Measure boundary value problem for semilinear elliptic equations with critical Hardy potentials

HAL Id: hal-01071455

https://hal.archives-ouvertes.fr/hal-01071455v2

Preprint submitted on 28 Oct 2014

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Measure boundary value problem for semilinear elliptic equations with critical Hardy potentials

Konstantinos Gkikas, Laurent Veron

To cite this version:

Konstantinos Gkikas, Laurent Veron. Measure boundary value problem for semilinear elliptic equa-

tions with critical Hardy potentials. 2014. �hal-01071455v2�

Partial Differential Equations

MEASURE BOUNDARY VALUE PROBLEM FOR SEMILINEAR ELLIPTIC EQUATIONS WITH CRITICAL HARDY POTENTIALS

Konstantinos T. Gkikas

, Laurent V´eron

R´esum´e. Let Ω ⊂ R^N be a bounded C² domain and L^κ = −∆−_d^κ2 the Hardy operator whered= dist (., ∂Ω) and 0< κ≤ ¹4. Letα±= 1±√

1−4κ be the two Hardy exponents,λκthe first eigenvalue ofL^κwith corresponding positive eigenfunctionφκ. Ifgis a continuous nondecreasing function satisfying R∞

1 (g(s) +|g(−s)|)s⁻²

2N−2+α+

2N−4+α+ds <∞, then for any Radon measuresν ∈ M_φ

κ(Ω) andµ∈M(∂Ω) there exists a unique weak solution to problemPν,µ: L^κu+g(u) =ν in Ω,u=µon∂Ω. Ifg(r) =|r|^q−1u(q >1) we prove that, in the supercritical range ofq, a necessary and sufficient condition for solving P0,µwithµ >0 is thatµis absolutely continuous with respect to the capacity associated to the Besov spaceB²⁻

2+α+ 2q′ ,q^′

(R^N−1). We also characterize the boundary removable sets in terms of this capacity. In the subcritical range of qwe classify the isolated singularities of positive solutions.

Probl` emes aux limites avec donn´ ees mesures pour des ´ equations semi lin´ eaires elliptiques avec des potentiels de Hardy critiques

R´esum´e.Soient Ω⊂R^Nun domaine de classeC²etL^κ=−∆−_d^κ2 l’op´erateur de Hardy o`ud= dist (., ∂Ω) et 0< κ≤ ¹4. Soientα±= 1±√

1−4κles deux exposants de Hardy,λκpremi`ere valeur propre deL^κetφκla fonction propre positive correspondante. Si g est une fonction continue croissante v´erifiant R∞

1 (g(s) +|g(−s)|)s⁻²

2N−2+α+

2N−4+α+ds <∞, alors pour toutes mesures de Radon ν ∈M_φκ(Ω) etµ∈M(∂Ω) il existe une unique solution faible au probl`eme Pν,µ :L^κu+g(u) = ν dans Ω, u = µsur ∂Ω. Sig(r) = |r|^q−1u (q > 1) nous d´emontrons qu’une condition n´ecessaire et suffisante pour r´esoudreP0,µ

avec µ > 0 est que µ soit absolument continue par rapport `a la capacit´e associ´ee `a l’espace de BesovB²⁻

2+α+ 2q′ ,q^′

(R^N−1). Nous caract´erisons les en- sembles ´eliminables pour les valeurs sur critiques deq. Dans le cas sous-critique nous classifions les singularit´es isol´ees au bord des solutions positives.

Version fran¸ caise abr´ eg´ ee. Soit Ω un domaine de R

de classe C

. On d´esigne par d(x) la distance de x ` a ∂Ω et on d´efinit l’op´erateur de Hardy dans Ω par

(1) L

u = − ∆u − κ

d

u o` u 0 < κ ≤

et ses exposants caract´eristiques

(2) α

= 1 + √

1 − 4κ α

= 1 − √ 1 − 4κ.

1. Centro de Modelamiento Matem`atico, Universidad de Chile, Santiago de Chile, Chile. E- mail : [email protected]. Supported by Fondecyt Grant 3140567

2. Laboratoire de Math´ematiques et Physique Th´eorique, CNRS UMR 7350, Facult´e des Sciences, 37200 Tours France. E-mail : [email protected]. Supported by MATH-Amsud pro- gram QUESP

1

2 SEMILINEAR EQUATIONS WITH HARDY POTENTIALS

On supposera Ω convexe si κ =

. Il est bien connu que sous ces conditions L

poss`ede une premi`ere valeur propre λ

> 0 d´efinie par

(3) λ

:= inf

u∈H₀¹(Ω)\{0}

Z

Ω

|∇ u |

dx Z

Ω

d

u

dx .

La premi`ere fonction propre positive associ´ee φ

n’appartient ` a H

(Ω) que si 0 <

κ <

, et dans tous les cas elle v´erifie φ

(x) ∼ (d(x))

au voisinage de ∂Ω. On d´enote par G

et K

les noyaux de Green et de Poisson de L

dans Ω et par ω

la mesure L

-harmonique dans Ω (x

∈ Ω). Si g est une fonction continue et croissante sur R telle que g(0) ≥ 0, nous ´etudions tout d’abord le probl`eme (P

) suivant :

(4) L

u + g(u) = ν in Ω

u = µ in ∂Ω, o` u ν, µ sont des mesures de Radon.

Th´ eor` eme 1. Supposons que g v´erifie

(5) Z

1

(g(s) + | g( − s) | ) s

−2^N−1+

α+

2 N−2+α+

2

ds < ∞ ;

alors pour toutes mesures de Radon ν et µ dans Ω et ∂Ω respectivement, ν v´erifiant en outre R

Ω

φ

d | ν | < ∞ , il existe une unique fonction u = u

∈ L

(Ω) telle que g ◦ u ∈ L

(Ω) v´erifiant

(6)

Z

Ω

(u L

ζ + ζg ◦ u) dx = Z

Ω

Z

Ω

G

(x, y)dν(y)ζ(x)dx +

Z

Ω

Z

∂Ω

K

(x, y)dµ(y) L

ζ(x)dx pour toute ζ ∈ X

(Ω) o` u

(7) X

(Ω) = { ζ ∈ H

(Ω) : (φ

)

ζ ∈ H

(Ω, φ

dx), (φ

)

L

ζ ∈ L

(Ω) } . En outre l’application (ν, µ) 7→ u

de M

(Ω) × M (∂Ω) dans L

(Ω) est croissante et stable pour la convergence faible des mesures.

La d´emonstration utilise des estimations des noyaux de Green et de Poisson d´eduits des propri´et´es de la mesure ω

. Dans le cas o` u g ◦ u = | u |

u, (5) est v´erifi´ee si 0 < q < q

:=

. Dans le cas q > 1 nous d´enotons par C

2−^2+α_2q′⁺,q^′

la capacit´e associ´ee ` a l’espace de Besov B

2+α+

2q′ ,q^′

( R

) et nous d´emontrons : Th´ eor` eme 2. Soit q ≥ q

et ν ∈ M

(∂Ω). Alors le probl`eme

(8) L

u + | u |

u = 0 in Ω

u = µ in ∂Ω

admet une unique solution u := u

si et seulement si pour tout bor´elien E ⊂ ∂Ω,

(9) C

2−^2+α_2q′⁺,q^′

(E) = 0 = ⇒ µ(E) = 0.

Nous caract´erisons aussi les sous ensembles du bord ´eliminables pour l’´equation

(10) L

u + | u |

u = 0 in Ω.

D´efinissons

(11) W (x) =

( (d(x))

if 0 < κ <

p d(x) ln | d(x) | if κ =

.

Th´ eor` eme 3. Soit q > 1 et K ⊂ ∂Ω un sous-ensemble compact. Toute solution u ∈ C(Ω \ { K } ) de (10) qui v´erifie

(12) lim

u(x)

W (x) = 0 ∀ y ∈ ∂Ω \ { K } , est identiquement nulle dans Ω si et seulement si C

2−^2+α+_2q′ ,q^′

(K) = 0.

Nous montrons que si q > 1, toute solution positive de (10) dans Ω admet une trace au bord repr´esent´ee par une mesure de Borel r´eguli`ere. En supposant que 0 ∈ ∂Ω et 1 < q < q

, nous ´etudions aussi le comportement au voisinage de 0 des solutions positives de (10) qui v´erifient (12) avec K = { 0 } .

———————————————————————————————

Let Ω be a bounded C

domain in R

, N ≥ 3 and d(x) = dist (x, Ω). We define λ

by (3). It is well known that λ

∈ (0,

]. Also we define the Hardy operator L

in Ω by (1) with 0 < κ < λ

if λ

<

or 0 < κ ≤

if λ

=

and the characteristic exponents by (2). We assume that Ω is convex if κ =

. It is well known that L

possesses a first eigenvalue λ

> 0 defined by

(13) λ

:= inf

u∈H₀¹(Ω)\{0}

Z

Ω

|∇ u |

dx − κ Z

Ω

d

u

dx

Z

Ω

u

dx

.

The first positive eigenfunction φ

> 0 may or may not belong to H

(Ω) according 0 < κ <

or κ =

, and φ

(x) ∼ (d(x))

, |∇ φ

(x) | ∼ (d(x))

as d(x) → 0.

Let G

(x, y) (resp. K

(x, y)) be the Green (resp. Poisson) kernel of L

, then

(14) G

(x, y) ∼ min



 

  1

| x − y |

, (d(x)) α

2 (d(y))

| x − y |



 

 

∀ (x, y) ∈ Ω × Ω, x 6 = y,

(15) K

(x, y) ∼ (d(x))

| x − y |

∀ (x, y) ∈ Ω × ∂Ω.

The corresponding Green and Poisson operators are denoted by G

[.] and K

[.].

We first consider the boundary value problem (4) where g is a continuous nonde- creasing function such that g(0) ≥ 0 and ν and µ are Radon measures in Ω and ∂Ω respectively. We say that g is a subcritical nonlinearity if it satisfies (5).

Theorem 1. Assume that g is a subcritical nonlinearity. Then for all (ν, µ) ∈

M

(Ω) × M (∂Ω) there exists a unique function u = u

∈ L

(Ω) such that

g ◦ u ∈ L

(Ω) verifying (6) for all ζ in the space of test functions X

(Ω) defined by

(7). Furthermore the mapping (ν, µ) 7→ u

from M

(Ω) × M (∂Ω) into L

(Ω) is

nondecreasing and stable for the weak convergence of measures.

4 SEMILINEAR EQUATIONS WITH HARDY POTENTIALS

When g(u) = | u |

u with q > 0, the inequality (6) means (16) 0 < q < q

:= 2N + α

2N + α

− 4 .

When q ≥ q

not all the measures µ are eligible for solving (8). We denote by C

2−^2+α_2q′⁺,q^′

the capacity associated to the Besov space B

2+α+ 2q′ ,q^′

( R

).

Theorem 2. Let q > 1 and ν ∈ M

(∂Ω). Then problem (8) admits a solution if and only if µ is absolutely continuous with respect to C

2−^2+α_2q′⁺,q^′

, i.e. for any Borel set E ⊂ ∂Ω, implication (9) holds.

We also characterize the boundary removable sets for (10).

Theorem 3. Let q > 1 and K ⊂ ∂Ω be compact. Any u ∈ C(Ω \ { K } ) solution of (10) which verifies (12) is identically zero in Ω if and only if C

2−^2+α_2q′⁺,q^′

(K) = 0.

When 1 < q < q

only the empty set has zero capacity. There exist singular solutions of (10) with an isolated singularity on the boundary, either solutions u

of (8) with µ = kδ

for k > 0 and a ∈ ∂Ω or solutions u

= lim

u

. This very singular solution is described by considering the following problem on the half upper-sphere S

= { x = (x

, ..., x

) ∈ R

: | x | = 1, x

> 0 }

(17) − ∆

ω − ℓ

ω −

ω + | ω |

ω = 0 in S

ω = 0 in ∂S

where ∆

is the Laplace-Beltrami operator on S

, (e

, ..., e

) is the canonic basis in R

, σ =

and

ℓ

= 2

q − 1

2q q − 1 − N

.

The spherical Hardy operator ω 7→ L

:= − ∆

ω −

ω on S

admits a first eigenvalue µ

defined by

(18) µ

= inf

ψ∈H¹₀(S₊^N−1)\{0}

Z

S^N−+ ¹

|∇

ψ |

− κ(e

.σ)

ω

dS

Z

Ω

(e

.σ)

ψ

dS

.

We prove that µ

=

N +

− 2

with corresponding positive eigenfunction ρ

= (e

.σ)

. There exists a second eigenvalue µ

= µ

+N +α

− 1 with N − 1 independent eigenfunctions ρ

= (e

.σ)

e

.σ for j = 1, ..., N − 1. We denote by E

the set of functions ω such that ρ

ω ∈ L

ρ^q+1κ

(S

) ∩ H

(S

, ρ

dS) which satisfy (17), and by E

the set of positive solutions.

Theorem 4. I- If q ≥ q

, E

= {∅} .

II- If 1 < q < q

, E

= { 0, ωκ } where ω

is the unique positive solution of (17).

III- If q

≤ q < q

, E

= { 0, ω

, − ω

} where q

:= 2N + 2 + α

2N − 2 + α

.

This allows us to describe the isolated boundary singularities of positive solutions

of (10). Assume 0 ∈ ∂Ω and e

is the outward normal unit vector to ∂Ω at 0.

Theorem 5. Assume , 1 < q < q

and u ∈ C(Ω \ { 0 } ) is a positive solution of (10) which verifies(12) with K = { 0 } . Then

(i) either there exists k ≥ 0 such that u = u

and lim

| x |

u(x) = c

k(e

.

)

,

(ii) or lim

| x |

u(x) = ω

(

).

The above two convergence hold locally uniformly on S

.

We can also define a boundary trace of any positive solution u of (10). For δ > 0 small enough, we denote by ω

δ

the harmonic measure relative to the operator L

in Ω

= { x ∈ Ω : d(x) > δ } where x

∈ Ω (with d(x

) ≥ δ

> δ) and set Σ

= ∂Ω

. Theorem 6. Assume q > 1 and u ∈ C(Ω \ { 0 } ) is a positive solution of (10) in Ω.

Then for any y ∈ ∂Ω, the following dichotomy occurs :

(i) Either there exist an open subset U ⊂ R

containing y and a positive Radon measure λ

on ∂Ω ∩ U such that

(19) lim

δ→0

Z

Σδ∩U

Z (x)u(x)dω

=

Z

∂Ω∩U

Zdλ

∀ Z ∈ C

(U).

(ii) Or for any open subset U ⊂ R

containing y, there holds

(20) lim

δ→0

Z

Σδ∩U

u(x)dω

= ∞ .

The set R

of y such that (i) holds is relatively open in ∂Ω and it carries a positive Radon measure µ

such that (19) occurs with U replaced by R

and λ

by µ

; its complement S

in ∂Ω has the property that (20) occurs for any open subset U such that U ∩ S

6 = {∅} .

Abridged proof of Theorem 1. Let (ν, µ) ∈ M

(Ω) × M (∂Ω). For λ > 0 we set (21) E

(ν) = { x ∈ Ω : G

[ | ν | ](x) > λ } , E

(ν) =

Z

Eλ(ν)

φ

dx,

and

(22) F

(ν) = { x ∈ Ω : K

[ | µ | ](x) > λ } , F

(µ) = Z

Eλ(ν)

dx, and prove

(23) E

(ν) + F

(µ) ≤ c k ν k

+ k µ k

λ

!

2N+α+

2N+α+−4

.

If g satisfies (5) and { (ν

, µ

) } is a sequence of smooth functions which converges in the weak-star topology of measures to (ν, µ), then the corresponding solutions { u

} of problem P

defined in (4) converges to some u and { g ◦ u

} converges to g ◦ u in L

by Vitali convergence theorem. This implies u = u

. Uniqueness holds by adapting Brezis estimates and using monotonicity.

Abridged proof of Theorem 2. Using estimate (15) and the harmonic lifting in Besov spaces introduced in [9, Sect. 3] we prove that for any µ ∈ M (∂Ω) there holds

(24) 1

c k µ k

B⁻²⁺

2+α+ 2q′ ,q

≤

Z

Ω

( K

[ | µ | ])

φ

dx ≤ c k µ k

B⁻²⁺

2+α+ 2q′ ,q

,

6 SEMILINEAR EQUATIONS WITH HARDY POTENTIALS

for some c = c(Ω, κ, q) > 0. If the above quantity is finite, we can solve (8) with such a µ. If µ ∈ B

2+α+ 2q′ ,q

(∂Ω) ∩ M

(∂Ω), it is absolutely continuous with respect to the capacity C

2−^2+α+_2q′ ,q^′

. Finally, if µ ∈ M

(∂Ω) is absolutely continuous with respect to the capacity C

2−^2+α_2q′⁺,q^′

, there exists an increasing sequence { µ

} ⊂ B

2+α+

2q′ ,q

(∂Ω) ∩ M

(∂Ω) which converges to µ. This implies that u

converges to u

in L

(Ω).

Conversely, if µ ∈ M

(∂Ω) is such that there exists a solution u

to (8), we use a variant of the optimal lifting R[.] defined in [7, Sect. 1] to prove that for any η ∈ C

(∂Ω) such that 0 ≤ η ≤ 1 there holds

(25) Z

∂Ω

ηdµ ≤ c Z

Ω

u

ζdx + c Z

Ω

u

ζdx

Z

Ω

φ

dx + k η k

B²⁻

2+α+ 2q′ ,q′

. Here ζ = φ

(R[η])

and R : C

(∂Ω) 7→ C

(Ω) is a linear mapping which satisfies 0 ≤ η ≤ 1 = ⇒ 0 ≤ R[η] ≤ 1 and R[η] ⌊

= η. If K ⊂ ∂Ω is a compact set with zero C

2−^2+α_2q′⁺,q

-capacity, there exists a sequence { η

} ⊂ C

(∂Ω) such that 0 ≤ η

≤ 1, η

= 1 on K and k η

k

B²⁻

2+α+

2q′ ,q′

→ 0. This implies φ

(R[η

])

→ 0 and finally µ(K) = 0.

Abridged proof of Theorem 3. If K ⊂ ∂Ω is compact with C

2−^2+α_2q′⁺,q

(K) > 0, its capacitary measure µ

belongs to B

2+α+

2q′ ,q

(∂Ω) ∩ M

(∂Ω) . Thus u

exists and K is not removable. Conversely by using again optimal lifting, and test functions of the form φ

(R[1 − η])

where 0 ≤ η ≤ 1 and η = 1 in a neighborhood of K, we prove first that that u ∈ L

κ

(Ω) and finally that u = 0.

Abridged proof of Theorems 4-5. Existence is obtained in minimizing the functional J

defined over L

ρ^q+1κ

(S

) ∩ H

(S

, ρ

dS) by (26) J

(w) :=

Z

S₊^N−1

|∇

w |

− (ℓ

− µ

)w

+ 2

q + 1 ρ

| w |

ρ

dS.

A non-trivial minimizer w exists if ℓ

> µ

(defined by (18)), i.e. 1 < q < q

, and ω = ρ

w satisfies (17). Nonexistence in standard since µ

< ℓ

if and only if 1 < q < q

. For uniqueness we assume that ω

(j = 1, 2) are positive solutions of (17) and we set w

=

κ

. Then

− div

.(ρ

∇

w

) + (µ

− ℓ

)ρ

w

+ ρ

w

= 0 on S

Since w

∼ ρ

α+

κ2

and |∇

w

| ∼ ρ

α+

2 −1

κ

near ∂S

, we use Green formula and get Z

S^N₊⁻¹

∇

w

w

− ∇

w

w

. ∇

(w

− w

) + ρ

(w

− w

)(w

− w

)

ρ

dS = 0,

thus w

= w

. For statement III we first prove, by the method used in [11, Th 3.1],

that any solution ω depends only on the azimuthal angle θ ∈ [0,

]. Then we show

that the corresponding ODE verified by ω admits only constant sign solutions. For

Theorem 5, we construct a barrier function as in [8, Appendix] and obtain (27) u(x) ≤ c | x |

(d(x))

∀ x ∈ Ω.

With this estimate we adapt the scaling method developed in [10, Sect. 3.3] to obtain the classification result.

R´ ef´ erences

[1] C.Bandle, V.Moroz and W.Reichel,Boundary blow up type sub-solutions to semilinear elliptic equations with Hardy potential, J. London Math. Soc.77, 503-523 (2008).

[2] G. Barbatis, S. Filippas and A. Tertikas,A unified approach to improvedL^pHardy inequalities with best constants.Trans. Amer. Math. Soc.356(2003), 2169–2196.

[3] H. Brezis and M. Marcus,Hardy’s inequalities revisited Ann. Sc. Norm. Super. Pisa Cl. Sci.

25(1997), 217-237.

[4] J. Davila, L. Dupaigne,Hardy-type inequalities,J. Eur. Math. Soc. (JEMS)6(2004), 335-365.

[5] S. Filippas, L. Moschini. and A. Tertikas,Sharp two-sided heat kernel estimates for critical Schr¨odinger operators on bounded domains.Comm. Math. Phys.273(2007), 237-281.

[6] M. Marcus & P. T. Nguyen, Moderate solutions of semilinear elliptic equations with Hardy potential. ArXiv :1407.3572v1 (2014).

[7] M. Marcus, L. V´eron,Removable singularities and boundary trace.J. Math. Pures Appl.80 (2001), 879-900.

[8] M. Marcus, L. V´eron, The boundary trace and generalized boundary value problem for se- milinear elliptic equations with coercive absorption. Comm. Pure Appl. Math.56(2003), 689-731.

[9] M. Marcus, L. V´eron, Boundary trace of positive solutions of supercritical semilinear el- liptic equations in dihedral domains. Ann. Sc. Norm. Super. Pisa Cl. Sci., to appear.

arXiv :1309.7778.

[10] Nguyen Phuoc T., L. V´eron, Boundary singularities of solutions to elliptic viscous Hamil- ton–Jacobi equations.J. Funct. Anal.263(2012), 1487–1538.

[11] L. V´eron,Geometric invariance of singular solutions of some nonlinear partial differential equations.Indiana Univ. Math. J.38(1989), 75-100.