Semilinear fractional elliptic equations with gradient nonlinearity involving measures

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Semilinear fractional elliptic equations with gradient nonlinearity involving measures

Huyuan Chen, Laurent Veron

To cite this version:

Huyuan Chen, Laurent Veron. Semilinear fractional elliptic equations with gradient nonlinearity involving measures. 2013. �hal-00856008v6�

Semilinear fractional elliptic equations with gradient nonlinearity involving measures

Huyuan Chen¹

Departamento de Ingenier´ıa Matem´atica Universidad de Chile, Chile

Laurent V´eron²

Laboratoire de Math´ematiques et Physique Th´eorique Universit´e Fran¸cois Rabelais, Tours, France

Abstract

We study the existence of solutions to the fractional elliptic equa- tion (E1) (−∆)^αu+ǫg(|∇u|) =ν in an open bounded regular domain Ω of R^N(N ≥ 2), subject to the condition (E2) u= 0 in Ω^c, where ǫ= 1 or−1, (−∆)^αdenotes the fractional Laplacian withα∈(1/2,1), ν is a Radon measure andg:R₊7→R₊ is a continuous function. We prove the existence of weak solutions for problem (E1)-(E2) wheng is subcritical. Furthermore, the asymptotic behavior and uniqueness of solutions are described whenǫ= 1,ν is Dirac mass andg(s) =s^pwith p∈(0,_N ^N

−2α+1).

Contents

1 Introduction 2

2 Preliminaries 6

2.1 Marcinkiewicz type estimates . . . 6 2.2 Classical solutions . . . 9

3 Proof of Theorems 1.1 and 1.2 11

3.1 The absorption case . . . 11 3.2 The source case . . . 15

4 The case of the Dirac mass 17

Key words: Fractional Laplacian, Radon measure, Green kernel, Dirac mass.

MSC2010: 35R11, 35J61, 35R06

1chenhuyuan@yeah.net

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

1 Introduction

Let Ω⊂R^N(N ≥2) be an open boundedC² domain andg:R₊7→R₊be a continuous function. The purpose of this paper is to study the existence of weak solutions to the semilinear fractional elliptic problem withα∈(1/2,1),

(−∆)^αu+ǫg(|∇u|) =ν in Ω,

u= 0 in Ω^c, (1.1)

where ǫ = 1 or −1 and ν ∈ M(Ω, ρ^β) with β ∈ [0,2α−1). Here ρ(x) = dist(x,Ω^c) andM(Ω, ρ^β) is the space of Radon measures in Ω satisfying

Z

Ω

ρ^βd|ν|<+∞. (1.2)

In particular, we denoteM^b(Ω) =M(Ω, ρ⁰). The associated positive cones are respectively M₊(Ω, ρ^β) and M^b₊(Ω). According to the value of ǫ, we speak of an absorbing nonlinearity the case ǫ= 1 and a source nonlinearity the caseǫ=−1. The operator (−∆)^α is the fractional Laplacian defined as

(−∆)^αu(x) = lim

ε→0⁺(−∆)^α_εu(x), where forε >0,

(−∆)^α_εu(x) =− Z

R^N

u(z)−u(x)

|z−x|^N+2αχ_ε(|x−z|)dz (1.3) and

χ_ε(t) =

(0, if t∈[0, ε], 1, if t> ε.

In a pioneering work, Brezis [7] (also see B´enilan and Brezis [1]) stud- ied the existence and uniqueness of the solution to the semilinear Dirichlet elliptic problem

−∆u+h(u) =ν in Ω,

u= 0 on ∂Ω, (1.4)

where ν is a bounded measure in Ω and the function h is nondecreasing, positive on (0,+∞) and satisfies that

Z +∞

1

(h(s)−h(−s))s^{−2N−1N−2}ds <+∞.

Later on, V´eron [29] improved this result in replacing the Laplacian by more general uniformly elliptic second order differential operator, where ν∈M(Ω, ρ^β) with β ∈[0,1] and his a nondecreasing function satisfying

Z _+∞

1

(h(s)−h(−s))s^{−2N+β−1N+β−2}ds <+∞.

The general semilinear elliptic problems involving measures such as the equa- tions involving boundary measures have been intensively studied; it was ini- tiated by Gmira and V´eron [16] and then this subject has being extended in various ways, see [4, 6, 18, 19, 20, 21] for details and [22] for a general panorama. In a recent work, Nguyen-Phuoc and V´eron [24] obtained the existence of solutions to the viscous Hamilton-Jacobi equation

−∆u+h(|∇u|) =ν in Ω,

u= 0 on ∂Ω, (1.5)

when ν ∈ M^b(Ω), h is a continuous nondecreasing function vanishing at 0 which satisfies

Z +∞

1

h(s)s^{−2N−1N−1} ds <+∞.

During the last years there has also been a renewed and increasing inter- est in the study of linear and nonlinear integro-differential operators, espe- cially, the fractional Laplacian, motivated by great applications in physics and by important links on the theory of L´evy processes, refer to [8, 12, 13, 10, 14, 26, 28, 27]. Many estimates of its Green kernel and generation for- mula can be found in the references [3, 11]. Recently, Chen and V´eron [13]

studied the semilinear fractional elliptic equation (−∆)^αu+h(u) =ν in Ω,

u= 0 in Ω^c, (1.6)

whereν∈M(Ω, ρ^β) withβ ∈[0, α]. We proved the existence and uniqueness of the solution to (1.6) when the function his nondecreasing and satisfies

Z _+∞

1

(h(s)−h(−s))s^{−1−kα,β}ds <+∞, where

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]. (1.7) Our interest in this article is to investigate the existence of weak solutions to fractional equations involving nonlinearity in the gradient term and with Radon measure. In order the fractional Laplacian be the dominant operator in terms of order of differentiation, it is natural to assume thatα∈(1/2,1).

Definition 1.1 We say that u is a weak solution of (1.1), if u ∈ L¹(Ω),

|∇u| ∈L¹_loc(Ω), g(|∇u|) ∈L¹(Ω, ρ^αdx) and Z

Ω

[u(−∆)^αξ+ǫg(|∇u|)ξ]dx= Z

Ω

ξdν, ∀ ξ ∈X_α, (1.8)

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 ε₀>0such that |(−∆)^α_εξ| ≤ϕa.e. in Ω, for all ε∈(0, ε₀].

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

G_α[ν](x) = Z

Ω

G_α(x, y)dν(y), ∀ ν ∈M(Ω, ρ^α). (1.9) Using bounds ofG_α[ν], we obtain in section 2 some crucial estimates which will play an important role in our construction of weak solutions. Our main result in the caseǫ= 1 is the following.

Theorem 1.1 Assume that ǫ= 1 and g :R₊ 7→ R₊ is a continuous func- tion verifyingg(0) = 0 and

Z +∞

1

g(s)s^{−1−p∗α}ds <+∞, (1.10) where

p^∗_α= N

N −2α+ 1. (1.11)

Then for any ν ∈ M₊(Ω, ρ^β) with β ∈ [0,2α−1), problem (1.1) admits a nonnegative weak solution uν which satisfies

u_ν ≤G_α[ν]. (1.12)

As in the caseα= 1, uniqueness remains an open question. We remark that the critical value p^∗_α is independent of β. A similar fact was first ob- served when dealing with problem (1.6) where the critical valuek_α,β defined by (1.7) does not depend onβ whenβ ∈[0,^N−2α_N α].

Whenǫ=−1, we have to consider the critical valuep^∗_α,β which depends truly onβ and is expressed by

p^∗_α,β= N

N−2α+ 1 +β. (1.13)

We observe thatp^∗_α,0 =p^∗_α and p^∗_α,β < p^∗_α when β > 0. In the source case, the assumptions ong are of a different nature from in the absorption case, namely

(G) g:R₊7→R₊ is a continuous function which satisfies

g(s)≤c₁s^p+σ₀, ∀s≥0, (1.14) for somep∈(0, p^∗_α,β), where c1 >0 andσ0>0.

Our main result concerning the source case is the following.

Theorem 1.2 Assume that ǫ= −1, ν ∈M(Ω, ρ^β) with β ∈ [0,2α−1) is nonnegative, g satisfies (G) and

(i) p∈(0,1), or

(ii) p= 1 and c1 is small enough, or

(iii) p∈(1, p^∗_α,β), σ₀ and kνkM(Ω,ρ^β) are small enough.

Then problem (1.1) admits a weak nonnegative solution uν which satisfies

uν ≥G_α[ν]. (1.15)

We note that Bidaut-V´eron, Garc´ıa-Huidobro and V´eron in [5] obtained the existence of a renormalized solution of

−∆_pu=|∇u|^q+ν in Ω,

whenν ∈M^b(Ω). We make use of some idea in [5] in the proof of Theorem 1.2 and extend some results in [5] to elliptic equations involving (−∆)^α with α∈(1/2,1) andν ∈M(Ω, ρ^β) withβ ∈[0,2α−1).

In the last section, we assume that Ω contains 0 and give pointwise estimates of the positive solutions

(−∆)^αu+|∇u|^p=δ₀ in Ω,

u= 0 in Ω^c, (1.16)

when 0< p < p^∗_α. Combining properties of the Riesz kernel with a bootstrap argument, we prove that any weak solution of (1.16) is regular outside 0 and is actually a classical solution of

(−∆)^αu+|∇u|^p = 0 in Ω\ {0},

u= 0 in Ω^c. (1.17)

These pointwise estimates are quite easy to establish in the case α = 1, but much more delicate when the diffusion operator is non-local. We give sharp asymptotics of the behaviour of u near 0 and prove that the solution of (1.16) is unique in the class of positive solutions.

The paper is organized as follows. In Section 2, we study the Green operator and prove the key estimate

k∇G_α[ν]k_Mp∗

α(Ω,ρ^αdx)≤c₂kνkM(Ω,ρ^β)

Section 3 is devoted to prove Theorem 1.1 and Theorem 1.2. In Section 4, we consider the case whereǫ= 1 in (1.1) and ν is a Dirac mass. We obtain precise asymptotic estimate and derive uniqueness.

Aknowledgements. The authors are grateful to Marie-Fran¸coise Bidaut- V´eron for useful discussions in the preparation of this work.

2 Preliminaries

2.1 Marcinkiewicz type estimates

In this subsection, we recall some definitions and properties of Marcinkiewicz spaces.

Definition 2.1 LetΘ⊂R^N be a domain andµbe a positive Borel measure in Θ. For κ >1, κ^′ =κ/(κ−1) and u∈L¹_loc(Θ, dµ), we set

kuk_M^κ_(Θ,dµ)= inf (

c∈[0,∞] : Z

E

|u|dµ≤c Z

E

dµ ¹

κ′

, ∀E⊂Θ, E Borel )

(2.1) and

M^κ(Θ, dµ) ={u∈L¹_loc(Θ, dµ) :kuk_M^κ_(Θ,dµ)<∞}. (2.2) M^κ(Θ, dµ) is called the Marcinkiewicz space of exponent κ, or weak L^κ-space andk.k_M^κ_(Θ,dµ) is a quasi-norm.

Proposition 2.1 [2, 9] Assume that 1 ≤q < κ <∞ and u ∈L¹_loc(Θ, dµ).

Then there exists c₃ >0 dependent of q, κsuch that Z

E

|u|^qdµ≤c₃kuk_M^κ_(Θ,dµ) Z

E

dµ 1−q/κ

,

for any Borel setE of Θ.

The next estimate is the key-stone in the proof of Theorem 1.1.

Proposition 2.2 Let Ω⊂R^N (N ≥2) be a bounded C² domain and ν ∈ M(Ω, ρ^β) withβ ∈[0,2α−1]. Then there existsc₂>0 such that

k∇G_α[|ν|]k_Mp∗

α(Ω,ρ^αdx) ≤c₂kνkM(Ω,ρ^β), (2.3) where ∇G_α[|ν|](x) =

Z

Ω

∇_xG_α(x, y)d|ν(y)|and p^∗_α is given by (1.11).

Proof. Forλ >0 and y∈Ω, we set

ω_λ(y) ={x∈Ω\ {y}:|∇_xG_α(x, y)|ρ^α(x)> λ}, m_λ(y) = Z

ωλ(y)

dx.

From [11], there existsc₄ >0 such that for any (x, y)∈Ω×Ω with x6=y, G_α(x, y) ≤c₄min

1

|x−y|^N−2α, ρ^α(x)

|x−y|^N−α, ρ^α(y)

|x−y|^N−α

, (2.4) Gα(x, y)≤c4

ρ^α(y) ρ^α(x)|x−y|^N−2α, and by Corollary 3.3 in [3], we have

|∇_xG_α(x, y)| ≤N G_α(x, y) max 1

|x−y|, 1 ρ(x)

. (2.5)

This implies that for anyτ ∈[0,1]

G_α(x, y)≤c₄( ρ^α(y)

|x−y|^N−α)^τ( ρ^α(x)

|x−y|^N−α)^1−τ =c₄ρ^ατ(y)ρ^α(1−τ)(x)

|x−y|^N−α , and then

|∇_xG_α(x, y)| ≤c₅max

( ρ^α(y)

ρ^α(x)|x−y|^N−2α+1,ρ^ατ(y)ρ^{α(1−τ)−1}(x)

|x−y|^N−α )

. (2.6) Lettingτ = ^2α−1_{α N−2α+1}^N−α ∈(0,1), we derive

|∇_xG_α(x, y)|ρ^α(x)≤c₅max







ρ^2α−1(y)ρ^1−α_Ω

|x−y|^N−2α+1,ρ(2α−1)(N−α) N−2α+1 (y)ρ

(2α−1)(1−α) N−2α+1

Ω

|x−y|^N−α





 .

whereρ_Ω= sup_z∈Ωρ(z). There exists some c₆ >0 such that ω_λ(y)⊂n

x∈Ω :|x−y| ≤c6ρ^{N−2α+12α−1} (y) max{λ^{−N−2α+11} , λ^−N−α1 }o .

By N−2α+ 1> N −α, we deduce that for any λ >1, there holds

ω_λ(y)⊂ {x∈Ω :|x−y| ≤c6ρ^{N−2α+12α−1} (y)λ^{−N−2α+11} }. (2.7) As a consequence,

m_λ(y)≤c7ρ^{(2α−1)p∗α}(y)λ^−p∗α, wherec₇ >0 independent of y and λ.

LetE ⊂Ω be a Borel set andλ >1, then Z

E

|∇_xG_α(x, y)|ρ^α(x)dx≤ Z

ωλ(y)

|∇_xG_α(x, y)|ρ^α(x)dx+λ Z

E

dx.

Noting that Z

ωλ(y)

|∇_xG_α(x, y)|ρ^α(x)dx = − Z ∞

λ

sdm_s(y)

= λm_λ(y) + Z ∞

λ

m_s(y)ds

≤ c₈ρ^{(2α−1)p∗α}(y)λ^1−p∗α, for somec₈ >0, we derive

Z

E

|∇xGα(x, y)|ρ^α(x)dx≤c8ρ^{(2α−1)p∗α}(y)λ^1−p∗α+λ Z

E

dx.

Choosingλ=ρ^2α−1(y)(R

Edx)^−p1∗α yields Z

E

|∇_xG_α(x, y)|ρ^α(x)dx≤(c₈+ 1)ρ^2α−1(y)(

Z

E

dx)

p∗ α−1

p∗

α , ∀y∈Ω.

Therefore, Z

E

|∇G_α[|ν|](x)|ρ^α(x)dx= Z

Ω

Z

E

|∇_xG_α(x, y)|ρ^α(x)dxd|ν(y)|

≤ Z

Ω

ρ^2α−1(y)

ρ^1−2α(y) Z

E

|∇_xG_α(x, y)|ρ^α(x)dx

d|ν(y)|

≤(c₈+ 1) Z

Ω

ρ^β(y)ρ^2α−1−β(y)d|ν(y)|

Z

E

dx ^p

∗α−1 p∗

α

≤(c₈+ 1)ρ^2α−1−β_Ω kνkM(Ω,ρ^β)

Z

E

dx ^p

∗α−1 p∗

α .

(2.8) As a consequence,

k∇G_α[|ν|]k_Mp∗

α(Ω,ρ^αdx) ≤c₂kνk_M(Ω,ρβ),

which ends the proof.

Proposition 2.3 [13] Assume that ν∈L¹(Ω, ρ^βdx) with 0≤β ≤α. Then for p∈(1,_N−2α+β^N ), there exists c₉>0 such that for any ν ∈L¹(Ω, ρ^βdx)

kG_α[ν]k_W^2α−γ,p_(Ω)≤c9kνk_L1(Ω,ρ^βdx), (2.9) where p^′ = _p−1^p , γ =β+^N_p′ if β >0 and γ > ^N_p′ if β = 0.

Proposition 2.4 If 0 ≤ β < 2α−1, then the mapping ν 7→ |∇G_α[ν]| is compact from L¹(Ω, ρ^βdx) into L^q(Ω) for any q ∈[1, p^∗_α,β) and there exists c₁₀>0 such that

Z

Ω

|∇G_α[ν](x)|^qdx ¹_q

≤c₁₀ Z

Ω

|ν(x)|ρ^β(x)dx, (2.10) where p^∗_α,β is given by (1.13).

Proof. For ν ∈ L¹(Ω, ρ^βdx) with 0 ≤ β < 2α−1 < α , we obtain from Proposition 2.3 that

G_α[ν]∈W^2α−γ,p(Ω),

where p ∈ (1, p^∗_α,β) and 2α−γ > 1. Therefore, |∇G_α[ν]| ∈ W^{2α−γ−1,p}(Ω) and

k∇G_α[ν]k_W^{2α−γ−1,p}_(Ω)≤c₉kνk_L1(Ω,ρ^βdx). (2.11) By [23, Corollary 7.2], the embedding ofW^{2α−γ−1,p}(Ω) into L^q(Ω) is com- pact forq ∈[1,N−(2α−γ−1)p^{N p} ). When β >0,

N p

N−(2α−γ −1)p = N p

N−(2α−β−N^p−1_p −1)p

= N

N−2α+ 1 +β =p^∗_α,β. Whenβ = 0,

lim

γ→(^N_p′)⁺

N p

N −(2α−γ−1)p = N p

N −(2α−N^p−1_p −1)p

= N

N −2α+ 1 =p^∗_α,0.

Then the mapping ν 7→ |∇G_α[ν]| is compact from L¹(Ω, ρ^βdx) into L^q(Ω) for any q ∈[1, p^∗_α,β). Inequality (2.10) follows by (2.11) and the continuity of the embedding ofW^{2α−γ−1,p}(Ω) intoL^q(Ω).

Remark. Ifν ∈L¹(Ω, ρ^βdx) with 0≤β <2α−1 and u is the solution of (−∆)^αu =ν in Ω,

u = 0 in Ω^c, then for anyq∈[1, p^∗_α,β),

Z

Ω

|∇u|^qdx ¹_q

≤c₁₀ Z

Ω

|ν(x)|ρ^β(x)dx.

2.2 Classical solutions

In this subsection we consider the question of existence of classical solutions to problem

(−∆)^αu+h(|∇u|) =f in Ω,

u= 0 in Ω^c. (2.12)

Theorem 2.1 Assume h ∈C^θ(R₊)∩L^∞(R₊) for some θ∈(0,1] and f ∈ C^θ( ¯Ω). Then problem (2.12) admits a unique classical solutionu. Moreover, (i) if f−h(0)≥0 in Ω, then u≥0;

(ii) the mappings h 7→ u and f 7→ u are respectively nonincreasing and nondecreasing.

Proof. We divide the proof into several steps.

Step 1. Existence. We define the operatorT by

T u=G_α[f−h(|∇u|)], ∀u∈W₀^1,1(Ω).

Using (2.6) withτ = 0 yields

kT uk_W1,1(Ω) ≤ kG_α[f]k_W1,1(Ω)+kG_α[h(|∇u|)]k_W1,1(Ω)

≤ kfk_L^∞_(Ω)+kh(|∇u|)k_L^∞_(Ω) k

Z

Ω

Gα(·, y)dyk_W^1,1_(Ω)

= c₁₁ kfk_L∞(Ω)+khk_L∞(R₊)

, (2.13)

where c₁₁ = kR

ΩG_α(·, y)dyk_W1,1(Ω). Thus T maps W₀^1,1(Ω) into itself.

Clearly, if un → u in W₀^1,1(Ω) as n → ∞, then h(|∇un|) → h(|∇u|) in L¹(Ω), thus T is continuous. We claim that T is a compact operator. In fact, foru∈W₀^1,1(Ω), we see thatf−h(|∇u|)∈L¹(Ω) and then, by Propo- sition 2.3, it implies that T u ∈ W₀^2α−γ,p(Ω) where γ ∈ (^N(p−1)_p ,2α−1) and 2α −1 > ^N(p−1)_p > 0 for p ∈ (1,_N−2α+1^N ). Since the embedding W₀^2α−γ,p(Ω)֒→W₀^1,1(Ω) is compact, T is a compact operator.

Let O = {u ∈ W₀^1,1(Ω) : kuk_W1,1(Ω) ≤ c₁₀(kfk_L^∞_(Ω) +khk_L^∞₍R₊))}, which is a closed and convex set ofW₀^1,1(Ω). Combining with (2.13), there holds

T(O)⊂ O.

It follows by Schauder’s fixed point theorem that there exists some u ∈ W₀^1,1(Ω) such that T u=u.

Next we show that u is a classical solution of (2.12). Let open set O satisfy O ⊂O¯ ⊂ Ω. By Proposition 2.3 in [26], for any σ ∈ (0,2α), there existsc₁₂>0 such that

kuk_C^σ_(O)≤c₁₂{kh(|∇u|)k_L^∞_(Ω)+kfk_L^∞_(Ω)}, and by choosing σ= ^2α+1₂ ∈(1,2α), then

k|∇u|k_Cσ−1(O)≤c₁₂{kh(|∇u|)k_L∞(Ω)+kfk_L∞(Ω)},

and then applied [26, Corollary 2.4],uisC^2α+ǫ0 locally in Ω for someǫ0 >0.

Thenu is a classical solution of (2.12). Moreover, from [13], we have Z

Ω

[u(−∆)^αξ+h(|∇u|)ξ]dx= Z

Ω

ξf dx, ∀ξ∈X_α. (2.14)

Step 2. Proof of (i). If u is not nonnegative, then there exists x0 ∈Ω such that

u(x₀) = min

x∈Ωu(x)<0,

then ∇u(x₀) = 0 and (−∆)^αu(x₀) <0. Since u is the classical solution of (2.12), (−∆)^αu(x₀) =f(x₀)−h(0)≥0, which is a contradiction.

Step 3. Proof of(ii). We just give the proof of the first argument, the proof of the second being similar. Leth₁ and h₂ satisfy our hypotheses for hand h₁ ≤ h₂. Denote u₁ and u₂ the solutions of (2.12) with h replaced by h₁ and h₂ respectively. If there existsx₀∈Ω such that

(u₁−u₂)(x₀) = min

x∈Ω{(u₁−u₂)(x)}<0.

Then

(−∆)^α(u1−u2)(x0)<0, ∇u1(x0) =∇u2(x0).

This implies

(−∆)^α(u₁−u₂)(x₀) +h₁(|∇u₁(x₀)|)−h₂(|∇u₂(x₀)|)<0. (2.15) However,

(−∆)^α(u₁−u₂)(x₀) +h₁(|∇u₁(x₀)|)−h₂(|∇u₂(x₀)|) =f(x₀)−f(x₀) = 0, contradiction. Thenu₁≥u₂.

Uniqueness follows from Step 3.

3 Proof of Theorems 1.1 and 1.2

3.1 The absorption case

In this subsection, we prove the existence of a weak solution to (1.1) when ǫ= 1. To this end, we give below an auxiliary lemma.

Lemma 3.1 Assume thatg:R₊ 7→R₊is continuous and (1.10) holds with p^∗_α. Then there is a sequence real positive numbers {Tn} such that

n→∞lim T_n=∞ and lim

n→∞g(T_n)T_n^−p∗α = 0.

Proof. Let {s_n} be a sequence of real positive numbers converging to ∞.

We observe Z 2sn

sn

g(t)t^{−1−p∗α}dt ≥ min

t∈[sn,2sn]g(t)(2s_n)^{−1−p∗α} Z 2sn

sn

dt

= 2^{−1−p∗α}s^−p_n ^∗α min

t∈[sn,2sn]g(t)

and by (1.10),

n→∞lim Z _2sn

sn

g(t)t^{−1−p∗α}dt= 0.

Then we choose T_n ∈[s_n,2s_n] such that g(T_n) = min_t∈[sn,2sn]g(t) and then

the claim follows.

Proof of Theorem 1.1. Letβ ∈[0,2α−1), we define the space C_β( ¯Ω) ={ζ ∈C( ¯Ω) :ρ^−βζ ∈C( ¯Ω)}

endowed with the norm

kζk_{Cβ( ¯Ω)}=kρ^−βζk_{C( ¯Ω)}.

Let{ν_n} ⊂C¹( ¯Ω) be a sequence of nonnegative functions such thatν_n→ν in sense of duality withC_β( ¯Ω), that is,

n→∞lim Z

Ω¯

ζν_ndx= Z

Ω¯

ζdν, ∀ζ ∈C_β( ¯Ω). (3.1) By the Banach-Steinhaus Theorem, kν_nkM(Ω,ρ^β) is bounded independently of n. We consider a sequence{g_n} of C¹ nonnegative functions defined on R₊ such thatg_n(0) = 0 and

g_n≤g_n+1 ≤g, sup

s∈R₊

g_n(s) =n and lim

n→∞kg_n−gk_L^∞

loc(R₊)= 0. (3.2) By Theorem 2.1, there exists a uniquenonnegative solution u_nof (1.1) with dataνn and gn instead of ν and g, and there holds

Z

Ω

(u_n+g_n(|∇u_n|)η₁)dx= Z

Ω

ν_nη₁dx≤CkνkM(Ω,ρ^β), (3.3) whereη₁ =G_α[1]. Therefore, kg_n(|∇u_n|)kM(Ω,ρ^α) is bounded independently of n. For ε > 0 and ξε = (η1 +ε)^βα −ε^βα ∈ X_α which is concave in the interval [0, η₁(¯ω)], whereη₁(¯ω) = max_x∈Ωη₁(x). By [13, Lemma 2.3 (ii)], we see that

(−∆)^αξ_ε = β

α(η₁+ε)^α1(−∆)^αη₁−β(β−α)

α² (η₁+ε)^β−2αα Z

Ω

(η₁(y)−η₁(x))²

|y−x|^N+2α dy

≥ β

α(η₁+ε)^β−αα , and ξ_ε∈X_α. Since

Z

Ω

(u_n(−∆)^αξ_ε+g_n(|∇u_n|)ξ_ε)dx= Z

Ω

ξ_εν_ndx,

we obtain Z

Ω

β

αu_n(η₁+ε)^β−αα +g_n(|∇u_n|)ξ_ε

dx≤ Z

Ω

ξ_εν_ndx.

If we letε→0, it yields Z

Ω

β αu_nη

β−α

1α +g_n(|∇u_n|)η

β

1α

dx≤

Z

Ω

η

β

1αν_ndx.

Using [13, Lemma 2.3], we derive the estimate Z

Ω

u_nρ^β−α+g_n(|∇u_n|)ρ^β

dx≤c₁₃kν_nkM(Ω,ρ^β)≤c₁₄kνkM(Ω,ρ^β). (3.4) Thus{g_n(|∇u_n|)} is uniformly bounded inL¹(Ω, ρ^βdx).Since u_n=G[ν_n− g_n(|∇u_n|)], there holds

k|∇u_n|k_Mp∗

α(Ω,ρ^αdx) ≤ kν_nkM(Ω,ρ^β)+kg_n(|∇u_n|)kM(Ω,ρ^β)

≤ c₁₅kνkM(Ω,ρ^β).

Sinceν_n−g_n(|∇u_n|) is uniformly bounded inL¹(Ω, ρ^βdx), we use Proposition 2.4 to obtain that the sequences {u_n}, {|∇u_n|} are relatively compact in L^q(Ω) for q ∈[1,_N−2α+β^N ) and q ∈[1, p^∗_α,β), respectively. Thus, there exist a sub-sequence{unk} and some u∈L^q(Ω) withq ∈[1,_N−2α+β^N ) such that (i) u_nk →ua.e. in Ω and in L^q(Ω) withq∈[1,_N−2α+β^N );

(ii) |∇u_nk| → |∇u|a.e. in Ω and inL^q(Ω) withq∈[1, p^∗_α,β).

Therefore,g_nk(|∇u_nk|)→g(|∇u|) a.e. in Ω. For λ >0, we denote S_λ ={x∈Ω :|∇u_nk(x)|> λ} and ω(λ) =

Z

Sλ

ρ^α(x)dx.

Then for any Borel setE ⊂Ω, we have that Z

E

gnk(|∇unk|)|ρ^α(x)dx≤ Z

E

g(|∇unk|)|ρ^α(x)dx

= Z

E∩S_λ^c

g(|∇u_nk|)ρ^α(x)dx+ Z

E∩Sλ

g(|∇u_nk|)ρ^α(x)dx

≤g(λ)˜ Z

E

ρ^α(x)dx+ Z

Sλ

g(|∇u_nk|)ρ^α(x)dx

≤˜g(λ) Z

E

ρ^α(x)dx− Z _∞

λ

g(s)dω(s),

where ˜g(s) = max_t∈[0,s]{g(t)}. But Z _∞

λ

g(s)dω(s) = lim

n→∞

Z _Tn

λ

g(s)dω(s).

where {Tn} is given by Lemma 3.1. Since |∇unk| ∈ M^p∗α(Ω, ρ^αdx), ω(s) ≤ c₁₆s^−p∗α and

− Z Tn

λ

g(s)dω(s) =−

g(s)ω(s) s=Tn

s=λ

+ Z Tn

λ

ω(s)dg(s)

≤g(λ)ω(λ)−g(T_n)ω(T_n) +c₁₆ Z _Tn

λ

s^−p∗αdg(s)

≤g(λ)ω(λ)−g(T_n)ω(T_n) +c₁₆

T_n^−p∗αg(T_n)−λ^−p∗αg(λ) + c₁₆

p^∗_α+ 1 Z _Tn

λ

s^{−1−p∗α}g(s)ds.

By assumption (1.10) and Lemma 3.1, it follows

n→∞lim T_n^−p∗αg(T_n) = 0. (3.5) Along with g(λ)ω(λ)≤c₁₆λ^−p∗αg(λ), we have

− Z ∞

λ

g(s)dω(s)≤ c16

p^∗_α+ 1 Z ∞

λ

s^{−1−p∗α}g(s)ds.

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

c₁₆ p^∗_α+ 1

Z ∞ λ

s^{−1−p∗α}g(s)ds≤ ǫ 2, and δ >0 such that

Z

E

ρ^α(x)dx≤δ=⇒g(λ)˜ Z

E

dx≤ ǫ 2.

This proves that{g_nk(|∇u_nk|)}is uniformly integrable inL¹(Ω, ρ^αdx). Then gnk(|∇unk|)→g(|∇u|) inL¹(Ω, ρ^αdx) by Vitali convergence theorem. Let- tingn_k→ ∞ in the identity

Z

Ω

(u_nk(−∆)^αξ+g_nk(|∇u_nk|)ξ)dx= Z

Ω

ν_nkξdx, ∀ξ ∈X_α,

it infers thatuis a weak solution of (1.1). Sinceu_nk is nonnegative, so isu.

Estimate (1.12) is a consequence of positivity and

u_nk =G_α[ν_nk]−G_α[g_nk(|∇u_nk|)]≤G_α[ν_nk].

Since limnk→∞unk =u, (1.12) follows.

3.2 The source case

In this subsection we study the existence of solutions to problem (1.1) when ǫ=−1.

Proof of Theorem 1.2. Let{ν_n}be a sequence of C² nonnegative functions converging to ν in the sense of (3.1), {gn} an increasing sequence of C¹, nonnegative bounded functions defined onR₊ satisfying (3.2) and converg- ing to g. We set p₀ = ^p+p

∗ α,β

2 ∈(p, p^∗_α,β), where p^∗_α,β is given by (1.13) and p < p^∗_α,β is the maximal growth rate of gwhich satisfies (1.14), and

M(v) = Z

Ω

|∇v|^p0dx _p¹

0 .

We may assume thatkν_nk_L1(Ω,ρ^βdx)≤2kνkM(Ω,ρ^β) for all n≥1.

Step 1. We claim that for n≥1,

(−∆)^αu_n=g_n(|∇u_n|) +ν_n in Ω,

u_n= 0 in Ω^c

admits a solution u_n such that

M(u_n)≤¯λ, where λ >¯ 0 independent of n.

To this end, we define the operators {T_n}by

T_nu=G_α[g_n(|∇u|) +ν_n], ∀u∈W₀^1,p0(Ω).

On the one hand, using (2.6) withτ = 0 yields

kTnuk_W^1,1_(Ω) ≤ kG_α[νn]k_W^1,1_(Ω)+kG_α[gn(|∇u|)]k_W^1,1_(Ω)

≤ c₁₁ kν_nk_L∞(Ω)+kg_nk_L∞(R₊)

,

where c₁₁ = kR

ΩG_α(·, y)dyk_W1,1(Ω). On the other hand, by (1.14) and Proposition 2.4, we have

Z

Ω

|∇(T_nu)|^p0dx _p¹

0 ≤ c₂kg_n(|∇u|) +ν_nk_L1(Ω,ρ^βdx)

≤ c₂[kg_n(|∇u|)k_L1(Ω,ρ^βdx)+ 2kνkM(Ω,ρ^β)] (3.6)

≤ c₂c₁ Z

Ω

|∇u|^pρ^βdx+c₁₇σ₀+ 2c₂kνkM(Ω,ρ^β), wherec17=c2R

Ωρ^βdx. Then we use H¨older inequality to obtain that Z

Ω

|∇u|^pρ^βdx ¹_p

≤( Z

Ω

ρ

βp0

p0−pdx)^1p−p10 Z

Ω

|∇u|^p0dx _p¹

0 , (3.7)

where R

Ωρ

βp0

p0−pdx is bounded, since _p^βp0

0−p ≥ 0. Along with (3.6) and (3.7), we derive

M(Tnu)≤c18M(u)^p+c19kνkM(Ω,ρ^β)+c17σ0, (3.8) wherec18=c2c1(R

Ωρ

βp0

p0−pdx)^1p−p10 >0 andc19>0 independent ofn. There- fore, if we assume that M(u)≤λ, inequality (3.8) implies

M(Tnu)≤c18λ^p+c19kνkM(Ω,ρ^β)+c17σ0. (3.9) Let ¯λ >0 be the largest root of the equation

c₁₈λ^p+c₁₉kνkM(Ω,ρ^β)+c₁₇σ₀=λ, (3.10) This root exists if one of the following condition holds:

(i)p∈(0,1), in which case (3.10) admits only one root;

(ii)p= 1 and c₁₇<1, and again (3.10) admits only one root;

(iii)p∈(1, p^∗_α) and there existsε₀ >0 such that maxn

kνkM(Ω,ρ^β), σ₀o

≤ε₀. In that case (3.10) admits usually two positive roots.

If we suppose that one of the above conditions holds, the definition of ¯λ >0 implies that it is the largest λ >0 such that

c18λ^p+c19kνkM(Ω,ρ^β)+c17σ0≤λ, (3.11) ForM(u)≤¯λ, we obtain that

M(T_nu)≤c₁₈λ¯^p+c₁₉kνkM(Ω,ρ^β)+c₁₇σ₀= ¯λ.

By the assumptions of Theorem 1.2, ¯λ exists and it is larger than M(u_n).

Therefore,

Z

Ω

|∇(T_nu)|^p0dx≤¯λ^p0. (3.12) Thus T_n maps W₀^1,p0(Ω) into itself. Clearly, if u_n → u in W₀^1,p0(Ω) as n → ∞, then gn(|∇un|) → gn(|∇u|) in L¹(Ω), thus T is continuous. We claim that T is a compact operator. In fact, for u ∈ W₀^1,p0(Ω), we see that ν_n−g_n(|∇u|) ∈ L¹(Ω) and then, by Proposition 2.3, it implies that T_nu∈W₀^2α−γ,p(Ω) where γ ∈(^N(p−1)_p ,2α−1) and 2α−1> ^N(p−1)_p >0 for p∈(1,_N−2α+1^N ). Since the embeddingW₀^2α−γ,p(Ω)֒→W₀^1,p0(Ω) is compact, T_n is a compact operator.

Let

G={u∈W₀^1,p0(Ω) :kuk_W1,1(Ω)≤c₁₁(kν_nk_L∞(Ω)+kg_nk_L∞(R₊)) and M(u)≤λ},¯

which is a closed and convex set ofW₀^1,p0(Ω). Combining with (2.13), there holds

T_n(G) ⊂ G.

It follows by Schauder’s fixed point theorem that there exists some u_n ∈ W₀^1,p0(Ω) such thatT_nu_n =u_nandM(u_n)≤λ,¯ where ¯λ >0 independent of n. By the same arguments as in Theorem 2.1,u_n belongs to C^2α+ǫ0 locally in Ω, and

Z

Ω

u_n(−∆)^αξ = Z

Ω

g_n(|∇u_n|)ξdx+ Z

Ω

ξν_ndx, ∀ξ∈X_α. (3.13) Step 2: Convergence. By (3.12) and (3.7), g_n(|∇u_n|) is uniformly bounded inL¹(Ω, ρ^βdx). By Proposition 2.3, {u_n} is bounded in W₀^2α−γ,q(Ω) where q∈(1, p^∗_α,β) and 2α−γ >1. By Proposition 2.4, there exist a subsequence {u_nk} and u such that u_nk → u a.e. in Ω and in L¹(Ω), and |∇u_nk| →

|∇u| a.e. in Ω and in L^q(Ω) for any q ∈ [1, p^∗_α,β). By assumption (G), g_nk(|∇u_nk|)→g(|∇u|) inL¹(Ω). Letting n_k→ ∞ to have that

Z

Ω

u(−∆)^αξ= Z

Ω

g(|∇u|)ξdx+ Z

Ω

ξdν, ∀ξ∈X_α,

thusuis a weak solution of (1.1) which is nonnegative as{un}are nonneg- ative. Furthermore, (1.15) follows from the positivity ofg(|∇u_n]).

4 The case of the Dirac mass

In this section we assume that Ω is an open, bounded and C² domain con- taining 0 and u a nonnegative weak solution of

(−∆)^αu+|∇u|^p =δ₀ in Ω,

u = 0 in Ω^c, (4.1)

where p ∈ (0, p^∗_α) and δ₀ is the Dirac mass at 0. We recall the following result dealing with the convolution operator ∗ in Lorentz spaces L^p,q(R^N) (see [25]).

Proposition 4.1 Let1≤p₁, q₁, p₂, q₂ ≤ ∞and suppose _p¹

1+_p¹

2 >1. Iff ∈ L^p1,q1(R^N) and g∈L^p2,q2(R^N), then f∗g∈L^r,s(R^N) with ¹_r = _p¹₁ +_p¹₂ −1,

1 q1 +_q¹

2 ≥ ¹_s and there holds

kf ∗gk_Lr,s(R^N)≤3rkfk_L^p₁^,q₁₍R^N)kgk_L^p₂^,q₂₍R^N). (4.2) In the particular case of Marcinkiewicz spaces L^p,∞(R^N) = M^p(R^N), the result takes the form

kf∗gk_M^r₍R^N)≤3rkfk_M^p1(R^N)kgk_M^p2(R^N). (4.3)

Proposition 4.2 Assume that 0 < p < p^∗_α and u is a nonnegative weak solution of (4.1). Then

0≤u≤G_α[δ₀], (4.4)

|∇u| ∈L^∞_loc(Ω\ {0}) and u is a classical solution of (−∆)^αu+|∇u|^p= 0 in Ω\ {0},

u= 0 in Ω^c. (4.5)

Proof. Since 0 < p < p^∗_α, (4.1) admits a solution. Estimate (4.4) is a particular case of (1.12). We pick a point a∈Ω\ {0} and consider a finite sequence {r_j}^κ_j=0 such that 0< r_κ < r_κ−1 < ... < r₀ and ¯B_r0(a)⊂Ω\ {0}.

We setd_j =r_j−1−r_j,j = 1, ...κ. By (3.4) withβ = 0, it follows that Z

Ω

(u+|∇u|^p)dx≤c₂₀. (4.6) Let {η_n} ⊂ C^∞

0 (R^N) be a sequence of radially decreasing and symmetric mollifiers such that supp(η_n) ⊂ B_εn(0) and ε_n ≤ ¹₂min{ρ(a)−r₀,|a| −r₀} and u_n=u∗η_n. Since

η_n∗(−∆)^αξ = (−∆)^α(ξ∗η_n) by Fourier analysis and

Z

R^N

(u(−∆)^α(ξ∗ηn)+ξ∗ηn|∇u|^p)dx= Z

R^N

(u∗ηn(−∆)^αξ+ηn∗ |∇u|^pξ)dx becauseη_n is radially symmetric, it follows thatu_n is a classical solution of

(−∆)^αu_n+|∇u|^p∗η_n=η_n in Ω_n,

un= 0 in Ω^c_n, (4.7) where Ω_n={x∈R^N : dist(x,Ω)< ε_n}. We denote byG_α,n(x, y) the Green kernel of (−∆)^αin Ω_nand byG_α,nthe Green operator. Setf_n=η_n−|∇u|^p∗ η_n, then u_n = G_α,n[f_n]. If we set f_n,r0 =f_nχ_Br

0(a), ˜f_n,r0 = f_n−f_n,r0, we have

∂_xiu_n(x) = Z

Ωn

∂_xiG_α,n(x, y)f_n(y)dy

= Z

Ωn

∂_xiG_α,n(x, y)f_n,r0(y)dy+ Z

Ωn

∂_xiG_α,n(x, y) ˜f_n,r0(y)dy

=v_n,r0(x) + ˜v_n,r0(x), where

vn,r0(x) = Z

Br0(a)

∂xiGα,n(x, y)fn(y)dy =− Z

Br0(a)

∂xiGα,n(x, y)|∇u|^p∗ηn(y)dy