C. R. Acad. Sci. Paris, t. 332, Série I, p. 705–710, 2001Équations aux dérivées partielles/Partial Differential Equations

On the asymptotic behavior of solutionsof a semilinear elliptic boundary problemin an unbounded coneYouri V. EGOROV a, Vladimir A. KONDRATIEV b

a Laboratoire des mathématiques pour l’industrie et la physique, UMR 5640, Université Paul-Sabatier,UFR MIG, 118, route de Narbonne, 31062, Toulouse cedex 4, France

b Mehmat Faculty, Lomonossov University, Vorob’evy Gory, 119899 Moscow, Russia

(Reçu le 10 novembre 2000, accepté après révision le 5 mars 2001)

Abstract. We consider solutions to the elliptic linear equation (1) of second order in an unboundeddomain Q in R

n supposing that Q⊂ x = (x′, xn) : 0 < xn <∞, |x′|< γ(xn), where1 γ(t) At+B, and that u satisfies the boundary condition (2). We show that any suchsolution u growing moderately at infinity is bounded and tends to 0 as |x| → ∞. Earlierwe showed in our notes [1,2] this theorem for the case γ(xn) = B, i.e., for a cylindricaldomain Q = Ω × (0,∞), Ω ⊂ R

n−1. In [3] the theorem was proved for the case whenA A0 with a constant A0 sufficiently small. Here we admit any value of A0. Our theoremis true even for the domain which is an outer part of a cone, in particular, for the half-spacexn > 0. Besides, we consider here more general operators L with lower order terms. It isworth noticing that the new proof is quite different of that in [3]. 2001 Académie dessciences/Éditions scientifiques et médicales Elsevier SAS

Sur le comportement asymptotique des solutions d’un problème au bordelliptique semi-linéaire dans un cone non borné

Résumé. On considère des solutions de l’équation elliptique linéaire (1) dans un domaine nonbornéQ de R

n en supposant queQ ⊂ x = (x′, xn) : 0 < xn < ∞, |x′| < γ(xn), où1 γ(t) At + B, |γ′(t)| cγ(t)/t et queu vérifie les conditions au bord(2). Nousmontrons que les solutionsu qui ne croissent pas trop vite à l’infini, sont bornées et tendentvers0 si xn →∞. Dans nos notes[1,2], nous avons consideré le casγ(xn) = B, i.e. lecas d’un domaine cylindriqueQ = Ω× (0,∞), Ω⊂ R

n−1.Nous avons démontré dans[3] ce théorème dans le cas oùA A0 avec une constanteA0 suffisamment petite. Ici, nous admettons une valeur deA0 arbitraire. Notre théorèmeest vrai même pour un domaine lequel est la partie extérieure d’un cône, en particulierpour le demi-espacexn > 0. En outre, nous considérons ici des opérateursL plusgénéraux que dans[3] avec des terms d’ordre inférieur. Notons que les démonstrationssont très différentes de celles en[3]. 2001 Académie des sciences/Éditions scientifiqueset médicales Elsevier SAS

Note présentée par Pierre-Louis LIONS.

S0764-4442(01)01930-9/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés 705

Yu.V. Egorov, V.A. Kondratiev

Version française abrégée

Nous considérons des solutions de l’équation elliptique linéaire :

Lu :=

n∑i,j=1

∂

∂xi

(aij(x)

∂u

∂xj

)+

n∑i=1

bi(x)∂u

∂xi+

n∑i=1

∂

∂xi

(ci(x)u

)+ d(x)u= 0 (1)

dans un domaine non borné Q de Rn en supposant que Q⊂ x = (x′, xn) : 0 < xn <∞, |x′|< γ(xn),

où 1 γ(t)At+B. Nous supposons que u vérifie la condition au bord :

∂u

∂νL+ k(x)u+ b(x)

∣∣u(x)∣∣p−1u(x) = 0 (2)

sur la surface latérale S = x ∈ ∂Q, 0 < xn < ∞, où p > 0, b(x) b0 > 0, γ ∈ C1(0,∞), |γ′(t)| cγ(t)/t et

∂u

∂νL=

n∑i,j=1

aij(x)∂u

∂xjcosθi, k(x) =

n∑i=1

ci(x) cosθi,

où θi est l’angle entre l’axe xi and le vecteur sortant normal extérieur. Nous supposons quen∑

i,j=1

aij(x)ξiξj c0|ξ|2, c0 > 0, x ∈Q,

et que |aij(x)| C , |bj(x)|+ |cj(x)| C/|x| pour i, j = 1, . . . , n, en tout point x ∈Q. Pour appliquer leprincipe du maximum nous supposons que

d(x) +

n∑j=1

∂cj(x)

∂xj 0

faiblement dans Q, c’est-à-dire,∫Q

(n∑

j=1

cj(x)∂v

∂xj− d(x)v

)dx−

∫S

k(x)v dS 0,

si v ∈C1(Q), v 0 et v(x) = 0 si xn = 1 ou xn =∞.Nous montrons que toute fonction u satisfaisant (1) et (2) qui ne croît pas trop vite à l’infini est bornée

et tend vers 0 si xn →∞. Dans nos notes [1,2], nous avons consideré le cas γ(xn) =B, c’est-à-dire, le casd’un domaine cylindrique Q=Ω× (0,∞), Ω⊂R

n−1.Nous avons démontré dans [3] le théorème dans le cas quand A A0 avec une constante A0

suffisamment petite. Ici, nous admettons la valeur de A0 arbitraire. Notre théorème est vrai même pourun domaine qui est le complémantaire d’un cône, en particulier pour le demi-espace xn > 0. Ceci nous aamenés à changer complètement la démonstration. En outre, nous considérons ici des opérateurs L plusgénéraux que dans [3] avec des termes d’ordre inférieur.

Nous considérons aussi les solutions de l’équation (1) satisfaisant la condition au bord

∂u

∂νL− b(x)

∣∣u(x)∣∣p−1u(x) = 0.

1. We study the solutions to the elliptic second order linear equation

Lu :=

n∑i,j=1

∂

∂xi

(aij(x)

∂u

∂xj

)+

n∑i=1

bi(x)∂u

∂xi+

n∑i=1

∂

∂xi

(ci(x)u

)+ d(x)u= 0 (1)

706

Semilinear elliptic boundary problems in an unbounded cone

in an unbounded domain Q in Rn supposing Q ⊂ x = (x′, xn) : |x′| < γ(xn), 0 < xn < ∞, where

1 γ(t) At+B, γ ∈C1(0,∞), |γ′(t)| cγ(t)/t and that u satisfies the boundary condition:

∂u

∂νL+ k(x)u+ b(x)

∣∣u(x)∣∣p−1u(x) = 0 (2)

on the lateral surface S = x∈ ∂Q, 0< xn <∞, where p > 0, b(x) b0 > 0, and

∂u

∂νL=

n∑i,j=1

aij(x)∂u

∂xjcosθi, k(x) =

n∑i=1

ci(x) cosθi,

θi is the angle between the axis xi and the outer normal vector.We assume that the surface S is of the class C1 with the Lipschitz constant independent of x and that

Q⊃ x= (x′, xn) : |x′|< cγ(xn), 0<xn <∞, 0< c 1.Suppose that

n∑i,j=1

aij(x)ξiξj c0|ξ|2, c0 > 0, B(x)≡n∑

j=1

∣∣bj(x)∣∣+ ∣∣cj(x)∣∣ C

|x| , x ∈Q,

and that |aij(x)| C for i, j = 1, . . . , n, and for all x ∈ Q. We don’t assume that the coefficients of Lare continuous. In order to apply the maximum principle we suppose that 0 d(x) +

∑ni=1 ∂ci(x)/∂xj

weakly, i.e., ∫Q

(n∑

j=1

cj∂v

∂xj− d(x)v

)dx−

∫S

k(x)v dS 0

if v ∈C1(Q), v 0 and v(x) = 0 as xn = 1 or xn =∞ (see[4], condition (8.8)).Let us denote ΩT and ΣT the sections of the domain Q and the boundary S by the plane xn = T , and

QT and ST the parts of Q and S between the planes xn = 1 and xn = T .We consider functions u satisfying (1) and (2) weakly. It means that u ∈H1

loc(Q)∩ Lp+1,loc(S) and∫Q

[n∑

i,j=1

aij(x)∂u

∂xj

∂ϕ

∂xi−

n∑i=1

bi(x)∂u

∂xiϕ+

n∑i=1

∂ϕ

∂xi

(ci(x)u

)− d(x)uϕ

]dx

+

∫S

b(x)∣∣u(x)∣∣p−1

u(x)ϕ(x)dS = 0 (3)

for all functions ϕ(x) ∈H1(Q), equal to 0 as xn = 0 and in a neighborhood of xn =∞.We will show that any solution of our problem growing moderately at infinity is bounded and tends to 0

as xn →∞.In our notes [1,2], we showed such a theorem for the case γ(xn) = B, i.e., for a cylindrical domain

Q=Ω× (0,∞), Ω⊂ Rn−1. We have studied in [3] the case when AA0 where A0 is sufficiently small.

The proof below is very different of that in [3]. It is based on the techniques developed in [6,4].

2. Consider firstly conical domains corresponding to the function γ(xn) =Axn +B.

THEOREM 1. – Let γ(xn) = Axn + B, p > 0. There exists a constanta0 such that a functionu,satisfying (1) and (2) and the inequality|u(x)| bxan in the domainQ = x ∈ R

n, |x′| < γ(xn),0< xn <∞, with some constantsb > 0, 0< a< a0, tends to0 asxn →∞ uniformly inQ.

The proof is based on the following lemma.

LEMMA 1. – LetLu= 0 in Q2 = x∈Q, 1< xn < 2, |u| 1 if xn = 1 or xn = 2 and the condition:

∂u

∂νL+ k(x)u+ λb(x)

∣∣u(x)∣∣p−1u(x) = 0

707

Yu.V. Egorov, V.A. Kondratiev

onS be satisfied withλ λ0. There exists a constantθ ∈ ]0,1[ independent ofu such that|u(x)| θ forx∈Q2, 4/3< xn < 5/3.

Proof. –Using the maximum principle we can assume that 0< u(x) 1. Let h(xn) be a smooth functionsuch that h(xn) = 1 as 7/6 < xn < 11/6, h(xn) = 0 for xn > 2 and for xn < 1. Substituting in (3) thefunction φ(x) = h(xn)u(x) we can show that∫

Q

h|∇u|2 dx+ λ0

∫S

h|u|p+1 dS C0.

There exist τ ∈ ]7/6,4/3[ and τ ′ ∈ ]5/3,11/6[ such that∫xn=τ

|∇u|2 dx′ +∫xn=τ ′

|∇u|2 dx′ 12C0.

Then ∫xn=τ

|∇u|dx′ +∫xn=τ ′

|∇u|dx′ C1.

Put Q=Qτ,τ ′ , S = Sτ,τ ′ . Put v(x) = 1− u(x). Using an even extension over the boundary we can obtaina function w such that w = v in Q and Lw 0 in a neighborhood of Q.

Let P be a point of Q and v(P ) =m. Then by the weak Harnack inequality (see[4], p. 184) we have∫Q

v dxC2m.

Since ∫S

v dS ε

(∫Q

|∇u|2 dx)1/2

+C3ε−2/n

∫Q

vdxC0ε+C3C2mε−2/n

for all ε > 0, we obtain choosing ε= (C0/C2C3m)n/(n+2) that∫S

vdS C4mn/(n+2).

Therefore,

measn−1 S −∫S

udS C4mn/(n+2).

As we have proved, λ0

∫Sv dS C5. If λ0 is large enough so that C5/λ0 <C4m

n/(n+2), then we see that

measn−1 S 2C4mn/(n+2), and mC6 > 0. Therefore, u(P ) 1−C6.

Proof of Theorem1. – Put Mλ = supxn=λ |u(x)|. By the condition of Theorem 1 we have

mλ ≡Mλ/(bλa) 1.

Applying the homothety y = x/λ to the domain Q2λ, and using Lemma 1 above, we see that

Mλ θmax(M1,M2λ), 0< θ < 1.

Therefore, mλ θmax(m1λ−a,2am2λ). Iterating this estimate k times we obtain that

mλ max(θm1λ

−a, (2aθ)km2kλ

) max

(θm1λ

−a, (2aθ)k),

since mµ 1 for all µ. If a is so small that 2aθ < 1, we can see after choosing k large enough that

mλ θm1λ−a.

708

Semilinear elliptic boundary problems in an unbounded cone

It implies the inequality Mλ θM1 for λ 2. In the same way we have M4 θM2 θ2M1, andM2k θkM1, so that Mλ θlog2 λM1 = λsM1, s= log2 θ < 0. 3. Let now |x′| γ(xn), 0< xn <∞, x= (x′, xn) ∈Q. Let

F (t) =

∫ t

0

ds

γ(s).

Suppose that γ(s) = o(s) as s→∞, γ(s)→∞ as s→∞, and γ(T + s) Cγ(T ) if 0 < s C1γ(T ).Note that F (t)→∞ as t→∞.

THEOREM 2. – If p > 1, u is a solution of equation(1) in Q, satisfying(2), and|u(x)| b eaF (xn) in Qwith a small enough positive constanta, thenu(x)→ 0 asxn →∞ uniformly inQ.

Remark. – If 0< p 1 and Q is a domain as above, in Theorem 2, the maximal admissible growth is asin Theorem 1, i.e., |u(x)| bxan. So the result of Theorem 1 at this case cannot be improved.

4. We consider now solutions to the equation (1) in the domain

Q=x= (x′, xn) : |x′|<Axσn +B, 0< xn <∞

, 0 σ 1,

in Rn. We study the weak solutions of (1) satisfying the nonlinear boundary condition:

∂u

∂νL+ k(x)u− b(x)

∣∣u(x)∣∣p−1u(x) = 0

on the lateral surface S where p > 0, b(x) b0 > 0. It means that u ∈H1loc(Q)∩ Lp+1,loc(S) and∫

Q

[n∑

i,j=1

aij(x)∂u

∂xj

∂ϕ

∂xi−

n∑i

bi(x)∂u

∂xiϕ+

n∑i=1

∂ϕ

∂xi(ci(x)u)− d(x)uϕ

]dx

−∫S

b(x)∣∣u(x)∣∣p−1

u(x)ϕ(x)dS = 0 (4)

for all functions ϕ(x) ∈H1(Q), equal to 0 as xn = 0 and in a neighborhood of xn =∞. We suppose in this

section that xn/2+δn B(x) C , xn/2+1+δ

n d(x) C as xn 1; δ > 0.We show that a global solution of the problem cannot exist for some values of parameters p, σ and

indicate these values. It generalizes some results of B. Hu in [5].

THEOREM 3. – Letn 3, Q= x= (x′, xn) : |x′|<Axσn +B, 1 < xn <∞, 1n−1 < σ 1. Suppose

that v(x) satisfies(4) and v(x) 0 in Q. If 1 < p < 2 + 1σ(n−2) and σ > 0; or σ = 0 and p > 1, then

v(x)≡ 0.

THEOREM 4. – LetQ= x= (x′, xn) : |x′|<Axσn+B, 1< xn <∞, 0 σ(n−1)< 1. Suppose thatv(x) satisfies(4) andv(x) 0 in Q. If p > 1, thenv(x)≡ 0.

Our proof of Theorem 3 is based on the following lemma.

LEMMA 2. – Let σ(n − 1) > 1, σ 1, n 3, Q = x = (x′, xn) : |x′| < Axσn + B, 1 < xn <∞,A> 0, B 0. There exists a weak solution to the following problem:

LE = 0 in Q,∂E

∂νL+ k(x)E = 0 onS, E = 1 for xn = 1,

709

Yu.V. Egorov, V.A. Kondratiev

from the classH1loc(Q), such thatE(x) cx

σ(2−n)n , c > 0. Moreover,

limxn→∞

E(x) = 0,

∣∣∣∣∣∫

Ω1

(n∑

j=1

anj(x)∂E

∂xj+ k(x)u

)dx′

∣∣∣∣∣= c0 = 0.

THEOREM 5. – LetQ= x= (x′, xn) : |x′|<Axσn+B, 1< xn <∞, 0 σ(n−1) 1. Suppose thatv(x) satisfies(4) andv(x) 0 in Q. Suppose that

n∑j=1

anj(x)∂v(x)

∂xj+ k(x)v(x) 0 asxn = 1

and∂v(x)/∂νL + k(x)v(x) 0 onS. Thenv(x)≡ 0.

Proof. –Let ε be a small positive number. Put in the definition of weak solutions ϕ(x) = h(xn)v(x)+ε , where

h(xn) = 1 for 1< xn < T , h(xn) = 0 for 2T < xn, h is a smooth function for xn > 1. We have

J(T )≡∫QT

h(xn)1

(v + ε)2

n∑i,j=1

aij(x)∂v(x)

∂xj

∂v(x)

∂xidx

∫QT

h′(xn)1

v+ ε

n∑j=1

anj(x)∂v(x)

∂xjdx−

∫QT

h(xn)d(x)v

v+ εdx

−∫QT

[h(xn)

1

v+ ε

n∑j=1

bj(x)∂v(x)

∂xj+

n∑j=1

cj(x)v(x)∂

∂xj

(h

v+ ε

)]dx,

and therefore, J(T )C∫QT

T−2 dxC1T−2+1+σ(n−1).

Letting T tend to infinity we see that v(x) ≡ 0 if σ(n − 1) < 1. If σ(n − 1) = 1 the proof is slightlymodified.

References

[1] Egorov Yu.V., Kondratiev V.A., On a Oleinik’s problem, Uspekhi Mat. Nauk 57 (1997) 159–160.[2] Egorov Yu.V., Kondratiev V.A., On asymptotic behavior in an infinite cylinder of solutions to an elliptic solution of

second order, Appl. Anal. 71 (1999) 25–41.[3] Egorov Yu.V., Kondratiev V.A., On the asymptotic behavior of solutions to a semi-linear elliptic boundary problem

in an unbounded domain, C. R. Acad. Sci. Paris, Série I 330 (2000) 785–790.[4] Gilbarg D., Trudinger H., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1977.[5] Hu B., Nonexistence of a positive solution of the Laplace operator with a nonlinear boundary condition, Differ. and

Integ. Eq. 7 (1994) 301–313.[6] Stampacchia G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus,

Ann. Inst. Fourier, Grenoble 15 (1965) 189–258.

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