Weak and strong singular solutions of semilinear fractional elliptic equations

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Weak and strong singular solutions of semilinear fractional elliptic equations

Huyuan Chen, Laurent Veron

To cite this version:

Huyuan Chen, Laurent Veron. Weak and strong singular solutions of semilinear fractional elliptic equations. 2013. �hal-00848582v3�

Weakly and strongly singular solutions of semilinear fractional elliptic equations

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 Letp∈(0,_N^N

−2α),α∈(0,1) and Ω⊂R^N be a boundedC²domain containing 0. Ifδ⁰is the Dirac measure at 0 andk >0, we prove that the weakly singular solutionuk of (Ek) (−∆)^αu+u^p=kδ⁰in Ω which vanishes in Ω^c, is a classical solution of (E_∗) (−∆)^αu+u^p= 0 in Ω\{0}

with the same outer data. When _N²₋^α2α ≤1 +²_N^α, p∈(0,1 +²_N^α] we show that theuk converges to∞in whole Ω when k→ ∞, while, for p∈(1 + ²_N^α,_N−^N2α), the limit of the uk is a strongly singular solution of (E_∗). The same result holds in the case 1 + ²_N^α< _N^2α

−2α excepted if

2α

N < p <1 + ²_N^α.

Contents

1 Introduction 2

2 Preliminaries 5

3 Regularity 7

4 The limit of weakly singular solutions 9

4.1 The casep∈(0,1 +^2α_N] . . . 10

4.2 The casep∈(1,1 +^2α_N] . . . 14

4.3 The case of p∈(1 +^2α_N,_N−^N_2α) . . . 15

4.4 Proof of Theorem 1.2 (ii) and (iii) . . . 19 Key words: Fractional Laplacian, Dirac measure, Isolated singularity, Weak solu- tion, Weakly singular solution, Strongly singular solution.

MSC2010: 35R11, 35J75, 35R06

1chenhuyuan@yeah.net

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

1 Introduction

Let Ω be a boundedC² domain of R^N(N ≥2) containing 0, α ∈(0,1) and letδ₀ denote the Dirac measure at 0. In this paper, we study the properties of the weak solution to problem

(−∆)^αu+u^p=kδ₀ in Ω

u= 0 in Ω^c, (1.1)

where k > 0 and p ∈ (0,_N−^N_2α) and (−∆)^α is the α-fractional Laplacian 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 and

χ_ǫ(t) =

(0, if t∈[0, ǫ]

1, if t> ǫ.

In 1980, Benilan and Brezis (see [2, 1]) studied the caseα= 1 in equation (1.1) and proved in particular that equation

−∆u+u^q=kδ₀ in Ω

u= 0 on ∂Ω (1.2)

admits a unique solutionu_kfor 1< q < N/(N−2), while no solution exists when q ≥ N/(N −2). Soon after, Brezis and V´eron [3] proved that the problem

−∆u+u^q = 0 in Ω\ {0}

u= 0 on ∂Ω (1.3)

admits only the zero solution whenq ≥N/(N−2). When 1< q < N/(N− 2), V´eron in [13] obtained the description of the all the possible singular behaviour of the positive solutions of (1.3). In particular he proved that this behaviour is always isotropic (when (N+ 1)/(N −1)≤q < N/(N−2) the assumption of positivity is unnecessary) and that two types of singular behaviour occur:

(i) either u(x)∼cNk|x|^2−N when x→0 andk can take any positive value;

u is said to have aweak singularity at 0, and actuallyu=u_k.

(ii) or u(x) ∼c_N,q|x|^−q−12 when x →0 and u has a strong singularity at 0, and u=u_∞:= lim_k→∞u_k.

A large series of papers has been devoted to the extension of semilinear problems involving the Laplacian to problems where the diffusion operator

is non-local, the most classical one being the fractional Laplacian, see e.g.

[4, 5, 9, 10, 11]. In a recent work, Chen and V´eron [7] considered the problem (−∆)^αu+u^p = 0 in Ω\ {0}

u= 0 in Ω^c, (1.4)

where 1 + ^2α_N < p < p^∗_α := _N−^N_2α. They proved that (1.4) admits a singular solution u_s which satisfies

xlim→0u_s(x)|x|^p−12α =c₀, (1.5) for some c₀ > 0. Moreover u_s is the unique positive solution of (1.4) such that

0<lim inf

x→0 u(x)|x|^p−12α ≤lim sup

x→0

u(x)|x|^p−12α <∞. (1.6) In this article we will call weakly singular solution a solution u of (1.4) which satisfies lim sup_x→0|u(x)||x|^N−2α <∞ and strongly singular solution if lim_x→0|u(x)||x|^N−2α =∞.

The existence of solutions of (1.1) is a particular case of the more general problem

(−∆)^αu+g(u) =ν in Ω

u= 0 in Ω^c (1.7)

which has been study by Chen and V´eron in [8] under the assumption thatg is a subcritical nonlinearity,ν being a positive and bounded Radon measure in Ω.

Definition 1.1 A functionu belonging toL¹(Ω)is a weak solution of (1.7) if g(u)∈L¹(Ω, ρ^αdx) and

Z

Ω

[u(−∆)^αξ+g(u)ξ]dx= Z

Ω

ξdν ∀ξ∈X_α, (1.8) where ρ(x) := dist(x,Ω^c) and 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)| ≤c1 for some c1 >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].

According to Theorem 1.1 in [8], problem (1.1) admits a unique weak solution u_k, moreover,

G_α[kδ₀]−G_α[(G_α[kδ₀])^p]≤u_k≤G_α[kδ₀] in Ω, (1.9)

whereG_α[·] is the Green operator defined by G_α[ν](x) =

Z

Ω

G_α(x, y)dν(y), ∀ ν ∈M(Ω, ρ^α), (1.10) withG_α is the Green kernel of (−∆)^α in Ω andM(Ω, ρ^α) denotes the space of Radon measures in Ω such thatR

Ωρ^αd|ν |<∞. By (1.9),

xlim→0u_k(x)|x|^N−2α =cα,Nk. (1.11) for somec_α,N >0. From Theorem 1.1 in [8], there holds

u_k(x)≤u_k+1(x), ∀x∈Ω; (1.12) then there exists

u_∞(x) = lim

k→∞u_k(x) ∀x∈R^N \ {0}, (1.13) and u_∞(x)∈R₊∪ {+∞}.

Motivated by these results and in view of the nonlocal character of the fractional Laplacian, in this article we analyse the connection between the solutions of (1.1) and the ones of (1.4). Our main result is the following Theorem 1.1 Assume that 1 + ^2α_N ≥ _N^2α_−2α and p∈ (0, p^∗_α). Then u_k is a classical solution of (1.4). Furthermore,

(i) if p∈(0,1 +^2α_N),

u_∞(x) =∞ ∀x∈Ω; (1.14)

(ii) ifp∈(1 +^2α_N, p^∗_α),

u_∞=u_s, where u_s is the solution of (1.4) satisfying (1.5).

Moreover, if 1 +^2α_N = _N^2α_−2α, (1.14) holds for p= 1 +^2α_N.

The result of part (i) indicates thateven if the absorption is superlinear, the diffusion dominates and there is no strongly singular solution to problem (1.4). On the contrary, part (ii) points out that the absorption dominates the diffusion; the limit functionu_s is the least strongly singular solution of (1.4). Comparing Theorem 1.1 with the results for Laplacian case, part (i) withp ∈(0,1] and (ii) are similar as the Laplacian case, but part (i) with p∈(1,1+^2α_N] is totally different from the one in the caseα= 1. This striking phenomenon comes comes from the fact that the fractional Laplacian is a nonlocal operator, which requires the solution to belong toL¹(Ω), therefore no local barrier can be constructed ifp is too close to 1.

At end, we consider the case where 1 + ^2α_N < _N^2α_−2α. It occurs when N = 2 and ^√5₂⁻¹ < α <1 orN = 3 and ³⁽

√5−1)

4 < α <1. In this situation, we have the following results.

Theorem 1.2 Assume that 1 + ^2α_N < _N^2α_−2α and p∈ (0, p^∗_α). Then u_k is a classical solution of (1.4). Furthermore,

(i) if p∈(0,_2α^N), then

u_∞(x) =∞ ∀x∈Ω;

(ii) if p∈(1 +^2α_N,_N^2α_−2α), then u_∞ is a classical solution of (1.4) and there exist ρ₀ >0 and c₂ >0 such that

c₂|x|^{−(N−2α)pp−1} ≤u_∞≤u_s ∀x∈B_ρ0\ {0}; (1.15) (iii) if p = _N^2α_−2α, then u_∞ is a classical solution of (1.4) and there exist ρ₀ >0 and c₃ >0 such that

c₃ |x|^{−(N−2α)pp−1} (1 +|log(|x|)|)^p−11

≤u_∞≤u_s ∀x∈B_ρ0 \ {0}; (1.16) (iv) ifp∈(_N^2α_−2α, p^∗_α), then

u_∞=u_s.

We remark that _2α^N <1 +^2α_N if 1 +^2α_N < _N^2α_−2α. Therefore Theorem 1.2 does not provide any description ofu_∞ in the region

U :=n

(α, p) ∈(0,1)×(1,_N^N₋₂) : _2α^N <1 +^2α_N,_2α^N < p <1 +^2α_No . Furthermore, in parts (ii) and (iii), we do not obtain that u_∞ =u_s, since (1.15) and (1.16) do not provide sharp estimates onu_∞in order it to belong to the uniqueness class characterized by (1.6).

The paper is organized as follows. In Section 2, we present some some estimates for the Green kernel and comparison principles. In Section 3, we prove that the weak solution of (1.1) is a classical solution of (1.4). Section 4 is devoted to analyze the limit of weakly singular solutions as k→ ∞.

2 Preliminaries

The purpose of this section is to recall some known results. We denote by B_r(x) the ball centered at xwith radius r and B_r :=B_r(0).

Lemma 2.1 Assume that 0< p < p^∗_α, then there exists c₄, c₅, c₆ > 1 such that

(i) if p∈(0,_N^2α_−2α), 1

c₄ ≤G_α[(G_α[δ₀])^p]≤c₄ in B_r\ {0};

(ii) ifp= _N^2α_−2α,

−1

c₅ln|x| ≤G_α[(G_α[δ₀])^p]≤ −c₅ln|x| in B_r\ {0};

(iii) if p∈(_N^2α_−2α, p^∗_α), 1

c₆|x|^{2α−(N−2α)p} ≤G_α[(G_α[δ₀])^p]≤c₆|x|^{2α−(N−2α)p} in B_r\ {0}, where r= ¹₄min{1, dist(0, ∂Ω)} andG_α is defined by (1.9).

Proof. The proof follows easily from Chen-Song’s estimates of Green func- tions [9], see [6, Theorem 5.2] for a detailled computation.

Theorem 2.1 Assume that O is a bounded domain of R^N and u₁, u₂ are continuous inO¯ and satisfy

(−∆)^αu+u^p = 0 in O.

Moreover, we assume that u1 ≥u2 in O^c. Then, (i) either u₁ > u₂ in O,

(ii) oru₁≡u₂ a.e. in R^N.

Proof. The proof refers to [5, Theorem 2.3] (see also [4, Theorem 5.2]).

The following stability result is proved in [5, Theorem 2.2].

Theorem 2.2 Suppose that O is a bounded C² domain and h :R → R is continuous. Assume {u_n} is a sequence of functions, uniformly bounded in L¹(O^c,₁₊ ^dy

|y|^N+2α), satisfying

(−∆)^αu_n+h(u_n)≥f_n(resp (−∆)^αu_n+h(u_n)≤f_n) in O in the viscosity sense, where the f_n are continuous in O. If there holds (i) u_n→u locally uniformly inO,

(ii) u_n→u in L¹(R^N,₁₊ ^dy

|y|^N+2α), (iii) f_n→f locally uniformly inO, then

(−∆)^αu+h(u)≥f (resp (−∆)^αu+h(u)≤f ) in O in the viscosity sense.

3 Regularity

In this section, we prove that any weak solution of (1.1) is a classical solution of (1.4). To this end, we introduce some auxiliary lemma.

Lemma 3.1 Assume that w∈C^2α+ǫ( ¯B₁) withǫ >0 satisfies (−∆)^αw=h in B₁,

where h∈C¹( ¯B₁). Then for β ∈(0,2α), there exists c₇ >0 such that kwk_Cβ( ¯B_1/4)≤c₇(kwk_L∞(B1)+khk_L∞(B1)+k(1+|·|)^−N−2αwk_L1(R^N)). (3.1) Proof. Letη :R^N →[0,1] be aC^∞ function such that

η= 1 in B3

4 and η = 0 in B₁^c. We denote v=wη, thenv∈C^2α+ǫ(R^N) and for x∈B1

2,ǫ∈(0,¹₄), (−∆)^α_ǫv(x) = −

Z

R^N\Bǫ

v(x+y)−v(x)

|y|^N+2α dy

= (−∆)^α_ǫw(x) + Z

R^N\Bǫ

(1−η(x+y))w(x+y)

|y|^N+2α dy.

Together with the fact of η(x+y) = 1 for y∈B_ǫ, we have Z

RN\Bǫ

(1−η(x+y))w(x+y)

|y|^N+2α dy = Z

RN

(1−η(x+y))w(x+y)

|y|^N+2α dy=:h1(x), thus,

(−∆)^αv=h+h₁ in B1 2. Forx∈B1

2 and z∈R^N \B3

4, there holds

|z−x| ≥ |z| − |x| ≥ |z| − 1 2 ≥ 1

16(1 +|z|) which implies

|h₁(x)|=|

Z

R^N

(1−η(z))w(z)

|z−x|^N+2α dz| ≤ Z

R^N\B3 4

|w(z)|

|z−x|^N+2αdz

≤ 16^N+2α Z

R^N

|w(z)|

(1 +|z|)^N+2αdz

= 16^N+2αk(1 +| · |)^−N−2αwk_L1(R^N).

By [11, Proposition 2.1.9], for β∈(0,2α), there exists c8 >0 such that kvk_Cβ( ¯B_1/4) ≤ c₈(kvk_L∞(RN)+kh+h₁k_L^∞_(B1/2))

≤ c₈(kwk_L∞(B1)+khk_L∞(B1)+kh₁k_L∞(B1/2))

≤ c₉(kwk_L∞(B1)+khk_L∞(B1)+k(1 +| · |)^−N−2αwk_L1(R^N)), wherec9 = 16^N+2αc8. Combining withw=v inB³

4, we obtain (3.1).

Theorem 3.1 Let α ∈ (0,1) and 0 < p < p^∗_α, then the weak solution of (1.1) is a classical solution of (1.4).

Proof. Letu_k be the weak solution of (1.1). By [8, Theorem 1.1], we have 0≤u_k =G_α[kδ₀]−G_α[u^p_k]≤G_α[kδ₀]. (3.2) We observe thatG_α[kδ₀] =kG_α[δ₀] =kG_α(·,0) is C_loc² (Ω\ {0}). Denote by O an open set satisfying ¯O⊂Ω\B_r withr >0. Then G_α[kδ₀] is uniformly bounded in Ω\B_r/2, so isu^p_k by (3.2).

Let {gn} be a sequence nonnegative functions in C₀^∞(R^N) such that g_n→ δ₀ in the weak sense of measures and let w_nbe the solution of

(−∆)^αu+u^p =kg_n in Ω

u= 0 in Ω^c. (3.3)

From [8], we obtain that

u_k= lim

n→∞w_n a.e.in Ω. (3.4)

We observe that 0 ≤ w_n = G_α[kg_n]−G_α[w^p_n] ≤ kG_α[g_n] and G_α[g_n] con- verges toG_α[δ₀] uniformly in any compact set of Ω\ {0} and inL¹(Ω); then there exists c₁₀>0 independent of nsuch that

kw_nk_L∞(Ω\B_r/2)≤c₁₀k and kw_nk_L1(Ω)≤c₁₀k.

By [10, Corollary 2.4] and Lemma 3.1, there exist ǫ > 0, β ∈ (0,2α) and positive constants c11, c12, c13>0 independent of nand k, such that

kw_nk_C^2α+ǫ_(O) ≤c₁₁(kw_nk^p_L∞(Ω\Br

2)+kkg_nk_L^∞_(Ω\Br

2)+kw_nk_Cβ(Ω\B3r 4 ))

≤c₁₂(kw_nk^p_L∞(Ω\Br

2)+kw_nk_L∞(Ω\Br

2)+kkg_nk_L∞(Ω\Br

2)+kw_nk_L1(Ω))

≤c₁₃(k+k^p).

Therefore, together with (3.4) and the Arzela-Ascoli Theorem, it follows that u_k ∈ C^2α+2ǫ(O). This implies that u_k is C^2α+2ǫ locally in Ω\ {0}.

Therefore,w_n→u_k andg_n→0 uniformly in any compact subset of Ω\ {0}

asn→ ∞. We conclude thatu_k is a classical solution of (1.4) by Theorem

2.2.

Corollary 3.1 Let u_k be the weak solution of (1.1) and O be an open set satisfying O¯ ⊂ Ω\B_r with r > 0. Then there exist ǫ > 0 and c₁₄ > 0 independent of k such that

ku_kk_C2α+ǫ(O) ≤c₁₄(ku_kk^p_L∞(Ω\Br

2)+ku_kk_L^∞_(Ω\Br

2)+ku_kk_L1(Ω)). (3.5) Proof. By Theorem 3.1, u_k is a solution of (1.4). Then the result follows from [10, Corollary 2.4] and Lemma 3.1 since there existǫ >0, β ∈(0,2α) and constantsc₁₅, c₁₆>0, independent of k, such that

ku_kk_C2α+ǫ(O) ≤ c₁₅(ku_kk^p_L∞(Ω\Br

2)+ku_kk_Cβ(Ω\B3r 4))

≤ c₁₆(ku_kk^p_L∞(Ω

\Br

2)+ku_kk_L∞(Ω\Br

2)+ku_kk_L1(Ω)).

Theorem 3.2 Assume that the weak solutions u_k of (1.1) satisfy

ku_kk_L1(Ω)≤c₁₇ (3.6)

for some c17 > 0 independent of k and that for any r ∈ (0, dist(0, ∂Ω)), there existsc₁₈>0 independent of k such that

ku_kk_L^∞_(Ω\Br

2)≤c18. (3.7)

Thenu_∞ is a classical solution of (1.4).

Proof. LetO be an open set satisfying ¯O ⊂Ω\B_rfor 0< r < dist(0, ∂Ω).

By (3.5), (3.6) and (3.7), there exist ǫ > 0 and c₁₉ > 0 independent of k such that

ku_kk_C2α+ǫ(O)≤c₁₉.

Together with (1.13) and the Arzela-Ascoli Theorem, it implies that u_∞ belongs to C^2α+ǫ2(O). Hence u_∞ is C^2α+ǫ2, locally in Ω\ {0}. Therefore, wn → u_k and gn → 0 uniformly in any compact set of Ω\ {0} asn → ∞.

Applying Theorem 2.2 we conclude thatu_∞ is a classical solution of (1.4).

4 The limit of weakly singular solutions

We recall thatu_kdenotes the weak solution of (1.1) andd= min{1, dist(0, ∂Ω)}.

4.1 The case p∈(0,1 + ^2α_N]

Proposition 4.1 Let p∈(0,1], then lim_k→∞u_k(x) =∞ for x∈Ω.

Proof. We observe thatG_α[δ₀],G_α[(G_α[δ₀])^p]>0 in Ω. Since by (1.9) u_k ≥kG_α[δ₀]−k^pG_α[(G_α[δ₀])^p],

this implies the claim whenp∈(0,1), for any x∈Ω. For p= 1, u_k =ku₁.

The proof follows sinceu₁ >0 in Ω.

Now we consider the case ofp∈(1,1+^2α_N]. Let{r_k} ⊂(0,^d₂] be a strictly decreasing sequence of numbers satisfying lim_k→∞r_k = 0. Denote by {z_k} the sequence of functions defined by

z_k(x) =

(−d^−N, x∈B_rk

|x|^−N −d^−N, x∈B_r^c_k. (4.1) Lemma 4.1 Let{ρ_k}be a strictly decreasing sequence of numbers such that

rk

ρk < ¹₂ and lim_k→∞ρ^rk

k = 0. Then

(−∆)^αz_k(x)≤ −c_1,k|x|^−N−2α, x∈B_ρ^c_k where c_1,k =−c₂₀log(^r_ρ^k

k) with c₂₀>0 independent of k.

Proof. For anyx∈B_ρ^c_k, there holds (−∆)^αz_k(x) =−1

2 Z

R^N

z_k(x+y) +z_k(x−y)−2z_k(x)

|y|^N+2α dy

=−1 2

Z

RN

|x+y|^−Nχ_B^c

rk(−x)(y) +|x−y|^−Nχ_B^c

rk(x)(y)−2|x|^−N

|y|^N+2α dy

=−1

2|x|^−N−2α Z

R^N

δ(x, z, r_k)

|z|^N+2α dz, whereδ(x, z, r_k) =|z+e_x|^−Nχ_B^c

rk|x|

(−ex)(z) +|z−e_x|^−Nχ_B^c

|x|rk

(ex)(z)−2 and ex = _|^x_x|.

We observe that _|^r_x^k_| ≤ ^r_ρ^k

k < ¹₂ and |z±e_x| ≥1− |z| ≥ ¹₂ for z ∈ B1 2. Then there exists c₂₁>0 such that

|δ(x, z, r_k)|=||z+e_x|^−N +|z−e_x|^−N −2| ≤c₂₁|z|². Therefore,

| Z

B1 2(0)

δ(x, z, r_k)

|z|^N+2α dz| ≤ Z

B1 2(0)

|δ(x, z, r_k)|

|z|^N+2α dz

≤ c₂₁ Z

B1 2(0)

|z|^2−N−2αdz ≤c₂₂,

wherec22>0 is independent of k.

Whenz∈B1

2(−e_x) there holds Z

B1 2(−ex)

δ(x, z, r_k)

|z|^N+2α dz ≥ Z

B^c₁

2

(−ex)

|z+ex|^−Nχ_Brk

|x|

(−ex)(z)−2

|z|^N+2α dz

≥ c₂₃ Z

B1 2(0)\B_rk

|x|

(0)

(|z|^−N −2)dz

≥ −c₂₄log(r_k

|x|)≥ −c₂₄log(r_k ρ_k), wherec₂₃, c₂₄>0 are independent of k.

Forz∈B1

2(e_x), we have Z

B1 2(ex)

δ(x, z, r_k)

|z|^N+2α dz= Z

B1 2(−ex)

δ(x, z, r_k)

|z|^N+2α dz.

Finally, for z∈O :=R^N \(B1

2(0)∪B1

2(−e_x)∪B1

2(e_x)), we have

| Z

O

δ(x, z, r_k)

|z|^N+2α dz| ≤c₂₅ Z

B^c₁

2

(0)

|z|^−N + 1

|z|^N+2α dz≤c₂₆, wherec₂₅, c₂₆>0 are independent of k.

Combining these inequalities we obtain that there existsc₂₀>0 independent ofk such that

(−∆)^αz_k(x)|x|^N+2α≤c₂₀log(r_k

ρ_k) :=c_1,k,

which ends the proof.

Proposition 4.2 Assume that 2α

N−2α <1 + 2α

N , max{1, 2α

N −2α}< p <1 + 2α

N (4.2)

andz_k is defined by (4.1) withr_k =k^{−N−(N−2α)pp−1} (logk)⁻². Then there exists k₀ >0 such that for any k≥k₀

u_k≥c

1 p−1

2,k z_k in B_d, (4.3)

where c_2,k = ln lnk.

Proof. For p ∈ (max{1,_N^2α_−2α},1 + ^2α_N), it follows by (1.9) and Lemma 2.1-(iii) that there exist ρ₀ ∈(0, d) and c₂₇, c₂₈ >0 independent of k such that, for x∈B¯_ρ0 \ {0},

u_k(x) ≥ kG_α[δ₀](x)−k^pG_α[(G_α[δ₀])^p](x)

≥ c₂₇k|x|^−N+2α−c₂₈k^p|x|^{−(N−2α)p+2α}

= c₂₇k|x|^−N+2α(1−c₂₈

c₂₇k^p−1|x|^{N−(N−2α)p}).

We choose

ρ_k=k^{−N−(N−2α)pp−1} (logk)⁻¹. (4.4) There exits k₁ >1 such that fork≥k₁

u_k(x) ≥ c₂₇k|x|^−N+2α(1−c28

c₂₇k^p−1ρ^N_k^{−(N−2α)p})

≥ c₂₇

2 k|x|^−N+2α, x∈B¯ρ_k \ {0}. (4.5) Sincep <1 + ^2α_N, 1−_N−^2α(p_(N−⁻_2α)p¹⁾ >0 and there existsk0≥k1 such that

c₂₇

2 kr^2α_k ≥(ln lnk)^p−11 , (4.6) fork≥k₀. This implies

c₂₇

2 k|x|^2α ≥(ln lnk)^p−11 , x∈B¯_ρk\B_rk. Together with (4.1) and (4.5), we derive

u_k(x)≥(ln lnk)^p−11 z_k(x), x∈B¯_ρk \B_rk, fork≥k₀. Furthermore, it is clear that

(ln lnk)^p−11 z_k(x)≤0≤u_k(x)

whenever x∈B_rk or x∈B_d^c. Setc_2,k= ln lnk, then by Lemma 4.1 (−∆)^αc

1 p−1

2,k z_k(x) +c

p p−1

2,k z_k(x)^p ≤c

p p−1

2,k |x|^−N−2α(−1 +|x|^{N+2α−N p})≤0, for anyx∈B_d\B_ρk, sinceN+ 2α−N p≥0 andd≤1. Applying Theorem 2.1, we infer that

c

1 p−1

2,k z_k(x)≤u_k(x) ∀x∈B¯_d,

which ends the proof.

Proposition 4.3 Assume 1< 2α

N−2α ≤1 + 2α

N and p= 2α

N −2α (4.7)

and letz_k be defined by (4.1) withr_k=k^{−N(N−2α)2α} (logk)⁻³ andk >2. Then there existsk₀>2 such that (4.3) holds fork≥k₀.

Proof. By (1.9) and Lemma 2.1-(ii), there existρ₀ ∈(0, d) andc₃₀, c₃₁>0 independent ofk, such that forx ∈B¯_ρ0 \ {0}

u_k(x) ≥ c₃₀k|x|^−N+2α+c₃₁k^plog|x|

= c₃₀k|x|^−N+2α(1 +c₃₁

c₃₀k^p−1|x|^N−2αlog|x|).

If we choose ρ_k = k^{−N(N−2α)2α} (logk)⁻² there exists k₁ > 1 such that for k≥k₁, we have 1 + ^c_c³¹

30k^p−1ρ^N_k^−2αlog(ρ_k)≥ ¹₂ and u_k(x)≥ c₃₀

2 k|x|^−N+2α ∀x∈B¯_ρk\ {0}. (4.8) Since _N^2α_−2α <1 + ^2α_N, there holds 1−_N(N^4α₋²_2α) >0 and there exists k₀ ≥k₁ such that

c₃₀

2 kr_k^2α= c₃₀

2 k^{1− 4α}

2

N(N−2α)(logk)^−6α ≥(ln lnk)^p−11

fork≥k0. The remaining of the proof is the same as in Proposition 4.2.

In the sequel, we point out the fact that the limit behavior of the u_k depends which of the following three cases holds:

2α

N −2α = 1 +2α

N = N

2α; (4.9)

2α

N −2α <1 +2α N < N

2α; (4.10)

2α

N −2α >1 +2α N > N

2α. (4.11)

Proposition 4.4 Assume 1< 2α

N −2α ≤1 + 2α

N and 1< p < 2α

N −2α, (4.12) or

1 +2α

N < 2α

N −2α and 1< p < N

2α, (4.13)

and z_k is defined by (4.1) with r_k = k^{−N−2αp−1} (logk)⁻¹. Then there exists k₀ >2 such that (4.3) holds for k≥k₀.

Proof. By (1.9) and Lemma 2.1-(i), there existρ0∈(0, d) andc33, c34>0 independent ofk such that forx∈B¯_ρ0 \ {0},

u_k(x) ≥ c₃₃k|x|^−N+2α−c₃₄k^p

= c₃₃k|x|^−N+2α(1−c₃₄ c33

k^p−1|x|^N−2α).

We choose ρ_k = k^{−N−2αp−1} . Then there exists k₁ > 1 such that for k ≥ k₁, 1−^c_c³⁴

33k^p−1ρ^N_k^−2α ≥ ¹₂ and u_k(x)≥ c₃₃

2 k|x|^−N+2α ∀x∈B¯_ρk\ {0}. (4.14) Clearlyp < _2α^N by assumptions (4.12), (4.13), together with relations (4.9)(4.10), (4.11), thus 1−(p−1)_N^2α_−2α >0. Therefore there existsk₀≥k₁ such that

c₃₃

2 kr^2α_k = c₃₃

2 k^{1−(p−1)N−2α2α} (logk)^−2α ≥(log logk)^p−11 =c

1 p−1

2,k

fork≥k₀. The remaining of the proof is similar to the one of Proposition

4.2.

4.2 The case p∈(1,1 + ^2α_N]

We give below the proof, in two steps, of Theorem 1.1 part (i) with p ∈ (1,1 + ^2α_N] and Theorem 1.2 part (i) with p∈(1,^2α_N].

Step 1: We claim thatu_∞=∞ in B_d. We observe that for _N^2α_−2α <1 +^2α_N, Propositions 4.2, 4.3, 4.4 cover the casep∈(max{1,_N^2α_−2α},1+^2α_N), the case 1< _N^2α_−2α <1 +^2α_N along with p= _N^2α_−2α and the case 1< _N^2α_−2α <1 +^2α_N along withp∈(1,_N^2α_−2α) respectively. For _N^2α_−2α = 1 + ^2α_N, Proposition 4.3, 4.4 cover the case p = _N^2α_−2α and the case p ∈ (1,_N^2α_−2α) respectively. So it covers p ∈ (1,1 + ^2α_N] in Theorem 1.1 part (i). When _N^2α_−2α > 1 + ^2α_N, Proposition 4.4 covers p ∈ (1,_2α^N) in Theorem 1.2 part (i). Therefore, we have

u_∞≥c

1 p−1

2,k z_k in B_d and since for any x∈B_d\ {0}, lim_k→∞c

1 p−1

2,k z_k(x) =∞,we deive u_∞=∞ in B_d.

Step 2: We claim that u_∞ = ∞ in Ω. By the fact of u_∞ = ∞ in B_d and u_k+1 ≥u_k in Ω, then for anyn > 1 there exists k_n > 0 such that u_kn ≥n inB_d. For anyx₀∈Ω\B_d, there existsρ >0 such that ¯B_ρ(x₀)⊂Ω∩B_d/2^c . We denote byw_n the solution of

(−∆)^αu+u^p = 0 in Bρ(x0) u= 0 in B_ρ^c(x₀)\B_d/2

u=n in B_d/2.

(4.15)

Then by Theorem 2.1, we have

u_kn ≥w_n. (4.16)

Letη₁ be the solution of

(−∆)^αu= 1 in B_ρ(x₀) u= 0 in B_ρ^c(x₀), and vn=wn−nχB_d/2,thenvn=wn inBρ(x0) and

(−∆)^αv_n(x) +v^p_n(x) = (−∆)^αw_n(x)−n(−∆)^αχ_Bd/2(x) +w_n^p(x)

= n

Z

B_d/2

dy

|y−x|^N+2α ∀x∈B_ρ(x₀).

This means that v_nis a solution of (−∆)^αu+u^p=n

Z

B_d/2

dy

|y−x|^N+2α in B_ρ(x₀),

u= 0 in B_ρ^c(x₀).

(4.17) It is clear that

1 c₃₅ ≤

Z

B_d/2

dy

|y−x|^N+2α ≤c35 ∀x∈Bρ(x0) for somec₃₅>1. Furhermore (_2c ⁿ

35maxη1)^1pη₁ is sub solution of (4.17) for n large enough. Then using Theorem 2.1, we obtain that

v_n≥( n 2c35maxη1

)^1pη₁ ∀x∈B_ρ(x₀), which implies that

w_n ≥( n

2c₃₅maxη₁)^1pη₁ ∀x∈B_ρ(x₀).

Then

nlim→∞w_n(x₀)→ ∞.

Sincex₀ is arbitrary and together with (4.16), it implies thatu_∞=∞in Ω,

which completes the proof.

4.3 The case of p∈(1 + ^2α_N,_N^N

−2α)

Proposition 4.5 Let α∈(0,1) and r₀=dist(0, ∂Ω). Then

(i)ifmax{1 +^2α_N,_N^2α_−2α}< p < p^∗_α, there existR₀∈(0, r₀) andc₃₆>0such that

u_∞(x)≥c₃₆|x|^−p−12α ∀x∈B_R0\ {0}, (4.18)

(ii) if _N^2α_−2α > 1 + ^2α_N and p = _N^2α_−2α, there exist R0 ∈ (0, r0) and c37 >0 such that

u_∞(x)≥ c37

(1 +|log(|x|)|)^p−11

|x|^{−p(N−2α)p−1} , ∀x∈B_R0 \ {0}, (4.19) (iii) if _N^2α_−2α >1 + ^2α_N and p∈(1 +^2α_N,_N^2α_−2α), there exist R₀ ∈(0, r₀) and c₃₈>0 such that

u_∞(x)≥c38|x|^{−p(N−2α)p−1} ∀x∈BR0\ {0}. (4.20) Proof. (i) Using (1.9) and Lemma 2.1(i) with max{1+^2α_N,_N^2α_−2α}< p < p^∗_α, we see that there existρ₀∈(0, r₀) and c₃₉, c₄₀>0 such that

u_k(x)≥c₃₉k|x|^−N+2α−c₄₀k^p|x|^{−(N−2α)p+2α} ∀x∈B_ρ0\ {0}. (4.21) Set

ρ_k= (2^{(N−2α)p−2α−1}c₄₀

c₃₉k^p−1)(N−2α)(p−1)−2α¹ . (4.22) Since (N−2α)(p−1)−2α <0, there holds lim_k→∞ρ_k= 0. Letk₀ >0 such thatρ_k0 ≤ρ₀, then forx∈B_ρk\Bρk

2 , we have c₄₀k^p|x|^{−(N−2α)p+2α} ≤ c₄₀k^p(ρ_k

2 )^{−(N−2α)p+2α}

= c39

2 kρ⁻_k^N+2α

≤ c₃₉

2 k|x|^−N+2α and

k= (2^{(N−2α)p−2α−1}c₄₀

c₃₉)^−p−11 ρ^N−2α−

2α p−1

k ≥c₄₁|x|^{N−2α−p−12α} , where c₄₁ = (2^{(N−2α)p−2α−1c}_c⁴⁰

39)^−p−11 2^{(N−2α)(p−1)−2α−1}. Combining with (4.18), we obtain

u_k(x) = c₃₉k|x|^−N+2α−c₄₀k^p|x|^{−(N−2α)p+2α}

≥ c₃₉

2 k|x|^−N+2α

≥ c₄₂|x|^−p−12α , (4.23)

forx ∈B_ρk \Bρk

2 , wherec₄₂ = ¹₂c₃₉c₄₁ is independent of k. By (4.22), we can choose a sequence{k_n} ⊂[1,+∞) such that

ρ_kn+1 ≥ 1 2ρ_kn,