On the Spectrum of a Nonlinear Planar Problem

www.elsevier.com/locate/anihpc

On the spectrum of a nonlinear planar problem

Francesca Gladiali

, Massimo Grossi

aStruttura dipartimentale di Matematica e Fisica, Università di Sassari,via Vienna 2, 07100 Sassari, Italy bDipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro 2, 00185 Roma, Italy

Received 28 March 2007; received in revised form 24 October 2007; accepted 25 October 2007 Available online 1 November 2007

Abstract

We consider the eigenvalue problem

⎧⎨

⎩

−v=λμe^uλv inΩ, v_∞=1

v=0 on∂Ω,

(0.1)

whereΩis a bounded smooth domain ofR²,λ >0 is a real parameter andu_λis a solution of −u_λ=λe^uλ inΩ,

u_λ=0 on∂Ω such thatλ

Ωe^uλ→8πasλ→0. In this paper we study the asymptotic behavior of the eigenvaluesμof (0.1) asλ→0. Moreover some explicit estimates for the four first eigenvalues and eigenfunctions are given.

Other related results as the Morse index of the solutionu_λwill be proved.

©2007 Elsevier Masson SAS. All rights reserved.

1. Introduction

Let us consider the Gelfand problem, −u=λe^u inΩ,

u=0 on∂Ω, (1.1)

whereΩ is a smooth bounded domain ofR²andλ >0 is a real parameter.

Eq. (1.1) has many applications. For example it arises in the contest of the statistical Mechanics as done in [5,6]

(see also [17,9] and references therein).

Another interesting field where (1.1) appears is in the Chern–Simon–Higgs model (see for example [23] and the references therein).

✩ Supported by M.I.U.R., project “Variational methods and nonlinear differential equations”.

* Corresponding author.

E-mail addresses:[email protected] (F. Gladiali), [email protected] (M. Grossi).

0294-1449/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2007.10.004

Throughout the paper we will consider a solutionu_λto (1.1) satisfying λ

Ω

e^uλ→8π asλ→0. (1.2)

The behavior of solutions of (1.1) satisfying (1.2) was largely studied by many authors. We can just mention here the papers [8,11,12,20,22,23] as well as many others.

Condition (1.2) corresponds to study the so-called one point blowing-up solution, i.e. solutions whose maximum is achieved exactly at one point where the solution goes to+∞.

For this class of solutions, denoted byu_λ, we consider the eigenvalue problem −v=λμe^uλv inΩ,

v_∞=1,

v=0 on∂Ω

(1.3) and we study some properties of the eigenvaluesμand of the corresponding eigenfunctions.

Problem (1.3) comes out from the eigenvalue problem related to the second derivative of the functional F (u)=1

2

Ω

|∇u|²−λ

Ω

e^u

in the Hilbert spaceH₀¹(Ω). The study of the spectrum ofFis crucial to calculate the Morse index of the solutionu_λ. One of the result of this paper will be the computation of the Morse index of the solutionu_λin some special cases.

Another interesting problem linked to (1.3) is the classical problem of the nodal line of the second eigenfunction. It was proved by A. Melas [19] that if we consider the second eigenfunction of the Laplace operator in a planar convex domains then its nodal line touches the boundary. This result is largely open for eigenfunctions of higher order. In this paper we describe some properties of the nodal line of the eigenfunctions to (1.3). For example we show that, if Ω is convex, the nodal line of the second and third eigenfunction touches the boundary. On the other hand, without any assumption onΩ, we prove that the nodal line of the fourth eigenfunction does not touch the boundary ofΩ. Moreover, the asymptotic behavior of these eigenfunctions is described.

A crucial tool in the study the eigenvalue problem (1.3) is given by the following “limit” problem, −v=_(1+|^μ_x^∞_|2/8)²v inR²,

v∈L^∞(R²). (1.4)

Roughly speaking, the eigenvalues μ_λ of (1.3) converge to the eigenvalues μ_∞ of the problem (1.4) asλ→0 and the same happens for the corresponding eigenfunctions (up to a scale argument). This will be stated precisely in Section 11.

The eigenfunctions of problem (1.4) can be explicitly computed using theLegendrefunction. In this way one can see that any eigenvalue has multiplicity greater than 1 and the corresponding eigenfunctions can be divided in two classes.

The first one is given by nonradial functions which go to zero at infinity and the second one is given by radial functions which converge to a nonzero constant at infinity. These two types of limiting eigenfunctions give rise to eigenfunctions of problem (1.3) which behave differently.

In this way we give some “global” results about the spectrum of (1.3) but the most important aim of this paper is to study with great attention the first fourth eigenvalues.

For example we will see that the second and the third eigenfunctions of (1.3) look like the nonradial eigenfunctions related to the eigenvalueμ_∞=1 and the fourth eigenfunction of (1.3) looks like the radial eigenfunction related again toμ_∞=1.

In this case we will compute asymptotic expansions for the eigenvalues and related eigenfunction. Note that, even in the case of the first eigenvalue, more work is needed.

The asymptotic estimate on the second and third eigenvalue enables to derive some results on the Morse index of the solution. The first one says that ifΩ is a convex domain then the Morse index of the solutionu_λof (1.1) is exactly 1 (see Corollary 2.8). Moreover, for a general domain we will derive that the Morse index of the solutionu_λ is at most 2. This last result appears differently from singular problems in higher dimensions (see [1] for example).

Finally, we observe that our results have some similarities with the corresponding in [16], where was considered a perturbed critical Sobolev exponent inR^N forN 3. But since inR² we do not have the Sobolev Embedding Theorem and some orthogonality properties of the eigenfunctions, here the problem seems harder.

The paper is organized as follows: in Section 2 we state our main results; in Section 3 we recall some known facts about problem (1.1); in Section 4 we consider the first eigenvalue and the first eigenfunction and in Section 5 we give an important estimate on the second eigenvalue; in Section 6 we study the behavior of the second eigenfunction;

Section 7 is devoted to the third eigenvalue and the third eigenfunction; in Section 8 the asymptotic behavior of the second and third eigenvalues of (1.1) is proved and, using a result of [4], we have that, for a convex domain, the Morse index of the solutionuλ is 1; in Section 9 we consider the nodal region for the second and third eigenfunctions; in Section 10 we treat the case of the fourth eigenfunction; in Section 11 finally we get the asymptotic behavior of the spectrum of problem (1.1).

2. Statement of the results

LetG(x, y)be the Green’s function of−inΩwith Dirichlet boundary conditions. Then G(x, y)= − 1

2πlog|x−y| +H (x, y), (2.1)

whereH (x, y)is the regular part of the Green function. LetR(x)=H (x, x)be the Robin function ofΩ.

Let us consider the solutionu_λof (1.1) satisfying (1.2). For such a solution we consider the eigenvalue problem −v=λμe^uλv inΩ,

v∞=1,

v=0 on∂Ω.

(2.2) It is well known that problem (2.2) admits a sequence of eigenvalues μ1,λ< μ2,λμ3,λ· · ·. Let vi,λ be the eigenfunction corresponding to the eigenvalueμi,λ, i.e.vi,λsolves

⎧⎨

⎩

−vi,λ=λμi,λe^uvi,λ inΩ, vi,λ∞=1,

v_i,λ=0 on∂Ω.

(2.3) In order to state our results we recall that ifx_λis a maximum point ofu_λ, i.e. a point such thatu_λ(x_λ)= u_λ∞, we have thatx_λconverges to a pointx₀∈Ω (see Section 3 for details). Let

δ_λ= 1 λe^uλ∞

1/2

(2.4) andv˜_i,λ(x)=v_i,λ(δ_λx+x_λ)be the rescaled eigenfunction defined in the domainΩ_λ=¹_{δ λ}(Ω−x_λ).

We start with some results concerning the eigenvalues and the eigenfunctions of (2.2).

Theorem 2.1.Letu_λbe a solution of (1.1)which satisfies(1.2), and letμ_1,λbe the first eigenvalue of (2.3)andv_1,λ be the first eigenfunction. Then

μ_1,λ= − 1 2 logλ

1+o(1)

; (2.5)

v_1,λ

μ_1,λ→8π G(x, x0) inC_loc¹ Ω¯ \ {x0}

λ→0; (2.6)

˜

v1,λ→1 inC_loc² (R²) λ→0. (2.7)

Theorem 2.2.In the same assumption of Theorem2.1, we have

˜

v_i,λ(x)→ ˜v_i=a₁ⁱx₁+a₂ⁱx₂

8+ |x|² asλ→0 (2.8)

inC_loc¹ (R²)fori=2,3and some vectorsaⁱ=(a₁ⁱ, a₂ⁱ) =0, v_i,λ(x)

δ_λ →2π 2 k=1

a_kⁱ∂G(x, x₀)

∂y_k asλ→0 (2.9)

inC_loc¹ (Ω¯ \ {x₀}), fori=2,3, where(a₁ⁱ, a₂ⁱ)are the same as in(2.8).

Theorem 2.3.Letc₁c₂be the eigenvalues of the Hessian matrixD²R(x₀)of the Robin function atx₀. Then 1−μ_i,λ

δ²_λ →24π ηi asλ→0 (2.10)

fori=2,3. Moreoverη₂=c₁andη₃=c₂, and the vectoraⁱof (2.8)are the eigenvectors corresponding toc_i−1. Next we study some properties of the nodal line of the eigenfunctions. Let us recall that the nodal set ofv_i,λis defined as

N_i,λ=

x∈Ω: v_i,λ(x)=0

. (2.11)

Theorem 2.4.In the same assumptions of Theorem2.1, we have:

(i) ifΩ be convex, thenNi,λ∩∂Ω = ∅, fori=2,3withλsmall enough;

(ii) the eigenfunctionsv_i,λ(x),i=2,3, have only two nodal regions, ifλis small enough.

Now we consider the fourth eigenvalue, getting the following results:

Theorem 2.5.In the same hypothesis of Theorem2.1we have

˜

v_4,λ→ ˜v₄=b8− |x|²

8+ |x|² inC¹_loc(R²)asλ→0; (2.12)

v_4,λ(x)logλ→4π bG(x, x0) inC_loc¹ Ω¯ \ {x₀}

asλ→0; (2.13)

1−μ_4,λ= − 1 logλ

c₁+o(1)

, (2.14)

whereb∈R,b =0,c0=^π₆ andc1=²⁽¹⁻_c0^{4π )}<0.

Theorem 2.6.The eigenvalueμ_4,λis simple and the corresponding eigenfunctionv_4,λhas only two nodal regions if λis small enough. Moreover the closure of the nodal set ofv_4,λdoes not touch the boundary.

Corollary 2.7.Letxλbe the maximum point ofuλinΩ, andlimxλ=x0∈Ω. Then ifx0is a nondegenerate critical point of the Robin functionR(x)ofΩ, denoting bym(x₀)the Morse index ofx₀as a critical point ofR(x), we find that the Morse index ofu_λis equal tom(x₀)+1.

The previous result enable us to compute the Morse index of the solutionuλ. We recall that the Morse index of a solutionuλis the number of eigenvalueμless than 1.

Corollary 2.8.LetΩ be a domain ofR². Then, (i) the Morse index ofu_λinΩis1or2;

(ii) ifΩ is a convex set then the Morse index ofuλinΩis exactly one.

Our final result concerns the convergence of thewholespectrum. A crucial role is played by the limit problem (1.4).

In order to state the precise result we need to introduce long and noisy notations. For this reason we prefer to state the results in Section 10. We just say that it will be proved that the whole spectrum converges to the corresponding one of the “limit problem” and an analogous convergence holds for the related eigenfunctions.

3. Preliminaries and known results Let us recall the following known facts:

Theorem 3.1.Letu_λbe a solution of (1.1)satisfying(1.2). Then ifx_λis a point such thatu_λ(x_λ)= u_λ∞, we have thatx_λconverges to a pointx₀∈Ω such that

∇R(x0)=0, (3.1)

uλ(x)→8π G(x, x0) inC_loc¹ Ω¯\ {x0}

, (3.2)

uλ(x)−log e^uλ(xλ)

(1+¹₈λe^uλ(xλ)|x−x_λ|²)²

C inΩ,¯ (3.3)

uλ∞= −2 logλ+C0−8π R(x0)+o(1) asλ→0, (3.4) whereC₀=2 log 8andR(x₀)=H (x₀, x₀).

Proof. Estimates (3.1) and (3.2) are proved in [20] while estimate (3.3) is proved in [15] using a result of [18]. In [15]

it is also proved (3.4). 2

Letδ_λbe as in (2.4). Then, from (3.4) it follows thatδ²_λ→0. Considering the rescaled function

˜

u_λ(x)=u(δ_λx+x_λ)− u_λ∞ (3.5)

forx∈Ω_λ=¹_{δ λ}(Ω−x_λ), the estimate (3.3) gives the following

˜

u_λ(x)C+log 1

(1+ |x|²/8)² inΩ_λ, (3.6)

whereC >0.

Theorem 3.2.Every solutionU∈C²(R²)of the problem −u=e^u inR²,

R²e^u<∞, (3.7)

is given by

U_δ,y(x)=log 8δ

(δ+ |x−y|²)² (3.8)

for any(δ, y)∈R⁺×R². Proof. See Chen and Li [10]. 2

In the sequel we write U (x)=U_8,0=log 1

(1+ |x|²/8)².

Theorem 3.3.Letv∈C²(R²)be a solution of the following problem −v=_(1+|x¹_|2/8)²v inR²,

v∈L^∞(R²).

(3.9) Then

v(x)= 2 i=1

aixi

8+ |x|²+b8− |x|²

8+ |x|² (3.10)

for somea_i, b∈R.

Proof. See [11] or also [13] for a more detailed proof. 2

Lemma 3.4.LetΩ be a smooth bounded domain of R². For anyy∈Ω we have

∂Ω

(x−y)·ν(x) ∂G(x, y)

∂νx

2

dσ_x= 1

2π, (3.11)

∂Ω

ν_i(x) ∂G(x, y)

∂νx

2

dσ_x= −∂R(y)

∂yi

, (3.12)

∂²R(y)

∂y_i∂y_j = −2

∂Ω

∂G(x, y)

∂x_i

∂

∂y_j

∂G(x, y)

∂ν_x

dσ_x, (3.13)

2

∂Ω

(x−y)·ν_x∂G

∂νx

(x, y) ∂²G

∂yi∂νx

(x, y) dσ_x= −∂R

∂yi

(y), (3.14)

∂Ω

∂²G

∂ν_x∂y_i(x, y) dσ_x=0 (3.15)

fori, j=1,2.

Proof. See [15] for the proof of (3.11)–(3.13). Let us prove (3.14). Differentiating (3.11) we obtain 2

∂Ω

(x−y)·ν_x∂G

∂νx

(x, y) ∂²G

∂yi∂νx

(x, y) dσ_x=

∂Ω

ν_i ∂G

∂νx

(x, y) 2

dσ_x.

The claim follows from (3.12).

To prove (3.15) is sufficient to note that

∂Ω

∂G

∂ν_x(x, y) dσ_x=

Ω

G(x, y) dx≡ −1

and differentiate. 2

Lemma 3.5.Letuλbe a solution of (1.1). Then the functionu˜λ:Ωλ=(Ω−xλ)/δλ→R,

˜

u_λ(x)=u_λ(δ_λx+x_λ)− u_λ (3.16)

verifies

˜

uλ(x)→log 1

(1+ |x|²/8)² inC_loc² (R²). (3.17)

Proof. The result is standard (see [15] for example). 2 4. On the first eigenvalue and the first eigenfunction

Letμ1,λbe the first eigenvalue of problem (2.2) andv1,λthe corresponding eigenfunction which solves

⎧⎨

⎩

−v1,λ=λμ1,λe^uλv1,λ inΩ, v1,λ∞=1,

v_1,λ=0 on∂Ω.

(4.1) The first eigenvalue is given by the classical (Rayleigh–Ritz) variational formula, namely

μ1,λ= inf

v∈H₀¹(Ω) v ≡0

Ω|∇v|²dx λ

Ωe^uλv²dx. (4.2)

Lemma 4.1.Letμ_1,λbe as stated before. Thenμ_1,λ→0asλ→0.

Proof. We want to estimateμ_1,λusing formula (4.2). Consider the functionu_λ∈H₀¹(Ω). Then, by (1.1) μ_1,λ

Ω|∇u_λ|² λ

Ωe^uλu²_λ =

Ωe^uλu_λ

Ωe^uλu²_λ. (4.3)

Using the rescaled functionu˜_λ(y)=u_λ(δ_λy+x_λ)− u_λ∞we have λ

Ω

e^uλu_λ=λ

Ω

e^uλ

u_λ− u_λ∞

+λu_λ∞

Ω

e^uλ (4.4)

while λ

Ω

e^uλu²_λ=λ

Ω

e^uλ

u_λ− u_λ∞2

+λu_λ²_∞

Ω

e^uλ+2λu_λ∞

Ω

e^uλ

u_λ− u_λ∞

. (4.5)

Inserting (4.4) and (4.5) into (4.3), and then rescaling, we get μ_1,λ

Ωλe^u˜λu˜_λ+ u_λ∞λ

Ωe^uλ

Ωλe^u˜λu˜²_λ+2u_λ∞

Ωλe^u˜λu˜_λ+ u_λ²_∞λ

Ωe^uλ. (4.6)

By the estimate (3.6), we see that e^u˜λu˜_λCC−log(1+ |y|²/8)²

(8+ |y|²)² ; (4.7)

e^u˜λu˜²_λC(C−log(1+ |y|²/8)²)²

(8+ |y|²)² . (4.8)

By (4.7) and (4.8), we can pass into the limit into (4.6) getting μ_1,λ C₁+o(1)+ u_λ∞(8π+o(1))

C2+o(1)+2uλ∞(C1+o(1))+(8π+o(1))uλ²_∞= 1 u_λ∞

1+o(1)

(4.9) whereC₁=

R²e^UU= −16πandC₂=

R²e^UU²=64π. The claim follows using (3.4). 2

Lemma 4.2.Letμ_1,λandv_1,λbe as stated before and letv˜_1,λ(x)=v_1,λ(δ_λx+x_λ). Thenv˜_1,λ→cinC_loc² (R²), where c =0is a constant.

Proof. It is easy to see thatv˜_1,λsatisfies the equation

⎧⎨

⎩

−v˜_1,λ=μ_1,λe^u˜λv˜_1,λ inΩ_λ,

˜v_1,λ∞=1,

˜

v_1,λ=0 on∂Ω_λ,

(4.10) whereΩ_λ=(Ω−x_λ)/δ_λ. From (3.6) the right-hand side of equation (4.10) is bounded inL^∞. Moreover˜v_1,λ∞=1 implies that|∇ ˜v1,λ|is bounded inL²(R²). Then using the standard elliptic regularity theory,v˜1,λ→v1inC²_loc(R²), wherev1satisfies

v₁=0 inR², (4.11)

and since˜v_1∞=1 we infer thatv₁is a constant. We want to show thatv₁ ≡0. Letz_λbe the points ofΩ_λsuch that

˜

v_1,λ(z_λ)=1. Ifv₁≡0 thenz_λshould go to the infinity. So let us consider ˆ

v1,λ= ˜v1,λ

x

|x|²

, and

ˆ uλ= ˜uλ

x

|x|²

.

Thenvˆ_1,λsatisfies the equation

−vˆ_1,λ= 1

|x|⁴μ_1,λe^uˆλ(x)vˆ_1,λ(x) where (again by (3.3))

1

|x|⁴μ1,λe^uˆλ(x)μ1,λ

1

|x|⁴

C 1+1/(8|x|²)

Cμ_1,λ 64|x|⁴

|x|⁴(8|x|²+1)²64Cμ1,λ→0.

Moreoverˆv1,λ∞1 andvˆ1,λ→0 inC_loc² (R²\ {0}), so thatvˆ1,λ→0 inL²(B1(0)). Since the capacity of one point is zero, we can apply the regularity theory tovˆ_λ(see Theorem 8.17 in [14]) observing that

ˆv_1,λL^∞(B₁

2

(0))Cˆv_1,λL²(B1(0))→0.

This gives a contradiction sinceˆv_λL^∞(B₁

2

(0))= ˆv_λ(z_λ)=1. 2 Remark 4.3.In Lemma 4.5 we will show thatc=1.

Lemma 4.4.Letμ_1,λbe the first eigenvalue of problem(4.1). Then μ_1,λ= − 1

2 logλ(1+o(1)).

Proof. Multiplying Eq. (4.1) foru_λand integrating, we get

Ω

∇v_1,λ· ∇u_λdx=λμ_1,λ

Ω

e^uλu_λv_1,λdx; (4.12)

while using Eq. (1.1) we get

Ω

∇u_λ· ∇v_1,λdx=λ

Ω

e^uλv_1,λdx. (4.13)

Then λ

Ω

e^uλv_1,λdx=λμ_1,λ

Ω

e^uλu_λv_1,λdx.

Rescaling both sides we have

Ωλ

e^u˜λv˜_1,λdy=μ_1,λ

Ωλ

e^u˜λu˜_λv˜_1,λdy+μ_1,λu_λ∞

Ωλ

e^u˜λv˜_1,λdy. (4.14)

Using estimate (3.6) and˜v1,λ_∞=1, we can pass to the limit in (4.14) and then

R²

e^{U (y)}v₁(y) dy+o(1)=μ_1,λ

Ωλ

e^u˜λ(y)u˜_λ(y)v˜_1,λ(y) dy+μ_1,λu_λ∞

R²

e^{U (y)}v₁(y) dy+o(1)

.

Again by estimate (4.7) and the boundedness ofv1,λ∞we have 8π c+o(1)=μ_1,λ

R²

e^{U (y)}U (y)v₁(y) dy+o(1)

+μ_1,λu_λ∞

8π c+o(1)

=μ1,λ

−16π c+o(1)

+μ1,λuλ∞

8π c+o(1) .

Passing to the limit as λ→0, we infer that limλ→0μ_1,λu_λ∞=1 and the lemma is proved using the esti- mate (3.4). 2

Lemma 4.5.Letv_1,λandμ_1,λbe as stated before. Then v_1,λ(x)

μ_1,λ →8π G(x, x0) inC_loc¹ Ω¯\ {x0}

asλ→0. (4.15)

Proof. Letx∈ ¯Ω\ {x₀}. Using the Green’s identity formula we have from (4.1), v_1,λ(x)

μ_1,λ =λ

Ω

G(x, y)e^uλ(y)v1,λ(y) dy

=λG(x, x0)

Ω

e^uλ(y)v1,λ(y) dy+λ

Ω

G(x, y)−G(x, x0)

e^uλ(y)v1,λ(y) dy

=G(x, x0)

˜ Ω_λ

e^u˜λ(y)v˜1,λ(y) dy+I1,λ

=8π cG(x, x0)+o(1)+I_1,λ. (4.16)

The passage into the limit is done by using the estimate (3.6) and|v1,λ|1.

To prove (4.15), we have to show thatI_1,λ→0. We can chooseρ >0 such thatB_ρ(x₀)⊂Ωandx /∈B_ρ(x₀). Then I_1,λ=λ

Ω

G(x, y)−G(x, x₀)

e^uλ(y)v_1,λ(y) dy

=λ

Ω\B_ρ(x₀)

G(x, y)−G(x, x₀)

e^uλ(y)v_1,λ(y) dy+λ

B_ρ(x₀)

G(x, y)−G(x, x₀)

e^uλ(y)v_1,λ(y) dy.

By estimate (3.3) and (3.4) we have λe^uλ(y)v1,λ(y) Cλ

(cλ+¹₈|y−x_λ|²)².

Recalling thatx_λ→x₀we have that|y−x_λ|^ρ₂ inΩ\B_ρ(x₀), ifλis small enough. Hence we get λ

Ω\B_ρ(x₀)

G(x, y)−G(x, x₀)

e^uλ(y)v_1,λ(y) dy Cλ (cλ+ρ²/32)²

Ω\B_ρ(x₀)

G(x, y)−G(x, x₀) dy

Cλ

(cλ+ρ²/32)². (4.17)

Choosingρ=λ^k withk <¹₄ we get λ

Ω\B_ρ(x₀)

G(x, y)−G(x, x0)

e^uλ(y)v1,λ(y) dy→0.

On the other hand we have λ

Bρ(x0)

G(x, y)−G(x, x₀)

e^uλ(y)v_1,λ(y) dy sup

y∈Bρ(x₀)

G(x, y)−G(x, x₀)λ

Bρ(x0)

e^uλv_1,λdy→0

because x /∈B_ρ(x₀)andλe^uλ(y)v_1,λ(y)∈L¹(Ω). In this way we have that I_1,λ→0 and from estimate (4.16) we get (4.15).

The same proof applies for the derivatives ofv_1,λ.

Now we prove thatc=1. We already know from Lemma 4.2 thatc =0. Letz_λ∈Ω such thatv_1,λ(z_λ)=1. This can be done since, by the definition ofv1,λ, we havev1,λ>0 inΩ. Up to a subsequencezλ→z∈ ¯Ω. Ifz =x0using equation (4.15), we have

1

μ_1,λ=v_1,λ(z_λ)

μ_1,λ →8π cG(z, x0) (4.18)

and this is not possible since the left-hand side goes to infinity while the right-hand side is bounded.

Hence we have that z=x₀. We have the following alternative: either |z_λ−x_λ|> δ_λR for any R >0 or z_λ∈ B_δλR(x_λ)forR >0 andλsufficiently small.

Case 1. We consider first the case where |z_λ−x_λ|> δ_λR for any R >0. Set w_λ(x)=v_1,λ(r_λx+x_λ)−γ_λ where γ_λ=_2π¹ μ_1,λlog_r¹

λ

Ωλe^u˜λ(y)v˜_1,λ(y) dy andr_λ= |z_λ−x_λ|. The functionw_λ(x)is defined in the set Ωˆ_λ= (Ω−xλ)/rλ. Sincezλ→x0we have thatrλ→0 asλ→0 and soΩˆλ→R².

Using the Green’s representation formula we can write w_λ(x)=μ_1,λ

Ωλ

G(r_λx+x_λ, δ_λy+x_λ)e^u˜λ(y)v˜_1,λ(y) dy−γ_λ

and by the standard decomposition of the Green’s function we get w_λ(x)=μ_1,λ

Ω_λ

1

2πlog 1

|rλx−δλy|e^u˜λ(y)v˜_1,λ(y) dy−γ_λ+μ_1,λ

Ω_λ

H (r_λx+x_λ, δ_λy+x_λ)e^u˜λ(y)v˜_1,λ(y) dy

= 1 2πμ1,λ

Ω_λ

log 1

|x−(δ_λ/r_λ)y|e^u˜λ(y)v˜1,λ(y) dy+μ1,λ

Ω_λ

H (r_λx+x_λ, δ_λy+x_λ)e^u˜λ(y)v˜1,λ(y) dy

=μ1,λ8π c log 1

|x|+H (x0, x0)+o(1)

, (4.19)

where we used (3.6), and the boundedness of H (r_λx+x_λ, δ_λy+x_λ)becauser_λx+x_λ is an interior point ofΩ_λ. Hence we obtain

w_λ(x)

μ_1,λ →w(x)=8π c log 1

|x|+R(x0)

inCloc

R²\ {0}

. (4.20)

The same can be shown for the derivatives ofw_λto derive the convergence inC_loc¹ (R²\ {0}).

This gives us a contradiction since we have a sequence of pointszˆ_n=(z_n−x_n)/r_nsuch that∇w_n(zˆ_n)=0, which converge to a pointzˆsuch that|ˆz| =1 and∇w(z)ˆ =0. This is a contradiction with (4.20).

Case 2. Here we assume that z_λ ∈B_δλR(x_λ) for some R >0. Let z˜_λ =(z_λ−x_λ)/δ_λ. Then z˜_λ ∈B_R(0) and

˜

v1,λ(z˜λ)=1. Reasoning as in the proof of Lemma 4.2 we have that v˜1,λ→c uniformly in B2R(0) and since

˜

v1,λ(z˜λ)=1 this implies thatc=1. 2

Remark 4.6.We observe here that the proof of Lemma 4.5 implies that the maximum points ofv_1,λare inside the ballB_δλR(x_λ)for someR >0 and hence they converge tox₀.

Proof of Theorem 2.1. This is derived from Lemmas 4.2, 4.4 and 4.5. 2 5. Estimates for the second eigenvalue

Lemma 5.1.For any eigenfunctionv_i,λwe have the following integral identity

∂Ω

∂u_λ

∂x_j

∂v_i,λ

∂ν dσ_x=λ(1−μ_i,λ)

Ω

e^uλv_i,λ∂u_λ

∂x_j dx (5.1)

forj=1,2.

Proof. Differentiating Eq. (1.1) with respect toxj we get

−∂u_λ

∂x_j =λe^uλ∂u_λ

∂x_j inΩforj=1,2. (5.2)

Multiplying (5.2) byv_i,λand integrating we get

Ω

∇ ∂uλ

∂xj

· ∇v_i,λdx=λ

Ω

e^uλ∂uλ

∂xj

v_i,λdx (5.3)

while multiplying Eq. (2.3) by ^∂u_∂x^λ

j we have

Ω

∇v_i,λ· ∇ ∂u_λ

∂x_j

dx−

∂Ω

∂u_λ

∂x_j

∂v_i,λ

∂ν dσ_x=λμ_i,λ

Ω

e^uλv_i,λ∂u_λ

∂x_j. (5.4)

Using (5.3) and (5.4) we get (5.1). 2

Letu_λbe a solution of (1.1) satisfying (1.2), and letx_λ∈Ωsuch thatu_λ(x_λ)= u_λ∞. By Theorem 3.1x_λ→x0∈ Ω, hence there existsρ >0 such thatB(x_λ,2ρ)⊂Ω. LetΦ˜∈C₀^∞(B(0,2ρ))such thatΦ˜=1 inB(0, ρ); 0Φ˜1 inB(0,2ρ)and let

Φ(x)= ˜Φ(x−x_λ). (5.5)

Proposition 5.2.We have

μ_2,λ1+Cδ_λ², (5.6)

μ_2,λ→1. (5.7)

Proof. We estimate the second eigenvalue using again the variational formula μ_2,λ= inf

v∈H₀¹(Ω), v =0, v⊥v1,λ

Ω|∇v|² λ

Ωe^uλv². (5.8)

To this end letψ₁(x)=^∂u_∂x^λ₁(x)andv=Φψ₁+a_1,λv_1,λ. We take a_1,λ= −λ

Ωe^uλΦψ₁v_1,λ λ

Ωe^uλv_1,λ² = −N1,λ

D1,λ

(5.9) so thatv⊥v_1,λinH₀¹(Ω).

Step 1: Here we show thata_1,λ=o(1). We estimateD_1,λin (5.9) as follows D_1,λ=λ

Ω

e^uλv_1,λ² =

Ωλ

e^u˜λv˜_1,λ² =8π+o(1).

This implies thatD_1,λ7π >0 ifλis small enough. We only have to prove thatN_1,λ=o(1). To do this, we observe that, sinceΦ=1 onB_ρ(x_λ),

N_1,λ=λ

Ω

e^uλ∂u_λ

∂x₁v_1,λΦ dx=λ

Ω\B_ρ(x_λ)

e^uλ∂u_λ

∂x₁v_1,λΦ dx+λ

B_ρ(x_λ)

e^uλ∂u_λ

∂x₁v_1,λdx=I₁+I₂. (5.10) By the convergence results (3.2) and (4.15) it is easy to see that

I₁=λ

Ω\Bρ(xλ)

e^uλ∂uλ

∂x1

v_1,λΦ dx=O(λμ1,λ)=o(1). (5.11)

To estimateI₂we use Eq. (5.1) wherei=1, i.e.

(1−μ_1,λ)λ

Ω

e^uλ∂u_λ

∂x1

v_1,λ=

∂Ω

∂u_λ

∂x1

∂v_1,λ

∂νx

dσ_x. (5.12)