Bubbling nodal solutions for a large perturbation of the Moser-Trudinger equation on planar domains

TL;DR

This paper constructs a family of nodal solutions for a perturbed Moser-Trudinger equation in planar domains, demonstrating existence beyond symmetric cases as the parameter approaches criticality.

Contribution

It introduces the first construction of sign-changing solutions for a Moser-Trudinger critical equation on arbitrary, non-symmetric domains as the perturbation parameter tends to criticality.

Findings

Existence of blowing-up nodal solutions as p approaches 1+

Construction valid for arbitrary domains with small λ

First such solutions on non-symmetric domains

Abstract

In this work we study the existence of nodal solutions for the problem $$ -\Delta u = \lambda u e^{u^2+|u|^p} \text{ in }\Omega, \; u = 0 \text{ on }\partial \Omega, $$ where $\Omega\subseteq \mathbb R^2$ is a bounded smooth domain and $p\to 1^+$. If $\Omega$ is ball, it is known that the case $p=1$ defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions. In this work we construct a blowing-up family of nodal solutions to such problem as $p\to 1^+$, when $\Omega$ is an arbitrary domain and $\lambda$ is small enough. As far as we know, this is the first construction of sign-changing solutions for a Moser-Trudinger critical equation on a non-symmetric domain.

Equations

$$-\Delta u=\lambda u{e}^{u^2+|u{|}^p}\text{ in }\Omega ,u=0\text{ on }\partial \Omega ,$$

$$\Delta u+\lambda u{e}^{u^2+a|u{|}^p}=0in\Omega ,u=0on\partial \Omega ,$$

$$\sup \left\{\underset{\Omega }{\int }{e}^{u^2}dx:u\in H_0^1(\Omega ),\Vert u{\Vert }_{H_0^1(\Omega )}^2\le 4\pi \right\}<+\infty ,$$

$$J_{\lambda }(u):=\frac{1}{2}\underset{\Omega }{\int }|\nabla u{|}^{2}dx-\lambda \underset{\Omega }{\int }F(u)dx,u\in H_0^1(\Omega ),$$

$$\{\begin{array}{ll}-\Delta u=\lambda u{e}^{u^2+|u{|}^{1+\epsilon }}& \text{ in }\Omega ,\\ u=0& \text{ on }\partial \Omega ,\end{array}$$

$$f_{\epsilon }(t)=t{e}^{t^2+|t{|}^{1+\epsilon }}.$$

$$\{\begin{array}{ll}-\Delta u_0=\lambda f_0(u_0)& \text{ in }\Omega ,\\ u_0>0& \text{ in }\Omega ,\\ u_0=0& \text{ on }\partial \Omega ,\end{array}$$

$$-\Delta \phi =\lambda f_0^{\prime }(u_0)\phi in\Omega ,\phi =0on\partial \Omega .$$

$$U_{\delta ,\mu ,\xi }(x)=\log \left(\frac{8{\mu }^2{\delta }^2}{({\mu }^2{\delta }^2+|x-\xi {|}^2{)}^2}\right),$$

$$-\Delta U=e^U\text{in }{\mathbb{R}}^2,$$

$$\{\begin{array}{ll}-\Delta P{U}_{\delta ,\mu ,\xi }=-\Delta U_{\delta ,\mu ,\xi }=e^{U_{\delta ,\mu ,\xi }}& \text{ in }\Omega ,\\ P{U}_{\delta ,\mu ,\xi }=0& \text{ on }\partial \Omega .\end{array}$$

$$v_{\epsilon }\to u_0\text{ in }{C}^1(\overline{\Omega }),$$

$$V_{\epsilon ,\alpha ,\xi }:=v_{\epsilon }+\alpha w_{\epsilon ,\xi }+{\alpha }^2{z}_{\epsilon ,\xi },$$

$$\{\begin{array}{ll}\Delta w_{\epsilon ,\xi }+\lambda f_{\epsilon }^{\prime }(v_{\epsilon })w_{\epsilon ,\xi }=8\pi \lambda G_{\xi }{f}_{\epsilon }^{\prime }(v_{\epsilon })& \text{ in }\Omega ,\\ w_{\epsilon ,\xi }=0& \text{ on }\partial \Omega ,\end{array}$$

$$\{\begin{array}{ll}\Delta z_{\epsilon ,\xi }+\lambda f_{\epsilon }^{\prime }(v_{\epsilon })z_{\epsilon ,\xi }=\frac{\lambda }{2}{f}_{\epsilon }^{\prime \prime }(-v_{\epsilon })(8\pi G_{\xi }-w_{\epsilon }{)}^2& \text{ in }\Omega ,\\ z_{\epsilon }=0& \text{ on }\partial \Omega ,\end{array}$$

$$\{\begin{array}{ll}-\Delta G_{\xi }={\delta }_{\xi }& \text{ in }\Omega ,\\ G_{\xi }=0& \text{ on }\partial \Omega .\end{array}$$

$${\omega }_{\epsilon ,\mu ,\xi }:=\alpha P{U}_{\delta ,\mu ,\xi }-V_{\epsilon ,\alpha ,\xi }$$

$$-\Delta \phi -\lambda f_{\epsilon }^{\prime }({\omega }_{\epsilon ,\mu ,\xi })\phi =R+N(\phi ),$$

$$R=R_{\epsilon ,\mu ,\xi }:=\Delta {\omega }_{\epsilon ,\mu ,\xi }+\lambda f_{\epsilon }({\omega }_{\epsilon ,\mu ,\xi }),$$

$$N(\phi )=N_{\epsilon ,\mu ,\xi }(\phi ):=\lambda \left(f_{\epsilon }({\omega }_{\epsilon ,\mu ,\xi }+\phi )-f_{\epsilon }({\omega }_{\epsilon ,\mu ,\xi })-f_{\epsilon }^{\prime }({\omega }_{\epsilon ,\mu ,\xi })\phi \right).$$

$$L\phi =L_{\epsilon ,\mu ,\xi }\phi :=\phi -(-\Delta {)}^{-1}(\lambda f^{\prime }({\omega }_{\epsilon ,\mu ,\xi })\phi ),$$

$$L\phi =(-\Delta {)}^{-1}\left(R+N(\phi )\right).$$

$$Z_{0,\delta ,\mu ,\xi }(x)=\frac{{\delta }^2{\mu }^2-|x-\xi {|}^2}{|x-\xi {|}^2+{\delta }^2{\mu }^2},Z_{i,\delta ,\mu ,\xi }(x)=\frac{2\delta \mu (x_i-{\xi }_i)}{|x-\xi {|}^2+{\delta }^2{\mu }^2},i=1,2.$$

$$\{\begin{array}{l}\phi =({\pi }^{\perp }L{)}^{-1}{\pi }^{\perp }{\left(-\Delta \right)}^{-1}\left(R+N(\phi )\right),\\ \pi L\phi =\pi {\left(-\Delta \right)}^{-1}\left(R+N(\phi )\right).\end{array}$$

$$\underset{B({\xi }_0,\sigma )}{\inf }{u}_0(\xi )>\frac{1}{2}.$$

$$H(x,\xi ):=G_{\xi }(x)-\frac{1}{2\pi }\log \frac{1}{|x-\xi |}.$$

$$\begin{array}{rl}P{U}_{\delta ,\mu ,\xi }(x)=& U_{\delta ,\mu ,\xi }(x)-\log (8{\mu }^2{\delta }^2)+8\pi H(x,\xi )+{\psi }_{\delta ,\mu ,\xi }(x),\end{array}$$

$$\Vert {\psi }_{\delta ,\mu ,\xi }{\Vert }_{C^1(\overline{\Omega })}=O({\delta }^2),$$

$$P{U}_{\delta ,\mu ,\xi }\to 8\pi G_{\xi }in{C}_{loc}^1(\overline{\Omega }\setminus \{\xi \}).$$

$$F(\epsilon ,u)=F_{\epsilon }(u):=u-(-\Delta {)}^{-1}(\lambda f_{\epsilon }(u)),$$

Full text

[TABLE]

where $\Omega\subseteq\mathbb{R}^{2}$ is a bounded smooth domain and $p\to 1^{+}$.

If $\Omega$ is ball, it is known that the case $p=1$ defines a critical threshold between the existence and the non-existence of radially symmetric sign-changing solutions. In this work we construct a blowing-up family of nodal solutions to such problem as $p\to 1^{+}$, when $\Omega$ is an arbitrary domain and $\lambda$ is small enough. As far as we know, this is the first construction of sign-changing solutions for a Moser-Trudinger critical equation on a non-symmetric domain.

1 Introduction

Let us consider the equation

[TABLE]

where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{2}$, $\lambda$ is a positive parameter and the nonlinear term $h(u):=ue^{a|u|^{p}}$, with $a\in\mathbb{R}$ and $p\in[0,2)$, is a lower-order perturbation of $e^{u^{2}}$ according to the definition given by Adimurthi in [2].

The nonlinearity $f(u)=h(u)e^{u^{2}}$ is critical from the view point of the Trudinger imbedding. Indeed, in view of the Moser-Trudinger inequality (see [25, 29, 24])

[TABLE]

the functional

[TABLE]

where $F(t)=\int\limits_{0}^{t}f(s)ds,$ is well defined and its critical points are solutions to problem (1). Adimurthi in [2] proved that $J_{\lambda}$ satisfies the Palais-Smale condition in the infinite energy range $(-\infty,2\pi)$ but, as observed by Adimurthi and Prashant in [5], the critical nature of $f(u)$ reflects in the failure of the Palais-Smale condition at the sequence of energy levels $2\pi k$ with $k\in\mathbb{N}$ (see also [7]).

In [2] Adimurthi proved the existence of a critical point of $J_{\lambda}$ if the perturbation $h$ is large, i.e. $a\geq 0$, and if $0<\lambda<\lambda_{1}(\Omega),$ where $\lambda_{1}(\Omega)$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary condition ((see also [1])). Such a critical point is a positive solution to problem (1). Successively, Adimurthi and Prashant in [6] showed that the condition $a\geq 0$ is necessary to get a positive solution to (1). Indeed, they proved that if the perturbation $h$ is small, i.e. $a<0$, then there are no positive solutions to problem (1) when the domain $\Omega$ is a ball provided $\lambda$ is small. The case $a=0$ in a general domain $\Omega$ has been studied by Del Pino, Musso and Ruf [14] using a perturbative approach. Indeed they find multiplicity of positive solutions which blow-up in one or more points of $\Omega$ (depending on the geometry) as $\lambda\to 0.$ We point out that a general qualitative analysis of blowing-up families of positive solutions to problem (1) has been obtained by Druet in [15] (see also [3, 17, 16]).

As far as it concerns the existence of sign-changing solutions, Adimurthi and Yadava in [8] proved that problem (1) has a nodal solution when $\lambda$ is small if there is the further restriction $p>1$ on the growth of the large perturbation $h$ (i.e. $a>0$). Actually, this condition turns out to be optimal for the existence of nodal radial solutions in a ball. Indeed Adimurthi and Yadava in [9] proved that if $a>0$ and $\Omega$ is a ball, problem (1) does not have any radial sign-changing solution when $\lambda$ is small and $p\in[0,1]$. If one drops the radial requirement, Adimurthi and Yadava in [8] proved the existence of infinitely many sign-changing solutions in a ball whatever $\lambda>0$ is. We point out that, in the case $a=0$, the approach of Del Pino, Musso and Ruf [14] allows to find sign-changing solutions which blow-up positively and negatively at least at two different points in any domain $\Omega$ as $\lambda\to 0$ (even if this is not explicitly said in their work).

According to the previous discussion, it turns out that when $a>0$ the case $p=1$ defines a critical threshold for the existence of radial sign-changing solutions in the ball. Indeed, when $\Omega=B(0,1)$, (1) has radially symmetric sign-changing solutions which blow-up as $p\to{1^{+}}$. The precise behavior of such solutions was studied by Grossi and Naimen in [19]. Therefore, when $a>0$, it is natural to ask whether it is possible to find sign-changing solutions to problem (1) on an arbitrary planar domain $\Omega$ which blow-up at one point in $\Omega$ as $p\to{1^{+}}$.

In this paper we give a positive answer. More precisely, let us consider the problem

[TABLE]

where $\varepsilon$ is a positive small parameter. Set

[TABLE]

For a given $0<\lambda<\lambda_{1}(\Omega)$, let $u_{0}$ be a positive solution of the problem

[TABLE]

whose existence has been established by Adimurthi in [2]. We make the following assumptions:

1. (A1)

$u_{0}$ is non-degenerate, i.e. there is no non-trivial solution $\varphi\in H^{1}_{0}(\Omega)$ of the equation

[TABLE] 2. (A2)

$u_{0}$ has a $C^{1}-$stable critical point $\xi_{0}\in\Omega$ such that $u_{0}(\xi_{0})>\frac{1}{2}$.

Then, we will show that (4) admits a family of sign-changing solutions which blow-up at $\xi_{0}$ with residual mass $-u_{0}$ as $\varepsilon\to 0$, namely:

Theorem 1.1

For $0<\lambda<\lambda_{1}(\Omega)$, let $u_{0}$ be a solution of (6) such that (A1) and (A2) are satisfied. Let also $\xi_{0}$ be as in (A2). Then there exist $\varepsilon_{0}>0$ and a family $(u_{\varepsilon})_{0<\varepsilon<\varepsilon_{0}}$ of sign-changing solutions to (4) such that:

• •

$\displaystyle{\max_{\overline{B(\xi_{0},r)}}u_{\varepsilon}\to+\infty}$* as $\varepsilon\to 0$, for any $0<r<d(\xi_{0},\partial\Omega)$.*

• •

$u_{\varepsilon}\to-u_{0}$* weakly in $H^{1}_{0}(\Omega)$ and in $C^{1}(\overline{\Omega}\setminus\{\xi_{0}\})$.*

Let us make some comments about assumtpions (A1) and (A2).

Remark 1.2

• •

The solution $u_{0}$ to problem (6) turns out to be non-degenerate when $\Omega$ is the ball as proved by Adimurthi, Karthik and Giacomoni in **[4]**. In a work in progress, Grossi and Naimen are going to prove that the solution is also non-degenerate when $\Omega$ is convex and symmetric (see **[20]**). Actually, we believe that the non-degeneracy condition holds true for most domains $\Omega$ and positive parameters $\lambda.$ Indeed, one could use similar arguments to those used by Micheletti and Pistoia in **[23]** for a class of singularly perturbed equations.

• •

We remind that $\xi_{0}$ is a $C^{1}-$stable critical point of $u_{0}$ if the Brouwer degree ${\rm deg}\left(\nabla u_{0},B(\xi_{0},r),0\right)\not=0.$ In particular, any strict local maximum point of $u_{0}$ is $C^{1}-$stable. We point out that by Adimurthi and Druet **[3] we can deduce that assumption (A2) holds true when the parameter $\lambda$ is small enough.

• •

We strongly believe that the condition $u_{0}(\xi_{0})>\frac{1}{2}$ is not purely technical, but it is necessary to build a solution which blows-up at $\xi_{0}.$ Indeed, we conjecture that, if $u_{0}(\xi_{0})\leq\frac{1}{2}$, there does not exist any sign-changing solution which blows-up at $\xi_{0}$ with non-trivial residual mass $u_{0}$ as $\varepsilon\to 0.$ We point out that, in a different setting, a similar condition was proved by Mancini and Thizy **[22]** for problem (1) on a ball with $p=1$ and $a<0$: in fact, they show that the value at the origin of the residual mass of any non-compact sequence of radially symmetric positive solutions must be equal to $-\frac{a}{2}$ (and we get $\frac{1}{2}$, when $a=-1$).

Actually, we can give a more precise description of the asymptotic behavior of the solution $u_{\varepsilon}$ as $\varepsilon\to 0,$ since it is build via a Lyapunov-Schmidt procedure. For $\delta,\mu>0$, and $\xi\in\mathbb{R}^{n}$, let us consider the functions

[TABLE]

which describe the set of all the solutions to the Liouville equation

[TABLE]

under the condition $e^{U}\in L^{1}(\mathbb{R}^{2})$ (see [21, 12]). We further consider the projection $PU_{\delta,\mu,\xi}:=(-\Delta)^{-1}e^{U_{\delta,\mu,\xi}}$, where $(-\Delta)^{-1}:L^{2}(\Omega)\rightarrow H^{1}_{0}(\Omega)$ is the inverse of $-\Delta$. Namely, $PU_{\delta,\mu,\xi}$ is defined as the unique solution to

[TABLE]

Intuitively, we want to look for solutions of (4) that look like $\alpha PU_{\delta,\mu,\xi}-u_{0}$ for suitable choices of the parameters $\alpha,\delta,\mu,\xi$. Unfortunately, in order to succesfully perform Lyapunov-Schmidt reduction, a more precise ansatz is necessary and we are forced to replace $u_{0}$ with a better approximation of the solutions. First, the non-degeneracy assumption (A1) allows to find a positive solution $v_{\varepsilon}\in C^{1}(\overline{\Omega})$ of (4) such that

[TABLE]

as $\varepsilon\to 0$. Then, we consider the function

[TABLE]

where $\alpha\in(0,1)$ is a small positive parameter depending on $\varepsilon,\mu,\xi$ such that $\alpha\to 0$ as $\varepsilon\to 0$, and $w_{\varepsilon,\xi}$ and $z_{\varepsilon,\xi}$ are defined as the unique solutions to the couple of linear problems

[TABLE]

and

[TABLE]

with $G_{\xi}$ denoting the Green function of $\Omega$ with singularity at $\xi$, namely the distributional solution to

[TABLE]

Problems (12) and (13) are nothing but the linearization of problem (4) around the solution $v_{\varepsilon}$ and the R.H.S.’s are the terms of the second order Taylor’s expansion with respect to $\alpha$ of $f_{\varepsilon}(\alpha PU_{\delta,\mu,\xi}-{V_{\varepsilon,\alpha,\xi}})$ far away from the concentration point $\xi$ (indeed $PU_{\delta,\mu,\xi}\sim 8\pi G_{\xi}$ because of (23)).

Theorem 1.1 follows at once by the following result:

Theorem 1.3

Let $\lambda$, $u_{0}$, $\xi_{0}$ be as in Theorem 1.1. There exists $\epsilon_{0}>0$ and functions $\alpha,\delta,\mu:(0,\varepsilon_{0})\to(0,+\infty)$, $\xi:(0,\varepsilon_{0})\rightarrow\Omega$ and $\varphi:(0,\varepsilon_{0})\rightarrow H^{1}_{0}(\Omega)$ such that:

• •

$u_{\varepsilon}:=\alpha(\varepsilon)PU_{\delta(\varepsilon),\mu(\varepsilon),\xi(\varepsilon)}-V_{\varepsilon,\alpha(\varepsilon),\xi(\varepsilon)}+\varphi(\varepsilon)$* is a solution (4).*

• •

$\alpha(\varepsilon)\to 0$, $\delta(\varepsilon)\to 0$, $\mu(\varepsilon)\to\sqrt{8}e^{-1}$, $\xi(\varepsilon)\to\xi_{0}$, and $u_{\varepsilon}(\xi(\varepsilon))\to+\infty$ as $\varepsilon\to 0$.

• •

$\|\varphi(\varepsilon)\|_{H^{1}_{0}(\Omega)}+\|\varphi(\varepsilon)\|_{L^{\infty}(\Omega)}=O(e^{-\frac{\log(2u_{0}({\xi_{0}}))}{\varepsilon}})$.

Let us briefly sketch the main steps of the proof of Theorem 1.3. First, in Section 2, we choose $\alpha=\alpha(\varepsilon,\mu,\xi)$ and $\delta=\delta(\varepsilon,\mu,\xi)$ such that the function

[TABLE]

is an approximate solution of (4). Then, we look for solutions of (4) of the form $\omega_{\varepsilon,\mu,\xi}+\varphi$ with $\varphi\in H^{1}_{0}(\Omega)$. Clearly, (4) can be written in terms of $\varphi$ as

[TABLE]

where the error term $R$ is defined by

[TABLE]

and the higher order term $N$ by

[TABLE]

Equivalently, introducing the linear operator

[TABLE]

we need to solve

[TABLE]

A careful and delicate estimate of the error $R$ will be given in Section 3. The behaviour of the operator $L$ will be studied in Section 4. On the one hand, for functions supported away from a suitable schrinking neighborhood of $\xi$, we will show that $L$ is close to the operator $L_{1}\varphi:=\varphi-(-\Delta)^{-1}(\lambda f^{\prime}_{0}(u_{0})\varphi)$, which is invertible on $H^{1}_{0}(\Omega)$ because of the non-degeneracy assumption (A1). On the other hand, near the point $\xi$, $L$ is close to the operator $L_{0}\varphi:=\varphi-(-\Delta)^{-1}(e^{U_{\delta,\mu,\xi}}\varphi)$. This operator appears in the analysis of several critical problems in dimension $2$ (see for example [10, 13, 18]) and its behavior is well known: although $L_{0}$ is not invertible, it is possible to find an approximate three-dimensional kernel $K_{\delta,\mu,\xi}$ for $L_{0}$ by projecting on $H^{1}_{0}(\Omega)$ the three functions

[TABLE]

Such properties transfer to the operator $L$, which turns out to be invertible on the subspace $K_{\delta,\mu,\xi}^{\perp}$ orthogonal to $K_{\delta,\mu,\xi}$ in $H^{1}_{0}(\Omega)$. More precisely, denoting by $\pi$ and $\pi^{\perp}$ the projections of $H^{1}_{0}(\Omega)$ respectively on $K_{\delta,\mu,\xi}$ and $K_{\delta,\mu,\xi}^{\perp}$, we will show that $\pi^{\perp}L$ is invertible on $K_{\delta,\mu,\xi}^{\perp}$. Then, it is natural to split equation (20) as

[TABLE]

The first equation of (21) will be solved in Section 5, where for any $\mu>0$, $\xi$ close to $\xi_{0}$ and any small $\varepsilon>0$, we will find a solution $\varphi_{\varepsilon,\mu,\xi}$ via a contraction mapping argument on a sufficiently small ball in $K_{\delta,\mu,\xi}^{\perp}\cap L^{\infty}(\Omega)$. Then, recalling that $\dim K_{\delta,\mu,\xi}=3$ and using assumption (A2), we will show in Section 6 that it is possible to choose the three parameters $\mu=\mu(\varepsilon)$ and $\xi=\xi(\varepsilon)=(\xi_{1}(\varepsilon),\xi_{2}(\varepsilon))$ so that the second equation in (21) is also fullfilled. Clearly, for such choice of $\mu$ and $\xi$, the function $\varphi_{\varepsilon,\mu(\varepsilon),\xi(\varepsilon)}$ solves both the equations in (21) (or, equivalently (16) and (20)), and $u_{\varepsilon}:={\omega_{\varepsilon,\mu(\varepsilon),\xi(\varepsilon)}+\varphi_{\varepsilon,\mu(\varepsilon),\xi(\varepsilon)}}$ is a solution of (4).

It is important to point out that choice of the concentration point $\xi(\varepsilon)$ is extremely delicate since the scaling parameter $\delta$ turns out to be much smaller than the parameter $\alpha$, whose powers control all the error terms. To overcome this difficulty, we introduce a new argument based on a precise Pohozaev-type identity. This allows us to bypass global a priori gradient estimates on the solution $\varphi_{\varepsilon,\mu,\xi}$, which are hard to obtain for Moser-Trudinger critical problems. Our argument requires a very precise ansatz of the approximate solution $\omega_{\varepsilon,\mu,\xi}$. In particular, the presence of the correction terms $w_{\varepsilon,\xi}$ and $z_{\varepsilon,\xi}$ in the expression of $V_{\varepsilon,\alpha,\xi}$ is not merely technical, but plays a crucial role both in the estimates of the error term $R$ and in the choice of $\xi(\varepsilon)$.

2 Construction of the approximate solution

In this section we give the detailed construction of the approximate solution $\omega_{\varepsilon,\mu,\xi}$. Here and in the rest of the paper, we will assume that $(\mu,\xi)\in\mathcal{U}\times B(\xi_{0},\sigma)$, where $\mathcal{U}\Subset\mathbb{R}^{+}$ is an open interval containing ${\mu_{0}:=\sqrt{8}e^{-1}}$, $\xi_{0}$ is as in the assumption (A2), and $0<\sigma<\frac{1}{2}d(\xi_{0},\partial\Omega)$. By (A2), we can also assume

[TABLE]

2.1 The main terms of the ansatz

Let us introduce the main property of the projection of the bubble $PU_{\delta,\mu,\xi}$ defined in (10), which gives the main term of the approximate solution close to the blow-up point $\xi.$ Let $G_{\xi}(\cdot)=G(\cdot,\xi)$ be the Green’s function of $-\Delta$ with Dirichlet boundary conditions introduced in (14) and let $H(\cdot,\xi)$ be its regular part, i.e.

[TABLE]

Lemma 2.1

We have

[TABLE]

where

[TABLE]

*uniformly with respect to $\mu\in\mathcal{U}$, $\xi\in B(\xi_{0},\sigma)$.

In particular,*

[TABLE]

Proof.

See for example [11, Proosition 5.1]. ∎

Next, let us define the main term of the approximate solution in the whole domain as $\alpha PU_{\delta,\mu,\xi}-v_{\varepsilon}$ where $\alpha$ is a positive parameter approaching zero as $\varepsilon\to 0$ and $v_{\varepsilon}$ is a non-degenerate solution to (4), whose existence is proved in the following lemma.

Lemma 2.2

Let $\lambda$ and $u_{0}$ be as in Theorems 1.1 and 1.3. There exists $\varepsilon_{0}>0$, and a family of functions $(v_{\varepsilon})_{0<\varepsilon<\varepsilon_{0}}\subseteq{C^{1}(\overline{\Omega})}$ such that:

1. i.

$v_{\varepsilon}$* is a non-degenerate weak solution of (4) for any $\varepsilon\in(0,\varepsilon_{0})$.* 2. ii.

$v_{\varepsilon}\to u_{0}\;\text{ in }{C^{1}(\overline{\Omega})}$* as $\varepsilon\to 0$.* 3. iii.

There exists $c>0$ such that $v_{\varepsilon}(x)\geq cd(x,\partial\Omega)$ for any $x\in\Omega$, $\varepsilon\in(0,\varepsilon_{0})$.

Proof.

Let $F:(-1,1)\times H^{1}_{0}(\Omega)\to H^{1}_{0}(\Omega)$ be defined by

[TABLE]

where $f_{\varepsilon}$ is defined as in (5). $F$ is well defined because the Moser-Trudinger inequality (2) implies that $f_{\varepsilon}(u)\in L^{p}(\Omega)$ for any $1\leq p<+\infty$ and $u\in H^{1}_{0}(\Omega)$. Moreover, it is a $C^{1}$-map and its partial derivative $DF_{\varepsilon}(u):H^{1}_{0}(\Omega)\to H^{1}_{0}(\Omega)$ defined by

[TABLE]

is a Fredholm operator of index [math] (since the embedding $H^{1}_{0}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact).

Now, let $u_{0}$ be a non-degenerate weak solution of (6) such that (A1) holds true. In particular, $F_{0}(u_{0})=0$ and $DF_{0}(u_{0})$ is invertible. Therefore, by the implicit function theorem, we can construct a $C^{1}$ curve $\varepsilon\mapsto v_{\varepsilon}\in H^{1}_{0}(\Omega)$, defined for $|\varepsilon|<\varepsilon_{0}$ such that $v_{0}=u_{0}$, $F_{\varepsilon}(v_{\varepsilon})=0$, and $DF_{\varepsilon}(v_{\varepsilon})$ is invertible for $|\varepsilon|<\varepsilon_{0}$. Then i. holds.

Applying the Moser-Trudinger inequality (2) and standard elliptic estimates, we obtain ii..

Hopf’s lemma and the compactness of $\partial\Omega$ give $\frac{\partial u_{0}}{\partial\nu}\leq-2c$ on $\partial\Omega$, for some $c>0$. Then, for $\varepsilon$ sufficiently small, we have $\frac{\partial v_{\varepsilon}}{\partial\nu}\leq-c$, which in turn gives $v_{\varepsilon}(x)\geq cd(x,\partial\Omega)$ for $x$ in a neighborhood of $\partial\Omega$. Finally, since $v_{\varepsilon}\to u_{0}$ uniformly in $\overline{\Omega}$, and $u_{0}>0$ in $\Omega$, we get iii.. ∎

2.2 The correction of the ansatz

We need to correct the ansatz in the whole domain by solving the following two linear problems (12) and (13):

[TABLE]

and

[TABLE]

Lemma 2.3

For any $0<\varepsilon<\varepsilon_{0}$ and any $\xi\in\Omega$, there exist $w_{\varepsilon,\xi}$, $z_{\varepsilon,\xi}$ such that (12) and (13) hold. Moreover, there exists $C>0$ such that

[TABLE]

for $\varepsilon\in(0,\varepsilon_{0})$, $\xi\in\Omega$.

Proof.

The existence of the solutions immediately follows from the non-degeneracy of the function $v_{\varepsilon}$ proved in Lemma 2.2. Moreover, since for any $p\in[1,+\infty)$ one has

[TABLE]

(25) follows by standard elliptic estimates. ∎

Finally, we introduce the corrected ansatz as

[TABLE]

with

[TABLE]

where $v_{\varepsilon}$ is defined in Lemma 2.2 and $w_{\varepsilon,\xi}$ and $z_{\varepsilon,\xi}$ as in Lemma 2.3.

2.3 The choice of parameters

It will be necessary to choose the parameters $\alpha=\alpha(\varepsilon,\mu,\xi)$ and $\delta=\delta(\varepsilon,\mu,\xi)$ such that $\lambda f_{\varepsilon}(\omega_{\varepsilon,\mu,\xi})\sim\alpha e^{U_{\delta,\mu,\xi}}$ when ${|x-\xi|}\sim\delta.$ We point out that one of the main difficulties in this problem is that this estimates holds true only at a very small scale.

Let us fix the values of $\alpha$ and $\delta$ according to the next lemma. The proof is based on the contraction mapping theorem and is postponed to the appendix.

Lemma 2.4

There exist $\varepsilon_{0}>0$ and functions $\alpha=\alpha(\varepsilon,\mu,\xi)$, $\beta=\beta(\varepsilon,\mu,\xi)$ and $\delta=\delta(\varepsilon,\mu,\xi)$, defined in $(0,\varepsilon_{0})\times\mathcal{U}\times B(\xi_{0},\sigma)$ and continuous with respect to $\mu$ and $\xi$, such that

[TABLE]

where $c_{\mu,\xi}:=-\log(8\mu^{2})+8\pi H(\xi,\xi)$ and $V_{\varepsilon,\alpha,\xi}$ is defined in (11).

Moreover, as $\varepsilon\to 0$, we have that

[TABLE]

[TABLE]

[TABLE]

where $o(1)\to 0$ as $\varepsilon\to 0$, uniformly for $\mu\in\mathcal{U}$ and $\xi\in B(\xi_{0},\sigma)$.

Remark 2.5

Note that (29)-(31) and (22) give $\alpha(\varepsilon,\mu,\xi),\delta(\varepsilon,\mu,\xi)\to 0$ and $\beta(\varepsilon,\mu,\xi)\to+\infty$ as $\varepsilon\to 0$, uniformly for $\mu\in\mathcal{U}$ and $\xi\in B(\xi_{0},\sigma)$.

From now on we let $\alpha=\alpha(\varepsilon,\mu,\xi)$, $\beta=\beta(\varepsilon,\mu,\xi)$ and $\delta=\delta(\varepsilon,\mu,\xi)$ be as in Lemma 2.4.

It will be convenient to work on the scaled domain $\frac{\Omega-\xi}{\delta}:=\left\{\frac{x-\xi}{\delta},\quad x\in\Omega\right\}.$ Note that we have the scaling relation

[TABLE]

where

[TABLE]

Lemma 2.6

As $\varepsilon\to 0$, we have

[TABLE]

*uniformly for $y\in B(0,\frac{\sigma}{\delta})$, $\mu\in\mathcal{U}$ and $\xi\in B(\xi_{0},\sigma)$.

Moreover, for any $R>0$ it holds also true that*

[TABLE]

as $\varepsilon\to 0$ uniformly for $y\in B(0,R)$, $\mu\in\mathcal{U}$ and $\xi\in B(\xi_{0},\sigma)$.

Proof.

Lemma 2.1 and the scaling relation (32) show that, as $\delta\to 0$, we have the following expansion uniformly for $\varepsilon\in(0,\varepsilon_{0})$, $\mu\in\mathcal{U}$, $\xi\in B(\xi_{0},\sigma)$ and $y\in B(0,\frac{\sigma}{\delta})$:

[TABLE]

By Lemmas 2.2 and 2.3, we know that $V_{\varepsilon,\alpha,\mu}$ is uniformly bounded in $C^{1}(\overline{\Omega})$. Thus

[TABLE]

Similarly, since $H\in C^{1}(\overline{\Omega}\times B(\xi_{0},\sigma))$, we have

[TABLE]

Then estimate (34) is proved.

Now, let us prove (35). Note that (29)-(31) yield $\beta=O(\frac{1}{\alpha})$, $\delta=O(e^{-\frac{1+o(1)}{8\alpha^{2}}})$, and $\beta^{\varepsilon}=2u_{0}(\xi)+o(1)=O(1)$. For $|y|\leq R$, (34) implies

[TABLE]

In particular

[TABLE]

and

[TABLE]

Then, using (28) we get

[TABLE]

which proves (35). ∎

It is also useful to point out the following result which will be used in the next sections.

Remark 2.7

Lemma 2.1 and Lemma 2.4 give

[TABLE]

and

[TABLE]

uniformly for $x\in\Omega$, $\varepsilon\in(0,\varepsilon_{0})$, $\mu\in\mathcal{U}$, $\xi\in B(\xi_{0},\sigma)$.

Notation: In order to simplify the notation, we will write $U_{\varepsilon}$, $\bar{U}$, $V_{\varepsilon}$, $\omega_{\varepsilon}$, $w_{\varepsilon}$ and $z_{\varepsilon}$ instead of $U_{\delta,\mu,\xi}$, $\bar{U}_{\mu}$, $V_{\varepsilon,\alpha,\xi}$, $\omega_{\varepsilon,\mu,\xi}$, $w_{\varepsilon,\xi}$ and $z_{\varepsilon,\xi}$, without specifying explicitly the dependence on the parameters. It is important to point out that all the estimates of the next sections will be uniform with respect to $\mu\in\mathcal{U}$ and $\xi\in B(\xi_{0},\sigma)$. This will allow us to choose freely the values of $\mu$ and $\xi$ in Section 6. Consistently, the notation $O(f(x,\varepsilon,\alpha,\beta,\delta))$ and $o(f(x,\varepsilon,\alpha,\beta,\delta))$ will be used for quantities depending on $\varepsilon,\xi,\mu$ (and the parameters $\alpha,\beta,\delta$ of Lemma 2.4) and satisfying respectively

[TABLE]

as $\varepsilon\to 0$, uniformly for $\mu\in\mathcal{U}$ and $\xi\in B(\xi_{0},\sigma)$.

3 The estimate of the error term

In this section we give estimates for the error term $R$ defined in (17)

[TABLE]

It will be convenient to split $\Omega$ into four different regions:

[TABLE]

where $\rho_{0}=\rho_{0}(\varepsilon,\mu,\xi)$, $\rho_{1}=\rho_{1}(\varepsilon,\mu,\xi)$, $\rho_{2}=\rho_{2}(\varepsilon,\mu,\xi)$, are defined by

[TABLE]

Note that

[TABLE]

by (29) and (31). Roughly speaking, we have to split the error into four parts: in $B(\xi,\rho_{0})$ we have $\lambda f_{\varepsilon}(\omega_{\varepsilon})=\alpha e^{U_{\varepsilon}}(1+o(1))$ (see (35)) and we can use a blow-up argument to get a uniform weighted estimate on $R$. This estimate does not hold anymore in the set $\Omega\setminus B(\xi,\rho_{0})$, which we further split into three parts: the region $\Omega\setminus B(\xi,\rho_{2})$, where $\alpha G_{\xi}=O(\varepsilon)$ and a uniform estimate on $R$ can be obtained via a Taylor expansion of $f_{\varepsilon}(\omega_{\varepsilon})$ (using that $\omega_{\varepsilon}=-V_{\varepsilon}+8\pi\alpha G_{\xi}+o(\alpha^{2})$), and the two annuli $B(\xi,\rho_{1})\setminus B(\xi,\rho_{0})$ and $B(\xi,\rho_{2})\setminus B(\xi,\rho_{1})$, where we give quite delicate integral estimates. The last two regions are treated separately since $\omega_{\varepsilon}\geq c_{0}>0$ in $B(\xi,\rho_{1})\setminus B(\xi,\rho_{0})$, while $\omega_{\varepsilon}$ changes sign in $B(\xi,\rho_{2})\setminus B(\xi,\rho_{1})$ (cfr. Lemma 3.2 and Lemma 3.11).

3.1 A uniform expansion in $B(\xi,\rho_{1})$

In this section we give a more precise version of the expasions in (36)-(37).

Lemma 3.1

For any $\varepsilon\in(0,1)$ and $x\geq-1$, we have

[TABLE]

Proof.

According to Bernoulli’s inequality we have

[TABLE]

and

[TABLE]

Since $x\geq-1$, thanks to (40) we have that

[TABLE]

Then, the conclusion follows from (41) and (42). ∎

Lemma 3.2

Set $\displaystyle{c_{0}:=\frac{1}{2}\inf_{\xi\in B(\xi_{0},\sigma)}u_{0}(\xi)}$. For $x\in B(\xi,\rho_{1})$, we have that

[TABLE]

for sufficiently small $\varepsilon$. In particular, we have

[TABLE]

Proof.

The definitons of $\overline{U}$ and $\rho_{1}$ (see (33) and (39)), and (30)-(31) give

[TABLE]

which implies (43) for sufficiently small $\varepsilon$. To get (44), it is sufficient to apply Lemma 2.6 and Remark 2.7. ∎

Lemma 3.3

For $x\in B(\xi,\rho_{1})$, we have

[TABLE]

Proof.

Set $y=\frac{x-\xi}{\delta}\in B(0,\frac{\rho_{1}}{\delta})$. Noting that $\overline{U}(y)=O(\alpha^{-2})$ and using Lemma 2.6, we get

[TABLE]

Similarly, since Lemma 3.2 gives $\frac{\alpha}{\beta}\bar{U}(y)\geq-1+\frac{c_{0}}{\beta}\geq-1$, by Lemma 3.1 we infer

[TABLE]

Then the conclusion follows from the second equation in (28). ∎

3.2 Expansions in $B(\xi,\rho_{0})$

Let us now restrict our attention to the smaller ball $B(\xi,\rho_{0})$. This allows to control the term $\alpha^{2}\bar{U}^{2}$ appearing in the expansion of Lemma 3.3. Indeed, since $|\bar{U}(y)|=-4\log|y|+O(1)$ as $|y|\to+\infty$, we have that

[TABLE]

Lemma 3.4

For $x\in B(\xi,\rho_{0})$, we have

[TABLE]

Proof.

Set $y=\frac{x-\xi}{\delta}$. First by Lemma 2.6, Lemma 3.3, and (28)-(32), we get that

[TABLE]

Now, by (45), we can expand the last exponential term, and find

[TABLE]

We can so conclude that

[TABLE]

Moreover, by (10)-(13), and Lemmas 2.2-2.3 we have

[TABLE]

where in the last equality we used that

[TABLE]

for $x\in B(\xi,\rho_{0})$. Thanks to (46) and (47), we conclude that

[TABLE]

∎

As an immediate consequence of the previous lemma we obtain the estimate:

Corollary 3.5

We have that

[TABLE]

in $B(\xi,\rho_{0})$.

3.3 Estimates on $B(\xi,\rho_{1})\setminus B(\xi,\rho_{0})$

In this region, it is diffcult to provide pointwise estimates of $R$ because the term $\alpha^{2}\bar{U}^{2}$ appearing in the expansion of Lemma 3.3 becomes very large. Then, we will look for integral estimates. Specifically we will show that $R$ is (very) small in $L^{p}(B(\xi,\rho_{1})\setminus B(\xi,\rho_{0}))$, for a suitable choice of $p=p(\alpha)>1$, such that $p\to 1$ as $\varepsilon\to 0$, uniformly with respect to $\xi\in B(\xi_{0},\sigma)$, $\mu\in\mathcal{U}$.

Lemma 3.6

There exists $c_{1}>0$ such that

[TABLE]

in $B(\xi,\rho_{1})\setminus B(\xi,\rho_{0})$.

Proof.

Since $0\leq\omega_{\varepsilon}\leq\beta$ in $B(\xi,\rho_{1})\setminus B(\xi,\rho_{0})$, from Lemma 3.3 and (28) we get

[TABLE]

∎

For $c_{1}$ as in Lemma 3.6, let us consider the function

[TABLE]

Lemma 3.7

Set $p:=1+\alpha^{2}$. There exists $c_{2}>0$ such that

[TABLE]

Proof.

First of all, we observe that for $q\in(\frac{1}{2},+\infty)$, $R>0$, one has

[TABLE]

For $x\in B(\xi,\rho_{1})\setminus B(\xi,\rho_{0})$, set $y=\frac{x-\xi}{\delta}\in B(0,\frac{\rho_{1}}{\delta})\setminus B(0,\frac{\rho_{0}}{\delta})$. Clearly we have

[TABLE]

Set $\bar{\rho}=\delta e^{\frac{1}{\alpha^{\frac{3}{2}}}}$, so that $\rho_{0}\ll\bar{\rho}\ll\rho_{1}$. For $\frac{\rho_{0}}{\delta}\leq|y|\leq\frac{\bar{\rho}}{\delta}$, we have

[TABLE]

Then, for $\varepsilon$ small enough, (49) yields

[TABLE]

For $\frac{\bar{\rho}}{\delta}\leq|y|\leq\frac{\rho_{1}}{\delta}$, by (30) and Lemma 3.2, we have

[TABLE]

Hence, we get

[TABLE]

Thus, by (50),(51),(52), we obtain

[TABLE]

Since (29)-(31) give

[TABLE]

we get the conclusion. ∎

Lemma 3.8

Let $p$ and $c_{2}$ be as in Lemma 3.7, then

[TABLE]

Proof.

By Lemma 3.6 and Lemma 3.7 we get that

[TABLE]

On the other hand, we have

[TABLE]

so that

[TABLE]

∎

3.4 Estimates in $B(\xi,\rho_{2})\setminus B(\xi,\rho_{1})$

In $B(\xi,\rho_{2})\setminus B(\xi,\rho_{1})$ we can only say that $\omega_{\varepsilon}$ and $R$ are uniformly bounded. Since $\rho_{2}$ is very small, we still get integral bounds for $R$.

Lemma 3.9

We have $\omega_{\varepsilon}=O(1)$ and $R=O(1)$ in $\Omega\setminus B(\xi,\rho_{1})$. In particular,

[TABLE]

Proof.

Let us recall that $\omega_{\varepsilon}=\alpha PU_{\varepsilon}-V_{\varepsilon}$ with $V_{\varepsilon}=V_{\varepsilon,\alpha,\xi}$ defined as in (11). According to Lemma 2.2 and Lemma 2.3, we have $V_{\varepsilon}=O(1)$ in $\Omega$. Besides Lemma 2.1 gives

[TABLE]

for $x\in\Omega\setminus B(\xi,\rho_{1})$. Then, $\omega_{\varepsilon}=O(1)$ and $f_{\varepsilon}(\omega_{\varepsilon})=O(1)$ in $\Omega\setminus B(\xi,\rho_{1})$. Similarly

[TABLE]

Therefore $R=O(1)$. ∎

3.5 Estimates in $\Omega\setminus B(\xi,\rho_{2})$

In $\Omega\setminus B(\xi,\rho_{2})$ we will use that ${\omega_{\varepsilon}}\sim 8\pi\alpha G_{\xi}-V_{\varepsilon}$. Our choice of $V_{\varepsilon}$ will make $R$ uniformly small, namely of order $\alpha^{3}$. Note further that the choice of $\rho_{2}$ gives $\alpha{G_{\xi}}=O(\varepsilon)$ on $\Omega\setminus B(\xi,\rho_{2})$.

Lemma 3.10

As $\varepsilon\to 0$ we have

[TABLE]

Proof.

By Lemma 2.1 we have

[TABLE]

Since $\|\psi_{\delta,\mu,\xi}\|_{C^{1}(\overline{\Omega})}=O(\delta^{2})$ as $\varepsilon\to 0$, it is sufficent to observe that

[TABLE]

∎

Lemma 3.11

There exists a constant $c>0$ such such that

[TABLE]

for any $x\in\Omega\setminus B(\xi,\rho_{2})$, provided $\varepsilon$ is sufficiently small.

Proof.

By Lemma 2.2, Lemma 2.3 and (11) we have

[TABLE]

for some $c>0$. Then, Lemma 3.10 implies that

[TABLE]

in a neighborhood of $\partial\Omega$. By definiton of $\rho_{2}$, we have that $PU_{\varepsilon}=G_{\xi}+o(1)=O(\frac{\varepsilon}{\alpha})$ in $\Omega\setminus B(\xi,\rho_{2})$. Then, using again Lemma 2.2 and Lemma 2.3, we get $\omega_{\varepsilon}=-u_{0}+o(1)$ uniformly in $\Omega\setminus B(\xi,\rho_{2})$. Since $u_{0}>0$ in $\Omega$, this toghether with (53) yields the conclusion. ∎

Lemma 3.12

In $\Omega\setminus B(\xi,\rho_{2})$, we have $R=O(\alpha^{3}(1+{G_{\xi}^{3}}))$. In particular,

[TABLE]

Proof.

Since $v_{\varepsilon}>0$ in $\Omega$, $\omega_{\varepsilon}<0$ in $\Omega\setminus B(\xi,\rho_{2})$, and $f_{\varepsilon}\in C^{3}((-\infty,0))$, for any $x\in\Omega\setminus B(\xi,\rho_{2})$ we can find $\theta(x)\in[0,1]$ such that

[TABLE]

According to Lemma 2.3 and Lemma 3.10, we have

[TABLE]

Thus we get

[TABLE]

A direct computation shows the existence of a constant $C>0$ such that

[TABLE]

Since $-v_{\varepsilon}+\theta(\alpha PU_{\varepsilon}-\alpha w_{\varepsilon}-\alpha^{2}z_{\varepsilon})=O(1)$ uniformly in ${\Omega\setminus B(\xi,\rho_{2})}$, and since Lemma 3.10 implies $-v_{\varepsilon}+\theta(\alpha PU_{\varepsilon}+\alpha w_{\varepsilon}+\alpha^{2}z_{\varepsilon})\leq-cd(\cdot,\partial\Omega)$ in a neighborhood of $\partial\Omega$, we get

[TABLE]

Since $G_{\xi}=O(d(\cdot,\partial\Omega))$ near $\partial\Omega$, we deduce that

[TABLE]

Since by construction we have $\Delta\omega_{\varepsilon}=-\alpha e^{U_{\varepsilon}}-\Delta v_{\varepsilon}-\alpha\Delta w_{\varepsilon}-\alpha^{2}\Delta z_{\varepsilon}$, with $v_{\varepsilon}$, $w_{\varepsilon}$, $z_{\varepsilon}$ solving (4) and (12)-(13), we conclude that

[TABLE]

∎

3.6 The final estimate of the error in a mixed norm

We can summarize the estimates of the previous sections as follows:

In $B(\xi,\rho_{0})$, Corollary 3.5 gives $|R|\leq\alpha^{3}j_{\varepsilon},$ where

[TABLE]

In $B(\xi,\rho_{1})\setminus B(\xi,\rho_{0})$, Lemma 3.8 shows that the norm of $R$ in $L^{1+\alpha^{2}}$ is exponentially small in $\alpha$.

Finally, in $\Omega\setminus B(\xi,\rho_{1})$, Lemma 3.9 and Lemma 3.12 give $L^{2}$ estimates on $R$. This suggests to introduce the norm

[TABLE]

The coefficient $\frac{1}{\alpha^{2}}$ is chosen in order to match the norm of $(-\Delta)^{-1}$ as a linear operator from $L^{1+\alpha^{2}}(B(\xi,\rho_{1})\setminus B(\xi,\rho_{0}))$ into $L^{\infty}(B(\xi,\rho_{1})\setminus B(\xi,\rho_{0}))$ (see Corollary B.4).

According to the estimates above we have:

Proposition 3.13

There exists $D_{1}>0$, $\varepsilon_{0}>0$ such that

[TABLE]

for any $\varepsilon\in(0,\varepsilon_{0})$, $\mu\in\mathcal{U}$, $\xi\in B(\xi_{0},\sigma)$.

We conclude this section by stating some simple properties of the norm $\|\cdot\|_{\varepsilon}$ and the weight $j_{\varepsilon}$.

Lemma 3.14

There exists a constant $C>0$ such that

[TABLE]

for any $\varepsilon>0$, $\mu\in\mathcal{U}$, $\xi\in B(\xi_{0},\sigma)$.

Proof.

Let $f:\Omega\rightarrow\mathbb{R}$ be a Lebesgue measurable function. Then

[TABLE]

By Hölder’s inequality

[TABLE]

and

[TABLE]

Hence, the conclusion follows. ∎

Lemma 3.15

For any $\varepsilon>0$ let $\rho_{\varepsilon}$, $\sigma_{\varepsilon}$ be such that $\rho_{2}\leq\sigma_{\varepsilon}\leq\sigma$ and $\delta\ll\rho_{\varepsilon}\leq\rho_{0}$ as $\varepsilon\to 0$. Let $\varphi_{\varepsilon}$ of be the solution to

[TABLE]

As $\varepsilon\to 0$, we have

[TABLE]

Proof.

Let us first note that there exists a constant $c>0$, such that

[TABLE]

in $\mathbb{R}^{2}$. Then, by the maximum principle, we have

[TABLE]

where $\psi$ satisfies

[TABLE]

Since the function $W:=-\log(\mu+\sqrt{|\cdot|^{2}+\mu^{2}})$ satisfies $-\Delta W=\frac{\mu}{(\mu^{2}+|\cdot|^{2})^{\frac{3}{2}}}$, we have

[TABLE]

for suitable constants $a,b\in\mathbb{R}$. Denoting $R_{1}=\frac{\rho_{\varepsilon}}{\delta}$ and $R_{2}=\frac{\sigma_{\varepsilon}}{\delta}$ one can verify that

[TABLE]

Since

[TABLE]

uniformly in $\overline{A_{\varepsilon}}$, one has $a=O\left({\frac{\log R_{2}}{R_{1}(\log R_{2}-\log R_{1})}}\right)$ and $b=1+O\left({\frac{1}{R_{1}(\log R_{2}-\log R_{1})}}\right)$. Then

[TABLE]

Since

[TABLE]

we conclude that $\psi_{\mu}=O(\frac{1}{R_{1}})=o(1)$, uniformly in $A_{\varepsilon}$. Then, the conclusion follows by (56). ∎

4 The Linear Theory

Let us consider the linear operator

[TABLE]

introduced in (19). In this section we give a priori estimates for the operator $L$ and we prove its invertibility on a suitable subspace of $H^{1}_{0}(\Omega)$.

Lemma 4.1

The following expansions hold:

$\lambda f_{\varepsilon}^{\prime}(\omega_{\varepsilon})=e^{U_{\varepsilon}}(1+O(\varepsilon^{2}))$* in $B(\xi,\rho_{0})$.* 2. 2.

$\lambda f_{\varepsilon}^{\prime}(\omega_{\varepsilon})=O(\Gamma_{\varepsilon})$* in $B(\xi,\rho_{1})$, with $\Gamma_{\varepsilon}$ as in (48).* 3. 3.

$\lambda f_{\varepsilon}^{\prime}(\omega_{\varepsilon})=O(1)$* in $\Omega\setminus B(\xi,\rho_{1})$.* 4. 4.

${\|\lambda f_{\varepsilon}^{\prime}(\omega_{\varepsilon})\chi_{B(\xi,\rho_{1})}-e^{U_{\varepsilon}}\|_{\varepsilon}=o(1)}$* as $\varepsilon\to 0$.*

Proof.

For $x\in B(\xi,\rho_{0})$, using (28)-(32), Lemma 3.3, (34), and (45), we have that

[TABLE]

For $x\in B(\xi,\rho_{1})$, using Remark 2.7, Lemma 3.3 we have

[TABLE]

Claim 3 follows directly from Lemma 3.9. Finally, claim 4 follows by claims 1 and 2, using also Lemma 3.7 and the estimates

[TABLE]

∎

According to Lemma 4.1, for $|x-\xi|\leq\rho_{0}$, $L$ approaches the operator $L_{0}\varphi:=\varphi-(-\Delta)^{-1}(e^{U_{\varepsilon}}\varphi)$. Note that

[TABLE]

Let us recall the following known fact about $L_{0}$ (see for example [10]).

Proposition 4.2

All bounded weak solutions of the problem

[TABLE]

have the form

[TABLE]

where $c_{0},c_{1},c_{2}\in\mathbb{R}$ and

[TABLE]

Remark 4.3

The functions $Z_{0},Z_{1},Z_{2}$ are orthogonal in $D^{1,2}(\mathbb{R}^{2})$, that is

[TABLE]

In the following we denote

[TABLE]

Lemma 4.4

It holds true that

[TABLE]

uniformly with respect to $\mu\in\mathcal{U}$, $\xi\in B(\xi_{0},\sigma)$.

Proof.

See for example Appendix A in [18]. ∎

Lemma 4.4 shows the smallness of $PZ_{i,\varepsilon}-Z_{i,\varepsilon}$ for $i=1,2$, but not for $i=0$. For this reason, in many cases it is convenient to replace $PZ_{0,\varepsilon}$ with the funtion

[TABLE]

Lemma 4.5

The function $\widetilde{Z}_{\varepsilon}$ satisfies the following properties:

• •

$\widetilde{Z}_{\varepsilon}\in H^{1}_{0}(\Omega)$* and $|\widetilde{Z}_{\varepsilon}|\leq 1$ in $\Omega$.*

• •

$\|\nabla(\widetilde{Z}_{\varepsilon}-Z_{0,\varepsilon})\|_{L^{2}(\Omega)}\to 0$, uniformly for $\mu\in\mathcal{U}$ and $\xi\in B(\xi_{0},\sigma)$.

Proof.

The first property follows trivially from the definition. Moreover we have

[TABLE]

as $\varepsilon\to 0$. ∎

We will denote by $K_{\varepsilon}$ the subspace of $H^{1}_{0}(\Omega)$ spanned by $PZ_{i,\varepsilon}$, $i=0,1,2$ and by $K_{\varepsilon}^{\perp}$ the subspaces of $H^{1}_{0}(\Omega)$ orthogonal to $K_{\varepsilon}$, i.e.

[TABLE]

Let $\pi$ and $\pi^{\perp}$ be the projections of $H^{1}_{0}(\Omega)$ respectively on $K_{\varepsilon}$ and $K_{\varepsilon}^{\perp}$. Finally, we denote

[TABLE]

Proposition 4.6

There exist $\varepsilon_{0}>0$ and a constant $D_{0}>0$ such that

[TABLE]

for any $\varepsilon\in(0,\varepsilon_{0})$, $\mu\in\mathcal{U}$, $\xi\in B(\xi_{0},\sigma)$, $h\in Y_{\varepsilon}$ and $\varphi\in K_{\varepsilon}^{\perp}$ satisfying

[TABLE]

Proof.

We assume by contradiction that there exists $\varepsilon_{n}\to 0$, $\mu_{n}\in\mathcal{U}$, $\xi_{n}\in B(\xi_{0},\sigma)$, $h_{n}\in Y_{\varepsilon}$ and a solution $\varphi_{n}\in K_{\varepsilon_{n}}^{\perp}$ of (61) such that

[TABLE]

Let $\delta_{n},\alpha_{n},\beta_{n}$ be the parameters in Lemma 2.4 corresponding to $\varepsilon_{n}$, $\mu_{n}$ and $\xi_{n}$. Let also $\rho_{0,n},\rho_{1,n}$, $\rho_{2,n}$ be defined as in (39). We denote $\omega_{n}:=\omega_{\varepsilon_{n}}$, $U_{n}:=U_{\varepsilon_{n}}$, $Z_{i,n}:=Z_{i,\varepsilon_{n}}$ and $f_{n}:=f_{\varepsilon_{n}}$. W.l.o.g we can assume that $\|\varphi_{n}\|_{H^{1}_{0}(\Omega)}+\|\varphi_{n}\|_{L^{\infty}(\Omega)}=1$ and $\|h_{n}\|_{\varepsilon_{n}}\to 0$. Since $\varphi_{n}$ satisfies (61), there exist $c_{i,n}\in\mathbb{R}$, $i=0,1,2$, such that

[TABLE]

Step 1

We have $c_{i,n}\to 0$ as $n\to+\infty$, $i=0,1,2$.

Let $\widetilde{Z}_{n}:=\widetilde{Z}_{\varepsilon_{n}}$ be the function defined in (59). Testing equation (62) against $\widetilde{Z}_{n}$, we get

[TABLE]

Since $\|\varphi_{n}\|_{H^{1}_{0}(\Omega)}\leq 1$ and $\varphi_{n}\in K_{\varepsilon_{n}}^{\perp}$, using Lemma 4.5 we get

[TABLE]

as $n\to+\infty$. By Lemma 4.1 and Lemma 3.7, we find

[TABLE]

Finally, Lemma 4.5 and Lemma 3.14 give

[TABLE]

Then (63) rewrites as

[TABLE]

With similar arguments, testing equation (62) against $PZ_{i,n}$ for $i=1,2$, we get that

[TABLE]

Note that, as in (58), we have

[TABLE]

for $j=0,1,2$. Similarly

[TABLE]

for $i=1,2$, $j=0,1,2$. Then, (63) and (64) rewrite as

[TABLE]

which implies the conclusion.

Step 2

If $\widetilde{h}_{n}:=h_{n}+{\left(\lambda f_{n}^{\prime}(\omega_{n})\chi_{B(\xi_{n},\rho_{1,n})}-e^{U_{n}}\right)\varphi_{n}}+\sum_{j=0}^{2}c_{j,n}e^{U_{n}}Z_{j,n}$, then

[TABLE]

Since $\|h_{n}\|_{\varepsilon_{n}}\to 0$, $|Z_{i,n}|\leq 1$, and $\|\lambda f_{n}^{\prime}(\omega_{n})\chi_{B(\xi_{n},\rho_{1,n})}-e^{U_{n}}\|_{\varepsilon_{n}}\to 0$ by Lemma 4.1, it is sufficient to observe that $\|e^{U_{n}}\|_{\varepsilon_{n}}=O(1)$ and apply Step 1.

Step 3

There exists $\delta_{n}\ll\rho_{n}\leq\rho_{0,n}$ such that, up to a subsequence, $\|\varphi_{n}\|_{L^{\infty}(B(\xi_{n},\rho_{n}))}\to 0$ as $n\to+\infty$.

Let us consider the sequence $\Phi_{n}(y):=\varphi_{n}(\xi_{n}+\delta_{n}y)$, $y\in\frac{\Omega-\xi_{n}}{\delta_{n}}$. By (66) $\Phi_{n}$ satisfies

[TABLE]

We know that

[TABLE]

and, for $y\in B(0,\frac{\rho_{0,n}}{\delta_{n}})$, that

[TABLE]

In particular $\Phi_{n}$ and $\Delta\Phi_{n}$ are uniformly bounded in $B(0,\frac{\rho_{0,n}}{\delta_{n}})$. By standard elliptic estimates, we can find $\Phi_{0}\in{C(\mathbb{R}^{2})\cap H^{1}_{loc}(\mathbb{R}^{2})}$ and a sequence $R_{n}\to+\infty$, $R_{n}\leq\frac{\rho_{0,n}}{\delta_{n}}$, such that, up to a subsequence, $\|\Phi_{n}-\Phi_{0}\|_{L^{\infty}(B(0,R_{n}))}\to 0$. Moreover, $|\Phi_{0}|\leq 1$ and $\Phi_{0}$ is a weak solution to

[TABLE]

According to Proposition 4.2, we must have $\Phi_{0}=\kappa_{0}Z_{0}+\kappa_{1}Z_{1}+\kappa_{2}Z_{2}$, for some $\kappa_{i}\in\mathbb{R}$, $i=0,1,2$. Keeping in mind (58) and using that $e^{\bar{U}}\in L^{1}(\mathbb{R}^{2})$, we obtain

[TABLE]

for $i=0,1,2$. This implies $\kappa_{i}=0$, $i=0,1,2$. Then $\Phi_{0}\equiv 0$ and we get the conclusion with $\rho_{n}=\delta_{n}R_{n}$.

Step 4

Up to a subsequence, $\xi_{n}\to\overline{\xi}\in\Omega$ and $\varphi_{n}\to 0$ in $L^{\infty}_{loc}(\Omega\setminus\{\bar{\xi}\})$, as $n\to\infty$.

We know that $\varphi_{n}$ satisfies (66) in $\Omega$. Since $|\varphi_{n}|\leq 1$, $\|e^{U_{n}}\|_{L^{\infty}(\Omega\setminus B(\xi_{n},{\rho_{1,n}}))}\to 0$, $\|h_{n}\|_{L^{2}(\Omega\setminus B(\xi,\rho_{1,n}))}\to 0$, and $\|f^{\prime}_{n}(\omega_{n})\|_{L^{\infty}(\Omega\setminus B(\xi,\rho_{1,n}))}=O(1)$, by ellpitic estimates we find that $\varphi_{n}$ is bounded in $C^{0,\gamma}_{loc}(\overline{\Omega}\setminus\{\overline{\xi}\})$, for some $\gamma\in(0,1)$. Therefore, there exists $\varphi_{0}\in C(\overline{\Omega})\cap H^{1}_{0}(\Omega)$, such that $\varphi_{n}\to\varphi_{0}$ locally uniformly on $\overline{\Omega}\setminus\{\overline{\xi}\}$ and weakly in $H^{1}_{0}(\Omega)$. Noting that $\omega_{n}\to-u_{0}$ locally uniformly in $\overline{\Omega}\setminus\{\xi\}$ and that $f^{\prime}_{n}$ is even, we see that $\varphi_{0}$ satisfies $\Delta\varphi_{0}+f^{\prime}_{0}(u_{0})\varphi_{0}$ in $\Omega\setminus\{\bar{\xi}\}$. Actually, since $\varphi_{0},\Delta\varphi_{0}\in L^{\infty}(\Omega)$, $\varphi_{0}$ is a weak solution of $\Delta\varphi_{0}+f^{\prime}_{0}(u_{0})\varphi_{0}=0$ in $\Omega$. Then, the non-degeneracy of $u_{0}$ implies $\varphi_{0}\equiv 0$.

Step 5

$\|\varphi_{n}\|_{L^{\infty}(\Omega)}\to 0$.

By Step 4, we can find a sequence $\sigma_{n}\geq\rho_{2,n}$ such that $\|\varphi_{n}\|_{L^{\infty}(\Omega\setminus B(\xi_{n},\sigma_{n}))}\to 0$ as $n\to+\infty$, up to a subsequence. Then, it is sufficient to show that $\|\varphi_{n}\|_{L^{\infty}(A_{n})}\to 0$, where $A_{n}:=B(\xi_{n},\sigma_{n})\setminus B(\xi_{n},\rho_{n})$ and $\rho_{n}$ is as in Step 3. We can split $\varphi_{n}=\varphi_{n}^{(0)}+\varphi_{n}^{(1)}+\varphi_{n}^{(2)}+\varphi_{n}^{(3)}$, where

[TABLE]

with

[TABLE]

By the maximum principle

[TABLE]

Since

[TABLE]

we get that $|\varphi_{n}^{(1)}|\leq 2\psi_{n}$, where $\psi_{n}$ satisfies

[TABLE]

Lemma 3.15 implies $\|\psi_{n}\|_{L^{\infty}(A_{n})}\to 0$, hence $\|\varphi_{n}^{(1)}\|_{L^{\infty}(A_{n})}\to 0$. Finally, since $|A_{n}|$ is uniformly bounded, elliptic estimates (see Corollaries B.3 and B.4) give

[TABLE]

and

[TABLE]

Step 6

Conclusion of the proof.

By Step 5, we have that $\|\varphi_{n}\|_{H^{1}_{0}(\Omega)}=1-\|\varphi_{n}\|_{L^{\infty}(\Omega)}\to 1$. But (66) gives

[TABLE]

Then, we get a contadiction. ∎

As a consequence we have that $\pi^{\perp}L$ is invertible on $K_{\varepsilon}^{\perp}$.

Corollary 4.7

$\pi^{\perp}L:K_{\varepsilon}^{\perp}\mapsto K_{\varepsilon}^{\perp}$* is invertible.*

Proof.

This follows by standard Fredholm theory. Indeed, for any $\varepsilon>0$ the map $F(\varphi):=\pi^{\perp}(-\Delta)^{-1}(f^{\prime}(\omega_{\varepsilon})\varphi)$ defines a compact operator on $K_{\varepsilon}^{\perp}$ (in fact on $H^{1}_{0}(\Omega)$). Then $\pi^{\perp}L=Id_{K_{\varepsilon}^{\perp}}-F$ is a Fredholm operator of index $0.$ Proposition 4.6 implies that $\pi^{\perp}L$ is injective, hence it is invertible on $K_{\varepsilon}^{\perp}$. ∎

5 The reduction to a finite dimensional problem

This section is devoted to reduce the problem to a finite dimensional one. More precisely, we prove:

Proposition 5.1

There exist $\varepsilon_{0}>0$ and a map $(\varepsilon,\mu,\xi)\to\varphi_{\varepsilon,\mu,\xi}\in K_{\varepsilon}^{\perp}\cap L^{\infty}(\Omega)$ defined in $(0,\varepsilon_{0})\times\mathcal{U}\times B(\xi_{0},\sigma)$ and continuous with respect to $\mu$ and $\xi$, such that for some $D>0$

[TABLE]

and

[TABLE]

where the linear operator $L$ is defined in (19), the error term $R$ is defined in (17) and the quadratic term $N$ is defined in (18).

5.1 Estimates on $N(\varphi)$

For a function $\varphi\in H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$, let $N(\varphi)$ be defined as in (18), i.e.

[TABLE]

Let us estimate $\|N(\varphi)\|_{\varepsilon}$, where $\|\cdot\|_{\varepsilon}$ is defined as in (55). Let us define

[TABLE]

Lemma 5.2

There exists $D_{2}>0$ such that

[TABLE]

for any $\varphi_{1},\varphi_{2}\in\mathcal{B}_{\alpha}$.

Proof.

First, for any $x\in\Omega$ we can find $\theta_{1}=\theta_{1}(x)\in[0,1]$ such that

[TABLE]

where $\varphi_{3}:=\theta_{1}\varphi_{2}+(1-\theta_{1})\varphi_{1}$. Furthermore, there exists $\theta_{2}=\theta_{2}(x)$ such that

[TABLE]

Thus, we obtain

[TABLE]

Then, in order to conclude the proof, we shall bound $\|f^{\prime\prime}_{\varepsilon}(\omega_{\varepsilon}+\theta_{2}\varphi_{3})\|_{\varepsilon}$. Note that, there exists a universal constant $C_{0}>0$ such that

[TABLE]

By Remark 2.7 we have $\omega_{\varepsilon}=O(\beta)=O(\alpha^{-1})$. Since $|\varphi_{3}|\leq|\varphi_{1}|+|\varphi_{2}|\leq 2\alpha$, we get

[TABLE]

By convexity, we also have

[TABLE]

In $B(\xi,\rho_{1})$ we have $\omega_{\varepsilon}\geq c_{0}$ by Lemma 3.2, so that

[TABLE]

Clearly (71)-(73) yield the existence of a constant $C_{1}>0$ such that

[TABLE]

in $B(\xi,\rho_{1})$. Arguing as in Lemma 3.4 (see (46)) we get

[TABLE]

Lemma 3.6 and Lemma 3.7 yield

[TABLE]

Finally, thanks to Lemma 3.9, we know that

[TABLE]

Thanks to (74)-(76) we infer

[TABLE]

and the conclusion follows from (70). ∎

Remark 5.3

Applying Lemma 5.2 with $\varphi_{2}=0$, we obtain that

[TABLE]

for any $\varphi\in\mathcal{B}_{\alpha}$.

Remark 5.4

The proof of Proposition 5.2 and Lemma 3.9 also shows that

[TABLE]

for any $\varphi\in\mathcal{B}_{\alpha}$.

5.2 Proof of Proposition 5.1: a fixed point argument

Let us consider the operator

[TABLE]

on the space $X:=K_{\varepsilon}^{\perp}\cap L^{\infty}(\Omega)$, which is a Banach space with respect to the norm

[TABLE]

Let $D_{1}$ and $D_{0}$ be the constants defined in Proposition 3.13 and Proposition 4.6. Let us set

[TABLE]

Proposition 5.1 is an immediate consequence of the following result.

Proposition 5.5

There exists $\varepsilon_{0}>0$ such that, for any $\varepsilon\in(0,\varepsilon_{0})$, $\mu\in\mathcal{U}$, $\xi\in B(\xi_{0},\sigma)$, $\mathcal{T}$ has a fixed point $\varphi_{\varepsilon,\mu,\xi}\in E_{\varepsilon}$, which depends continuosly on $\mu$ and $\xi$.

Proof.

Since $E_{\varepsilon}$ is a closed subspace of $X$ and $\mathcal{T}$ depends continuously on $\mu$ and $\xi$, it is sufficient to verifry that

$\mathcal{T}$ maps $E_{\varepsilon}$ into itself. 2. 2.

$\mathcal{T}$ is a contraction, i.e. $||\mathcal{T}(\varphi_{1})-\mathcal{T}(\varphi_{2})||_{H^{1}_{0}(\Omega)}\leq\theta||\varphi_{1}-\varphi_{2}||_{H^{1}_{0}(\Omega)}$ for some positive constant $\theta<1$ and for any $\varphi_{1},\varphi_{2}\in E_{\varepsilon}$.

Then the conclusion follows by the contraction mapping theorem.

Step 1

$\mathcal{T}$* maps $E_{\varepsilon}$ into itself.*

Let us denote $C_{0}:=D_{0}(D_{1}+1)$. Take $\varphi\in E_{\varepsilon}$ and set

[TABLE]

If $\varepsilon$ is small enough, we have that $\alpha^{2}C_{0}\leq 1$, so that $E_{\varepsilon}\subseteq\mathcal{B}_{\alpha}$ (see (69)). By Proposition 3.13 and Remark 5.3 we get

[TABLE]

for any $\varphi\in E_{\varepsilon}$. Then, if we take $\varepsilon$ small enough so that ${C_{0}^{2}}D_{2}\alpha^{2}\leq 1$, we get that

[TABLE]

Since by definition

[TABLE]

we have by Proposition 4.6 that

[TABLE]

that is $\mathcal{T}(\varphi)\in E_{\varepsilon}$.

Step 2

$\mathcal{T}$* is a contraction mapping in $E_{\varepsilon}$.*

Let us take $\varepsilon$ small enough so that $D_{0}D_{2}C_{0}\alpha^{2}\leq\frac{1}{4}$ and $E_{\varepsilon}\subseteq\mathcal{B}_{\alpha}$. By Propositions 4.6 and 5.2 we have

[TABLE]

for any $\varphi_{1},\varphi_{2}\in E_{\varepsilon}$. Then, $\mathcal{T}$ is a contraction mapping on $E_{\varepsilon}$. ∎

6 The reduced problem: proof of Theorem 1.3 completed

Let $\varphi_{\varepsilon}:=\varphi_{\varepsilon,\mu,\xi}$ be as in Proposition 5.1. By (68), we can find $\kappa_{\varepsilon,i}=\kappa_{\varepsilon,i}(\mu,\xi)$, $i=0,1,2$ (which depend continuously on $\mu$, and $\xi$), such that

[TABLE]

Equivalently, setting $u_{\varepsilon}:=\omega_{\varepsilon}+\varphi_{\varepsilon},$

[TABLE]

Our aim is to find the parameter $\mu=\mu(\varepsilon)$ and the point $\xi=\xi(\varepsilon)$ so that the $\kappa_{\varepsilon,i}$’s are zero provided $\varepsilon$ is small enough.

Proposition 6.1

It holds true that

[TABLE]

and

[TABLE]

*as $\varepsilon\to 0$ uniformly with respect to $\mu\in\mathcal{U}$ and $\xi\in B(\xi_{0},\sigma).$ Here, the $a_{i,\varepsilon}$’s are continuous functions of $\mu$ and $\xi$ and $a_{i,\varepsilon}=O(\alpha^{2})$ uniformly for $(\mu,\xi)\in\mathcal{U}\times B(\xi_{0},\sigma)$. *

Proof.

Step 1

Let us prove that

[TABLE]

and

[TABLE]

First, since (67) gives $\|\phi\|_{L^{\infty}(\Omega)}=O(\alpha^{3})$, Proposition 3.13, Lemma 3.14, Remark 5.3 and Lemma 4.1 yield

[TABLE]

Recalling that

[TABLE]

by Lemma 4.4 and (58), we get (82) by testing equation (78) with $PZ_{i,n}$, $i=0,1,2$.

By Lemma 3.12, Remark 5.4, and Lemma 4.1, one has

[TABLE]

uniformly in $\Omega\setminus{B(\xi,\frac{\sigma}{2})}$. Then

[TABLE]

and we get (83) by standard elliptic estimates.

Step 2

Proof of (80).

Let $\widetilde{Z}_{\varepsilon}$ be the function defined in (59). We shall test equation (78) against $\widetilde{Z}_{\varepsilon}$. With the same arguments of the proof of Proposition 4.6 (Step 1), we obtain

[TABLE]

Moreover

[TABLE]

and

[TABLE]

By Lemma 3.4 and Lemma 3.8, we get

[TABLE]

Finally, we have that

[TABLE]

Then, testing (78) against $\widetilde{Z}_{\varepsilon}$ and using (82), one gets

[TABLE]

from which we get (80).

Step 3

Let us prove

[TABLE]

We multiply (79) and $\frac{\partial u_{\varepsilon}}{\partial x_{i}}$. Applying the Pohozaev identity (see e.g. [27, Proposition 2, Proof of Step 1]), we obtain

[TABLE]

Since $u_{\varepsilon}=0$ on $\partial\Omega$, the divergence theorem yields

[TABLE]

By (83), the definition of $u_{\varepsilon}$ and $\omega_{\varepsilon}$, Lemma 2.3, Lemma 3.10, we have

[TABLE]

on $\partial\Omega$. Thus, keeping in mind that $|\nabla v_{\varepsilon}|$, $|\nabla w_{\varepsilon}|$ and $|\nabla G_{\xi}|$ are uniformly bounded on $\partial\Omega$ (see Lemma (2.2) and (2.3)) and that $\frac{\partial u_{\varepsilon}}{\partial x_{i}}=\frac{\partial u_{\varepsilon}}{\partial\nu}\nu_{i}$, we obtain

[TABLE]

Applying the Pohozaev identity to $v_{\varepsilon}$ and arguing as in (86), we get that

[TABLE]

Integrating by parts and noting that $-\Delta\frac{\partial v_{\varepsilon}}{\partial x_{i}}=\lambda f^{\prime}_{\varepsilon}(v_{\varepsilon})\frac{\partial v_{\varepsilon}}{\partial x_{i}}$ in $\Omega$, we get

[TABLE]

This together with (87)-(88) gives

[TABLE]

Finally, (84) follows by (85)-(86) and (89).

Step 4

For $i=1,2$, $j=0,1,2$, we have

[TABLE]

For $i=1,2$ and $j=0,1,2$. Note that we have the identity

[TABLE]

Setting $\Psi_{ij}:=\delta_{ij}(Z_{0}+1)-\delta_{j0}Z_{i}-3Z_{i}Z_{j}$ and applying the divergence theorem, we find

[TABLE]

where in the last equality we used that

[TABLE]

for $|y|\geq\frac{\sigma}{\delta}$. By Lemma 2.6 we have

[TABLE]

for $y\in B(0,\frac{\sigma}{\delta})$. Using again (91), we get that

[TABLE]

Similarly, we have

[TABLE]

and (90) is proved.

Step 5

Proof of (81).

Let us set

[TABLE]

According to Step 4, we have $a_{i0,\varepsilon}=O(\alpha^{2})$ if $i=1,2$. Moreover the matrix $A=(a_{ij,\varepsilon})_{i,j\in\{1,2\}}$ is invertible and its inverse $A^{-1}=(a^{ij}_{\varepsilon})_{ij\in\{1,2\}}$ satisfies

[TABLE]

Then (81) follows by (84), just setting

[TABLE]

∎

It is important to point out that (81) cannot be considered a precise uniform expansion of $\kappa_{i,\varepsilon}$. Indeed, (80) and the rough (but difficult to improve) estimate $a_{i,\varepsilon}=O(\alpha^{2})$ yield only $\kappa_{0,\varepsilon}a_{i,\varepsilon}=O(\alpha^{5})$. Since $\delta\ll\alpha^{5}$ it is not possible to identify the leading term in the RHS of (81). However, it is clear that the term involving $\frac{\partial v_{\varepsilon}}{\partial x_{i}}$ becomes dominant when $\kappa_{0,\varepsilon}$ vanishes. This is enough for our argument.

**Proof of Theorem 1.3 completed

**

Proof.

Let us consider the vector field

[TABLE]

By construction, for any $\varepsilon>0$, $B_{\varepsilon}$ depends continuously on $\mu$ and $\xi$. Moreover, thanks to (80), (81) and Lemma 2.2, we have

[TABLE]

as $\varepsilon\to 0$, uniformly for $\mu\in\mathcal{U}$ and $\xi\in B(\xi_{0},\sigma)$. By assumption (A2), $\bar{B}$ has a $C^{0}$-stable zero at the point $(\mu_{0},\xi_{0})$, with $\mu_{0}=\sqrt{8}e^{-1}$. Then, for $\varepsilon$ small enough, there exist $\xi=\xi(\varepsilon)\to\xi_{0}$, $\mu=\mu(\varepsilon)\to\mu_{0}$ as $\varepsilon\to 0$ such that $B_{\varepsilon}(\mu(\varepsilon),\xi(\varepsilon))=0$. Clearly, this is equivalent to $\kappa_{i,\varepsilon,\mu(\varepsilon),\xi(\varepsilon)}=0$, $i=0,1,2$. That concludes the proof. ∎

Appendix Appendix A. The proof of Lemma 2.4

Proof.

The third equation in (28) allows to write $\delta$ as a function of $\alpha,\beta,\varepsilon,\mu,\xi$:

[TABLE]

and the second equation in (28) gives $\alpha$ as a function of $\beta,\varepsilon,\mu,\xi$:

[TABLE]

Then, (after a simple computation) it is sufficient to prove that there exists $\beta=\beta(\varepsilon,\mu,\xi)$ such that

[TABLE]

Now, we choose $\beta^{\varepsilon}:=2u_{0}(\xi)+\theta_{\varepsilon}(\xi,\mu)$ with $\|\theta_{\varepsilon}\|_{C^{0}(\overline{B(\xi_{0},\sigma)}\times\overline{\mathcal{U}})}$ so small that

[TABLE]

This is possible because of (22). With this choice we have $\frac{1}{\beta}=O\left(\eta^{-\frac{1}{\varepsilon}}\right)$. It is easy to show that (93) has a solution $\theta_{\varepsilon}$ because of a simple fixed point argument. Indeed (93) rewrites as $\theta_{\varepsilon}=\mathcal{T}(\theta_{\varepsilon})$ where $\mathcal{T}$ is a contraction mapping on the ball

[TABLE]

where $\rho_{\varepsilon}:=\rho\min\left\{\frac{1}{\varepsilon}\eta^{-\frac{1}{\varepsilon}},\|v_{\varepsilon}-u_{0}\|_{C^{0}(\overline{\Omega})}\right\}$ for some $\rho>0$ and $\rho_{\varepsilon}\to 0$ as $\varepsilon\to 0.$ Here we use the expression of $V_{\varepsilon,\alpha,\xi}(\xi)$ in (11) and (ii) of Lemma 2.2. ∎

Appendix Appendix B. A Stampacchia type estimate

In this section we prove domain-independent estimates for solutions of the Poisson equation $-\Delta u=f$, under Dirichlet boundary conditions, with $f\in L^{p}(\Omega)$ and $p$ approaching $1$. Our strategy is based on the Stampacchia method.

Lemma B.1** **([28], Lemma 4.1)

Let $\psi:\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$ be a nonincreasing function. Assume that there exist $M>0,\gamma>0$, $\delta>1$ such that

[TABLE]

Then $\psi(d)=0$, where $d=M^{\frac{1}{\gamma}}\psi(0)^{\frac{\delta-1}{\gamma}}2^{\frac{\delta}{\delta-1}}$.

Let $\Omega\subseteq\mathbb{R}^{2}$ be a bounded smooth domain. For any $q>1$, let $S_{q}(\Omega)$ be the Sobolev’s constant for the embedding of $H^{1}_{0}(\Omega)$ in $L^{q}(\Omega)$, namely

[TABLE]

It is known that $0<S_{q}(\Omega)<+\infty$ and that (see [26] Lemma 2.2)

[TABLE]

Theorem B.2

Let $\Omega$ be a bounded smooth domain. For $p>1$, $f\in L^{p}(\Omega)$, the unique solution $u\in H^{1}_{0}(\Omega)$ of the equation $-\Delta u=f$ satisfies

[TABLE]

Proof.

We want to apply the previous lemma to the function

[TABLE]

For any $k>0$, let us consider the function

[TABLE]

Note that $v_{k}\in H^{1}_{0}(\Omega)$ and $|\nabla v_{k}|=|\nabla u|\chi_{A_{k}}$. If we test the equation against $v_{k}$ we get

[TABLE]

For any $q\in(1,p)$ Hölder’s inequality gives

[TABLE]

By Sobolev’s inequality, we have that

[TABLE]

By (94)-(96), we have

[TABLE]

Now, for any $h>k$, we have that $A_{h}\subseteq A_{k}$ and $v_{k}\geq(h-k)$ in $A_{h}$, hence

[TABLE]

In conlcusion, we find

[TABLE]

or, equivalently,

[TABLE]

Then, we are in position to apply Lemma B.1 to $\psi$ with $M=S_{\frac{q}{q-1}}(\Omega)^{-\frac{2q}{q-1}}\|f\|_{L^{p}}^{\frac{q}{q-1}}$, $\gamma=\frac{q}{q-1}$, and $\delta=\frac{p-q}{p(q-1)}$. For this, we need to impose that $\delta=\frac{p-q}{p(q-1)}$, that is $q<\frac{2p}{p+1}$. Note that $1<\frac{2p}{p+1}<p$. According to Stampacchia’s Lemma, we have

[TABLE]

This implies that

[TABLE]

This is true for any choice of $q\in(1,\frac{2p}{p+1})$. If we take for example $p$ the midpoint of $(1,\frac{2p}{p+1})$, that is $q=\frac{1}{2}+\frac{p}{p+1}=\frac{3p+1}{2(p+1)}$, then we find that

[TABLE]

and we get the conclusion. ∎

Corollary B.3

Given $K>0$ and $p>1$, there exists a constant $C=C(K,p)$ such that, for any domain $\Omega\subseteq\mathbb{R}^{2}$ with $|\Omega|\leq K$ and any $f\in L^{p}(\Omega)$ the unique solution $u\in H^{1}_{0}(\Omega)$ of $-\Delta u=f$ satisfies

[TABLE]

Corollary B.4

Given $K>0$, there exist $p_{0}=p_{0}(K)$ and $C=C(K)$ such that, for any $1<p<p_{0}$, any domain $\Omega\subseteq\mathbb{R}^{2}$ with $|\Omega|\leq K$, and any $f\in L^{p}(\Omega)$, the unique solution $u\in H^{1}_{0}(\Omega)$ of $-\Delta u=f$ satisfies

[TABLE]