{1}{2 π} ({\tilde{γ}}_{j j} - γ_{j}^{*} γ_{j j}) = - \frac{α_{j} (α_{j} - 2)}{3} + O (\frac{1}{| log ρ |})$ . Furthermore, we have that $\begin{array}{l} \int_{Ω_{ϵ}} (W - | x - ξ_{j} |^{α_{j} - 2} e^{w_{j}}) P_{ϵ} Z_{j} ϕ \\ = \int_{\frac{B_{j} - ξ_{j}}{δ_{j}}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} O (δ_{j} | y | + ρ^{\tilde{σ}}) Φ_{j, n} P_{ϵ} Z_{j} (ξ_{j} + δ_{j} y) d y \\ + \sum_{i = 1, i \neq j}^{m} \int_{\frac{B_{i} - ξ_{i}}{δ_{i}}} \frac{2 α_{i}^{2} | y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} [1 + O (δ_{i} | y | + ρ^{\tilde{σ}})] Φ_{i, n} P_{ϵ} Z_{j} (ξ_{i} + δ_{i} y) d y \\ + O (ρ^{\tilde{σ}} | log ρ |) \\ = O (ρ^{\tilde{σ}} | log ρ | \int_{\frac{B_{i} - ξ_{i}}{δ_{i}}} \frac{2 α_{j}^{2} | y |^{α_{j} - 2}}{{(1 + | y |^{α_{j}})}^{2}} (| y | + 1) | Φ_{i, n} | d y) + o (1) \\ = o (1), \end{array}$ since from (4.11), $P_{ϵ} Z_{j} (ξ_{j} + δ_{j} y) = O (| log ρ |)$ , and for $i \neq j$ and $y \in δ_{i}^{- 1} (B_{i} - ξ_{i})$ it holds $\begin{array}{rcl} P_{ϵ} Z_{j} (ξ_{i} + δ_{i} y) & = & \frac{8 π}{3} α_{j} G (ξ_{i}, ξ_{j}) + O (δ_{j}^{α_{j}} + δ_{i} | y |) \\ - ({\tilde{γ}}_{i j} - γ_{i j} γ_{j}^{*}) (- \frac{1}{2 π} log | δ_{i} y | + H (ξ_{i} + δ_{i} y, ξ_{i})) \\ - \sum_{k = 1, k \neq i}^{m} ({\tilde{γ}}_{k j} - γ_{k j} γ_{j}^{*}) (G (ξ_{i}, ξ_{k}) + O (δ_{i} | y |)) \\ = & \frac{1}{2 π} ({\tilde{γ}}_{i j} - γ_{i j} γ_{j}^{*}) log | δ_{i} y | + \frac{8 π}{3} α_{j} G (ξ_{i}, ξ_{j}) - ({\tilde{γ}}_{i j} - γ_{i j} γ_{j}^{*}) H (ξ_{i}, ξ_{i}) \\ - \sum_{k = 1, k \neq i}^{m} ({\tilde{γ}}_{k j} - γ_{k j} γ_{j}^{*}) G (ξ_{i}, ξ_{k}) + O (δ_{i} | y |) \end{array}$ and $\begin{array}{l} \int_{\frac{B_{i} - ξ_{i}}{δ_{i}}} \frac{2 α_{i}^{2} | y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} [1 + O (δ_{i} | y | + ρ^{\tilde{σ}})] Φ_{i, n} P_{ϵ} Z_{j} (ξ_{i} + δ_{i} y) d y \\ = \frac{1}{2 π} ({\tilde{γ}}_{i j} - γ_{i j} γ_{j}^{*}) log δ_{i} \int_{\frac{B_{i} - ξ_{i}}{δ_{i}}} \frac{2 α_{i}^{2} | y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} Φ_{i, n} d y \\ + \frac{1}{2 π} ({\tilde{γ}}_{i j} - γ_{i j} γ_{j}^{*}) \int_{\frac{B_{i} - ξ_{i}}{δ_{i}}} \frac{2 α_{i}^{2} | y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} Φ_{i, n} log | y | d y \\ - (bounded constant) \int_{\frac{B_{i} - ξ_{i}}{δ_{i}}} \frac{2 α_{i}^{2} | y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} [1 + O (δ_{i} | y | + ρ^{\tilde{σ}})] Φ_{i, n} d y \\ + O (δ_{i} \int_{\frac{B_{i} - ξ_{i}}{δ_{i}}} \frac{2 α_{i}^{2} | y |^{α_{i} - 2}}{{(1 + | y |^{α_{i}})}^{2}} [| y | + δ_{i} | y |^{2}] d y) = o (1) . \end{array}$ Therefore, we conclude (4.9) and hence, $a_{j} = 0$ for all $j = 1, \dots, m$ . □

Now, by using Claims 1 and 2 we deduce that $Ψ_{j, n}$ converges to zero weakly in $H_{α_{j}} (R^{2})$ and strongly in $L_{α_{j}} (R^{2})$ as $n \to + \infty$ . Thus, we arrives at a contradiction with (4.6) and it follows the priori estimate $‖ ϕ ‖ ⩽ C | log ρ | ‖ h ‖_{p}$ . It only remains to prove the solvability assertion. To this purpose consider the space $H = H_{0}^{1} (Ω_{ϵ})$ endowed with the usual inner product $[ϕ, ψ] = \int_{Ω_{ϵ}} \nabla ϕ \nabla ψ$ . Problem (3.4) expressed in weak form is equivalent to that of finding a $ϕ \in H$ such that $\begin{matrix} [ϕ, ψ] = \int_{Ω_{ϵ}} [W ϕ - h] ψ, for all ψ \in H . \end{matrix}$ With the aid of Riesz’s representation theorem, this equation gets rewritten in H in the operator form $ϕ = K (ϕ) + \tilde{h}$ , for certain $\tilde{h} \in H$ , where $K$ is a compact operator in H. Fredholm’s alternative guarantees unique solvability of this problem for any h provided that the homogeneous equation $ϕ = K (ϕ)$ has only the zero solution in H. This last equation is equivalent to (3.4) with $h \equiv 0$ . Thus, existence of a unique solution follows from the a priori estimate (3.5). This finishes the proof. □

Footnotes

Acknowledgements

The author would like to thank Professor Angela Pistoia (U. Roma “La Sapienza”, Italia) and Professor Pierpaolo Esposito (U. Roma Tre, Italia) for introduce him to mean field type models and for many interesting discussions about these type of problems. The author has been supported by grant Fondecyt Regular Nº 1201884, Chile.

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