Local Lipschitz continuity of the inverse of the fractional p -Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains

Nonlinear Analysis 135 (2016) 129–157

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Local Lipschitz continuity of the inverse of the fractional p-Laplacian, H¨older type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains Mahamadi Warma University of Puerto Rico, Rio Piedras Campus, College of Natural Sciences, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377, USA

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Article history: Received 29 October 2015 Accepted 27 January 2016 Communicated by S. Carl MSC: 35R11 35J60 35K55 35B65 47J30 Keywords: Fractional p-Laplace operator Nonlocal quasi-linear and nonlinear elliptic type equations Existence and regularity of solutions Local Lipschitz continuity Parabolic equations Continuous dependence of solutions Nonlinear submarkovian semigroups Ultracontractivity

abstract Let p ∈ (1, ∞), s ∈ (0, 1) and Ω ⊂ RN an arbitrary bounded open set. In the first part we consider the inverse Φs,p := [(−∆)sp,Ω ]−1 of the fractional p-Laplace operator (−∆)sp,Ω with the Dirichlet boundary condition. We show that in the singular case p ∈ (1, 2), the operator Φs,p is locally Lipschitz continuous on L∞ (Ω ) and that global Lipschitz continuity cannot be achieved. We use this result to show p < p < 2, 0 ≤ q ≤ NN−sp − 2 and α, β are small that in the case N > sp, if N2N +2s constants, then the nonlinear problem Np

(−∆)sp u = α | u |q u + β | u | N −sp

−2

u+h

in Ω , u = 0 on RN \ Ω ,

has at least one weak solution. In the second part of the paper, we prove that the operator −(−∆)sp,Ω generates a (nonlinear) submarkovian semigroup (Ss,p (t))t≥0 on L2 (Ω ). If p ∈ [2, ∞) and sp < N , we obtain that Ss,p (t) satisfies the following (Lq − L∞ )-H¨ older type estimate: there exists a constant C > 0 such that for every t > 0 and u, v ∈ Lq (Ω ) (q ≥ 2), we have γ(s)

∥Ss,p (t)u − Ss,p (t)v∥L∞ (Ω) ≤ C | Ω |β(s) t−δ(s) ∥u − v∥Lq (Ω) where β(s), δ(s) and γ(s) are explicit constants depending only on N, s, p, q. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Let Ω ⊂ RN be an arbitrary bounded open set, p ∈ (1, ∞) and 0 < s < 1. In the present paper, we consider the quasi-linear elliptic type Dirichlet boundary value problem (−∆)sp u = f

in Ω ,

u=0

E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.na.2016.01.022 0362-546X/© 2016 Elsevier Ltd. All rights reserved.

on RN \ Ω ,

(1.1)

130

M. Warma / Nonlinear Analysis 135 (2016) 129–157

the nonlinear Dirichlet boundary value problem (−∆)sp u = g(x, u) + h in Ω ,

on RN \ Ω ,

u=0

and the nonlocal quasi-linear diffusion equation  ∂u   (t, x) + (−∆)sp u(t, x) = 0 in (0, ∞) × Ω ,  ∂t u =0 on (0, ∞) × (RN \ Ω ),    u(0, x) = u0 (x) x ∈ Ω.

(1.2)

(1.3)

Here, (−∆)sp denotes the fractional p-Laplace operator, f , h are given distributions, g : Ω × R → R is a measurable function satisfying a certain growth condition and u0 ∈ L2 (Ω ) is given. The main concerns in the article are the following. • Let f ∈ Lr (Ω ) for some r ∈ [1, ∞]. Denoting the unique solution u of (1.1) by u := [(−∆)sp,Ω ]−1 (f ) = Φs,p (f ), we would want to show that the mapping Φs,p is globally H¨older continuous on L∞ (Ω ) if p > 2 and locally Lipschitz continuous on L∞ (Ω ) if 1 < p < 2, where locally means on bounded subsets. • Use the local Lipschitz continuity of the mapping Φs,p (p ∈ (1, 2)) to show that the nonlinear boundary value problem (1.2) has at least one weak solution. • Investigate the existence, regularity and some fine estimates of solutions to the diffusion equation (1.3) and show that the corresponding operator solution is also H¨older continuous for every t > 0. For the convenience of the reader, we introduce the fractional p-Laplace operator. Let 0 < s < 1, p ∈ (1, ∞) and set    |u(x)|p−1 p−1 N N Ls (R ) := u : R → R measurable, dx < ∞ . N +ps RN (1 + |x|) For u ∈ Lp−1 (RN ), x ∈ RN and ε > 0, we write s  (−∆)sp,ε u(x) = CN,p,s

|u(x) − u(y)|p−2

{y∈RN ,|y−x|>ε}

u(x) − u(y) dy, |x − y|N +ps

where the normalized constant CN,p,s is given by s22s Γ CN,p,s =



ps+p+N −2 2



N

π 2 Γ(1 − s)

and Γ is the usual Gamma function (see e.g. [6–8,18,31,37] for the linear case p = 2 and [38] for the general case 1 < p < ∞). The fractional p-Laplacian (−∆)sp u of the function u is defined by  u(x) − u(y) (−∆)sp u(x) = CN,p,s P.V. |u(x) − u(y)|p−2 dy = lim(−∆)sp,ε u(x), x ∈ RN , (1.4) N +ps ε↓0 |x − y| N R provided that the limit exists. We mention that Lp−1 (RN ) is the natural space on which (−∆)sp,ε u exists for s every ε > 0 and is continuous where the function u is continuous. We refer to [38, Section 4] for the class of functions for which the limit in (1.4) exists. The nonlocal operator (−∆)sp is linear if and only if p = 2, hence, nonlinear if and only if p ̸= 2, degenerate if p > 2 and singular if 1 < p < 2. Caffarelli et al. [6–8] have intensively studied the linear operator (−∆)s2 on RN and on subsets of RN with the Dirichlet boundary condition and they have obtained some fundamental and beautiful results. For parabolic problems associated with the linear case, that is, p = 2, we refer to [5,9,37] and their references. The fractional Laplacians being the generator of s-stable processes (Levy flights in some of the physical literature) are widely used to model systems of stochastic dynamics with applications in operation research,

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queuing theory, mathematical finance, risk estimate and others (see e.g. [2,3,32]). We also notice that nonlocal models have been recently used in many applications, including continuum mechanics, graph theory, jump processes, image analyses, machine learning, kinetic equations, phase transitions, nonlocal heat conduction, and the peridynamic model for mechanics. The fractional Laplacian and fractional derivative operators are commonly used to model anomalous diffusion. Physical phenomena exhibiting this property cannot be modeled accurately by the usual advection–dispersion equation; among others, we mention turbulent flows and chaotic dynamics of classical conservative systems. For more applications and details on these facts we refer to [2,3,25,32] and the references therein. At our knowledge there is a few references where the fractional p-Laplacian (−∆)sp for p ̸= 2 has been deeply studied. For example in [26] the authors have investigated the existence and the global regularity of weak solutions to the quasi-linear elliptic problem (3.1). In particular, they have obtained that if Ω is a bounded open set of class C 1,1 and f ∈ L∞ (Ω ), then the unique weak solution of (3.1) belongs to C 0,α (Ω ) for some 0 < α < 1. A finer regularity for solutions of the linear case p = 2 has been obtained in [31]. The eigenvalues associated with (−∆)sp with Dirichlet boundary condition have been also recently investigated in [21]. In [16,17,28] more general nonlocal operators where |x − y|−N −sp is replaced by a general symmetric kernel K(x, y) have been investigated. In particular the corresponding Eq. (3.1) where f is a measure has been studied. Finally in [38] an integration by parts formula for the fractional p-Laplace operator defined on an open set Ω (and called regional fractional p-Laplace operator) has been obtained. We also mention that the normalized constant does not appear in [21,26]. It has been introduced in [38] and justified in [38, Remark 4.2]. But this is not a restriction in the framework of [21,26]. In fact the normalized constant is used only if one needs to approach the classical p-Laplace operator as s ↑ 1. The first main objective of the present article is to investigate the properties of the operator solution Φs,p to the Dirichlet problem (1.1) on arbitrary bounded open sets. We note that if N < sp, then using Sobolev embedding (see (2.6)) one immediately obtains that weak solutions of (1.1) are H¨older continuous on Ω (we refer to Section 3 for our definition of weak solutions). For this reason, we shall concentrate on the case N N ≥ sp. More precisely, we show that in the degenerate case p ∈ [2, ∞), if f, g ∈ Lr (Ω ) for some r > sp , then there is a constant C > 0 such that 1

∥Φs,p (f ) − Φs,p (g)∥L∞ (Ω) ≤ C∥f − g∥Lp−1 r (Ω) . In the singular case p ∈ (1, 2), if N2N +2s < p < 2 and Ω is an arbitrary bounded open set, or if 1 < p ≤ and Ω satisfies a certain regularity condition (see Definition 3.9), then we have the following: if   1 p pq with p := N − and for some q ∈ [p, ∞] ∩ ( p, ∞], r> (2 − p)(q − p) s 2s

(1.5) 2N N +2s

then there is a constant C > 0 such that for all f, g ∈ Lr (Ω ), ∥Φs,p (f ) − Φs,p (g)∥L∞ (Ω)

  2−p p−1 ≤ C ∥f ∥  + ∥g∥ ∥f − g∥Lr (Ω) , q (Ω) Lq (Ω) L

(1.6)

p for some q ∈ [(p⋆ )′ , ∞] where p⋆ := NN−sp is the Sobolev exponent and (p⋆ )′ the index conjugate of p⋆ . We also obtain that if 1 < p < ∞, 0 < s < 1 are such that sp < N , and if p1 ≥ (p⋆ )′ , then the operator Φs,p : Lp1 (Ω ) → W0s,p (Ω ) ∩ Lr (Ω ) is compact for every r ∈ (1, p⋆ ), where W0s,p (Ω ) is the fractional order Sobolev space defined in Section 2.1. In addition, we use the local Lipschitz continuity of the mapping Φs,p , ⋆ that is, (1.6), to show that if N2N +2s < p < 2, q ∈ [0, p − 2] and the measurable function g : Ω × R → R satisfies    ∂g(x, τ )  q p⋆ −2   , ∀ x ∈ Ω , τ ∈ R,  ∂τ  ≤ α|τ | + β|τ |

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for some constants α, β, then the nonlinear problem (1.2) has at least one weak solution provided that α, β are small. We mention that the case s = 1, that is, the classical p-Laplace operator has been studied in [10] where the author has shown that if p ∈ (1, 2) then Φ1,p satisfies the following estimate: there exists a constant ′ C > 0 such that for every f, g ∈ W −1,p (Ω ),  2−p  p−1 ∥Φ1,p (f ) − Φ1,p (g)∥W 1,p (Ω) ≤ C ∥f ∥W −1,p′ (Ω) + ∥g∥W −1,p′ (Ω) ∥f − g∥W −1,p′ (Ω) , 0

−1,p′

where W (Ω ) denotes the dual of the first order Sobolev space W01,p (Ω ). The properties of Φ1,p as an operator from Lr (Ω ) to L∞ (Ω ) have not been considered in [10]. In the same direction, an interpolation argument has been used in [29] to show that if p ∈ (1, 2) and Ω is a bounded open set of class C 1,γ (for some 0 < γ ≤ 1), then for every M > 0, there exist two constants C(M ) > 0 and r ∈ (0, 1) such that for all f, g ∈ BL∞ (Ω) (0, M ), ∥Φ1,p (f ) − Φ1,p (g)∥C 1 (Ω) ≤ C(M )∥f − g∥rL∞ (Ω) . But we notice that there is no way to prove local Lipschitz continuity of Φs,p (including the case s = 1) by using some interpolation arguments. Our second main goal is to study the existence, regularity and find some fine estimates of solutions to the diffusion equation (1.3). To be more precise, we show that a realization of the operator (−∆)sp with the Dirichlet boundary condition on L2 (Ω ) generates a (nonlinear) submarkovian semigroup Ss,p = (Ss,p (t))t≥0 , that is, Ss,p is order preserving and non-expansive on Lq (Ω ) for every q ∈ [2, ∞] (see Section 5). This shows that the system (1.3) has a unique strong solution u which is given by the semigroup Ss,p , that is, u(t, x) = Ss,p (t)u0 (x) for every t ≥ 0 and x ∈ Ω . Moreover, the semigroup Ss,p can be extended to orderpreserving and non-expansive semigroup on Lq (Ω ) for every q ∈ [1, ∞). Hence, (1.3) can be solved for each initial data u0 ∈ Lq (Ω ). We also obtain that if p ∈ [2, ∞) and s ∈ (0, 1) are such that sp < N , then Ss,p (t) maps Lq (Ω ) (q ≥ 2) into L∞ (Ω ) and the following (Lq − L∞ )-H¨older type estimate holds: there exists a constant C > 0 such that for every t > 0 and u0 , v0 ∈ Lq (Ω ) (q ≥ 2), we have γ(s)

∥Ss,p (t)u0 − Ss,p (t)v0 ∥L∞ (Ω) ≤ C|Ω |β(s) t−δ(s) ∥u0 − v0 ∥Lq (Ω) ,

(1.7)

where  N    sp N − sp q β(s) := 1− , N q−2+p N   sp q γ(s) := . q−2+p

 N    sp 1 q δ(s) := 1− , p−2 q−2+p (1.8)

The estimate (1.7) shows in particular that solutions of (1.3) depend continuously on the initial data. The corresponding local problem (s = 1), that is, the case of the p-Laplace operator ∆p has been done in [11,12], where the authors have shown the corresponding estimate (1.7) for the (nonlinear) submarkovian semigroup generated by the realization on L2 (Ω ) of ∆p with Dirichlet boundary condition on bounded open sets. The cases of the Robin and Wentzell boundary conditions for the operator ∆p have been studied in [35,36]. The rest of the paper is organized as follows. In Section 2 we introduce the function spaces and give some tools that are needed to investigate our problems. In Section 3 we study the operator solution to the quasi-linear elliptic type problem (1.1) where in particular the estimates (1.5) and (1.6) have been obtained. In Section 4 we investigate the existence of solutions to the nonlinear problem (1.2). Section 5 contains the main results related to the parabolic problem. We first define the realization on L2 (Ω ) of the fractional p-Laplace operator with the Dirichlet boundary condition by using the method of proper, convex and lower semi-continuous functional. We then prove that the associated operator generates a (nonlinear) submarkovian semigroup on L2 (Ω ) and we conclude the paper by showing the estimate (1.7).

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2. Preliminaries In this section we collect some well-known results on Sobolev spaces and give some tools as they are needed to prove our main results. 2.1. The functional setup Let Ω ⊂ RN be an arbitrary bounded open set. For p ∈ (1, ∞) and s ∈ (0, 1), we denote by     |u(x) − u(y)|p dx dy < ∞ W s,p (Ω ) := u ∈ Lp (Ω ) : N +ps Ω Ω |x − y| the fractional order Sobolev space endowed with the norm   p1   CN,p,s |u(x) − u(y)|p p |u| dx + ∥u∥W s,p (Ω) := dx dy . N +ps 2 Ω Ω Ω |x − y| We let W s,p (Ω)

W0s,p (Ω ) = D(Ω )

.

By definition, W0s,p (Ω ) is the smallest closed subspace of W s,p (Ω ) containing D(Ω ). We let W0s,p (Ω ) = {u ∈ W s,p (RN ) : u = 0 a.e. on RN \ Ω }. Then  ∥u∥W s,p (Ω) = 0

CN,p,s 2

 RN

 RN

|u(x) − u(y)|p dxdy |x − y|N +ps

 p1 (2.1)

defines an equivalent norm on W0s,p (Ω ). It is clear that W0s,p (Ω ) and W0s,p (Ω ) are both subspaces of W s,p (Ω ), but there is no obvious inclusion between W0s,p (Ω ) and W0s,p (Ω ). It has been proved in [23, Theorem 1.4.2.2] (see also [20] for some more general spaces) that if Ω has a continuous boundary, then D(Ω ) is dense in W0s,p (Ω ). In addition if Ω has a Lipschitz continuous boundary and s ̸= 1/p, then W0s,p (Ω ) = W0s,p (Ω ) with equivalent norm. In that case,   p1   CN,p,s |u(x) − u(y)|p ∥u∥W0s,p (Ω) = dxdy , (2.2) N +ps 2 Ω Ω |x − y| defines an equivalent norm on W0s,p (Ω ) = W0s,p (Ω ). In fact, it is well-known (see e.g. [18]) that there is a constant C > 0 such that for every u ∈ W0s,p (Ω ),   |u(x) − u(y)|p ∥u∥pLp (Ω) ≤ C dxdy, N +ps Ω Ω |x − y| and this shows that (2.2) defines an equivalent on W0s,p (Ω ) = W0s,p (Ω ). For (2.1), let u ∈ W0s,p (Ω ). Then       |u(x) − u(y)|p |u(x) − u(y)|p |u(x)|p dxdy = dxdy + dxdy N +sp N +sp N +sp RN RN |x − y| Ω Ω |x − y| Ω RN \Ω |x − y|   |u(y)|p + dxdy N +sp RN \Ω Ω |x − y|     |u(x) − u(y)|p |u(x)|p = dxdy + 2 dxdy. (2.3) N +sp N +sp Ω Ω |x − y| Ω RN \Ω |x − y| Since Ω has a Lipschitz continuous boundary, then by [23, Formula (1.3.2.12) p.19], there exist two constants 0 < C1 ≤ C2 such that  C1 dy C2 ≤ ≤ , x ∈ Ω. (2.4) sp N +sp dist(x, ∂Ω ) dist(x, ∂Ω )sp RN \Ω |x − y|

134

M. Warma / Nonlinear Analysis 135 (2016) 129–157

It follows from (2.4) and the fractional Hardy inequality that there exists a constant C > 0 such that      |u(x)|p |u(x) − u(y)|p |u(x)|p dxdy ≤ C dx ≤ C dxdy. N +sp sp N +sp Ω dist(x, ∂Ω ) Ω Ω |x − y| Ω RN \Ω |x − y| We have shown that (2.1) also defines an equivalent on W0s,p (Ω ) = W0s,p (Ω ). We notice that in the case 1

,p

1

,p

s = 1/p, even if Ω has a Lipschitz continuous boundary, then the spaces W0p (Ω ) and W0p (Ω ) are different. Now, we assume that Ω ⊂ RN is an arbitrary bounded open set. It is well-known (see e.g. [18,37]) that if N > sp, then there exists a constant C > 0 such that for every u ∈ W0s,p (Ω ), ∥u∥Lq (Ω) ≤ C∥u∥W s,p (Ω) ,

∀ q ∈ [1, p⋆ ], p⋆ :=

0

Np . N − sp

(2.5)

If N < sp, then N

W0s,p (Ω ) ↩→ C 0,s− p (RN ).

(2.6)

Remark 2.1. We notice that it also follows from (2.5) that the embedding W0s,p (Ω ) ↩→ Lq (Ω ) is compact for ⋆ every q ∈ [1, p⋆ ). But the continuous embedding W0s,p (Ω ) ↩→ Lp (Ω ) is not compact. Throughout the rest of the paper, if we write u ∈ W0s,p (Ω ), we shall always mean that u ∈ W s,p (RN ) and satisfies u = 0 a.e. on RN \ Ω . For more details on the fractional order Sobolev spaces we refer to [1,18,23,27,37] and their references. 2.2. Some tools We give some lemmas and inequalities that are needed to obtain our main results in the subsequent sections. At our knowledge some of these tools are not included in the available literature. We start with the Logarithm-Sobolev type inequality for the space W0s,p (Ω ) whose proof follows the lines of the proof of [11, Proposition 2.1] by making the necessary modifications. Lemma 2.2. Let Ω ⊂ RN be a bounded open set, s ∈ (0, 1) and p ∈ (1, ∞) such that sp < N . Let f ∈ W0s,p (Ω ), f ≥ 0, ∥f ∥Lp (Ω) = 1 and C the constant appearing in (2.5) with q = p⋆ . Then, for every ε > 0, we have      N |f (x) − f (y)|p CN,p,s f p log(f ) dx ≤ 2 − log(ε) + εC dxdy . (2.7) N +sp sp 2 Ω RN RN |x − y| The following result may be of interest in its own independently of the application given in this paper. Lemma 2.3. Let p, r ∈ [2, ∞), 0 < s < 1 and define   (u(x) − u(y))(v(x) − v(y)) CN,p,s |u(x) − u(y)|p−2 Ap (u, v) := dxdy, 2 |x − y|N +sp N N R R

u, v ∈ W0s,p (Ω ).

Then for every u, v ∈ W0s,p (Ω ) ∩ L∞ (RN ) we have,  r−2+p   r−2  r−2+p r−2 r−2 CAp |u| p , |u| p ≤ CAp |u| p u, |u| p u ≤ Ap (u, |u| u),  p p where C := (r − 1) p+r−2 .

(2.8)

r−2

Proof. First, we notice that if u ∈ W0s,p (Ω ) ∩ L∞ (RN ), then it follows from [37, Remark 2.5] that |u| p u, r−2+p r−2+p r−2 |u| p , |u|r−2 u ∈ W0s,p (Ω ) ∩ L∞ (RN ). Since |u| p = | |u| p u|, then the first inequality in (2.8) follows from [37, Lemma 2.6]. Therefore, we just have to prove the second inequality. Define the function

M. Warma / Nonlinear Analysis 135 (2016) 129–157

135

g : R × R → R by g (z, t) = g (z, t) = |z − t|

p−2

p  r−2   r−2   r−2 r−2 (z − t) |z| z − |t| t − C  |z| p z − |t| p t .

Using the definition of Ap , we first notice that the second inequality in (2.8) is equivalent to showing that g(z, t) ≥ 0

for all (z, t) ∈ R2 .

(2.9)

We now prove our claim (2.9). First, we observe that g(z, t) = g(t, z),

g(z, 0) ≥ 0,

g(0, t) ≥ 0

and g(z, t) = g(−z, −t).

Therefore, without any restriction, we may assume that z ≥ t. A simple calculation shows that  r−2   z r−2 r−2 p p p |z| z − |t| t = |τ | p dτ. p+r−2 t

(2.10)

Since the function ξ : R → R given by ξ(τ ) = |τ |p (p ≥ 2) is convex, then using the Jensen inequality, we get from (2.10) that  p  p  p  r−2 r−2 r−2 p p   r−2 |z| p z − |t| p t   |z| p z − |t| p t =  p+r−2 p+r−2  z  z p p    r−2 r−2 dτ  |τ | p dτ  = (z − t)p  =  |τ | p z − t t t  z  z dτ ≤ (z − t)p = (z − t)p−1 |τ |r−2 |τ |r−2 dτ z−t t t  (z − t)p−1  r−2 r−2 |z| z − |t| t . = r−1 

We have shown the claim (2.9) and this completes the proof of lemma.

The following result on convex analysis taken from [34, Lemma 3.3] will be useful. Lemma 2.4. Let F : R2 → (−∞, ∞] be a convex and lower semi-continuous functional with effective domain D(F ). Assume that U := Int(D(F )) ̸= ∅. Then the following assertions are equivalent. (i) For all (x0 , x1 ), (y0 , y1 ) ∈ R2 with (x0 − y0 )(x1 − y1 ) < 0, F (x0 , y1 ) + F (y0 , x1 ) ≤ F (x0 , x1 ) + F (y0 , y1 ). (ii)

∂2F ∂x0 ∂x1

≤ 0 on D′ (U ) and ∀ (x0 , x1 ), (y0 , y1 ) ∈ D(F ) with (x0 − y0 )(x1 − y1 ) < 0;

(x0 , y1 ), (y0 , x1 ) ∈ D(F ).

The following result taken from [33, Lemma 4.1(i)] is a useful tool to obtain an a priori estimate of weak solutions of quasi-linear elliptic equations. Lemma 2.5. Let ψ : [0, ∞) → R be a non-negative and non-increasing function such that there are positive constants k, γ, δ (δ > 1) such that (β − α)γ ψ(β) ≤ kψ(α)δ , Then ψ(d) = 0 for d = k 1/γ ψ(0)(δ−1)/γ 2δ/(δ−1) .

∀ β > α ≥ 0.

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The proof of the following inequalities are contained in [4, Section 4] (see also [10]). Lemma 2.6. Let p ∈ [2, ∞). Then for every a, b ∈ R,  p−2  |a| a − |b|p−2 b (a − b) ≥ 2−2−p 3−p/2 |a − b|p ,

(2.11)

and this implies that 

 |a|p−2 a − |b|p−2 b sgn(a − b) ≥ 2−2−p 3−p/2 |a − b|p−1 .

(2.12)

Lemma 2.7. Let p ∈ (1, 2]. Then the following assertions hold. (a) For every a, b ∈ R, 

 p−2 |a|p−2 a − |b|p−2 b (a − b) ≥ (p − 1)|a − b|2 (|a| + |b|) .

(2.13)

(b) For every a, b ∈ R with a ̸= b, 

p−2

|a|

p−2  min{|a|, |b|} |a − b|p , b (a − b) ≥ (p − 1) 1 + |a − b|

(2.14)

 p−2 min{|a|, |b|} b sgn(a − b) ≥ (p − 1) 1 + |a − b|p−1 . |a − b|

(2.15)

p−2

a − |b|



and this implies that 

|a|

p−2

p−2

a − |b|



(c) Let ε > 0. Then for every a, b ∈ R with |a − b| ≥ ε min{|a|, |b|},  p−2  p−2  1 p−2 |a| a − |b| b (a − b) ≥ (p − 1) 1 + |a − b|p , ε

(2.16)

and this implies that 

|a|

p−2

a − |b|

p−2

p−2   1 |a − b|p−1 . b sgn(a − b) ≥ (p − 1) 1 + ε

(2.17)

We conclude the section with the following result. Lemma 2.8. Let A, B, K, τ, ρ ∈ [0, +∞), p ∈ [1, ∞) and assume that A ≤ τ ε−ρ B + εp K

for all ε ∈ (0, 1].

(2.18)

Then   A ≤ (τ + 1) B p/(p+ρ) K ρ/(p+ρ) + B .

(2.19)

Proof. Let A, B, K, τ, ρ ∈ [0, +∞), p ∈ [1, ∞) and assume that (2.18) holds. • Assume first that 0 < B ≤ K and let α := 1/(p + ρ) and ε1 := (B/K)α ∈ (0, 1]. Note that 1 − ρα = αp. Then using (2.18) we get that p −ρα A ≤ τ ε−ρ B + (B/K)pα K = τ B 1−ρα K ρα + B pα K 1−pα 1 B + ε1 K = τ (B/K)

= (τ + 1)B pα K ρα = (τ + 1)B p/(p+ρ) K ρ/(p+ρ) . Hence, (2.19) holds. • If B = 0, then by (2.18) we have that 0 ≤ A ≤ εp K for all ε ∈ (0, 1]. This shows that A = 0 and hence (2.19) holds.

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• Finally, if B > K, let ε2 := 1. Then it follows from (2.18) that p A ≤ τ ε−ρ 2 B + ε2 K = τ B + K ≤ τ B + B = (τ + 1)B

and hence, (2.19) holds. The proof is finished.



3. The quasi-linear problem q Throughout the rest of the paper for a number q ∈ [1, ∞] we denote by q ′ := q−1 its conjugate exponent ′ ′ N with the convention that q = ∞ if q = 1 and q = 1 if q = ∞. Let Ω ⊂ R be an arbitrary bounded ′ open set, 1 < p < ∞ and 0 < s < 1. We shall denote by W −s,p (Ω ) the dual of the reflexive Banach space W0s,p (Ω ). Here, we consider the quasi-linear elliptic type problem

(−∆)sp u = f

in Ω ,

u=0

on RN \ Ω .

(3.1)

′

Definition 3.1. Let f ∈ W −s,p (Ω ). A function u ∈ W0s,p (Ω ) is said to be a weak solution of (3.1) if   CN,p,s (u(x) − u(y))(v(x) − v(y)) |u(x) − u(y)|p−2 dxdy = ⟨f, v⟩, 2 |x − y|N +ps RN RN

(3.2)

′

for every v ∈ W0s,p (Ω ) where ⟨·, ·⟩ denotes the duality pair between W −s,p (Ω ) and W0s,p (Ω ). Throughout the rest of this section, for p ∈ (1, ∞), 0 < s < 1 and u, v ∈ W0s,p (Ω ), we set   CN,p,s (u(x) − u(y))(v(x) − v(y)) Ap (u, v) := |u(x) − u(y)|p−2 dxdy, 2 |x − y|N +ps N N R R and Fp (x, y, u, v) := |u(x) − u(y)|p−2 (u(x) − u(y))(v(x) − v(y)). We have the following result of existence of weak solutions. ′

Proposition 3.2. Let p ∈ (1, ∞) and 0 < s < 1. Then for every f ∈ W −s,p (Ω ) the Dirichlet problem (3.1) has a unique weak solution u ∈ W0s,p (Ω ). Moreover, 1 p−1 ∥u∥W s,p (Ω) ≤ ∥f ∥W −s,p′ (Ω) .

(3.3)

0

Proof. First, let u ∈ W0s,p (Ω ) be fixed. Using the H¨older inequality we get that for every v ∈ W0s,p (Ω ), p−1 ∥v∥W s,p (Ω) . |Ap (u, v)| ≤ ∥u∥W s,p (Ω) 0

(3.4)

0

′

Since Ap (u, ·) is linear (in the second variable), it follows from (3.4) that Ap (u, ·) ∈ W −s,p (Ω ). Next, let u, v ∈ W0s,p (Ω ). Then using (2.11) if p ∈ [2, ∞) and (2.13) if p ∈ (1, 2), we get that Ap (u, u − v) − Ap (v, u − v) ≥ 0. Moreover, Ap (u, u − v) − Ap (v, u − v) > 0 if u ̸= v. Hence, Ap is strictly monotone. By the continuity of the norm function we have that for all u, v, w ∈ W0s,p (Ω ), lim Ap (u + tv, w) = Ap (u, w). t↓0

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We have shown that Ap is hemi-continuous. Since (using (2.1)) ∥u∥pW s,p (Ω) Ap (u, u) p−1 0 ≥C lim =C lim lim = ∞, ∥u∥W s,p (Ω) ∥u∥W s,p (Ω) →∞ ∥u∥W s,p (Ω) ∥u∥W s,p (Ω) →∞ ∥u∥W s,p (Ω) →∞ ∥u∥W s,p (Ω) 0 0

0

0

0

0

′

W0s,p (Ω ),

there exists Ap (u) ∈ W −s,p (Ω ) it follows that Ap is coercive. We have shown that for every u ∈ ′ s,p such that Ap (u, v) = ⟨Ap (u), v⟩ for every v ∈ W0 (Ω ). This defines an operator Ap : W0s,p (Ω ) → W −s,p (Ω ) which is strictly monotone, continuous, coercive and bounded (the boundedness of Ap follows from (3.4)). ′ Applying Browder’s theorem (see e.g. [19, Theorem 5.3.22]) we deduce that Ap (W0s,p (Ω )) = W −s,p (Ω ). ′ Therefore, for every f ∈ W −s,p (Ω ), there exists a unique u ∈ W0s,p (Ω ) such that Ap (u) = f , that is, for every v ∈ W0s,p (Ω ), Ap (u, v) = ⟨Ap (u), v⟩ = ⟨f, v⟩. ′

We have shown that for every f ∈ W −s,p (Ω ), the problem (3.1) has a unique weak solution u ∈ W0s,p (Ω ). Finally, taking v = u in (3.2) and using (2.1) we get that ∥u∥pW s,p (Ω) = ⟨f, u⟩ ≤ ∥f ∥W −s,p′ (Ω) ∥u∥W s,p (Ω) 0

0

and this implies the estimate (3.3). The proof is finished.



Remark 3.3. We mention that the result of existence and uniqueness of weak solutions and the estimate (3.3) given in Proposition 3.2 is easy to obtain and classical. More general existence results to similar nonlocal quasi-linear elliptic type problems where f is replaced by a measure are contained in [16,17,26,28] and their references. We have decided to include the proof of Proposition 3.2 in the present paper for the seek of completeness. ′

Throughout the rest of this section, for f ∈ W −s,p (Ω ), we denote the unique weak solution u of (3.1) by u := [(−∆)sp,Ω ]−1 f = Φs,p (f ). ′

Then Φs,p is an operator from W −s,p (Ω ) into W0s,p (Ω ). The first main result in this section is the following theorem which shows in particular that if p ∈ (1, 2), ′ then the mapping Φs,p : W −s,p (Ω ) → W0s,p (Ω ) is locally Lipschitz continuous. Theorem 3.4. Let Ω ⊂ RN be an arbitrary bounded open set, 1 < p < ∞ and 0 < s < 1. Then the following assertions hold. ′

(a) If 2 ≤ p < ∞, then there exists a constant C1 > 0 such that for every f, g ∈ W −s,p (Ω ), 1 p−1 ∥Φs,p (f ) − Φs,p (g)∥W s,p (Ω) ≤ C1 ∥f − g∥W −s,p′ (Ω) .

(3.5)

0

′

(b) If 1 < p < 2, then there exists a constant C2 > 0 such that for every f, g ∈ W −s,p (Ω ),  2−p  p−1 ∥Φs,p (f ) − Φs,p (g)∥W s,p (Ω) ≤ C2 ∥f ∥W −s,p′ (Ω) + ∥g∥W −s,p′ (Ω) ∥f − g∥W −s,p′ (Ω) .

(3.6)

0

Remark 3.5. We mention that in Theorem 3.4, the estimate (3.5) is not difficult to establish. We have decided to include it for the seek of completeness. Our focus is to establish (3.6) which is not obvious and requires a careful estimate of the solution, and will also be used to prove the existence of solutions of the nonlinear problem (1.2). Moreover, the nonlocal nature of our operator makes that one cannot get directly such an estimate by simply following the proof of the local case s = 1.

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To prove Theorem 3.4, we need some intermediate lemmas. Lemma 3.6. Let p ∈ [2, ∞), 0 < s < 1, u, v ∈ W0s,p (Ω ), α ≥ 0 and wα := (|u − v| − α)+ sgn(u − v). Then wα ∈ W0s,p (Ω ) and Ap (u, wα ) − Ap (v, wα ) ≥ C(p)∥wα ∥pW s,p (Ω) ,

(3.7)

0

where C(p) := 2−2−p 3−p/2 is the constant appearing in (2.11). Proof. Let p ∈ [2, ∞), 0 < s < 1, u, v ∈ W0s,p (Ω ), α ≥ 0 and wα := (|u − v| − α)+ sgn(u − v). Let Aα := {x ∈ RN : |u − v| > α} and Bα := RN \ Aα = {x ∈ RN : 0 ≤ |(u − v)(x)| ≤ α}. Let N A+ α := {x ∈ R : (u − v)(x) > α},

N A− α := {x ∈ R : (u − v)(x) < −α}

− so that Aα = A+ α ∪ Aα .

Let Bα+ := {x ∈ RN : 0 ≤ (u − v)(x) ≤ α},

Bα− := {x ∈ RN : −α ≤ (u − v)(x) ≤ 0}

so that Bα = Bα+ ∪ Bα− . First, we note that since u, v ∈ W0s,p (Ω ), it follows from [37, Lemma 2.6] that wα ∈ W0s,p (Ω ). Next, since wα = 0 on Bα , we have that   CN,p,s Fp (x, y, u, wα ) − Fp (x, y, v, wα ) Ap (u, wα ) − Ap (v, wα ) := dxdy 2 |x − y|N +sp N N R R Fp (x, y, u, wα ) − Fp (x, y, v, wα ) CN,p,s dxdy = 2 |x − y|N +sp Aα Aα   Fp (x, y, u, wα ) − Fp (x, y, v, wα ) + CN,p,s dxdy. |x − y|N +sp Aα Bα We claim that on RN × RN , Fp (x, y, u, wα ) − Fp (x, y, v, wα ) ≥ C(p)|wα (x) − wα (y)|p .

(3.8)

− − + Indeed, using the estimate (2.11), we get that on A+ α × Aα and Aα × Aα ,

Fp (x, y, u, wα ) − Fp (x, y, v, wα )   = |u(x) − u(y)|p−2 (u(x) − u(y)) − |v(x) − v(y)|p−2 (v(x) − v(y)) [(u − v)(x) − (u − v)(y)] ≥ C(p)|(u − v)(x) − (u − v)(y)|p = C(p)|wα (x) − wα (y)|p . − − + Similarly, using (2.12), we get that on A+ α × Aα and Aα × Aα ,

Fp (x, y, u, wα ) − Fp (x, y, v, wα )   = |u(x) − u(y)|p−2 (u(x) − u(y)) − |v(x) − v(y)|p−2 (v(x) − v(y)) [(u − v)(x) − (u − v)(y) − 2α] ≥ C(p)|(u − v)(x) − (u − v)(y)|p−1 [(u − v)(x) − (u − v)(y) − 2α] ≥ C(p)|(u − v)(x) − (u − v)(y) − 2α|p−1 [(u − v)(x) − (u − v)(y) − 2α] = C(p)|(u − v)(x) − (u − v)(y) − 2α|p = C(p)|wα (x) − wα (y)|p . + + − − + − − A similar estimate using (2.12) shows that, on the sets A+ α × Bα , Aα × Bα , Aα × Bα , Aα × Bα ,

Fp (x, y, u, wα ) − Fp (x, y, v, wα ) ≥ C(p)|wα (x) − wα (y)|p , and this completes the proof of the claim (3.8). Now (3.7) follows from (3.8) and the proof is finished.



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Throughout this section, for a set A ⊂ Ω ⊂ RN , B := RN \ A, 0 < s < 1, p ∈ (1, ∞) and u ∈ W0s,p (Ω ), we let  p1    CN,p,s |u(x) − u(y)|p dxdy , Ns,p,A (u) := N +sp 2 A A |x − y| and  Ns,p,A,B (u) :=

  CN,p,s A

B

|u(x) − u(y)|p dxdy |x − y|N +sp

 p1 .

The following result is the version of Lemma 3.6 for the singular case p ∈ (1, 2). Lemma 3.7. Let 1 < p < 2 and 0 < s < 1. For u, v ∈ W0s,p (Ω ) and α ≥ 0, we let wα , Aα and Bα be as in Lemma 3.6. Then wα ∈ W0s,p (Ω ) and there is a constant C = C(p) > 0 such that, p

Ns,p,Aα (wα )p + Ns,p,Aα ,Bα (wα )p ≤ C [Ap (u, wα ) − Ap (v, wα )] 2

p 1− p · [Ns,p,Aα (u) + Ns,p,Aα (v) + Ns,p,Aα ,Bα (u) + Ns,p,Aα ,Bα (v)] ( 2 ) .

(3.9)

− + − Proof. Let 1 < p < 2 and 0 < s < 1. For u, v ∈ W0s,p (Ω ) and α ≥ 0, let wα , A+ α , Aα , Bα and Bα be as in s,p the proof of Lemma 3.6. It also follows from [37, Lemma 2.6] that wα ∈ W0 (Ω ). Set

M (x, y, u, v) :=

min{|u(x) − u(y)|, |v(x) − v(y)|} χ{(x,y)∈RN ×RN : (u−v)(x)̸=(u−v)(y)} . |(u − v)(x) − (u − v)(y)|

(3.10)

We claim that on RN × RN , p−2

Fp (x, y, u, wα ) − Fp (x, y, v, wα ) ≥ (p − 1) [1 + M (x, y, u, v)]

|wα (x) − wα (y)|p .

(3.11)

First we notice that if (u − v)(x) = (u − v)(y), then both sides of (3.11) are zero and hence, there is nothing to prove. Therefore we may assume that (u − v)(x) ̸= (u − v)(y). Using the estimate (2.14), we get that on − − + A+ α × Aα and Aα × Aα , Fp (x, y, u, wα ) − Fp (x, y, v, wα )   = |u(x) − u(y)|p−2 (u(x) − u(y)) − |v(x) − v(y)|p−2 (v(x) − v(y)) [(u − v)(x) − (u − v)(y)] p−2 ≥ (p − 1) [1 + M (x, y, u, v)] |(u − v)(x) − (u − v)(y)|p p−2 = (p − 1) [1 + M (x, y, u, v)] |wα (x) − wα (y)|p . − − + Next using (2.15) we get that on A+ α × Aα and Aα × Aα ,

Fp (x, y, u, wα ) − Fp (x, y, v, wα )   = |u(x) − u(y)|p−2 (u(x) − u(y)) − |v(x) − v(y)|p−2 (v(x) − v(y)) [(u − v)(x) − (u − v)(y) − 2α] p−2 ≥ (p − 1) [1 + M (x, y, u, v)] |(u − v)(x) − (u − v)(y)|p−1 [(u − v)(x) − (u − v)(y) − 2α] p−2 ≥ (p − 1) [1 + M (x, y, u, v)] |(u − v)(x) − (u − v)(y) − 2α|p−1 [(u − v)(x) − (u − v)(y) − 2α] p−2 = (p − 1) [1 + M (x, y, u, v)] |(u − v)(x) − (u − v)(y) − 2α|p p−2 = (p − 1) [1 + M (x, y, u, v)] |wα (x) − wα (y)|p . + On A+ α × Bα , we have that

Fp (x, y, u, wα ) − Fp (x, y, v, wα ) p−2 ≥ (p − 1) (1 + M (x, y, u, v)) |(u − v)(x) − (u − v)(y)|p−1 [(u − v)(x) − α] p−2 ≥ (p − 1) (1 + M (x, y, u, v)) |(u − v)(x) − α|p−1 [(u − v)(x) − α] p−2 = (p − 1) (1 + M (x, y, u, v)) |wα (x) − wα (y)|p .

M. Warma / Nonlinear Analysis 135 (2016) 129–157

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− − − − + Proceeding similarly, we get that (3.11) holds on each of the sets A+ α × Bα , Aα × Bα , Aα × Bα . This completes the proof of the claim (3.11).

Next, for ε ∈ (0, 1] we let Gε := {(x, y) ∈ RN × RN : |(u(x) − u(y)) − (v(x) − v(y))| ≥ ε|u(x) − u(y)|}. For almost all (x, y) ∈ Aα × Bα we have that 0 ≤ |wα (x)| = |wα (x) − wα (y)| = |(u − v)(x)| − α ≤ |(u − v)(x)| − |(u − v)(y)| ≤ |(u − v)(x) − (u − v)(y)|.

(3.12)

The estimate (3.12) implies that on (Aα × Bα ) \ Gε we have, 0 ≤ |wα (x)| ≤ ε|u(x) − u(y)|.

(3.13)

0 ≤ |wα (x) − wα (y)| ≤ | |(u − v)(x)| − |(u − v)(y)|| ≤ |(u − v)(x) − (u − v)(y)|.

(3.14)

On Aα × Aα we also have that,

We notice that Aα ⊂ Ω for every α ≥ 0. Using (2.16) and the fact that wα = 0 on Bα , we get from (3.11) that Ap (u, wα ) − Ap (v, wα )   CN,p,s Fp (x, y, u, wα ) − Fp (x, y, v, wα ) = dxdy 2 |x − y|N +sp RN RN   p CN,p,s p−2 |wα (x) − wα (y)| [1 + M (x, y, u, v)] dxdy ≥ (p − 1) 2 |x − y|N +sp RN RN  p−2  1 CN,p,s |wα (x) − wα (y)|p ≥ (p − 1) 1 + dxdy ε 2 |x − y|N +sp (Aα ×Aα )∩Gε  p−2  1 CN,p,s |wα (x)|p + 2(p − 1) 1 + dxdy. N +sp ε 2 (Aα ×Bα )∩Gε |x − y|

(3.15)

Using (3.13) and (3.14) we get from (3.15) that Ap (u, wα ) − Ap (v, wα )



≥ (p − 1) 1 +

1 ε

p−2

1 + 2(p − 1) 1 + ε



1 ≥ (p − 1) 1 + ε







≥ (p − 1) 1 +





+2 Aα

Bα

1 ε

p−2

p−2

+ 2(p − 1) 1 +

1 ε

CN,p,s 2

p−2

CN,p,s 2 p

Aα







(Aα ×Bα )\Gε

Bα

|wα (x) − wα (y)|p dxdy − εp |x − y|N +sp

Aα

p



Aα

Aα

(Aα ×Aα )\Gε



Aα





|wα (x)|p dxdy − |x − y|N +sp





Aα

CN,p,s 2

|wα (x) − wα (y)|p dxdy − |x − y|N +sp



Aα

CN,p,s 2

CN,p,s 2

p−2



Bα

|wα (x)| dxdy − εp |x − y|N +sp

Aα

|wα (x)| dxdy − εp 2 |x − y|N +sp

|wα (x) − wα (y)| dxdy − εp |x − y|N +sp



p



Aα

Bα

1+

1 ε

 (Aα ×Aα )\Gε

(Aα ×Bα )\Gε

 Aα

p−2

≤ 22−p εp−2 ,

 Aα





|u(x) − u(y)|p dxdy |x − y|N +sp

|u(x) − u(y)|p dxdy |x − y|N +sp





|u(x) − u(y)|p dxdy |x − y|N +sp



|u(x) − u(y)| dxdy . |x − y|N +sp

Since 

|wα (x)|p dxdy |x − y|N +sp



p



|wα (x) − wα (y)|p dxdy |x − y|N +sp

(3.16)

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142

it follows from (3.16) that Ns,p,Aα (wα )p + Ns,p,Aα ,Bα (wα )p 22−p p−2 p ε [Ap (u, wα ) − Ap (v, wα )] + εp [Ns,p,Aα (u) + Ns,p,Aα ,Bα (u)] p−1 22−p p−2 ≤ ε [Ap (u, wα ) − Ap (v, wα )] p−1

≤

p

+ εp [Ns,p,Aα (u) + Ns,p,Aα (v) + Ns,p,Aα ,Bα (u) + Ns,p,Aα ,Bα (v)] .

(3.17)

Using Lemma 2.8 with A := Ns,p,Aα (wα )p + Ns,p,Aα ,Bα (wα )p ,

B := Ap (u, wα ) − Ap (v, wα ),

τ :=

22−p , p−1

ρ := 2 − p, (3.18)

and p

K := [Ns,p,Aα (u) + Ns,p,Aα (v) + Ns,p,Aα ,Bα (u) + Ns,p,Aα ,Bα (v)] ,

(3.19)

we get from (3.17) that A≤

 22−p + p − 1  p 1− p B2K 2 +B . p−1

(3.20)

As in (2.3) we have that   |u(x) − u(y)|p−2 (u(x) − u(y))(wα (x) − wα (y)) CN,p,s dxdy 2 |x − y|N +sp RN RN   CN,p,s |u(x) − u(y)|p−2 (u(x) − u(y))(wα (x) − wα (y)) = dxdy 2 |x − y|N +sp Aα Aα   |u(x) − u(y)|p−2 (u(x) − u(y))wα (x) dxdy. + CN,p,s |x − y|N +sp Aα Bα

Ap (u, wα ) =

(3.21)

We also notice the following inequality: for all a, b ≥ 0 and p ∈ (1, 2), we have that ap−1 b ≤

p−1 p 1 p a + b . p p

Using (3.14) and (3.22), we get from (3.21) that there exists a constant C > 0 such that   CN,p,s |u(x) − u(y)|p−1 |(u − v)(x) − (u − v)(y)| |Ap (u, wα )| ≤ dxdy 2 |x − y|N +sp Aα Aα    |u(x) − u(y)|p−1 |(u − v)(x) − (u − v)(y)| +2 dxdy |x − y|N +sp Aα Bα   CN,p,s |u(x) − u(y)|p + |v(x) − v(y)|p ≤ C dxdy 2 |x − y|N +sp Aα Aα    |u(x) − u(y)|p + |v(x) − v(y)|p +2 dxdy . |x − y|N +sp Aα Bα

(3.22)

(3.23)

Similarly, we have that   CN,p,s |v(x) − v(y)|p + |u(x) − u(y)|p |Ap (v, wα )| ≤ C dxdy 2 |x − y|N +sp Aα Aα    |v(x) − v(y)|p + |u(x) − u(y)|p +2 dxdy . |x − y|N +sp Aα Bα

(3.24)

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143

Combining (3.23) and (3.24) we get that there is a constant C = C(N, p, Ω ) > 0 such that B ≤ CK, where we recall that B and K are given in (3.18) and (3.19), respectively. This estimate together with (3.20) imply that  22−p + p − 1  p 1− p B2K 2 +K . A≤C p−1 We have shown that (3.9) holds for every α ≥ 0 and the proof is finished.



Remark 3.8. We notice that if α = 0, then the estimate (3.9) becomes Ns,p,Ω (u − v)p + Ns,p,Ω,RN \Ω (u − v)p   CN,p,s |(u − v)(x) − (u − v)(y)|p dxdy = 2 |x − y|N +sp RN RN p

≤ C [Ap (u, u − v) − Ap (v, u − v)] 2 1− p2      |u(x) − u(y)|p CN,p,s |v(x) − v(y)|p CN,p,s dxdy + dxdy · . N +sp N +sp 2 2 RN RN |x − y| RN RN |x − y|

(3.25)

Now, we are ready to give the proof of the first main result of this section. ′

Proof of Theorem 3.4. Let 1 < p < ∞, 0 < s < 1, f, g ∈ W −s,p (Ω ), u := Φs,p (f ) and v := Φs,p (g). Then by definition, for every ϕ ∈ W0s,p (Ω ), we have that Ap (u, ϕ) − Ap (v, ϕ) = ⟨f − g, ϕ⟩.

(3.26)

(a) Let 2 ≤ p < ∞. Taking ϕ = u − v in (3.26) and using (3.7) with α = 0, we get that there exists a constant C > 0 such that C∥u − v∥pW s,p (Ω) ≤ ⟨f − g, u − v⟩ ≤ ∥f − g∥W −s,p′ (Ω) ∥u − v∥W s,p (Ω) , 0

0

and we have shown the estimate (3.5). (b) Now let 1 < p < 2. Taking ϕ = u − v in (3.26), using (3.25) and recalling that the left hand side of (3.25) defines an equivalent norm on W0s,p (Ω ) (by (2.1)), we get that there is a constant C > 0 such that  p−2 C∥u − v∥2W s,p (Ω) ∥u∥W s,p (Ω) + ∥v∥W s,p (Ω) ≤ Ap (u, u − v) − Ap (v, u − v) = ⟨f − g, u − v⟩ 0

0

0

≤ ∥f − g∥W −s,p′ (Ω) ∥u − v∥W s,p (Ω) .

(3.27)

0

Since (by the estimate (3.3)) 1 p−1 ∥u∥W s,p (Ω) ≤ ∥f ∥W −s,p′ (Ω) 0

1 p−1 and ∥v∥W s,p (Ω) ≤ ∥g∥W −s,p′ (Ω) , 0

it follows from (3.27) that C∥u − v∥W s,p (Ω) 0

 2−p 1 1 p−1 p−1 ≤ ∥f ∥W −s,p′ (Ω) + ∥g∥W −s,p′ (Ω) ∥f − g∥W −s,p′ (Ω)

and this implies the estimate (3.6). The proof of the theorem is finished.



Next, we assume that N > sp. The case N = sp can be done by a simple modification. Throughout the rest of this section, we let the number p be defined by   pp⋆ (2 − p) 2p⋆ − p⋆ p 1 p p := = p = N − . (3.28) 2(p⋆ − p) 2p⋆ − 2p s 2s

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Note that p > p ⇐⇒ p >

2N ⇐⇒ p⋆ > 2 N + 2s

and p ≥ 0 ⇐⇒ p ∈ [1, 2].

Definition 3.9. Let Ω ⊂ RN be a bounded open set, 0 < s < 1 and p ∈ (1, 2). We say that Ω is (s, p)-regular, if there exist 0 < s ≤ t < 1, q ∈ [p, ∞] ∩ ( p, ∞] and q ∈ [(p⋆ )′ , ∞] such that N N +t= + s, q p

(3.29)

q and there is a constant C > 0 such that for every f ∈ L (Ω ),   1 |Φs,p (f )(x) − Φs,p (f )(y)|q dxdy ≤ C∥f ∥ p−1 . N +tq  q |x − y| L (Ω) RN RN

(3.30)

⋆ N is (s, p)-regular Remark 3.10. If N2N +2s < p < 2, then p > 2 and hence, every bounded open set Ω ⊂ R 2N ⋆ ′ with 0 < t = s < 1, q := p and q := (p ) . If 1 < p ≤ N +2s , then Ω is (s, p)-regular if and only if for q f ∈ L (Ω ), the unique solution u of (3.1) belongs to W t,q (Ω ) for some t satisfying 0 < s < t < 1 and for 0

some q > p. This may require a regularity on the open set Ω . Since elliptic type regularity results (except the ones given in the present article) for solutions of (3.1) are not available in the literature, we do not know the regularity needed on Ω for (3.29) and (3.30) to be true. But we conjecture that if Ω is a bounded open set of class C 1,1 , then it is (s, p)-regular. The following theorem is the second main result of this section. Theorem 3.11. Let Ω ⊂ RN be an arbitrary bounded open set, 1 < p < ∞, 0 < s < 1 and f, g ∈ Lr (Ω ) for some r ≥ (p⋆ )′ . Then the following assertions hold. (a) If 2 ≤ p < ∞ and r >

N sp ,

then there exists a constant C1 > 0 such that 1

∥Φs,p (f ) − Φs,p (g)∥L∞ (Ω) ≤ C1 ∥f − g∥Lp−1 r (Ω) .

(3.31)

(b) Let 1 < p < 2 and assume that Ω is (s, p)-regular. Let q, p, q be as in Definition 3.9 and let r satisfy r> with the convention that

q q− p

q p p⋆ pq = · · ⋆ , (2 − p)(q − p) q − p 2 p − p

(3.32)

:= 1 if q = ∞. Then there is a constant C2 > 0 such that

∥Φs,p (f ) − Φs,p (g)∥L∞ (Ω)

  2−p p−1 ≤ C2 ∥f ∥  + ∥g∥  ∥f − g∥Lr (Ω) . q q L (Ω) L (Ω)

(3.33)

Proof. Let 1 < p < ∞, 0 < s < 1, f, g ∈ Lr (Ω ) for some r ≥ (p⋆ )′ , u := Φs,p (f ) and v := Φs,p (g). Let α ≥ 0 and wα := (|u − v| − α)+ sgn(u − v). Then we already know that wα ∈ W0s,p (Ω ). Let Aα := {x ∈ RN : |u − v| > α}

and Bα := RN \ Aα = {x ∈ RN : 0 ≤ |(u − v)(x)| ≤ α}.

(a) Let 2 ≤ p < ∞ and assume that r >

p⋆ p⋆ −p

=

N sp .

Let p1 ∈ [1, ∞] be such that

1 r

1 1 1 p⋆ 1 p⋆ − p p−1 p⋆ =1− ⋆ − > ⋆ − ⋆ − = =⇒ p < . 1 p1 p r p p p⋆ p⋆ p−1

+

1 p1

+

1 p⋆

= 1. Then

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145

Using the classical H¨ older inequality and the fact that wα = 0 on RN \ Aα , we get that   |Ap (u, wα ) − Ap (v, wα )| ≤ |f − g| |wα | dx = |f − g| |wα | dx Ω

Aα 1

≤ ∥f − g∥Lr (Ω) ∥wα ∥Lp⋆ (Ω) |Aα | p1 .

(3.34)

Using (3.7), (3.34) and the Sobolev embedding (2.5), we get that there exists a constant C > 0 such that ∥wα ∥pLp⋆ (Aα ) = ∥wα ∥pLp⋆ (Ω) ≤ C∥wα ∥pW s,p (Ω) ≤ C [Ap (u, wα ) − Ap (v, wα )] 0

1

≤ C∥f − g∥Lr (Ω) ∥wα ∥Lp⋆ (Aα ) |Aα | p1 . This estimate implies that 1

∥wα ∥p−1 = ∥wα ∥p−1 ≤ C∥f − g∥Lr (Ω) |Aα | p1 . Lp⋆ (Aα ) Lp⋆ (Ω) For β > α ≥ 0, we have that |wα | ≥ (β − α)χAβ on Ω . This implies that ⋆

(β − α)p−1 |Aβ |(p−1)/p ≤ ∥wα ∥p−1 . Lp⋆ (Ω) ⋆

Let C2 := |Ω |1/p and δ :=

p⋆ >p−1 p1

δ > 1. p−1

and δ0 :=

Then    − pp⋆  C 1 χAα   2 

p⋆

Lp1 (Ω) p⋆

This shows that for CΩ := C2p1 C3 > 0 such that

= ∥C2−1 χAα ∥Lp1p⋆ (Ω) = ∥C2−1 χAα ∥δLp⋆ (Ω) = ∥χAα ∥δLp⋆ (Ω) C2−δ .

−δ

we have ∥χAα ∥Lp1 (Ω) = CΩ ∥χAα ∥δLp⋆ (Ω) . Hence, there exists a constant

∥χAβ ∥p−1 ≤ C3 (β − α)−(p−1) ∥f − g∥Lr (Ω) ∥χAβ ∥δLp⋆ (Ω) Lp⋆ (Ω)  δ0 = C3 (β − α)−(p−1) ∥f − g∥Lr (Ω) ∥χAα ∥p−1 . ⋆ p L (Ω) It follows from Lemma 2.5 with ψ(β) := ∥χAβ ∥p−1 that there is a constant C4 > 0 (independent of C3 ) Lp⋆ (Ω) such that ∥χAd ∥p−1 =0 Lp⋆ (Ω)

1

1

with d := C4 · C3p−1 ∥f − g∥Lp−1 r (Ω) .

The proof of part (a) is finished. (b) Let p ∈ (1, 2) and assume that Ω is (s, p)-regular. Let t, q and q be as in Definition 3.9. Let p1 ∈ [1, ∞] be such that 1r + p11 + p1⋆ = 1. The estimate (3.34) also holds in this case. Using (3.34), (3.9) and (2.5) we get that there exists a constant C > 0 such that p

∥wα ∥pLp⋆ (Ω) ≤ ∥wα ∥pW s,p (Ω) ≤ C [Ap (u, wα ) − Ap (v, wα )] 2 0

p 1− p · [Ns,p,Aα (u) + Ns,p,Aα (v) + Ns,p,Aα ,Bα (u) + Ns,p,Aα ,Bα (v)] ( 2 ) p/2

p/2

p

≤ C∥f − g∥Lr (Ω) ∥wα ∥Lp⋆ (Ω) |Aα | 2p1 p 1− p · [Ns,p,Aα (u) + Ns,p,Aα (v) + Ns,p,Aα ,Bα (u) + Ns,p,Aα ,Bα (v)] ( 2 ) .

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This estimate implies that p/2 p 1− p ∥wα ∥Lp⋆ (Ω) ≤ C [Ns,p,Aα (u) + Ns,p,Aα (v) + Ns,p,Aα ,Bα (u) + Ns,p,Aα ,Bα (v)] ( 2 ) p

p/2

· ∥f − g∥Lr (Ω) |Aα | 2p1 . Let q1 ∈ [1, ∞] be such that

1 q

+

1 q1

 Ns,p,Aα (u) =

(3.35)

= p1 . Then using the H¨older inequality, we get that CN,p,s 2

 Aα

1

 p1 |u(x) − u(y)|p dxdy N +sp Aα |x − y|   q1   CN,p,s |u(x) − u(y)|q dxdy . Nq 2 Ω Ω |x − y| p +sq



1

≤ |Aα | q1 |Aα | q1

Using the (s, p)-regularity of Ω , that is, (3.29) and (3.30), we get that (3.36) we obtain that 1

1

1

1

Nq p

(3.36)

+ sq = N + tq and hence, from

Ns,p,Aα (u) ≤ C|Aα | q1 ∥f ∥ p−1 . q (Ω) L

(3.37)

Similarly, we get that Ns,p,Aα (v) ≤ C|Aα | q1 ∥g∥ p−1 . q (Ω) L

(3.38)

Let U ⊂ RN be a bounded open set such that Ω b U, that is, Ω ⊂ U. Since dist(Ω , RN \ U) = δ > 0, we have that there exists a constant C = C(Ω , U) > 0 such that |x − y| ≥ C(1 + |y|),

∀ x ∈ Ω , ∀ y ∈ RN \ U.

(3.39)

Using the H¨ older inequality and (3.39), we get that (recall that u = 0 on RN \ Ω )   |u(x) − u(y)|p dxdy Ns,p,Aα ,Bα (u)p = CN,p,s N +sp Aα Bα |x − y|     |u(x) − u(y)|p |u(x)|p = CN,p,s dxdy + C dxdy N,p,s N +sp N +sp Aα Bα ∩U |x − y| Aα Bα ∩(RN \U ) |x − y|     |u(x) − u(y)|p |u(x)|p ≤ CN,p,s dxdy + C dydx N,p,s N +sp N +sp Aα Bα ∩U |x − y| Aα RN \U (1 + |y|)  p/q    p p CN,p,s |u(x) − u(y)|q ≤ 2|Aα | q1 |Bα ∩ U | q1 dxdy + C |u(x)|p dx, (3.40) N +tq 2 |x − y| Ω Ω Aα where we have also used that  RN \U

1 dy < ∞. (1 + |y|)N +sp

It follows from (3.40) and the (s, p)-regularity of Ω that there is a constant C > 0 (independent of α) such that   1 1 p−1 q1 Ns,p,Aα ,Bα (u) ≤ C |Aα | ∥f ∥ + ∥u∥Lp (Aα ) . (3.41) q (Ω) L Using the H¨ older inequality and the (s, p)-regularity of Ω again, we get that there is a constant C > 0 such that  1/q   1 1 CN,p,s |u(x) − u(y)|q q q ∥u∥Lp (Aα ) ≤ |Aα | 1 ∥u∥Lq (Ω) ≤ C|Aα | 1 dxdy N +tq 2 Ω Ω |x − y| 1

1

≤ C|Aα | q1 ∥f ∥ p−1 . q (Ω) L

(3.42)

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147

It follows from (3.41) and (3.42) that there exists a constant C > 0 such that for every α ≥ 0, 1

1

Ns,p,Aα ,Bα (u) ≤ C|Aα | q1 ∥f ∥ p−1 . q (Ω) L

(3.43)

Similarly we have that there exists a constant C > 0 such that for every α ≥ 0, 1

1

Ns,p,Aα ,Bα (v) ≤ C|Aα | q1 ∥f ∥ p−1 . q (Ω) L Combining (3.35)–(3.44), we get that there exists a constant C > 0 such that for every α ≥ 0,  p′ (1−p/2) p−2 p p/2 p/2 ∥wα ∥Lp⋆ (Ω) ≤ C ∥f ∥  + ∥g∥ ∥f − g∥Lr (Ω) |Aα | 2p1 + q1 . q  q L (Ω) L (Ω)

(3.44)

(3.45)

For β > α ≥ 0, we have that |wα | ≥ (β − α)χAβ on Ω . This implies that ⋆

p/2

(β − α)p/2 |Aβ |(p/2)/p ≤ ∥wα ∥Lp⋆ (Ω) .

(3.46)

⋆

For α ≥ 0, we define the function ψ(α) := |Aα |(p/2)/p . It follows from (3.45) and (3.46) that  p′ (1−p/2) p/2 (β − α)p/2 ψ(β) ≤ C ∥f ∥  + ∥g∥ ∥f − g∥Lr (Ω) ψ(α)δ , q  q L (Ω)

L (Ω)

where δ := p⋆



(2 − p) 1 + p1 q1

 > 1.

We mention that δ > 1 if and only if (3.32) holds. It follows from Lemma 2.5 with  p′ (1−p/2) p p/2 and k := ∥f ∥  + ∥g∥  ∥f − g∥Lr (Ω) , γ := Lq (Ω) Lq (Ω) 2 that ψ(d) = 0 with d := k 1/γ ψ(0)(δ−1)/γ 2δ/(δ−1) , that is, p′ ( p2 −1)  ∥u − v∥L∞ (Ω) ≤ C ∥f ∥Lq˜(Ω) + ∥g∥Lq˜(Ω) ∥f − g∥Lr (Ω) . The proof of the theorem is finished.



Remark 3.12. We mention that in (3.33), global Lipschitz continuity cannot be achieved. In fact, let 0 < s < 1 and p ∈ (1, 2). For α ≥ 0 we let fα : Ω → R be the function fα (x) := α. Then a simple calculation shows 1 that Φs,p (fα ) = α p−1 Φs,p (f1 ). Therefore,  1  1   ∥Φs,p (fα ) − Φs,p (fβ )∥L∞ (Ω) = α p−1 − β p−1  ∥Φs,p (f1 )∥L∞ (Ω) . If there exists a constant C > 0 such that for all f, g ∈ L∞ (Ω ), ∥Φs,p (f ) − Φs,p (g)∥L∞ (Ω) ≤ C∥f − g∥L∞ (Ω) , then for all α, β ≥ 0, we have that  1  1   p−1 − β p−1  ≤ α

C |α − β|, ∥Φs,p (f1 )∥L∞ (Ω)

and it is clear that this is not possible. We conclude this section with a compactness result. Proposition 3.13. Let Ω ⊂ RN be an arbitrary bounded open set. Let 1 < p < ∞, 0 < s < 1 be such that sp < N and let p1 ≥ (p⋆ )′ . Then the operator Φs,p : Lp1 (Ω ) → W0s,p (Ω ) ∩ Lr (Ω ) is compact for every r ∈ (1, p⋆ ).

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148

Proof. Let 1 < r < p⋆ . Since the injection W0s,p (Ω ) ↩→ Lr (Ω ) is compact, then by duality, the injection ′ ′ ′ Lr (Ω ) ↩→ W −s,p (Ω ) is compact for every r′ > (p⋆ )′ . This, together with the fact that Φs,p : W −s,p (Ω ) → W0s,p (Ω ) is continuous and bounded, imply that Φs,p : Lp1 (Ω ) → W0s,p (Ω ) is compact for every p1 > (p⋆ )′ . It remains to show that Φs,p is also compact as a map into Lr (Ω ) for every r ∈ (1, p⋆ ). Since Φs,p is bounded, we have to show that the image of every bounded set B ⊂ Lp1 (Ω ) is relatively compact in Lr (Ω ) for every r ∈ (1, p⋆ ). Let un be a sequence in Φs,p (B). Let Fn := (−∆)sp,Ω (un ) ∈ B. Since B is bounded, it follows that the sequence Fn is bounded. Since Φs,p is compact as a map into W0s,p (Ω ), we have that there is a subsequence Fnk such that Φs,p (Fnk ) converges to u ∈ W0s,p (Ω ). We may assume that un = Φs,p (Fn ) converges to u in W0s,p (Ω ) and hence, in Lp (Ω ). It remains to show that un → u in Lr (Ω ). Let r ∈ [p, p⋆ ). ⋆ Since un is bounded in Lp (Ω ), a standard interpolation inequality shows that there exists τ ∈ (0, 1) such that ≤ C∥un − um ∥τLp (Ω) . ∥un − um ∥Lr (Ω) ≤ ∥un − um ∥τLp (Ω) ∥un − um ∥1−τ Lp⋆ (Ω)

(3.47)

Since un converges in Lp (Ω ), it follows from (3.47) that un is a Cauchy sequence in Lr (Ω ) and therefore converges in Lr (Ω ). Hence, Φs,p : Lp1 (Ω ) → W0s,p (Ω ) ∩ Lr (Ω ) is compact for every r ∈ [p, p⋆ ). The case r ∈ (1, p) follows from the fact that Lp (Ω ) ↩→ Lr (Ω ) and the proof is finished.  4. The nonlinear problem In this section we apply the results obtained in Section 3 to study the existence of solutions to the nonlinear elliptic type problem (−∆)sp u = g(x, u) + h in Ω ,

on RN \ Ω .

u=0

(4.1)

Definition 4.1. Let p ∈ (1, ∞) and 0 < s < 1. A function u ∈ W0s,p (Ω ) is called a weak solution of (4.1) if for every v ∈ W0s,p (Ω ), Ap (u, v) = ⟨g(x, u) + h, v⟩,

(4.2)

and the right hand-side of (4.2) makes sense, where we recall that   CN,p,s (u(x) − u(y))(v(x) − v(y)) Ap (u, v) := |u(x) − u(y)|p−2 dxdy. 2 |x − y|N +ps N N R R We assume that the measurable function g : Ω × R → R satisfies the following growth condition:    ∂g(x, τ )    ≤ α|τ |q + β|τ |p⋆ −2 , ∀ x ∈ Ω , τ ∈ R, g(x, 0) = 0 and  ∂τ  for some constants α, β and q ≥ 0, where we recall that p⋆ = the following equation (−∆)sp u = α|u|q u + β|u|p

⋆

−2

u + h in Ω ,

Np N −sp .

(4.3)

A typical example of problem (4.1) is

u=0

on RN \ Ω .

⋆

We mention that since the embedding W0s,p (Ω ) ↩→ Lp (Ω ) is only continuous, so the weak lower-semicontinuity of the corresponding functional is in question. Here we shall use the contraction mapping theorem and some properties of the operator Φs,p obtained in the previous section, to establish the existence of at least one solution to (4.1). The following theorem is our main result of this section. Theorem 4.2. Let Ω ⊂ RN be an arbitrary bounded open set. Let 0 < s < 1, N2N +2s < p ≤ 2 and assume ⋆ −s,p′ that (4.3) holds with q ∈ [0, p − 2]. Then for every 0 ̸≡ h ∈ W (Ω ), the problem (4.1) has at least one weak solution provided that α, β are small.

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Proof. Let 0 < s < 1 and

2N N +2s

149

< p ≤ 2. Then p⋆ > 2. Let Φs,p : W0s,p (Ω ) → W0s,p (Ω ) be defined by ∀ u ∈ W0s,p (Ω ).

Φs,p (u) = [(−∆)sp,Ω ]−1 (g(x, u) + h),

It follows from (2.5) and Proposition 3.2 that Φs,p is well-defined and is also continuous. To show that (4.1) has a solution, it suffices to prove that Φs,p has a fixed point. For this, we show that Φs,p is a contraction map in a neighborhood of u0 := Φs,p (0) = [(−∆)sp,Ω ]−1 (h). Indeed, let u, v ∈ W0s,p (Ω ) and set u1 := g(x, u) + h ⋆ p⋆ −q−2 1 and v1 := g(x, v) + h. Since p p−1 + p1⋆ = 1 and q+1 = 1, then using (4.3) and the H¨older ⋆ p⋆ + p⋆ + p⋆ inequality we get that there exists a constant C > 0 such that for every ϕ ∈ W0s,p (Ω ),     ⋆ |g(x, u)ϕ| dx ≤ α|u|q+1 |ϕ| + β|u|p −1 |ϕ| dx Ω

Ω

 p⋆ −2−q ≤ α|Ω | p⋆ ∥u∥Lp⋆ (Ω) + β∥u∥Lp⋆ (Ω) ∥ϕ∥Lp⋆ (Ω)   p⋆ −2−q ≤ C α|Ω | p⋆ ∥u∥Lp⋆ (Ω) + β∥u∥Lp⋆ (Ω) ∥ϕ∥W s,p (Ω) . 

0

′

We have shown that u1 , v1 ∈ W −s,p (Ω ). Therefore, using (3.5) we obtain that 2−p   p−1 ∥u1 − v1 ∥W −s,p′ (Ω) . ∥Φs,p (u) − Φs,p (v)∥W s,p (Ω) ≤ C ∥u1 ∥W −s,p′ (Ω) + ∥v1 ∥W −s,p′ (Ω)

(4.4)

0

Let CS be the best Sobolev constant in (2.5) with q = p⋆ . Let t = 2τ and τ p−1 = ∥h∥W −s,p′ (Ω) . It follows from (3.3) that ∥u0 ∥W s,p (Ω) ≤ τ . Let ϕ ∈ W0s,p (Ω ) be such that ∥ϕ∥W s,p (Ω) ≤ 1. Since 0

0

q 1 1 p⋆ − 2 − q + + + =1 p⋆ p⋆ p⋆ p⋆

and

p⋆ − 2 1 1 + ⋆ + ⋆ = 1, ⋆ p p p

then using (4.3), the Sobolev embedding (2.5) and the H¨older inequality, we obtain that there exists a constant C1 = C1 (α, β) > 0 such that for every u, v ∈ W0s,p (Ω ), with ∥u∥W s,p (Ω) ≤ t and ∥v∥W s,p (Ω) ≤ t, 0 0  |⟨u1 − v1 , ϕ⟩| ≤ |(g(x, u) − g(x, v)) ϕ| dx Ω

≤

 

α(|u| + |v|)q + β(|u| + |v|)p

⋆

−2



|u − v| |ϕ| dx

Ω

≤ αCS2+q |Ω |

p⋆ −2−q p⋆



∥u∥W s,p (Ω) + ∥v∥W s,p (Ω)

q

0

0

∥u − v∥W s,p (Ω) 0

p⋆ −2

 ⋆

∥u − v∥W s,p (Ω) + βCsp ∥u∥W s,p (Ω) + ∥v∥W s,p (Ω) 0 0 0   ⋆ p −2−q ⋆ ⋆ ≤ αCS2+q |Ω | p⋆ (2t)q + βCSp (2t)p −2 ∥u − v∥W s,p (Ω) 0

= C1 (α, β)∥u − v∥W s,p (Ω) .

(4.5)

0

If ∥u∥W s,p (Ω) ≤ t and ∥v∥W s,p (Ω) ≤ t, then it follows from (4.5) that 0

0

∥u1 − v1 ∥W −s,p′ (Ω) ≤ C1 (α, β)∥u − v∥W s,p (Ω) . 0

Since q+1 1 p⋆ − 2 − q + ⋆+ =1 ⋆ p p p⋆

p⋆ − 2 1 1 + ⋆ + ⋆ = 1, p⋆ p p

and

then proceeding as in (4.5), we have that for any ϕ ∈ W0s,p (Ω ) with ∥ϕ∥W s,p (Ω) ≤ 1, 0  |⟨u1 , ϕ⟩| ≤ ∥h∥W −s,p′ (Ω) ∥ϕ∥W0s,p (Ω) + |g(x, u)ϕ| dx Ω

M. Warma / Nonlinear Analysis 135 (2016) 129–157

150

≤τ

p−1

  + Ω

 β α q+1 p⋆ −1 |u| |u| + ⋆ |ϕ| dx q+1 p −1 ⋆

≤τ

p−1

p⋆ −2−q ⋆ βC p αCS2+q + |Ω | p⋆ tq+1 + ⋆ S tp −1 = C2 (α, β). q+1 p −1

We have shown that ∥u1 ∥W −s,p′ (Ω) ≤ C2 (α, β). Similarly, we get that ∥v1 ∥W −s,p′ (Ω) ≤ C2 (α, β). We deduce from (4.4) that if u, v ∈ W0s,p (Ω ) with ∥u∥W s,p (Ω) ≤ t and ∥v∥W s,p (Ω) ≤ t, then 0

0

∥Φs,p (u) − u0 ∥W s,p (Ω) = ∥Φs,p (u) − Φs,p (0)∥W s,p (Ω) ≤ C(α, β)t, 0

0

and from (4.5) we have that ∥Φs,p (u) − Φs,p (v)∥W0s,p (Ω) ≤ C(α, β)∥u − v∥W s,p (Ω) , 0

where 2−p

C(α, β) = C(2C2 (α, β)) p−1 C1 (α, β). Choosing α, β small so that C(α, β) < 1

and C(α, β) ≤ τ,

and since {u ∈ W0s,p (Ω ) : ∥u − u0 ∥W s,p (Ω) ≤ τ } ⊂ {u ∈ W0s,p (Ω ) : ∥u∥W s,p (Ω) ≤ τ }, we have that the 0 0 mapping Φs,p : {u ∈ W0s,p (Ω ) : ∥u − u0 ∥W s,p (Ω) ≤ τ } → {u ∈ W0s,p (Ω ) : ∥u − u0 ∥W s,p (Ω) ≤ τ } 0

is a contraction. We have shown that Φs,p has a fixed point and the proof is finished.

0



Remark 4.3. We notice that in Theorem 4.2, since N2N +2s < p ≤ 2, then in view of Remark 3.10, there is no regularity assumption on the open set, that is, Theorem 4.2 holds for an arbitrary bounded open set Ω . 5. The parabolic problem Let Ω ⊂ RN be an arbitrary bounded open set, 0 < s < 1 and p ∈ (1, ∞). In this section we consider the parabolic problem  ∂u s    ∂t (t, x) + (−∆)p u(t, x) = 0 in (0, ∞) × Ω , (5.1) on (0, ∞) × (RN \ Ω ), u = 0   u(0, x) = u0 (x) x ∈ Ω, where u0 is a given function. First, we introduce the realization in L2 (Ω ) of (−∆)sp with the Dirichlet boundary condition. Let Φ be the functional with domain D(Φ) = W0s,p (Ω ) ∩ L2 (Ω ) and defined on L2 (Ω ) by    |u(x) − u(y)|p   CN,p,s dxdy, if u ∈ W0s,p (Ω ), N +ps 2p |x − y| N N Φ(u) := (5.2) R R  s,p 2 ∞, if u ∈ L (Ω ) \ W0 (Ω ). It has been shown in [38, Theorem 6.3], that Φ is proper, convex and lower semi-continuous. Moreover, if f ∈ L2 (Ω ), u ∈ W0s,p (Ω ) ∩ L2 (Ω ) and ∂Φ is the single-valued subgradient of Φ, then f = ∂Φ(u) if and only if for every v ∈ W0s,p (Ω ) ∩ L2 (Ω ),    CN,p,s (u(x) − u(y))(v(x) − v(y)) |u(x) − u(y)|p−2 dxdy = f v dx. (5.3) 2 |x − y|N +ps RN RN Ω

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151

In addition, we have that 

D(∂Φ) = {u ∈ W0s,p (Ω ) ∩ L2 (Ω ), (−∆)sp u ∈ L2 (Ω )} ∂Φ(u) = (−∆)sp u.

(5.4)

We set (−∆)sp,Ω := ∂Φ and call it the realization in L2 (Ω ) of the fractional p-Laplace operator (−∆)sp with the Dirichlet boundary condition. 5.1. The submarkovian property The following theorem is the first main result of this section. Theorem 5.1. Let p ∈ (1, ∞), 0 < s < 1 and (−∆)sp,Ω the operator introduced in (5.4). Then −(−∆)sp,Ω generates a (nonlinear) submarkovian semigroup Ss,p = (Ss,p (t))t≥0 on L2 (Ω ). In particular, we have that the system (5.1) is well-posed on Lq (Ω ) for every q ∈ [1, ∞), that is, for every u0 ∈ Lq (Ω ), (5.1) has a unique solution. Proof. Let p ∈ (1, ∞) and 0 < s < 1. Recall that ∂Φ = (−∆)sp,Ω where Φ is the functional defined in (5.2). Since Φ is proper, convex, lower semi-continuous and D(Φ) is dense in L2 (Ω ), it follows from a classical result by Minty [30] that −∂Φ = −(−∆)sp,Ω generates a (nonlinear) strongly continuous semigroup Ss,p = (Ss,p (t))t≥0 on L2 (Ω ). Hence, for every u0 ∈ L2 (Ω ), the function u(·, ·) := Ss,p (·)u0 (·) satisfies:  1,∞ 2 2  u ∈ C([0, ∞); L (Ω )) ∩ Wloc ((0, ∞); L (Ω ));   s u(t, ·) ∈ D((−∆)p,Ω ) for a.e. t ∈ R+ ; (5.5) ∂u  + (−∆)sp,Ω u = 0 for a.e. t ∈ R+ ;   ∂t   u(0, x) = u0 (x) x ∈ Ω. That is, u is the unique strong solution of (5.1). Next, we show that the semigroup Ss,p is order preserving, that is, Ss,p (t)u ≤ Ss,p (t)v,

∀ t ≥ 0 whenever u, v ∈ L2 (Ω ),

u ≤ v.

(5.6)

By [12, Theorem 1.4] (see also [13, Theorem 3.6]), the estimate (5.6) is equivalent to u ∧ v, u ∨ v ∈ D(Φ) for all u, v ∈ D(Φ) and Φ(u ∧ v) + Φ(u ∨ v) ≤ Φ(u) + Φ(v). Let A := {x ∈ RN : u(x) ≤ v(x)} and B := RN \ A = {x ∈ RN : u(x) > v(x)}. Then   |(u ∧ v)(x) − (u ∧ v)(y)|p dxdy |x − y|N +ps RN  RN   |u(x) − u(y)|p |u(x) − v(y)|p dxdy + dxdy = N +ps N +ps A A |x − y| A B |x − y|     |v(x) − u(y)|p |v(x) − v(y)|p + dxdy + dxdy N +ps N +ps B A |x − y| B B |x − y|

(5.7)

(5.8)

and |(u ∨ v)(x) − (u ∨ v)(y)|p dxdy |x − y|N +ps RN  RN   |v(x) − v(y)|p |v(x) − u(y)|p = dxdy + dxdy N +ps N +ps A A |x − y| A B |x − y|     |u(x) − v(y)|p |u(x) − u(y)|p + dxdy + dxdy. N +ps N +ps B A |x − y| B B |x − y|





(5.9)

152

M. Warma / Nonlinear Analysis 135 (2016) 129–157

Define the functional F : R2 → [0, ∞) by F (ξ, τ ) := |ξ − τ |p . It is clear that F satisfies the hypothesis of Lemma 2.4. Since p > 1, we have that ∂2F 2 = −p(p − 1)|ξ − τ |p−2 (sgn(ξ − τ )) ≤ 0. ∂ξ∂τ Hence, Lemma 2.4(ii) is satisfied. Applying this lemma, we get that on A × B and on B × A, |u(x) − v(y)|p + |v(x) − u(y)|p ≤ |u(x) − u(y)|p + |v(x) − v(y)|p .

(5.10)

Combining (5.8), (5.9) and using (5.10) we get (5.7). Hence, the semigroup Ss,p is order preserving. Next, we show that the semigroup is non-expansive on Lq (Ω ) for every q ∈ [2, ∞], that is, Ss,p (t) maps Lq (Ω ) into Lq (Ω ) for all t > 0 and for all u, v ∈ Lq (Ω ).

(5.11)

By [12, Theorem 1.4] (see also [13, Theorem 3.6]), the estimate (5.11) is equivalent to     Φ v + gα (u, v) + Φ u − gα (u, v) ≤ Φ(u) + Φ(v),

(5.12)

∥Ss,p (t)u − Ss,p (t)v∥Lq (Ω) ≤ ∥u − v∥Lq (Ω) ,

for all u, v ∈ L2 (Ω ) and α > 0 where   1 (u − v + α)+ − (u − v − α)− . gα (u, v) := 2 If u or v does not belong to D(Φ), then the inequality (5.12) is trivial. Therefore, we may assume without any restriction that u, v ∈ D(Φ). It then follows from [37, Lemma 2.6] that gα (u, v) ∈ D(Φ) for every α > 0. (u,v) Hence, v + gα (u, v), u − gα (u, v) ∈ D(Φ). For α > 0, we let λ := χ{u̸=v} gαu−v . A simple calculation shows that λ ∈ [0, 1]. Moreover, u − gα (u, v) = λv + (1 − λ)u and v + gα (u, v) = λu + (1 − λ)v. Using the convexity of the functional Φ, we get that     Φ v + gα (u, v) + Φ u − gα (u, v) = Φ(λu + (1 − λ)v) + Φ(λv + (1 − λ)u) ≤ λΦ(u) + (1 − λ)Φ(v) + λΦ(v) + (1 − λ)Φ(u) = Φ(u) + Φ(v), and we have shown (5.12). Hence, the semigroup Ss,p is non-expansive on Lq (Ω ) for every q ∈ [2, ∞]. We have shown that Ss,p is a (nonlinear) submarkovian semigroup on L2 (Ω ). Since Φ is nonnegative, it follows from [13, Theorem 2.4] that Ss,p can be extended to a strongly continuous, non-expansive semigroup on Lq (Ω ) for every q ∈ [1, ∞) and to a non-expansive semigroup on L∞ (Ω ), and each of such semigroups is order preserving. Hence, the Cauchy problem (5.1) is well posed in Lq (Ω ) for every q ∈ [1, ∞). The proof of the theorem is finished.  5.2. The (Lq − L∞ )-H¨ older type continuity The following theorem is the second main result of this section. Theorem 5.2. Let Ω ⊂ RN be a bounded open set, p ∈ [2, ∞), 0 < s < 1 such that sp < N and Ss,p = (Ss,p (t))t≥0 the (nonlinear) submarkovian semigroup on L2 (Ω ) generated by −(−∆)sp,Ω . Let q ∈ [2, ∞] and β(s), δ(s), γ(s) given by (1.8). Then there exists a constant C > 0 such that for every u0 , v0 ∈ Lq (Ω ) and t > 0, γ(s)

∥Ss,p (t)u0 − Ss,p (t)v0 ∥L∞ (Ω) ≤ C|Ω |β(s) t−δ(s) ∥u0 − v0 ∥Lq (Ω) .

(5.13)

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153

Proof. Let p ∈ [2, ∞) and 0 < s < 1 be such that sp < N . We prove the theorem in several steps. Step 1: In the first step we assume that the initial data are bounded. We show that solutions of (5.1) are differentiable a.e. in the time variable. Let u0 , v0 ∈ L∞ (Ω ), u(τ ) := Ss,p (τ )u0 and v(τ ) := Ss,p (τ )v0 , where τ > 0. Let r ≥ 2 be a real number and consider the function Gr defined by Gr : (0, ∞) → [0, ∞),

Gr (τ ) := ∥u(τ ) − v(τ )∥rLr (Ω) .

First, we notice that Gr is well-defined since u and v are bounded in Ω × (0, ∞) and Ω is a bounded open set. Second, it follows from (5.5) that Gr is differentiable a.e. Throughout the remainder of the proof, we set U (τ ) = U (τ, x) := u(τ, x) − v(τ, x),

 (τ ) = U  (τ, x, y) = u(τ, x) − u(τ, y), U

and V (τ ) = V (τ, x, y) = v(τ, x) − v(τ, y) and in our notation, we sometimes omit the dependence of u, v in the space variable. By Leibniz rule, using that u(τ ) and v(τ ) are solutions of the Cauchy problem (5.1) with initial data u0 and v0 , respectively, and using (5.3) we have that for a.e. τ > 0,  d ∂U r ∥U (τ )∥Lr (Ω) = r |U (τ )|r−1 sgn(U (τ )) (τ ) dx dτ ∂τ Ω  = −r |u(τ ) − v(τ )|r−1 sgn(u(τ ) − v(τ ))∂Φ(u(τ )) dx Ω +r |u(τ ) − v(τ )|r−1 sgn(u(τ ) − v(τ ))∂Φ(v(τ )) dx Ω

   (τ )|p−2 U  (τ ) − |V (τ )|p−2 V (τ ) |U CN,p,s 2 |x − y|N +ps RN RN   × |U (τ, x)|r−2 U (τ, x) − |U (τ, y)|r−2 U (τ, y) dxdy.

= −r

(5.14)

Step 2: Fine estimates of the first derivative of solutions. Using (2.12), we get from (5.14) that there is a constant C1 > 0 such that   |U (τ, x) − U (τ, y)|p−1 d CN,p,s ∥U (τ )∥rLr (Ω) ≤ −C1 r |U (τ, x) − U (τ, y)|r−1 dxdy N +ps dτ 2 |x − y| N N R R   |U (τ, x) − U (τ, y)|r−2+p CN,p,s dxdy. (5.15) = −C1 r 2 |x − y|N +ps RN RN Since u(τ ), v(τ ) ∈ W0s,p (Ω ) ∩ L∞ (RN ), it follows that the integral in the right hand side of (5.15) exists. Next, let r : [0, ∞) → [2, ∞) be an increasing differentiable function. Using the above argument, one has r(τ ) that the function τ → ∥U (τ )∥Lr(τ ) (Ω) is differentiable a.e. and from (5.15) we get that  d r(τ ) ′ |U (τ )|r(τ ) log |U (τ )| dx ∥U (τ )∥Lr(τ ) (Ω) ≤ r (τ ) dτ Ω   CN,p,s |U (τ, x) − U (τ, y)|r(τ )−2+p − C1 r(τ ) dxdy. (5.16) 2 |x − y|N +ps RN RN Using (5.16) and calculating, we obtain the following estimate:    d r′ (τ ) |U (τ )|r(τ ) |U (τ )| log ∥U (τ )∥Lr(τ ) (Ω) ≤ log dx ) dτ r(τ ) Ω ∥U (τ )∥r(τ ∥U (τ )∥Lr(τ ) (Ω) r(τ ) L (Ω)   C1 CN,p,s |U (τ, x) − U (τ, y)|r(τ )−2+p − dxdy. r(τ ) 2 |x − y|N +ps RN RN ∥U (τ )∥ r(τ ) L

(Ω)

(5.17)

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154

Step 3: In this step we use the fractional Logarithmic-Sobolev inequality. Let |U (τ, x)|

F (τ ) = F (τ, x) :=

∥U (τ, ·)∥

r(τ )−2+p p

r(τ )−2+p p Lr(τ )−2+p (Ω)

.

Since U (τ ) ∈ W0s,p (Ω )∩L∞ (RN ), it follows from [37, Remark 2.5] that F (τ ) ∈ W0s,p (Ω ). Moreover, F (τ ) ≥ 0 and ∥F (τ, ·)∥Lp (Ω) = 1. Using (2.7), (2.8) and (5.17), we have that there is a constant C2 > 0 such that for every ε > 0,    d r′ (τ ) |U (τ )|r(τ ) |U (τ )| log ∥U (τ )∥Lr(τ ) (Ω) ≤ log dx ) dτ r(τ ) Ω ∥U (τ )∥r(τ ∥U (τ )∥Lr(τ ) (Ω) r(τ ) L (Ω) r(τ )−2+p

sp2 (r(τ ) − 1) ∥U (τ )∥Lr(τ )−2+p (Ω) − r(τ ) N C2 ε ∥U (τ )∥Lr(τ ) (Ω)    |U (τ )| |U (τ )|r(τ )−2+p log × dx r(τ )−2+p ∥U (τ )∥Lr(τ )−2+p (Ω) Ω ∥U (τ )∥ r(τ )−2+p 

p r(τ ) − 2 + p

p−1

L

 −

p r(τ ) − 2 + p

(Ω)

p

r(τ )−2+p

r(τ ) − 1 ∥U (τ )∥Lr(τ )−2+p (Ω) log(ε) . r(τ ) C2 ε ∥U (τ )∥ r(τ ) L

(5.18)

(Ω)

Letting  K(r(τ ), U (τ )) := Ω

|U (τ )|r(τ ) r(τ )

∥U (τ )∥Lr(τ ) (Ω)

 log

|U (τ )| ∥U (τ )∥Lr(τ ) (Ω)

 dx

and sp2 r(τ )(r(τ ) − 1) ε(τ ) := N C2 r′ (τ )



p r(τ ) − 2 + p

p−1 ∥U ∥r(τ )−2+p Lr(τ )−2+p (Ω) r(τ )

∥U ∥Lr(τ ) (Ω)

we get from (5.18) that d r′ (τ ) log ∥U (τ )∥Lr(τ ) (Ω) ≤ [K(r(τ ), U (τ )) − K(r(τ ) − 2 + p, U (τ ))] dτ r(τ )   r(τ )−2+p  p−1 2 ∥U (τ )∥Lr(τ )−2+p (Ω) p sp r(τ )(r(s) − 1) N r′ (τ ) . log  − r(τ ) sp(r(τ ) − 2 + p)r(τ ) r(τ ) − 2 + p N Cr′ (τ ) ∥U ∥Lr(τ ) (Ω) Step 4: Since the map q →

d dq

(5.19)

log ∥U ∥qLq (Ω) is non-decreasing, we have that for every q2 ≥ q1 ≥ 1, K(q1 , U ) − K(q2 , U ) = log

∥U ∥Lq2 (Ω) . ∥U ∥Lq1 (Ω)

(5.20)

Applying (5.20) with q1 = r(τ ) and q2 = r(τ ) − 2 + p and calculating, we get from (5.19) that d (p − 2) N r′ (τ ) log ∥U (τ )∥Lr(τ ) (Ω) ≤ − log ∥U (τ )∥Lr(τ ) (Ω) dτ sp r(τ ) (r(τ ) + p − 2)   ∥U (τ )∥Lr(τ )−2+p (Ω) N r′ (τ ) + 1− log sp r(τ ) ∥U (τ )∥Lr(τ ) (Ω)   p−1 2 N r′ (τ ) p sp r(τ )(r(τ ) − 1) − log . (5.21) sp(r(τ ) − 2 + p)r(τ ) r(τ ) − 2 + p N Cr′ (τ )

M. Warma / Nonlinear Analysis 135 (2016) 129–157

Since |Ω | < ∞ and sp < N , then using the classical H¨older inequality we have that     ∥U (τ )∥Lr(τ )−2+p (Ω) N r′ (τ ) p−2 N 1− log ≤ − 1 log |Ω |. sp r(τ ) ∥U (τ )∥Lr(τ ) (Ω) r(τ )(r(τ ) − 2 + p) sp

155

(5.22)

Using (5.22), we get from (5.21) that d N r′ (τ ) (p − 2) log ∥U (τ )∥Lr(τ ) (Ω) ≤ − log ∥U (τ )∥Lr(τ ) (Ω) dτ sp r(τ ) (r(τ ) + p − 2)   p−2 N r′ (τ ) − 1 log |Ω | + r(τ ) (r(τ ) − 2 + p) sp   p−1 2 N r′ (τ ) sp r(τ )(r(τ ) − 1) p . (5.23) − log sp(r(τ ) − 2 + p)r(τ ) r(τ ) − 2 + p N Cr′ (τ ) Set A(τ ) :=

N r′ (τ ) (p − 2) , sp r(τ ) (r(τ ) + p − 2)

Y (τ ) := log ∥U (τ )∥Lr(τ ) (Ω)

and r′ (τ ) p−2 r(τ ) (r(τ ) − 2 + p)

 N − 1 log |Ω | sp   p−1 2 ′ N r (τ ) p sp r(τ )(r(τ ) − 1) + log . sp(r(τ ) − 2 + p)r(τ ) r(τ ) − 2 + p N Cr′ (τ )

B(τ ) := −



It follows from (5.23) that Y (τ ) satisfies the differential inequality d Y (τ ) + A(τ )Y (τ ) + B(τ ) ≤ 0, dτ

for all τ > 0.

Step 5: We integrate the differential inequality. Let Y satisfy (5.24). Integrating we get that   τ   σ    τ Y (τ ) ≤ X(τ ) := exp − A(σ) dσ Y (0) − B(σ) exp A(z) dz dσ . 0

Let r(τ ) := lim X(τ ) =

(5.25)

0

qt t−τ

 τ →t−

0

(5.24)

with q ≥ 2 and 0 ≤ τ < t. A simple calculation gives   N N  N   sp  sp  sp   1 N − sp q q q Y (0) − log(t) + |Ω |. 1− 1− q−2+p p−2 q−2+p N q−2+p

Using the submarkovian property of the semigroup Ss,p and (5.25), we get that for all 0 < τ < t, ∥U (t)∥Lr(τ ) (Ω) = ∥u(t) − v(t)∥Lr(τ ) (Ω) ≤ ∥u(τ ) − v(τ )∥Lr(τ ) (Ω) = eY (τ ) ≤ eX(τ ) .

(5.26)

Since limτ →t− r(τ ) = ∞ and Y (0) := log ∥u(0) − v(0)∥Lr(0) (Ω) = log ∥u0 − v0 ∥Lq (Ω) , then taking the limit of (5.26) as τ → t− , we get that for all t > 0, γ(s)

∥u(t) − v(t)∥L∞ (Ω) ≤ C|Ω |β(s) t−δ(s) ∥u0 − v0 ∥Lq (Ω) ,

(5.27)

with the constants β(s), δ(s) and γ(s) given by (1.8). Step 6: In this final step we consider more general initial data. Let u0 , v0 ∈ Lq (Ω ) and un,0 , vn,0 ∈ L∞ (Ω ) be sequences which converge respectively to u0 and v0 in Lq (Ω ). Let un (t) := Ss,p (t)un,0 , u(t) := Ss,p (t)u0 , vn (t) := Ss,p (t)vn,0 and v(t) := Ss,p (t)v0 . Using (5.27) with first vn,0 = 0 and then un,0 = 0, we obtain that for every t > 0, un (t) and vn (t) converge in L∞ (Ω ) to u(t) and v(t), respectively. Hence, for every u0 , v0 ∈ Lq (Ω ) and t > 0, we have the estimate (5.13) and the proof is finished. 

M. Warma / Nonlinear Analysis 135 (2016) 129–157

156

We conclude the paper with the following observations. Remark 5.3. We mention the following situations. (a) A simple calculation shows that lim β(s) = 0,

p→2

lim δ(s) =

p→2

N 2sq

and

lim γ(s) = 1

p→2

so that if p = 2 (that is the linear case), the estimate (5.13) reads N

∥Ss,2 (t)u0 − Ss,2 (t)v0 ∥L∞ (Ω) = ∥Ss,2 (t)(u0 − v0 )∥L∞ (Ω) ≤ C1 t− 2sq ∥u0 − v0 ∥Lq (Ω) .

(5.28)

Since Ss,2 is a linear semigroup, (5.28) shows that Ss,2 is ultracontractive in the classical sense. (b) In the linear case p = 2, the proof of the corresponding result in Theorem 5.2 follows from the ideas described in [14, Chapter 2] by using some results on classical Logarithmic-Sobolev inequalities contained in [24]. In the quasi-linear case, the strategy is inspired from the linear case and the works of Cipriani and Grillo [11,12] on the classical p-Laplace operator. (c) Finally, instead of the use of the Logarithmic-Sobolev inequality in the proof of Theorem 5.2, a classical Moser iterative technique used in [22, Proof of Theorem 3.2] (see also [15, Proof of Theorem 8.6]) to estimate solutions of the Porous Media Equation, may be also used here to obtain the estimate (5.13). Acknowledgments • The work of the author is partially supported by the Air Force Office of Scientific Research under the Award No: FA9550-15-1-0027. • We would like to thank both anonymous referees for the careful reading of the manuscript and for their helpful comments and suggestions. References [1] D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory, in: Grundlehren der Mathematischen Wissenschaften, vol. 314, Springer-Verlag, Berlin, 1996. [2] R.J. Adler, R.E. Feldman, M.S. Taqqu (Eds.), A Practical Guide to Heavy Tails. Statistical Techniques and Applications, Birkh¨ auser, 1998, Papers from the workshop held in Santa Barbara, CA, December 1995. [3] D. Applebaum, L´ evy Processes and Stochastic Calculus, in: Cambridge Studies in Advanced Mathematics, vol. 93, 2004. [4] M. Biegert, A priori estimates for the difference of solutions to quasi-linear elliptic equations, Manuscripta Math. 133 (2010) 273–306. [5] K. Bogdan, K. Burdzy, Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields 127 (2003) 89–152. [6] L. Caffarelli, J.-M. Roquejoffre, Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. 12 (2010) 1151–1179. [7] L. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008) 425–461. [8] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007) 1245–1260. [9] Z.-Q. Chen, T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stochastic Process. Appl. 108 (2003) 27–62. [10] Y. Cheng, H¨ older continuity of the inverse of p-Laplacian, J. Math. Anal. Appl. 221 (1998) 734–748. [11] F. Cipriani, G. Grillo, Uniform bounds for solutions to quasilinear parabolic equations, J. Differential Equations 177 (2001) 209–234. [12] F. Cipriani, G. Grillo, Lq − L∞ H¨ older continuity for quasilinear parabolic equations associated to Sobolev derivations, J. Math. Anal. Appl. 270 (2002) 267–290. [13] F. Cipriani, G. Grillo, Nonlinear Markov semigroups, nonlinear Dirichlet forms and application to minimal surfaces, J. Reine Angew. Math. 562 (2003) 201–235. [14] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.

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