Lane–Emden problems: Asymptotic behavior of low energy nodal solutions

Ann. I. H. Poincaré – AN 30 (2013) 121–140

www.elsevier.com/locate/anihpc

Lane–Emden problems: Asymptotic behavior of low energy nodal solutions

Massimo Grossi

, Christopher Grumiau

, Filomena Pacella

aDipartimento di Matematica, Universita’ di Roma “Sapienza”, P. le A. Moro 2, 00185 Roma, Italy bInstitut de Mathématique, Université de Mons, 20, Place du Parc, B-7000 Mons, Belgium

Received 3 February 2012; accepted 14 June 2012 Available online 24 July 2012

Abstract

We study the nodal solutions of the Lane–Emden–Dirichlet problem −u= |u|^p−1u, inΩ,

u=0, on∂Ω,

whereΩis a smooth bounded domain inR²andp >1. We consider solutionsu_psatisfying p

Ω

|∇up|²→16π e asp→ +∞ (∗)

and we are interested in the shape and the asymptotic behavior asp→ +∞.

First we prove that (∗) holds for least energy nodal solutions. Then we obtain some estimates and the asymptotic profile of this kind of solutions. Finally, in some cases, we prove thatpupcan be characterized as the difference of two Green’s functions and the nodal line intersects the boundary ofΩ, for largep.

©2012 Elsevier Masson SAS. All rights reserved.

MSC:primary 35J91; secondary 35B32

Keywords:Superlinear elliptic boundary value problem; Least energy nodal solution; Asymptotic behavior; Variational methods

✩ This work has been done while the second author was visiting the Mathematics Department of the University of Roma “Sapienza” supported by INDAM-GNAMPA. He also acknowledges the national bank of Belgium and the program “Qualitative study of solutions of variational elliptic partial differential equations. Symmetries, bifurcations, singularities, multiplicity and numerics” of the FNRS, project 2.4.550.10.F of the Fonds de la Recherche Fondamentale Collective for the partial support.

* Corresponding author.

E-mail addresses:grossi@mat.uniroma.it(M. Grossi),Christopher.Grumiau@umons.ac.be(C. Grumiau),pacella@mat.uniroma.it (F. Pacella).

0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.06.005

1. Introduction

We consider the superlinear elliptic boundary value problem −u= |u|^p−1u, inΩ,

u=0, on∂Ω, (Pp)

whereΩis a smooth bounded domain inR²andp >1.

By standard variational methods we know that problem (Pp) has a positive ground state solution. Moreover many other results about the multiplicity and the qualitative properties of positive solutions in various types of domains have been obtained in the last decades.

In this paper we are interested in studying sign changing solutions of (Pp). In contrast with the case of positive solutions not much is known on nodal solutions of (Pp), in particular about their qualitative behavior. Let us therefore recall some recent results. In the paper[10]A. Castro, J. Cossio and J.M. Neuberger proved the existence of a nodal solution with least energy among nodal solutions, which is therefore referred to as theleast energy nodal solutionof problem (Pp). T. Bartsch and T. Weth showed that these solutions possess exactly two nodal regions and have Morse index two (see[3]). Since positive ground state solutions have the symmetries of the domainΩ, ifΩ is convex, by the classical result of[14], a natural question is whether least energy nodal solutions also inherit the symmetries of the domainΩ. In[2]A. Aftalion and F. Pacella proved that, in a ball or in an annulus, a least energy nodal solution cannot be radial. In fact, in dimensionN, they cannot be even with respect to more thanN−1 orthogonal directions.

They also proved that the nodal set touches the boundary. On the other hand, T. Bartsch, T. Weth and M. Willem in[4]

and F. Pacella and T. Weth in[19], with different methods, obtained partial symmetry results: they showed that on a radial domain, a least energy nodal solutionuhas the so-called foliated Schwarz symmetry, i.e.ucan be written asu(x)= ˜u(|x|, ξ ·x), where ξ ∈R^N andu(r,˜ ·)is nondecreasing for everyr >0. In fact, as they are not radial,

˜

u(r,·)is increasing. In dimensionN, it implies that the least energy nodal solutions are even with respect toN−1 orthogonal directions. Concerning the “last direction”, in[8,15], D. Bonheure, V. Bouchez, C. Grumiau, C. Troestler and J. Van Schaftingen proved that forp close to 1 the least energy nodal solution must be odd with respect to this direction. Moreover, it is unique up to a rotation. For general open bounded domains, they prove that least energy nodal solutions must respect the symmetries of their orthogonal projection on the second eigenspace of−whenp is close to 1.

In this paper we study the profile and other qualitative properties of low energy nodal solutions of problem (Pp) asp→ +∞andΩ⊆R²is any bounded smooth domain. For ground state positive solutions the same analysis has been done by X. Ren and J. Wei in[21]and[20], obtaining, in particular,L^∞estimates. This result has been improved by Adimurthi and M. Grossi in[1](see also[11]) who computed the exact value of theL^∞-norm at the limit, by a different approach.

Here by low energy we mean that we are interested in the families of nodal solutions(up)p>1satisfying p

Ω

|∇u_p|²→16π e asp→ +∞. (A)

Note that as a consequence of[21]and as it will be clear later, this kind of solutions cannot have more than 2 nodal regions forplarge.

Let us observe that there are nodal solutions of (Pp) satisfying (A). In fact least energy nodal solutions are among those and we have:

Theorem 1.The condition(A)holds for any family of least energy nodal solutions.

To describe our results we need some notations. InH₀¹(Ω), we use the scalar product(u, v)=

Ω∇u· ∇v and denote by · qthe usual norm inL^q(Ω)and byd(x, D)the distance between a pointx∈R²and the setD⊆R². Let us consider a family of nodal solutions(u_p)_p>1. Throughout the paper, we assume thatu_pare low energy solutions, i.e. (A) holds. The positive partu⁺_p (resp. negative partu⁻_p) is defined asu⁺_p :=max(up,0)(resp.u⁻_p :=min(up,0)).

Let us define the families(x_p⁺)_p>1(resp.(x_p⁻)_p>1) of maximum (resp. minimum) points inΩofu_p, i.e.u_p(x_p⁺)= u⁺_p∞andu_p(x_p⁻)= −u⁻_p∞and assume w.l.o.g. thatu_p(x_p⁺)= u_p∞, i.e.u_p(x_p⁺)−u_p(x_p⁻). To start with,

we prove thatx_p⁺cannot go “too fast” to the boundary ofΩ which is the key point to make some rescaling aroundx_p⁺ and obtain a limit profile onR². More precisely we prove that ^d(xp+_ε^,∂Ω)

p → +∞(see Proposition3.1), where ε⁻_p²:=pu_p

x_p⁺p−1

. Then we get the following result.

Theorem 2.The scaling ofuparoundx_p⁺: z_p(x):= p

u_p(x_p⁺) u_p

ε_px+x_p⁺

−u_p x_p⁺

defined onΩ⁺(ε_p):=^Ω−_εp^x+p converges, asp→ ∞to a functionzinC_loc² (R²). Moreoverzmust solve the equation

−z=e^zonR²,z0,z(0)=0,

R²e^z=8πandz(x)=log ₁

(1+¹8|x|²)²

.

As a consequence of the previous theorem, we deduce thatε⁻_p¹d(x_p⁺,NLp)→ +∞asp→ ∞, whereNLpdenotes the nodal line ofu_p. So, in some sense, the rescaled solution aboutx⁺_p ignores the other nodal domain ofu_p. This im- plies that we can repeat the same kind of rescaling argument in the positive nodal domainΩ˜_p⁺:= {x∈Ω: u_p(x) >0}

ofu_p. Hence, definingΩ˜⁺(ε_p):=^Ω˜p+_ε⁻_p^xp+, we get the analogous of Theorem2:

Theorem 3.The functionzp: ˜Ω⁺(εp)→Rconverges, asp→ +∞, to a functionzinC_loc² (R²)asp→ ∞. Moreover zmust solve the equation−z=e^zonR²,z0,z(0)=0,

R²e^z=8πandz(x)=log ₁

(1+¹₈|x|²)²

.

At this point, to the aim of studying the negative partu⁻_p, let us observe that we can have two types of families of solutions satisfying the assumption (A), the ones which satisfy

(B) there existsK0 such thatp(u_p(x_p⁺)+u_p(x⁻_p))→K;

and the ones which satisfy (B) p(u_p(x_p⁺)+u_p(x_p⁻))→ ∞.

The meaning of(B)is that the speeds of convergence of the maximum and the minimum ofup(multiplied byp) are comparable. Instead the condition(B)implies that one of the two values converges faster than the other one.

Remark 4.It is easy to see that nodal solutions of type(B)exist. Indeed, ifΩis a ball, it is enough to consider the antisymmetric, with respect to a diameter, solution with two nodal regions. We believe that also solution of type(B) should exist and we conjecture that the radial solution in the ball, with two nodal regions, should be of type(B).

However, the complete characterization of low energy solutions in the ball will be analyzed in a subsequent paper.

In this paper we investigate the alternative(B)that we conjecture holding for the least energy nodal solutions.

First, we prove that, as forx_p⁺, the condition(B)implies thatε_p⁻¹d(x_p⁻, ∂Ω)→ +∞asp→ ∞. Then we get the following result.

Theorem 5.If(B)holds then the scaling ofuparoundx_p⁻ z⁻_p(x):= p

u_p(x_p⁺) −u_p

ε_px+x_p⁻

−u_p x_p⁺

defined onΩ⁻(ε_p):=^Ω−_εp^xp−converges, asp→ +∞, to a functionzinC_loc² (R²). Moreoverzmust solve the equation

−z=e^zonR²,z0,

R²e^z=8πandz(x)=log _μ

(1+^μ₈|x|²)²

for some0< μ1. WhenK=0in condition(B), we getμ=1.

As for the case ofx_p⁺, as a consequence of Theorem5, we get thatε⁻_p¹d(x_p⁻,NLp)→ +∞, which allows to do the same rescaling in the negative nodal domainΩ˜_p⁻:= {x∈Ω: u_p(x) <0}, obtaining the analogous of Theorem5.

Theorem 6.If(B)holds, the function z⁻_p(x):= p

up∞

−u⁻_p

ε_px+x_p⁻

− u_p∞

defined on Ω˜⁻(ε_p):= ^Ω˜p−_ε⁻_p^xp− converges, as p→ +∞, to a functionz in C_loc² (R²). Moreover z must solve the equation −z=e^z on R², z0,

R²e^z=8π and z(x)=log _μ

(1+^μ8|x|²)²

for some0< μ1. When K=0 in condition(B), we getμ=1.

Remark 7.Another natural condition to make the rescaling in the negative nodal domain without assuming condi- tion(B)could be to consider the parameter

˜

ε_p⁻²=pu⁻_p

x⁻_p^p−1

which is now just related to the negative part ofu(we are not using theL^∞-norm ofu_pbut theL^∞-norm ofu⁻_p) and assume thatε˜⁻_p¹d(x_p⁻,NLp)→ +∞(as beforeNLpis the nodal line ofu_p). This assumption is essentially equivalent to condition(B)and allows to prove thatε˜⁻_p¹d(x_p⁻, ∂Ω)→ +∞(see Proposition3.3). Then one could repeat the proof of Theorem6obtaining forz_p(x):=_u ^p

p(xp⁻)(u⁻_p(ε˜_px+x⁻_p)−u_p(x⁻_p))the same assertion as forz⁻_p.

If the positive part ofu, i.e.u⁺_p, as a solution of (Pp) inΩ˜⁺(εp), has Morse index one then the previous results allow to obtain the exact value of the limits ofu^±_p∞, asp→ +∞.

Theorem 8.Let us assume that the Morse index ofu⁺_p as a solution of (Pp) inΩ˜_p⁺is one. Then we have:u⁺_p∞→ e^1/2. If also(B)holds thenu⁻_p∞→e^1/2.

The result of the previous statement is similar to the one obtained in [1] for the least energy positive solution of (Pp).

Let us remark that the additional assumption on the Morse index ofu⁺_p holds for any nodal solutions with Morse index 2, hence, in particular, for least energy nodal solutions.

Our last result gives the asymptotic behavior of the nodal solutions in the whole domainΩ.

Let us denote byG(x, y)= −_2π¹ log|x−y| +H (x, y)the Green’s function ofΩand byHits regular part. Finally, letx^±be the limit point ofx_p^±asp→ +∞.

Theorem 9.Under the same hypothesis of Theorem8,pu_pconverges, asp→ +∞, to the function8π e^1/2(G(·, x⁺)− G(·, x⁻))inC_loc² (Ω¯\ {x⁻, x⁺})andx⁺=x⁻∈Ω. Moreover the limit pointsx⁺andx⁻satisfy the system

⎧⎪

⎪⎨

⎪⎪

⎩

∂G

∂x_i

x⁺, x⁻

−∂H

∂x_i

x⁺, x⁺

=0,

∂G

∂x_i

x⁻, x⁺

−∂H

∂x_i

x⁻, x⁻

=0,

fori=1,2. Finally, the nodal line ofu_pintersects the boundary ofΩforplarge.

The result of Theorem9gives a very accurate description of the profile of the low energy solutions of type(B) in terms of the Green function ofΩ and of its regular part. It is also remarkable that the property that the nodal line intersects∂Ωholds for this kind of solutions in any bounded domainΩ, extending so the result proved in[2]for least energy solutions in balls or annulus. It is also reminiscent of the property of the second eigenfunction of the laplacian in planar convex domains (see[18]), though we are not analyzing the case ofpclose to 1 as in[8,15].

Let us remark that nodal solutions with this property have been constructed in[13,12].

Finally we would like to point out that our analysis is similar to the one carried out in [5–7] for low energy nodal solutions of an almost critical problem or of the Brezis–Nirenberg problem in dimensionN3. However, the techniques and the proofs are completely different since in[5–7]the nodal solutions whose energy is close to 2SN

(SNis the best Sobolev constant inR^N) can be written almost explicitly.

The outline of the paper is as follows. In Section 2, we recall the variational characterization of the problem and we prove Theorem1and some useful asymptotic estimates. In Section 3, we show thatx_p⁺cannot go too fast to the boundary and then prove Theorem2and Theorem5using a rescaling argument on the whole domainΩ. Then, using a rescaling argument on the nodal domains, we prove Theorem3and Theorem6. In Section 4, we improve the bounds given in Section 2 to obtain Theorem8. Finally, in Section 5, we prove Theorem9.

2. Variational setting and estimates

We recall that solutions of problem (Pp) are the critical points of the energy functionalEpdefined onH₀¹(Ω)by Ep(u)=1

2

Ω

|∇u|²− 1 p+1

Ω

|u|^p+1.

The Nehari manifoldNpand the nodal Nehari setMpare defined by Np:=

u∈H₀¹(Ω)\ {0}:

dEp(u), u

=0

, Mp:=

u∈H₀¹(Ω): u^±∈Np

,

whereu⁺(x):=max(u(x),0)andu⁻(x):=min(u(x),0). Ifu∈H₀¹(Ω),u⁺=0 andu⁻=0 thenu∈Mp if and only if

Ω

∇u⁺²=

Ω

u^+p+1 and

Ω

∇u⁻²=

Ω

u^−p+1. (1) For anyu=0 fixed, there exists a unique multiplicative factorαsuch thatαu∈Np. Ifuchanges sign then there exists a unique couple(α₊, α₋)such thatα₊u⁺+α₋u⁻∈Mp.

The interest ofNp (resp.Mp) comes from the fact that it contains all the non-zero (resp. sign-changing) critical points ofEp. IfuminimizesEponNp(resp.Mp) thenuis a (resp. nodal) solution of problem (Pp) usually referred to as theground state solutions(resp.least energy nodal solutions). So, we need to solve

inf 1

2− 1

p+1

Ω

|∇u|²

on

Ω

∇u^±2=

Ω

u^±p+1

to characterize the least energy nodal solutions.

Theorem 2.1.(See T. Bartsch, T. Weth[3].) There exists a least energy nodal solution of problem(Pp)which has exactly two nodal domains and Morse index2.

To start with, we show that each family of least energy nodal solutions for problem (Pp) is a family of low energy nodal solutions, i.e. satisfies condition (A) of the Introduction. To this aim let us prove an upper bound and a control on the energy.

Lemma 2.2.Let(u_p)_p>1be a family of least energy nodal solutions of problem(Pp). For anyε >0, there existsp_ε such that, for anypp_ε,

pEp(u_p)=p 1

2− 1

p+1

Ω

|∇u_p|²8π e+ε.

Proof. Leta, b∈Ω. Let us consider 0< r <1 such thatB(a, r), B(b, r)⊆Ω andB(a, r)∩B(b, r)= ∅. Then, we define a cut-off functionϕ:Ω→ [0,1]inC₀^∞(Ω)such that

ϕ(x):=

1 if|x−a|< r/2, 0 if|x−a|r.

First we introduce the family of functionsW¯_p:Ω→Rwhich are defined onB(a, r)as W¯p(x):=ϕ(x)√

e

1+z(^x_ε^−a

p ) p

wherez(x)= −2 log(1+^|x₈^|2)andε_p²:= ¹

p√

e^p−1. The functionsW¯_pvanish outside the ballB(a, r). We claim that

Ω

| ¯W_p|^p+1=8π e

p +o(1/p),

Ω

|∇ ¯W_p|²=8π e

p +o(1/p).

Indeed, setting ^x_ε^−a

p =ψand using the fact that

R²e^z=8π,

Ω

| ¯W_p|^p+1=(√ e )^p+1ε_p²

Ω−aεp

ϕ(εψ+a)^p+1

1+z(ψ ) p

p+1

dψ

= e p

R²

e^z+o(1)

=8π e

p +o(1/p).

Concerning

Ω|∇ ¯W_p|², we get that

Ω

|∇ ¯Wp|²=

Ω

ϕ²(x) ∇

√ e

1+z((x−a)/ε_p) p

²+

Ω

∇ϕ(x)²√ e

1+z((x−a)/ε_p) p

2

+2

Ω

ϕ(x)√ e

1+z((x−a)/ε_p) p

∇ϕ(x)· ∇ √

e

1+z((x−a)/ε_p) p

.

The first term gives

Ω

ϕ²(x) ∇

√ e

1+z((x−a)/ε_p) p

²

= e p²

Ω

16ϕ²(x) |x−a|² (8ε_p²+ |x−a|²)²

=16e p²

B(a,r/2)

|x−a|² (8ε²_p+ |x−a|²)²+

Ω\B(a,r/2)

ϕ²(x) |x−a|² (8ε²_p+ |x−a|²)²

=16e p²

2π

r/2 0

ψ³

(8ε_p²+ψ²)²+O(1)

.

Settingψ²=t and integrating, we get r/2

0

ψ³

(8ε_p²+ψ²)²=1 2log

^r2+8ε²_p 8ε_p²

+1 2

8ε²_p 8ε_p²+^r₂ −1

= −log|ε_p| +O(1).

So, we get

Ω

ϕ²(x) ∇

√ e

1+z((x−a)/ε_p) p

²= −32π e p²

logε_p+O(1)

=−32eπ p²

−p−1

4 +o(p)+O(1)

=8π e

p +o(1/p).

The second term gives the existence of a constantK >0 such that

Ω

∇ϕ(x)²√ e

1+z((x−a)/ε_p) p

2

=

B(a,r)\B(a,r/2)

∇ϕ(x)²√ e

1+z((x−a)/ε_p) p

2

K(1+2 maxx∈Ω\B(a,1/p)|log(^|x−₈^a|2p)| +K)² p²

=o(1/p).

The third term can be treated with similar techniques. So, finally, we get

Ω

|∇ ¯Wp|²=8π e

p +o(1/p) which proves the claim.

Then, we define the family of test functionsWp:Ω→Rwhich are defined onB(a, r)asW¯pand onB(b, r)as the odd reflection ofW¯_p. The functionsW_pvanish outside the two ballsB(a, r)andB(b, r). So,∇W_p^±2₂=^{8π e}_p + o(1/p)andW_p^±p_p⁺₊¹₁=^{8π e}_p +o(1/p). Clearly, the unique multiplicative factorα_p:=α_p⁺such thatα⁺_pW_p⁺∈Np

equals the unique multiplicative factorα⁻_p such thatα⁻_pW_p⁻∈Np. To characterize it, we need to solve α_p²∇W_p^±2

2=α_p^p+1W_p^±p+1

p+1. It implies that

α_p= ^Ω|∇W_p^±|²

Ω|W_p^±|^{p+1 1}

p−1

→1. (2)

So, asu_pis a minimum for theH₀¹-norm onMpand

Ω|∇u_p|²=

Ω|∇u⁺_p|²+

Ω|∇u⁻_p|², we conclude that p

1

2 − 1

p+1

∇u_p²2p 1

2− 1

p+1

2(αp)²

Ω

∇W_p⁺².

As the right-hand side converges to 8π e, we get the assertion. 2

Lemma 2.3.Let(u_p)_p>1be a family of least energy nodal solutions of problem(Pp). For anyε >0, there existsp_ε such that, for anypp_ε,

pEp(u_p)=p 1

2− 1

p+1

Ω

|∇u_p|²8π e−ε.

Proof. To do this, we prove that for any sequencep_n→ +∞lim infn→+∞p_n(¹₂−_pn¹₊₁)

Ω|∇u^±_p

n|²4π e. On one hand, 1=^{Ω(u±pn)pn+1}

Ω|∇u^±_pn|² . On the other hand, in[21, p. 752], it is proved that, for anyt >1,utD_tt^1/2∇u2where D_t→(8π e)^−1/2is independent ofuinH₀¹(Ω).

So, we obtain 1D^p_pⁿ⁺¹

n+1(pn+1)^pn+21

Ω

∇u^±_p

n^2pn−₂¹ ,

i.e.

Ω|∇u^±_p

n|²D⁻²

pn+1 pn−1

pn+1 (p_n+1)^−pn+

1 pn−1. Thus, 1

2− 1

p_n+1

(p_n+1)^pn+

pn−11

Ω

∇u^±_p

n² 1

2− 1

p_n+1

D⁻²

pn+1 pn−1

p_n+1 . As ^pn

(pn+1)

pn+1pn−1 converges to 1 and the right-hand side converges to 4π e, we get the assertion. 2 Proof of Theorem1. It follows from Lemma2.2and Lemma2.3. 2

Remark 2.4.The proof of Lemma2.3does not depend on the fact thatu_pn is a least energy nodal solution. Indeed, for any(u_p)_p>1verifying (A), asp→ +∞, we get

• p(¹₂−_p¹)

Ω|∇u^±_p|²→4π e,p

Ω|∇u^±_p|²→8π eandp

Ω|∇u_p|²→16π e.

• Ep(u_p)→0,

Ω|∇u_p|²→0,

Ω|∇u⁻_p|²→0 and

Ω|∇u⁺_p|²→0.

Moreover the proof of Lemma2.3implies, as corollary, thatu_phas 2 nodal domains forplarge.

From now on, throughout the paper, we consider a family(u_p)_p>1of nodal solutions for which (A) holds. The following result shows an asymptotic lower bound for theL^∞-norms ofu⁺_p andu⁻_p. We denote byλ1(D)the first eigenvalue of−with Dirichlet boundary conditions in a domainDand byx_p^±both the maximum or the minimum point ofup, as defined in the Introduction.

Proposition 2.5.For anyp >1we have that|u_p(x_p^±)|λ

1 p−1

1 whereλ₁:=λ₁(Ω).

Proof. Using Poincaré’s inequality, we get 1=

Ω|u^±_p|^p+1

Ω|∇u^±_p|² |u_p(x^±_p)|^p−1

Ω(u^±_p)²

Ω|∇u^±_p|² u_p

x_p^±p−1λ⁻₁¹Ω˜_p^± ,

whereΩ˜_p^±are the nodal domains ofu_p. AsΩ˜_p^±⊆Ω, we haveλ₁(Ω˜_p^±)λ₁which ends the proof. 2 Remark 2.6.We have:

• For anyε >0,|u_p(x_p^±)|1−εforplarge. In particular this holds foru_pL^+∞.

• By Remark2.4, as|up(x_p^±)|^p−1is bounded from below,^|up(xp±)|

p−1

Ω|∇u^±_p|² and^|up(xp±)|

p−1

Ω|∇up|² converge to+∞whenp→ +∞.

The next result gives a direct argument to prove that theL^∞-norms ofu⁺_p andu⁻_p are bounded. It will be improved in the next sections.

Proposition 2.7.We have thatu_p(x_p^±)is bounded asp→ +∞.

Proof. Let us make the proof for the positive case. By Proposition2.5, we only have to prove thatu_p(x_p⁺)is bounded from above. Let us denote byGthe Green’s function onΩ. As|G(x, y)|C|log|x−y||for anyx, y∈Ωand some independent constantC >0, using the Hölder inequality we have

up

x_p⁺

=

Ω

G

x_p⁺, yu_p(y)^p−1up(y)dy

C

Ω

logx_p⁺−yu_p(y)^pdy

C

Ω

logx⁺_p −y^p+1dy _p+1¹

Ω

|u_p|^p+1 _p+1^p

.

Sincep

Ω|u_p|^p+1→16π easp→ +∞(see Remark2.4), it is enough to show the existence of a constantC >0 such that

Ω

logx_p⁺−y^p+1dyC(p+1)^p+2.

Let us considerR >0 such thatΩ⊆B(xp, R)for alln. Then there exists a constantK >0 such that

Ω

logx_p⁺−y^p+1dy

B(xp,R)

logx_p⁺−y^p+1dy=K R 0

|logr|^p+1rdr.

Integrating([p] +1)-times by parts, we get

Ω

logx_p⁺−y^p+1dyKlog(R)^p+1+(p+1)log(R)^p+ · · · +(p+1)· · ·

p− [p] +2

×log(R)^p−[p]+1

+K(p+1)p· · ·

p− [p] +1^R

0

|logr|^p−[p]rdr.

Thus, there existsCsuch that for largen

Ω

logx_p⁺−y^p+1dyC(p+1)^p+2,

which ends the proof. 2 3. Asymptotic behavior

For the rest of the paper, w.l.o.g., let us assume thatu_p∞=u_p(x_p⁺)for anyp >1.

In this section we use several rescaling arguments to characterize the asymptotic behavior ofu^±_p. Let us defineε²_p:=_pu ¹

p(x⁺_p)^p−1 →0 by Remark2.6.

3.1. Control close to the boundary

We prove thatx_p⁺cannot go to the boundary ofΩtoo fast.

Proposition 3.1.We have d(x_p⁺, ∂Ω)

ε_p → +∞ (3)

asp→ +∞.

Proof. Let us argue by contradiction and assume that, for a sequencep_n→ +∞, ^d(xpn+_ε^,∂Ω)

pn →l0 and thatx_p⁺

n→ x_∗∈∂Ω (i.e. ^d(x_ε^pn+,x∗)

pn →l).

First, we treat the case when∂Ω is flat aroundx_∗. We consider a semi-ballDcentered inx_∗with radiusRsuch that D⊆Ω and the diameter of D belongs to ∂Ω. For largen, let us remark that x_p⁺

n belongs to D. Then, on A:=B(x_∗, R), we consider the functionu^∗_p

nwhich is defined asu_pn onDand as the odd reflection ofu_pnonA\D.

It is a solution of−u= |u|^pn−1uonA. For largen, we consider z^∗_p

n(x):= p_n u^∗_pn(xp⁺_n)

u^∗_p

n

ε_pnx+x_p⁺

n

−u^∗_p

n

x_p⁺

n

(4)

onΩ_p^∗

n:=^A−_εpn^x+pn →R². OnΩ_p^∗

n, we get from (4)

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−z^∗_p

n= 1+z^∗_pn

pn

^pn−1 1+z^∗_pn

pn

, 1+z^∗_p

n

pn

1.

Let us fixR >0. For largen,B(0, R)⊆Ω_p^∗

nand we consider the problem

⎧⎪

⎨

⎪⎩

−w_pn= 1+z_p^∗

n

p_n ^pn−1

1+z^∗_p

n

p_n

, inB(0, R),

w_pn=0, on∂B(0, R).

Since, by (4),|1+^z_p^∗pn_n|1, we have that|w_pn|is uniformly bounded by a constant C independent ofnby the maximum principle and the regularity theory. Moreover, becausezp_n0, we have thatψp_n=zp_n−wp_nis a harmonic function which is uniformly bounded above. By Harnack’s inequality,ψp_nis bounded inL^∞(B(0, R))or tends to−∞

on each compact set ofB(0, R). Asψ_pn(0)=z_pn(0)−w_pn(0)−C, we get thatψ_pnandz_pnare uniformly bounded on each compact set ofB(0, R).

Since we are assuming that ^d(x_ε^pn,x∗)

pn →lwe get thaty_n:=^x∗_ε⁻_pn^xpn+ ∈B[0, l+1]for largenandz_pn(y_n)= −p_n→

−∞which is a contradiction.

Next, we treat the case when ∂Ω is not locally flat aroundx_∗ but is aC¹-curve. We consider aC¹-domainD which is the intersection of a fixed neighborhood ofx_∗ and Ω. Let us define the square Q:=(−1,1)², Q⁺:=

(−1,1)×(0,1)⊆QandS:=(−1,1)× {0}.

We consider the change of variablesϕ:D→Q⁺andϕ(D∩∂Ω)=S (see[9]to get thatϕ is well-defined and can be assumed to beC¹(D)). Moreover¯ ϕ⁻1∈C¹(Q¯⁺).

We fix a positive functionθ∈C²such thatθ◦ϕ⁻¹: ¯Q⁺→Requals 0 on∂Q⁺\Sand∂_νθ◦ϕ⁻¹=0 onSwhere

∂_ν denotes the normal derivative. We extendθ◦ϕ⁻¹onQby even symmetry with respect toS.

On Q, we define u˜_pn asθ (ϕ⁻¹(·))u_pn(ϕ⁻¹(·))onQ⁺ and the odd symmetric function onQ\Q⁺. Sinceθ u_pn solves

−u=θ|u_pn|^pn−1u_pn−2∇θ∇u_pn−(θ )u_pn=:g_pn (5) with Dirichlet boundary conditions on D, by the change of variables y=ϕ(x), we get thatu˜p_n solves for some matrixA_pn

−div(Apn∇u)=h_pn