Semilinear fractional elliptic equations involving measures

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Semilinear fractional elliptic equations involving measures

Huyuan Chen, Laurent Veron

To cite this version:

Huyuan Chen, Laurent Veron. Semilinear fractional elliptic equations involving measures. 2013.

�hal-00820401v2�

Semilinear fractional elliptic equations involving measures

Huyuan Chen¹ Laurent V´eron² Laboratoire de Math´ematiques et Physique Th´eorique

CNRS UMR 7350

Universit´e Fran¸cois Rabelais, Tours, France

Abstract

We study the existence of weak solutions to (E) (−∆)^αu+g(u) =ν in a bounded regular domain Ω inR^N(N ≥2) which vanish inR^N\Ω, where (−∆)^α denotes the fractional Laplacian with α∈ (0,1), ν is a Radon measure andgis a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case whereνis Dirac measure, we characterize the asymptotic behavior of the solution. When g(r) = |r|^k−1r with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.

Contents

1 Introduction 2

2 Linear estimates 5

2.1 The Marcinkiewicz spaces . . . 5 2.2 Non-homogeneous problem . . . 9

3 Proof of Theorem 1.1 16

4 Applications 21

4.1 The case of a Dirac mass . . . 21 4.2 The power case . . . 23 Key words: Fractional Laplacian, Radon measure, Dirac measure, Green kernel, Bessel capacities.

MSC2010: 35R11, 35J61, 35R06

1chenhuyuan@yeah.net

2Laurent.Veron@lmpt.univ-tours.fr

1 Introduction

Let Ω⊂R^N be an open boundedC² domain andg:R7→Rbe a continuous function. We are concerned with the existence of weak solutions to the semilinear fractional elliptic problem

(−∆)^αu+g(u) =ν in Ω,

u= 0 in Ω^c, (1.1)

where α ∈ (0,1), ν is a Radon measure such that R

Ωρ^βd|ν| <∞ for some β∈[0, α] andρ(x) =dist(x,Ω^c). The fractional Laplacian (−∆)^αis defined by

(−∆)^αu(x) = lim

ǫ→0⁺(−∆)^α_ǫu(x), where forǫ >0,

(−∆)^α_ǫu(x) =− Z

R^N

u(z)−u(x)

|z−x|^N+2αχ_ǫ(|x−z|)dz (1.2) and

χ_ǫ(t) =

(0, if t∈[0, ǫ], 1, if t> ǫ.

Whenα = 1, the semilinear elliptic problem

−∆u+g(u) =ν in Ω,

u= 0 on ∂Ω, (1.3)

has been extensively studied by numerous authors in the last 30 years. A fundamental contribution is due to Brezis [7], Benilan and Brezis [2], where ν is a bounded measure in Ω and the functiong:R→Ris nondecreasing, positive on (0,+∞) and satisfies the subcritical assumption:

Z _+∞

1

(g(s)−g(−s))s^{−2N−1N−2}ds <+∞.

They proved the existence and uniqueness of the solution for problem (1.3).

Baras and Pierre [1] studied (1.3) when g(u) = |u|^p−1u for p >1 and ν is absolutely continuous with respect to the Bessel capacityC_2, ^p

p−1, to obtain a solution. In [37] V´eron extended Benilan and Brezis results in replacing the Laplacian by a general uniformly elliptic second order differential operator with Lipschitz continuous coefficients; he obtained existence and uniqueness results for solutions, when ν ∈ M(Ω, ρ^β) with β ∈ [0,1] where M(Ω, ρ^β) denotes the space of Radon measures in Ω satisfying

Z

Ω

ρ^βd|ν|<+∞, (1.4)

M(Ω, ρ⁰) = M^b(Ω) is the set of bounded Radon measures and g is nonde- creasing and satisfies theβ-subcritical assumption:

Z +∞

1

(g(s)−g(−s))s^{−2N+β−1N+β−2}ds <+∞.

The study of general semilinear elliptic equations with measure data have been investigated, such as the equations involving measures boundary data which was initiated by Gmira and V´eron [20] who adapted the method introduced by Benilan and Brezis to obtain the existence and uniqueness of solution. This subject has been vastly expanded in recent years, see the papers of Marcus and V´eron [25, 26, 27, 28], Bidaut-V´eron and Vivier [5], Bidaut-V´eron, Hung and V´eron [4].

Recently, great attention has been devoted to non-linear equations in- volving fractional Laplacian or more general integro-differential operators and we mention the reference [8, 9, 10, 14, 15, 24, 30, 32]. In particular, the authors in [23] used the duality approach to study the equations of

(−∆)^αv=µ in R^N,

where µ is a Radon measure with compact support. In [14] the authors obtained the existence of large solutions to equation

(−∆)^αu+g(u) =f in Ω, (1.5) where Ω is a bounded regular domain. In [13] we considered the properties of possibly singular solutions of (1.5) in punctured domain . It is a well-known fact [36] that for α = 1 the weak singular solutions of (1.5) in punctured domain are classified according the type of singularities they admits: either weak singularities with Dirac mass, or strong singularities which are the upper limit of solutions with weak singularities. One of our interests is to extend these properties to anyα∈(0,1) and furthermore to consider general Radon measures.

In this paper we study the existence and uniqueness of solutions of (1.1) in a measure framework. Before stating our main theorem we make precise the notion of weak solution used in this article.

Definition 1.1 We say that u is a weak solution of (1.1), if u ∈ L¹(Ω), g(u)∈L¹(Ω, ρ^αdx) and

Z

Ω

[u(−∆)^αξ+g(u)ξ]dx= Z

Ω

ξdν, ∀ξ∈X_α, (1.6) where X_α⊂C(R^N) is the space of functions ξ satisfying:

(i) supp(ξ)⊂Ω,¯

(ii) (−∆)^αξ(x) exists for all x∈Ωand |(−∆)^αξ(x)| ≤C for some C >0, (iii) there exist ϕ∈L¹(Ω, ρ^αdx) and ǫ0 >0 such that |(−∆)^α_ǫξ| ≤ϕa.e. in Ω, for all ǫ∈(0, ǫ₀].

We notice that for α = 1, the test spaceX_α is used as C₀^1,L(Ω), which has similar properties like (i) and (ii). The counter part for the Lapla- cian of assumption (iii) would be that the difference quotient ∇_xj,h[u](.) :=

h⁻¹[∂xju(.+hej)−∂xju(.)] is bounded by anL¹-function, which is true since

∇_xj,h[u](x) =h⁻¹ Z h

0

∂_x²_j,xju(x+se_j)ds.

We denote by Gthe Green kernel of (−∆)^α in Ω and by G[.] the Green operator defined by

G[f](x) = Z

Ω

G(x, y)f(y)dy, ∀f ∈L¹(Ω, ρ^αdx). (1.7) ForN ≥2, 0< α <1 and β ∈[0, α], we define the critical exponent

k_α,β = ( _N

N−2α, if β ∈[0,^N−2α_N α],

N+α

N−2α+β, if β ∈(^N−2α_N α, α]. (1.8) Our main result is the following:

Theorem 1.1 Assume Ω⊂ R^N (N ≥ 2) is an open bounded C² domain, α ∈ (0,1), β ∈ [0, α] and k_α,β is defined by (1.8). Let g : R → R be a continuous, nondecreasing function, satisfying

g(r)r ≥0, ∀r ∈R and Z ∞

1

(g(s)−g(−s))s^{−1−kα,β}ds <∞. (1.9) Then for any ν∈M(Ω, ρ^β) problem (1.1) admits a unique weak solution u.

Furthermore, the mapping: ν 7→u is increasing and

−G(ν−)≤u≤G(ν+) a.e. in Ω (1.10) whereν₊andν₋ are respectively the positive and negative part in the Jordan decomposition of ν.

We note that forα= 1 and β ∈[0,1), we have k_1,β > N +β

N −2 +β, (1.11)

where k_1,β is given in (1.8) and the number in right hand side of (1.11) is from Theorem 3.7 in [37]. Inspired by [20, 37], the existence of solution could be extended in assuming thatg: Ω×R→Ris continuous and satisfiesthe (N, α, β)-weak-singularity assumption, that is, there existsr₀>0 such that

g(x, r)r ≥0, ∀(x, r)∈Ω×(R\(−r₀, r₀)),

and

|g(x, r)| ≤g(|r|),˜ ∀(x, r)∈Ω×R,

where ˜g: [0,∞)→[0,∞) is continuous, nondecreasing and satisfies Z _∞

1

˜

g(s)s^{−1−kα,β}ds <∞.

We also give a stability result which shows that problem (1.1) is weakly closed in the space of measures M(Ω, ρ^β). In the last section we characterize the behaviour of the solutionuof (1.1) whenν=δ_afor somea∈Ω. We also study the case where g(r) = |r|^k−1r when k ≥ k_α,β, which doesn’t satisfy (1.9). We show that a necessary and sufficient condition in order a weak solution to problem

(−∆)^αu+|u|^k−1u=ν in Ω,

u= 0 in Ω^c, (1.12)

to exist where ν is a positive bounded measure is that ν vanishes on compact subsets K of Ω with zero C_2α,k^′ Bessel-capacity.

The paper is organized as follows. In Section 2 we give some properties of Marcinkiewicz spaces and obtain the optimal indexk for which there holds kG(ν)k_Mk(Ω,ρ^γdx)≤CkνkM(Ω,ρ^β). (1.13) We also gives some integration by parts formulas and prove a Kato’s type inequalities. In Section 3, we prove Theorem 1.1. It Section 4 we give applications the cases where the measure is a Dirac mass and where the nonlinearity is a power function.

2 Linear estimates

2.1 The Marcinkiewicz spaces

We recall the definition and basic properties of the Marcinkiewicz spaces.

Definition 2.1 Let Ω⊂R^N be an open domain and µ be a positive Borel measure inΩ. For κ >1, κ^′ =κ/(κ−1) andu∈L¹_loc(Ω, dµ), we set kuk_Mκ(Ω,dµ)= inf{c∈[0,∞] :

Z

E

|u|dµ≤c Z

E

dµ _κ¹_′

, ∀E ⊂Ω Borel set}

(2.1) and

M^κ(Ω, dµ) ={u∈L¹_loc(Ω, dµ) :kuk_Mκ(Ω,dµ)<∞}. (2.2) M^κ(Ω, dµ) is called the Marcinkiewicz space of exponentκ or weak L^κ space andk.k_Mκ(Ω,dµ) is a quasi-norm. The following property holds.

Proposition 2.1 [3, 16] Assume 1 ≤ q < κ < ∞ and u ∈ L¹_loc(Ω, dµ).

Then there exists C(q, κ)>0 such that Z

E

|u|^qdµ≤C(q, κ)kuk_M^κ_(Ω,dµ) Z

E

dµ 1−q/κ

, for any Borel setE of Ω.

Forα ∈(0,1) andβ, γ∈[0, α] we set k₁(t) = γ

α +N−(N −2α)_α^γ

N −2α+t , k₂(t) =γ+N −(N−2α)_α^γ

N −2α+t t (2.3) and

t_α,β,γ = min{t∈[0, α] : k₂(t)

k₁(t) ≥β}. (2.4)

Remark 2.1 The quantity t_α,β,γ is well defined, since k₂(α)

k1(α) = γ+α^{N−(N−2α)}

γ

N−α α

γ

α+ ^{N−(N−2α)}

γ α

N−α

=α≥β.

Remark 2.2 The function t7→k₁(t) is decreasing in [0, α] with the follow- ing bounds

k₁(0) = N

N−2α and k₁(α) = N+γ N −α >1.

Remark 2.3 The function t7→ ^k_k^2(t)

1(t) is increasing in [0, α], since k₂(t)

k₁(t) ′

= [N −(N−2α)_α^γ](N +γ) k²₁(t) >0.

As a consequence (2.4) is equivalent to

t_α,β,γ= max{0, t_β}, (2.5)

where

t_β = βN −(N−2α)γ

N −(N −3α+β)^γ_α. (2.6)

is the solution of ^k_k^2(t)

1(t) =β.

Proposition 2.2 LetΩ⊂R^N (N ≥2) be an open bounded C² domain and ν∈M(Ω, ρ^β) withβ ∈[0, α]. Then

kG[ν]k_{Mkα,β,γ(Ω,ργdx)}≤CkνkM(Ω,ρ^β), (2.7) where γ ∈ [0, α], G[ν](x) = R

ΩG(x, y)dν(y) where G is Green’s kernel of (−∆)^α and

k_α,β,γ =

( _N+γ

N−2α+β, if γ ≤ _N−2α^Nβ ,

N

N−2α, if not. (2.8)

Proof. Forλ >0 andy∈Ω, we denote

A_λ(y) ={x∈Ω\ {y}:G(x, y)> λ} and m_λ(y) = Z

Aλ(y)

ρ^γ(x)dx.

From [11], there existsC >0 such that for any (x, y)∈Ω×Ω, x6=y, G(x, y)≤Cmin

1

|x−y|^N−2α, ρ^α(x)

|x−y|^N−α, ρ^α(y)

|x−y|^N−α

(2.9) and

G(x, y)≤C ρ^α(y)

ρ^α(x)|x−y|^N−2α. (2.10) Therefore, if γ∈[0, α] andx∈A_λ(y), there holds

ρ^γ(x)≤ Cρ^γ(y)

λ^αγ|x−y|^{(N−2α)γα}. (2.11) Lett∈[0, α] be such that ^k_k^2(t)

1(t) ≥β, wherek₁(t) andk₂(t) are given in (2.3), then

G(x, y)≤

C

|x−y|^N−2α

1−_α^t

Cρ^α(y)

|x−y|^N−α _α^t

= Cρ^t(y)

|x−y|^N−2α+t. We observe that

A_λ(y)⊂

x∈Ω\ {y}: Cρ(y)^t

|x−y|^N−2α+t > λ

⊂D_λ(y) where D_λ(y) := n

x∈Ω :|x−y|<(^Cρ_λ^t(y))^N−2α+t1 o

; together with (2.11), this implies

m_λ(y)≤ Z

Dλ(y)

Cρ^γ(y)

λ^γα|x−y|^{(N−2α)γα}dx≤Cρ(y)^k2(t)λ^−k1(t). For any Borel setE of Ω, we have

Z

E

G(x, y)ρ^γ(x)dx≤ Z

Aλ(y)

G(x, y)ρ^γ(x)dx+λ Z

E

ρ^γ(x)dx and

Z

Aλ(y)

G(x, y)ρ^γ(x)dx = − Z ∞

λ

sdm_s(y)

= λm_λ(y) + Z _∞

λ

m_s(y)ds

≤ Cρ(y)^k2(t)λ^1−k1(t).

Thus, Z

E

G(x, y)ρ^γ(x)dx≤Cρ(y)^k2(t)λ^1−k1(t)+λ Z

E

ρ^γ(x)dx.

By choosingλ= [ρ(y)^−k2(t)R

Eρ^γ(x)dx]^−k1(1t), we have Z

E

G(x, y)ρ^γ(x)dx≤Cρ(y)

k2(t) k1(t)(

Z

E

ρ^γ(x)dx)

k1 (t)−1 k1(t) . Therefore,

Z

E

G(|ν|)(x)ρ^γ(x)dx = Z

Ω

Z

E

G(x, y)ρ^γ(x)dxd|ν(y)|

≤ C Z

Ω

ρ(y)

k2(t)

k1(t)d|ν(y)|

Z

E

ρ^γ(x)dx

^k1(_k^t)−1

1(t)

≤ CkνkM(Ω,ρ^β)

Z

E

ρ^γ(x)dx

^k1(_k^t)−1

1(t)

, since by our choice oft, ^k_k^2(t)

1(t) ≥β, which guarantees that Z

Ω

ρ(y)

k2(t)

k1(t)d|ν(y)| ≤max

Ω ρ

k2(t) k1(t)−βZ

Ω

ρ(y)^βd|ν(y)|.

As a consequence,

kG(ν)k_Mk1(t)(Ω,ρ^γdx) ≤CkνkM(Ω,ρ^β). Therefore,

k_α,β,γ:= max{k₁(t) :t∈[0, α]}=k₁(t_α,β,γ),

wheret_α,β,γ is defined by (2.4) andk_α,β,γ is given by (2.8). We complete the

proof.

We choose the parameter γ in order to makek_α,β,γ the largest possible, and denote

k_α,β = max

γ∈[0,α]k_α,β,γ. (2.12)

Sinceγ 7→k_α,β,γ is increasing, the following statement holds.

Proposition 2.3 Let N ≥2 and k_α,β be defined by (2.12), then k_α,β =

( _N

N−2α, if β ∈[0,^N−2α_N α],

N+α

N−2α+β, if β ∈(^N−2α_N α, α]. (2.13)

2.2 Non-homogeneous problem

In this subsection, we study some properties of the solution of the linear non-homogeneous, which will play a key role in the sequel. We assume that Ω⊂R^N,N ≥2 is an open bounded domain with aC² boundary.

Lemma 2.1 (i) There exists C >0 such that for any ξ∈X_α there holds kξk_Cα( ¯Ω) ≤Ck(−∆)^αξk_L^∞_(Ω) (2.14) and

kρ^−αξk_Cθ( ¯Ω)≤Ck(−∆)^αξk_L∞(Ω). (2.15) where 0< θ <min{α,1−α}. In particular, for x∈Ω

|ξ(x)| ≤Ck(−∆)^αξk_L∞(Ω)ρ^α(x). (2.16) (ii) Let u be the solution of

(−∆)^αu=f in Ω,

u= 0 in Ω^c, (2.17)

where f ∈C^γ( ¯Ω) for γ >0. Then u∈X_α.

Proof. (i). Estimates (2.14) and (2.15) are consequences of [30, Prop 1.1]

and [30, Th 1.2] respectively. Furthermore, if η₁ is the solution of (2.17) withf ≡1 in Ω, then η₁ >0 in Ω and by follows [30, Th 1.2], there exists C >0 such that

C⁻¹≤ η₁

ρ^α ≤C in Ω. (2.18)

In this expression the right-side follows [30, Th 1.2] and the left-hand side inequality follows from the maximum principle and [11, Th 1.2]. Since

−k(−∆)^αξk_L∞(Ω)≤(−∆)^αξ ≤ k(−∆)^αξk_L∞(Ω) in Ω, it follows by the comparison principle,

−k(−∆)^αξk_L^∞_(Ω)η₁(x)≤ξ(x)≤ k(−∆)^αξk_L^∞_(Ω)η₁(x).

which, together with (2.18), implies (2.16).

(ii)For r >0, we denote

Ω_r={z∈Ω : dist(z, ∂Ω)> r}.

Sincef ∈C^γ( ¯Ω), then by Corollary 1.6 part (i) and Proposition 1.1 in [30], forθ∈[0,min{α,1−α, γ}), there exists C >0 such that for anyr >0, we have

kuk_C2α+θ(Ωr)≤Cr^−α−θ

and

kuk_Cα(RN)≤C.

Then forx∈Ω, letting r=ρ(x)/2,

|δ(u, x, y)| ≤Cr^−α−θ|y|^2α+θ, ∀y∈B_r(0) (2.19) and

|δ(u, x, y)| ≤C|y|^α, ∀y∈R^N, whereδ(u, x, y) =u(x+y) +u(x−y)−2u(x). Thus,

|(−∆)^α_ǫu(x)| ≤ 1 2

Z

R^N

|δ(u, x, y)|

|y|^N+2α χ_ǫ(|y|)dy

≤ 1 2

Z

Br(0)

|δ(u, x, y)|

|y|^N+2α dy+ 1 2

Z

B^c_r(0)

|δ(u, x, y)|

|y|^N+2α dy

≤ Cr^−α−θ 2

Z

Br(0)

1

|y|^N−θdy+C 2

Z

B^c_r(0)

1

|y|^N+αdy

≤ Cρ(x)^−α, x∈Ω,

for someC >0 independent of ǫ. Moreover, ρ^−α is inL¹(Ω, ρ^αdx). Finally, we prove (−∆)^α_ǫu → (−∆)^αu as ǫ → 0⁺ pointwise. For x ∈ Ω, choosing ǫ∈(0, ρ(x)/2), then by (2.19),

|(−∆)^αu(x)−(−∆)^α_ǫu(x)| ≤ 1 2

Z

Bǫ(0)

|δ(u, x, y)|

|y|^N+2α dy

≤ Cρ(x)^−α−θǫ^θ

→ 0, ǫ→ 0⁺.

The proof is complete.

The following Proposition is the Kato’s type estimate for proving the uniqueness of the solution of (1.1).

Proposition 2.4 If ν∈L¹(Ω, ρ^αdx), there exists a unique weak solution u of the problem

(−∆)^αu=ν in Ω,

u= 0 in Ω^c. (2.20)

For any ξ∈X_α, ξ ≥0, we have Z

Ω

|u|(−∆)^αξdx≤ Z

Ω

ξsign(u)νdx (2.21)

and Z

Ω

u+(−∆)^αξdx≤ Z

Ω

ξsign₊(u)νdx, (2.22)

We note here that for α= 1, the proof of Proposition 2.4 could be seen in [37, Th 2.4]. For α ∈ (0,1), we first prove some integration by parts formula.

Lemma 2.2 Assume u, ξ∈X_α, then Z

Ω

u(−∆)^αξdx= Z

Ω

ξ(−∆)^αudx. (2.23)

Proof. Denote

(−∆)^α_Ω,ǫu(x) =− Z

Ω

u(z)−u(x)

|z−x|^N+2αχ_ǫ(|x−z|)dz. (2.24) By the definition of (−∆)^α_ǫ, we have

(−∆)^α_ǫu(x) = − Z

Ω

u(z)−u(x)

|z−x|^N+2αχǫ(|x−z|)dz+u(x) Z

Ω^c

χǫ(|x−z|)

|z−x|^N+2αdz

= (−∆)^α_Ω,ǫu(x) +u(x) Z

Ω^c

χ_ǫ(|x−z|)

|z−x|^N+2αdz.

We claim that Z

Ω

ξ(x)(−∆)^α_Ω,ǫu(x)dx= Z

Ω

u(x)(−∆)^α_Ω,ǫξ(x)dx, foru, ξ∈X_α. (2.25) By using the fact of

Z

Ω

Z

Ω

[u(z)−u(x)]ξ(x)

|z−x|^N+2α χ_ǫ(|x−z|)dzdx= Z

Ω

Z

Ω

[u(x)−u(z)]ξ(z)

|z−x|^N+2α χ_ǫ(|x−z|)dzdx, we have

Z

Ω

ξ(x)(−∆)^α_Ω,ǫu(x)dx

=−1 2

Z

Ω

Z

Ω

[(u(z)−u(x))ξ(x)

|z−x|^N+2α +(u(x)−u(z))ξ(z)

|z−x|^N+2α ]χ_ǫ(|x−z|)dzdx

= 1 2

Z

Ω

Z

Ω

[u(z)−u(x)][ξ(z)−ξ(x)]

|z−x|^N+2α χǫ(|x−z|)dzdx.

Similarly, by the fact thatu∈X_α, Z

Ω

u(x)(−∆)^α_Ω,ǫξ(x)dx= 1 2

Z

Ω

Z

Ω

[u(z)−u(x)][ξ(z)−ξ(x)]

|z−x|^N+2α χǫ(|x−z|)dzdx.

Then (2.25) holds. In order to prove (2.23), we first notice that by (2.25), Z

Ω

ξ(x)(−∆)^α_ǫu(x)dx

= Z

Ω

ξ(x)(−∆)^α_Ω,ǫu(x)dx+ Z

Ω

u(x)ξ(x) Z

Ω^c

χǫ(|x−z|)

|z−x|^N+2αdzdx

= Z

Ω

u(x)(−∆)^α_Ω,ǫξ(x)dx+ Z

Ω

u(x)ξ(x) Z

Ω^c

χ_ǫ(|x−z|)

|z−x|^N+2αdzdx

= Z

Ω

u(x)(−∆)^α_ǫξ(x)dx. (2.26)

Since u and ξ belongs to X_α, (−∆)^α_ǫξ → (−∆)^αξ and (−∆)^α_ǫu → (−∆)^αu and |u(−∆)^α_ǫξ|+|ξ(−∆)^α_ǫu| ≤Cϕfor someC >0 and ϕ∈L¹(Ω, ρ^αdx). It follows by the Dominated Convergence Theorem

ǫ→0lim⁺ Z

Ω

ξ(x)(−∆)^α_ǫu(x)dx= Z

Ω

ξ(x)(−∆)^αu(x)dx and

lim

ǫ→0⁺

Z

Ω

(−∆)^α_ǫξ(x)u(x)dx= Z

Ω

(−∆)^αξ(x)u(x)dx.

Lettingǫ→0⁺ of (2.26) we conclude that (2.23) holds.

For 1≤p <∞ and 0< s <1,W^s,p(Ω) is the set ofξ ∈L^p(Ω) such that Z

Ω

Z

Ω

|ξ(x)−ξ(y)|^p

|x−y|^N+sp dydx <∞. (2.27) This space is endowed with the norm

kξk_W^s,p_(Ω) = Z

Ω

|ξ(x)|^pdx+ Z

Ω

Z

Ω

|ξ(x)−ξ(y)|^p

|x−y|^N+sp dydx ¹_p

. (2.28) Furthermore, if Ω is bounded, the following Poincar´e inequality holds [35, p 134].

Z

Ω

|ξ(x)|^pdx ¹_p

≤C Z

Ω

Z

Ω

|ξ(x)−ξ(y)|^p

|x−y|^N+sp dydx ¹_p

, ∀ξ ∈C_c^∞(Ω).

(2.29) Lemma 2.3 Letu∈X_αandγ beC² in the intervalu( ¯Ω)and satisfyγ(0) = 0, then u∈W^α,2(Ω),γ◦u ∈X_α and for all x∈Ω, there exists z_x ∈Ω¯ such that

(−∆)^α(γ◦u)(x) = (γ^′◦u)(x)(−∆)^αu(x)−γ^′′◦u(z_x) 2

Z

Ω

(u(y)−u(x))²

|y−x|^N+2α dy.

(2.30) Proof. Sinceu∈C( ¯Ω) vanishes in Ω^c,γ◦u shares the same properties. By (2.14), for any xand y in Ω

(u(x)−u(y))²≤C|x−y|^2αk(−∆)^αuk²_L∞(Ω). Thenu∈W^α,2(Ω). Similarly γ◦u∈W^α,2(Ω). Furthermore

(γ ◦u)(y)−(γ◦u)(x) = (γ^′◦u)(x) (u(y)−u(x)) + Z u(y)

u(x)

(u(y)−t)γ^′′(t)dt.

By the mean value theorem, there exists someτ ∈[0,1] such that Z u(y)

u(x)

(u(y)−t)γ^′′(t)dt= γ^′′(τ u(y) + (1−τ)u(x))

2 (u(y)−u(x))². Sinceγ^′′ is continuous andu is continuous in ¯Ω,

Z _u(y)

u(x)

(u(y)−t)γ^′′(t)dt

≤ kγ^′′◦uk_L∞( ¯Ω)

2 (u(y)−u(x))² and by (2.14),

Z

|y−x|>ǫ

Z u(y) u(x)

(u(y)−t)γ^′′(t)dt dy

|y−x|^N+2α

≤ kγ^′′◦uk_L^∞ 2

Z

Ω

(u(y)−u(x))² dy

|y−x|^N+2α. Notice also thatτ u(y) + (1−τ)u(x)∈u( ¯Ω) :=I, therefore

mint∈I γ^′′(t)≤γ^′′(τ u(y) + (1−τ)u(x))≤max

t∈I γ^′′(t), thus

min_t∈Iγ^′′(t) 2

Z

Ω

(u(y)−u(x))²

|y−x|^N+2α dy

≤ Z

Ω

Z u(y) u(x)

(u(y)−t)γ^′′(t)dt dy

|y−x|^N+2α

≤ max_t∈I γ^′′(t) 2

Z

Ω

(u(y)−u(x))²

|y−x|^N+2α dy.

Sinceγ^′′ is continuous, there exists t0∈I such that Z

Ω

Z u(y) u(x)

(u(y)−t)γ^′′(t)dt dy

|y−x|^N+2α = γ^′′(t₀) 2

Z

Ω

(u(y)−u(x))²

|y−x|^N+2α dy and sinceuis continuous in R^N and vanishes in Ω^c, there existszx∈Ω such¯

thatt₀ =u(z_x), which ends the proof.

Proof of Proposition 2.4. Uniqueness. Letw be a weak solution of (−∆)^αw= 0 in Ω

w= 0 in Ω^c. (2.31)

Ifω is a Borel subset of Ω and η_ω,n the solution of (−∆)^αη_ω,n=ζ_n in Ω

ηω,n= 0 in Ω^c, (2.32)

whereζn: ¯Ω7→[0,1] is a C¹( ¯Ω) function such that ζ_n→χ_ω inL^∞( ¯Ω) asn→ ∞.

Then by Lemma 2.1 part (ii), η_ω,n∈X_α and Z

Ω

wζ_ndx= 0.

Then passing the limit of n→ ∞, we have Z

ω

wdx= 0.

This implies w= 0.

Existence and estimate (2.21). Forδ >0 we define an even convex function φ_δ by

φ_δ(t) =

(|t| − ^δ₂, if |t| ≥δ,

t²

2δ, if |t|< δ/2. (2.33) Then for any t, s ∈ R, |φ^′_δ(t)| ≤ 1, φ_δ(t) → |t| and φ^′_δ(t) → sign(t) when δ→0⁺. Moreover

φ_δ(s)−φ_δ(t)≥φ^′_δ(t)(s−t). (2.34) Let{νn}be a sequence functions in C¹( ¯Ω) such that

n→∞lim Z

Ω

|νn−ν|ρ^αdx= 0.

Let u_n be the corresponding solution to (2.20) with right-hand side ν_n, then by Lemma 2.1, un ∈X_α and by Lemmas 2.2, 2.3, for any δ > 0 and ξ∈X_α, ξ ≥0,

Z

Ω

φ_δ(u_n)(−∆)^αξdx= Z

Ω

ξ(−∆)^αφ_δ(u_n)dx

≤ Z

Ω

ξφ^′_δ(u_n)(−∆)^αu_ndx

= Z

Ω

ξφ^′_δ(u_n)ν_ndx.

(2.35)

Lettingδ →0, we obtain Z

Ω

|u_n|(−∆)^αξdx≤ Z

Ω

ξsign(u_n)ν_ndx≤ Z

Ω

ξ|ν_n|dx. (2.36) If we take ξ=η1, we derive from Lemma 2.1

Z

Ω

|u_n|dx≤C Z

Ω

|ν_n|ρ^αdx. (2.37)

Similarly

Z

Ω

|un−um|dx≤C Z

Ω

|νn−νm|ρ^αdx. (2.38) Therefore{u_n}is a Cauchy sequence inL¹ and its limitu is a weak solution of (2.20). Letting n → ∞ in (2.36) we obtain (2.21). Inequality (2.22) is proved by replacing φ_δ by ˜φ_δ which is zero on (−∞,0] andφ_δ on [0,∞).

The next result is a higher order regularity result

Proposition 2.5 Let the assumptions of Proposition 2.2 be fulfilled and 0≤β ≤α. Then for p∈(1,_N+β−2α^N ) there exists cp >0 such that for any ν∈L¹(Ω, ρ^βdx)

kG[ν]k_W^2α−γ,p_(Ω)≤c_pkνk_L1(Ω,ρ^βdx) (2.39) where γ =β+^N_p′ if β >0 and γ > ^N_p′ if β= 0.

Proof. We use Stampacchia’s duality method [33] and put u = G[ν]. If ψ∈C_c^∞( ¯Ω), then

Z

Ω

ψ(−∆)^αudx

≤ Z

Ω

|ν||ψ|dx

≤sup

Ω

|ρ^−βψ|

Z

Ω

|ν|ρ^βdx

≤ kψk_Cβ( ¯Ω)kνk_L1(Ω,ρ^βdx).

(2.40)

By Sobolev-Morrey embedding type theorem (see e.g. [29, Th 8.2]), for any p∈(1,_N+β−2α^N ) andp^′= _p−1^p ,

kψk_Cβ( ¯Ω) ≤Ckψk_Wγ,p′

(Ω)

withγ =β+^N_p′ if β >0 andγ > ^N_p′ if β= 0. Therefore,

Z

Ω

ψ(−∆)^αudx

≤Ckψk_Wγ,p′

(Ω)kνk_L1(Ω,ρ^βdx), (2.41) which implies that the mappingψ7→R

Ωψ(−∆)^αudxis continuous onW^γ,p′(Ω) and thus

k(−∆)^αuk_W−γ,p(Ω)≤Ckνk_L1(Ω,ρ^βdx). (2.42) Since (−∆)^−α is an isomorphism fromW^−γ,p(Ω) intoW^2α−γ,p(Ω), it follows that

kuk_W^2α−γ,p_(Ω)≤Ckνk_L1(Ω,ρ^βdx). (2.43)

Proposition 2.6 Under the assumptions of Proposition 2.5 the mapping ν7→G[ν]is compact from L¹(Ω, ρ^βdx) into L^q(Ω) for any q∈[1,_N+β−2α^N ).

Proof. By [29, Th 6.5] the embedding ofW^2α−γ,p(Ω) intoL^q(Ω) is compact,

this ends the proof.

3 Proof of Theorem 1.1

Before proving the main we give a general existence result inL¹(Ω, ρ^αdx).

Proposition 3.1 Suppose thatΩis an open boundedC²domain ofR^N (N ≥ 2), α ∈(0,1) and the function g:R→ Ris continuous, nondecreasing and rg(r) ≥ 0 for all r ∈ R. Then for any f ∈ L¹(Ω, ρ^αdx) there exists a unique weak solution u of (1.1) with ν =f. Moreover the mapping f 7→u is increasing.

Proof. Step 1: Variational solutions. If w ∈ L²(Ω), we denote by w its extension by 0 in Ω^c and byW_c^α,2(Ω) the set of function inL²(Ω) such that

kwk²_Wα,2 c (Ω):=

Z

R^N

|ˆw|²(1 +|x|^α)dx <∞, where ˆw is the Fourier transform of w. Forǫ >0 we set

J(w) = 1 2

Z

R^N

(−∆)^α2w2

dx+ Z

Ω

(j(w) +ǫw²)dx,

with domain D(J) = {w ∈ Wc^α,2(R^N) s.t. j(w) ∈ L¹(Ω)} and j(s) = R_s

0 g(t)dt. Furthermore since there holds J(w)≥σkwk²

Wc^α,2 for someσ >0, the subdifferential∂J ofJ is a maximal monotone in the sense of Browder- Minty (see [6] and the references therein) which satisfies R(∂J) = L²(Ω).

Then for anyf ∈L²(Ω) there exists a uniqueu_ǫ in the domainD(∂J) such that∂J(u_ǫ) =f. Since for anyψ∈Wc^α,2(Ω)

Z

R^N

(−∆)^α2w(−∆)^α2ψdx= (4π)^α Z

R^N

ˆ

wψ|x|ˆ ^2αdx= Z

Ω

ψ(−∆)^αwdx,

∂J(u_ǫ) = (−∆)^αu_ǫ+g(u_ǫ) + 2ǫu=f,

withu_ǫ∈W_c^2α,2(Ω) such that g(u_ǫ)∈L²(Ω). This is also a consequence of [6, Cor 2.11]. Iff is assumed to be bounded, thenu∈C^α(Ω) by [30, Prop 1.1]. Note that more delicate variational formulations can be found in [21], [22].

Step 2: L¹ solutions. Forn∈N^∗ we denote by u_n,ǫ the solution of (−∆)^αu_n,ǫ+g(u_n,ǫ) + 2ǫu_n,ǫ =f_n in Ω

u_n,ǫ = 0 in Ω^c (3.1)

wherefn= sgn(f) min{n,|f|}. By (2.36) with ξ=η1, Z

Ω

(|u_n,ǫ|+ (2ǫ|u_n,ǫ|+|g(u_n,ǫ)|)η₁)dx≤ Z

Ω

|f_n|η₁dx≤ Z

Ω

|f|η₁dx, (3.2)

and, forǫ^′ >0 andm∈N^∗, Z

Ω

|u_n,ǫ−u_m,ǫ^′|+|g(u_n,ǫ)−g(u_m,ǫ^′)|η₁ dx

≤ Z

Ω

|f_n−f_m|+ 2ǫ|u_n,ǫ|+ 2ǫ^′|u_m,ǫ^′| η₁dx.

(3.3)

Since f_n → f in L¹(Ω, ρ^αdx), {u_n,ǫ} and {g◦u_n,ǫ} are Cauchy filters in L¹(Ω) and L¹(Ω, ρ^αdx) respectively. Set u = lim_{n→∞,ǫ→0}u_n,ǫ, we derive from the following identity valid for any ξ∈X_α

Z

Ω

(u_n,ǫ(−∆)^αξ+g(u_n,ǫ)ξ)dx= Z

Ω

(f_n−ǫu_n,ǫ)ξdx

thatu is a solution of (1.1). Uniqueness follows from (2.36)-(3.3), since for any f and f^′ in L¹(Ω, ρ^αdx), the any couple (u, u^′) of weak solutions with respective right-hand sidef and f^′ satisfies

Z

Ω

|u−u^′|+|g(u)−g(u^′)|η₁ dx≤

Z

Ω

|f −f^′|η₁dx. (3.4) Finally, the monotonicity of the mappingf 7→u follows from (2.22) thanks to which (3.4) is transformed into

Z

Ω

(u−u^′)₊+ (g(u)−g(u^′))₊η₁ dx≤

Z

Ω

(f−f^′)₊η₁dx. (3.5) Proof of Theorem 1.1. Uniqueness follows from (3.4). For existence we define

C_β( ¯Ω) ={ζ ∈C( ¯Ω) :ρ^−βζ ∈C( ¯Ω)}

endowed with the norm

kζk_{Cβ( ¯Ω)}=kρ^−βζk_{C( ¯Ω)}.

We consider a sequence {ν_n} ⊂C¹( ¯Ω) such that ν_n,± → ν_± in the duality sense withC_β( ¯Ω), which means

n→∞lim Z

Ω¯

ζν_n,±dx= Z

Ω¯

ζdν_±

for all ζ ∈ C_β( ¯Ω). It follows from the Banach-Steinhaus theorem that kν_nkM(Ω,ρ^β) is bounded independently of n, therefore

Z

Ω

(|u_n|+|g(u_n)|η₁)dx≤ Z

Ω

|ν_n|η₁dx≤C. (3.6)

Therefore kg(un)kM(Ω,ρ^α) is bounded independently of n. For ǫ > 0, set ξǫ= (η1+ǫ)^βα−ǫ^αβ, which is concave in the intervalη(¯ω). Then, by Lemma 2.3 part (ii),

(−∆)^αξ_ǫ = β

α(η₁+ǫ)^β−αα (−∆)^αη₁− β(β−α)

α² (η₁+ǫ)^β−2αα Z

Ω

(η₁(y)−η₁(x))²

|y−x|^N+2α dy

≥ β

α(η₁+ǫ)^β−αα , and ξ_ǫ∈X_α. Since

Z

Ω

(|u_n|(−∆)^αξ_ǫ+|g(u_n)|ξ_ǫ)dx≤ Z

Ω

ξ_ǫd|ν_n|, we obtain

Z

Ω

|u_n|β

α(η₁+ǫ)^β−αα +|g(u_n)|ξ_ǫ

dx≤ Z

Ω

ξ_ǫd|ν_n|.

If we letǫ→0, we obtain Z

Ω

|u_n|β αη

β−α

1α +|g(u_n)|η

β

1α

dx≤

Z

Ω

η

β

1αd|ν_n|.

By Lemma 2.3, we derive the estimate Z

Ω

|u_n|ρ^β−α+|g(u_n)|ρ^β

dx≤Ckν_nk_M(Ω,ρβ) ≤C^′. (3.7) Sinceu_n=G[ν_n−g(u_n)], it follows by (2.7), that

kunk_{Mkα,β(Ω,ρβdx)}≤ kνn−g(un)kM(Ω,ρ^β), (3.8) where k_α,β is defined by (2.13). By Corollary 2.6 the sequence {u_n} is relatively compact in the L^q(Ω) for 1≤q < _N+β−2α^N . Therefore there exist a sub-sequence {un_k} and some u ∈ L¹(Ω)∩L^q(Ω) such that un_k → u in L^q(Ω) and almost every where in Ω. Furthermore g(u_nk) → g(u) almost every where. Put ˜g(r) =g(|r|)−g(−|r|) and we note that|g(r)| ≤g(|r|) for˜ r∈Rand ˜gis nondecreasing. Forλ >0, we setS_λ ={x∈Ω :|un_k(x)|> λ}

and ω(λ) =R

Sλρ^βdx. Then for any Borel setE ⊂Ω, we have Z

E

|g(u_nk)|ρ^βdx= Z

E∩S_λ^c

|g(u_nk)|ρ^βdx+ Z

E∩Sλ

|g(u_nk)|ρ^βdx

≤˜g(λ) Z

E

ρ^βdx+ Z

S_λ

˜

g(|u_nk|)ρ^βdx

≤˜g(λ) Z

E

ρ^βdx− Z _∞

λ

˜

g(s)dω(s).

But

Z _∞

λ

˜

g(s)dω(s) = lim

T→∞

Z _T

λ

˜

g(s)dω(s).

Sinceun_k ∈M^kα,β(Ω, ρ^βdx),ω(s)≤cs^−kα,β and

− Z _T

λ

˜

g(s)dω(s) =−

˜

g(s)ω(s) s=T

s=λ

+ Z _T

λ

ω(s)d˜g(s)

≤˜g(λ)ω(λ)−g(T˜ )ω(T) +c Z T

λ

s^−kα,βd˜g(s)

≤˜g(λ)ω(λ)−g(T˜ )ω(T) +c

T^−kα,βg(T˜ )−λ^−kα,βg(λ)˜

+ c

k_α,β+ 1 Z T

λ

s^{−1−kα,β}g(s)ds.˜ By assumption (1.9) there exists {T_n} → ∞ such that Tn^−kα,βg(T˜ _n) → 0 whenn→ ∞. Furthermore ˜g(λ)ω(λ)≤cλ^−kα,βg(λ), therefore˜

− Z ∞

λ

˜

g(s)dω(s)≤ c k_α,β+ 1

Z ∞ λ

s^{−1−kα,β}˜g(s)ds.

Notice that the above quantity on the right-hand side tends to 0 when λ→ ∞. The conclusion follows: for any ǫ >0 there exists λ >0 such that

c k_α,β+ 1

Z _∞

λ

s^{−1−kα,β}g(s)ds˜ ≤ ǫ 2 and δ >0 such that

Z

E

ρ^βdx≤δ =⇒˜g(λ) Z

E

ρ^βdx≤ ǫ 2.

This proves that {g ◦u_nk} is uniformly integrable in L¹(Ω, ρ^βdx). Then g◦ u_nk → g◦u in L¹(Ω, ρ^βdx) by Vitali convergence theorem. Letting n_k→ ∞ in the identity

Z

Ω

(u_nk(−∆)^αξ+ξg◦u_nk)dx= Z

Ω

ν_nkξdx whereξ ∈X_α, it infers that uis a weak solution of (1.1).

The right-hand side of estimate (1.9) follows from the fact that v_n,+:=

G[νn,+] satisfies

(−∆)^αv_n,++g(v_n,+) =ν_n,++g(v_n,+)≥ν_n

Therefore v_n,+ ≥ u_n by Proposition 3.1. Letting n → ∞ yields to (1.10).

The left-hand side is proved similarly.

To prove the mapping ν 7→ u is increasing. Let ν1, ν2 ∈ M(Ω, ρ^β) and ν₁ ≥ ν₂, then there exist two sequences {ν_1,n} and {ν_2,n} in C^∞( ¯Ω) such thatν_1,n≥ν_2,n and

ν_i,n→ν_i as n→ ∞, i = 1,2.

Letu_i,nbe the unique solution of (1.1) withν_i,nandu_ibe the unique solution of (1.1) with ν_i where i= 1,2. Then u_1,n ≥u_2,n. Moveover, by uniqueness u_i,n convergence tou_i inL¹(Ω) for i= 1 andi= 2. Then we haveu₁≥u₂.

Corollary 3.1 Under the hypotheses of Theorem 1.1, we further assume that {ν_n} is a sequence of measures in M(Ω, ρ^β) and ν ∈ M(Ω, ρ^β) such that for any ξ∈C_β( ¯Ω),

Z

Ω

ξdν_n→ Z

Ω

ξdν as n→ ∞.

Then the sequence {u_n} of weak solutions to

(−∆)^αu_n+g◦u_n=ν_n in Ω,

u_n= 0 in Ω^c, (3.9)

converges to the solution u of (1.1) in L^q(Ω) for 1 ≤ q < _N+β−2α^N and {g◦u_n} converges tog◦u in L¹(Ω, ρ^βdx).

Proof. The method is an adaptation of [38]. Since ν_n → ν in the duality sense ofC_β(Ω), there existsM >0 such that

kν_nkM(Ω,ρ^β)≤M, ∀n∈N.

Therefore (3.7) and (3.8) hold (but with u_n solution of (3.9)). The above proof shows that {g◦u_n} is uniformly integrable in L¹(Ω, ρ^βdx) and {u_n} relatively compact inL^q(Ω) for 1≤q < _N+β−2α^N . Thus, up to a subsequence {u_nk} ⊂ {u_n}, u_nk → u, and u is the weak solution of (1.1). Since u is

unique, u_n→uas n→ ∞.

Remark 3.1 Under the hypotheses of Theorem 1.1, we assume ν≥0, then G(ν)−G(g(G(ν)))≤u≤G(ν). (3.10) Indeed, sinceg is nondecreasing and u≤G(ν), then

u = G(ν)−G(g(u))

≥ G(ν)−G(g(G(ν))).

4 Applications

4.1 The case of a Dirac mass

In this subsection we characterize the asymptotic behavior of a solution near a singularity created by a Dirac mass.

Theorem 4.1 Assume thatΩis an open, bounded andC²domain ofR^N (N ≥ 2) with 0 ∈ Ω, α ∈ (0,1), ν = δ₀ and the function g : [0,∞) → [0,∞) is continuous, nondecreasing and (1.9) holds for

k_α,0 = N

N −2α. (4.1)

Then problem (1.1) admits a unique positive weak solution u such that

x→0limu(x)|x|^N−2α =C, (4.2) for some C >0.

Remark 4.1 We note here that a weak solution u of (1.1) with ν = δ₀ satisfies

(−∆)^αu+g(u) = 0 in Ω\ {0},

u= 0 in R^N\Ω. (4.3)

The asymptotic behavior (4.2) is one of the possible singular behaviors of solutions of (4.3) given in [13].

Before proving Theorem 4.1, we give an auxiliary lemma.

Lemma 4.1 Assume that g: [0,∞) →[0,∞) is continuous, nondecreasing and (1.9) holds with k_α,β >1. Then

s→∞lim g(s)s^−kα,β = 0.

Proof. Since Z _2s

s

g(t)t^{−1−kα,β}dt≥g(s)(2s)^{−1−kα,β} Z _2s

s

dt= 2^{−1−kα,β}g(s)s^−kα,β and by (1.9),

s→∞lim Z _2s

s

g(t)t^{−1−kα,β}dt= 0.

Then

s→∞lim g(s)s^−kα,β = 0.

The proof is complete.