Citation: Tomas Godoy. A semilnear singular problem for the fractional laplacian[J]. AIMS Mathematics, 2018, 3(4): 464-484. doi: 10.3934/Math.2018.4.464
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Elliptic problems with singular nonlinearities appear in many nonlinear phenomena, for instance, in the study of chemical catalysts process, non-Newtonian fluids, and in the study of the temperature of electrical conductors whose resistance depends on the temperature (see e.g., [3,6,10,15] and the references therein). The seminal work [7] is the start point of a large literature concerning singular elliptic problems, see for instance, [1,3,5,6,8,9,10,13,15,17,18,21,22,23,24], and [30]. For additional references and a systematic study of singular elliptic problems see also [26].
In [10], Diaz, Morel and Oswald considered problems of the form
{−Δu=−u−γ+λh(x) in Ω,u=0 on ∂Ω,u>0 in Ω | (1.1) |
where Ω is a bounded and regular enough domain, 0<γ<1, λ>0 and h∈L∞(Ω) is a nonnegative and nonidentically zero function. They proved (see [10], Theorem 1, Corollary 1, Lemma 2 and Theorem 3) that there exists λ0>0 such that. for λ>λ0, problem (1.1) has a unique maximal solution u∈H10(Ω) and has no solution when λ<λ0.
Concerning nonlocal singular problems, Barrios, De Bonis, Medina, and Peral proved in [2] that if Ω is a bounded and regular enough domain in Rn, 0<s<1, n>2s, f is a nonnegative function in a suitable Lebesgue space, λ>0, M>0 and 1<p<n+2sn−2s, then the problem
{(−Δ)su=λf(x)u−γ+Mup in Ω,u=0 on Rn∖Ω,u>0 in Ω, | (1.2) |
has a solution, in a suitable weak sense whenever λ>0 and M>0, and that, if M=1 and f=1, then there exists Λ>0 such that (1.2) has at least two solutions when λ<Λ and has no solution when λ>Λ.
A natural question is to ask if an analogous of the quoted result of [10] hold in the nonlocal case, i.e., when −Δ is replaced by the fractional laplacian (−Δ)s, s∈(0,1), and with the boundary condition u=0 on ∂Ω replaced by u=0 on Rn∖Ω. Our aim in this paper is to obtain such a result. Note that the approach of [10] need to be modified in order to be used in the fractional case. Indeed, a step in [10] was to observe that, if φ1 denotes a positive principal eigenfunction for −Δ on Ω, with Dirichlet boundary condition, then
−Δφ21+γ1=21+γλ1φ21+γ1−2(1−γ)(γ+1)2|∇φ1|2φ−2γ1+γ1 in Ω, | (1.3) |
where λ1 is the corresponding principal eigenvalue. From this fact, and using the properties of a principal eigenfunction, Diaz, Morel and Oswald proved that, for ε positive and small enough, εφ21+γ1 is a subsolution of problem (1.1). Since formula (1.3), is not avalaible for the principal eigenfunction of (−Δ)s, the arguments of [10] need to be modified in order to deal with the fractional case.
Let us state the functional setting for our problem. For s∈(0,1) and n∈N, let
Hs(Rn):={u∈L2(Rn):∫Rn×Rn|u(x)−u(y)|2|x−y|n+2sdxdy<∞}, |
and for u∈Hs(Rn), let ‖u‖Hs(Rn):=(∫Rnu2+∫Rn×Rn|u(x)−u(y)|2|x−y|n+2sdxdy)12. Let Ω be a bounded domain in Rn with C1,1 boundary and let
Xs0(Ω):={u∈Hs(Rn):u=0 a.e. in Rn∖Ω}, |
and for u∈Xs0(Ω), let ‖u‖Xs0(Ω):=(∫Rn×Rn|u(x)−u(y)|2|x−y|n+2sdxdy)12.
With these norms, Hs(Rn) and Xs0(Ω) are Hilbert spaces (see e.g., [29], Lemma 7), C∞c(Ω) is dense in Xs0(Ω) (see [16], Theorem 6). Also, Xs0(Ω) is a closed subspace of Hs(Rn), and from the fractional Poincaré inequality (as stated e.g., in [11], Theorem 6.5; see Remark 2.1 below), if n>2s then ‖.‖Xs0(Ω) and ‖.‖Hs(Rn) are equivalent norms on Xs0(Ω). For f∈L1loc(Ω) we say that f∈(Xs0(Ω))′ if there exists a positive constant c such that |∫Ωfφ|≤c‖u‖Xs0(Ω) for any φ∈Xs0(Ω). For f∈(Xs0(Ω))′ we will write ((−Δ)s)−1f for the unique weak solution u (given by the Riesz theorem) of the problem
{(−Δ)su=f in Ω,u=0 in Rn∖Ω. | (1.4) |
Here and below, the notion of weak solution that we use is the given in the following definition:
Definition 1.1. Let s∈(0,1), let f:Ω→R be a Lebesgue measurable function such that fφ∈L1(Ω) for any φ∈Xs0(Ω). We say that u:Ω→R is a weak solution to the problem
{(−Δ)su=f in Ω,u=0 in Rn∖Ω |
if u∈Xs0(Ω), u=0 in Rn∖Ω and, for any φ∈ Xs0(Ω),
∫Rn×Rn(u(x)−u(y))(φ(x)−φ(y))|x−y|n+2sdxdy=∫Ωfφ. |
For u∈Xs0(Ω) and f∈L1loc(Ω), we will write (−Δ)su≤f in Ω (respectively (−Δ)su≥f in Ω) to mean that, for any nonnegative φ∈Hs0(Ω), it hold that fφ∈L1(Ω) and
∫Rn×Rn(u(x)−u(y))(φ(x)−φ(y))|x−y|n+2sdxdy≤∫Ωfφ (resp. ≥∫Ωfφ). |
For u,v∈Xs0(Ω), we will write (−Δ)su≤(−Δ)sv in Ω (respectively (−Δ)su≥(−Δ)sv in Ω), to mean that (−Δ)s(u−v)≤0 in Ω (resp. (−Δ)s(u−v)≥0 in Ω).
Let
E:={u∈Xs0(Ω):cdsΩ≤u≤c′dsΩ a.e. in Ω, for some positive constants c and c′} |
where, for x∈Ω, dΩ(x):=dist(x,∂Ω). With these notations, our main results read as follows:
Theorem 1.2. Let Ω be a bounded domain in Rn with C1,1 boundary, let s∈(0,1), and assume n>2s. Let h∈L∞(Ω) be such that 0≤h≢0 in Ω (i.e., |{x∈Ω:h(x)>0}|>0) and let g:Ω×(0,∞)→[0,∞) be a function satisfying the following conditions g1)-g5)
g1) g:Ω×(0,∞)→[0,∞) is a Carathéodory function, g(.,s)∈L∞(Ω) for any s>0 and limσ→∞‖(g(.,σ))‖∞=0.
g2) σ→g(x,σ) is non increasing on (0,∞) a.e. x∈Ω.
g3) g(.,σdsΩ)∈(Xs0(Ω))′ and d−sΩ((−Δ)s)−1(dsΩg(.,σdsΩ))∈L∞(Ω) for all σ>0.
g4) It hold that:
limσ→∞‖(σdsΩ)−1((−Δ)s)−1(dsΩg(.,σdsΩ))‖∞=0, and
limσ→∞‖d−sΩ((−Δ)s)−1(g(.,σ))‖L∞(Ω)=0.
g5) dsΩg(.,σdsΩ)∈L2(Ω) for any σ>0.
Consider the problem
{(−Δ)su=−g(.,u)+λhinΩ,u=0inRn∖Ω,u>0inΩ | (1.5) |
Then there exists λ∗≥0 such that:
i) If λ>λ∗ then (1.5) has a weak solution u(λ)∈E, which is maximal in the following sense: If v∈E satisfies (−Δ)sv≤−g(.,v)+λh in Ω, then u(λ)≥v a.e. in Ω .
ii) If λ<λ∗, no weak solution exists in E.
iii) If, in addition, there exists b∈L∞(Ω) such that 0≤b≢0 in Ω and g(.,s)≥bs−β a.e. in Ω for any s∈(0,∞), then λ∗>0.
Theorem 1.2 allows g(x,s) to be singular at s=0. In fact, in Lemma 3.2, using some estimates from [4] for the Green function of (−Δ)s in Ω (with homogeneous Dirichlet boundary condition on Rn∖Ω), we show that if g(x,s)=as−β with a a nonnegative function in L∞(Ω) and β∈[0,s), then g satisfies the assumptions of Theorem 1.2. Thus, as a consequence of Theorem 1.2, we obtain the following:
Theorem 1.3. Let Ω, s, and h be as in the statement of Theorem 1.2, and let g:Ω×(0,∞)→[0,∞). Then the assertions i)−iii) of Theorem 1.2 remain true if we assume, instead of the conditions g1)-g5), that the following conditions g6) and g7) hold:
g6) g:Ω×(0,∞)→[0,∞) is a Carathéodory function and s→g(x,s) is nonincreasing for a.e. x∈Ω.
g7) There exist positive constants a and β∈[0,s) such that g(.,s)≤as−β a.e. in Ω for any s∈(0,∞).
Let us sketch our approach: In Section 2 we consider, for ε>0, the following approximated problem
{(−Δ)su=−g(.,u+ε)+λh in Ω,u=0 in Rn∖Ω,u>0 in Ω. | (1.6) |
Let us mention that, in order to deal with problems involving the (p;q)-Laplacian and a convection term, this type of approximation was considered in [14] (see problem Pε therein).
Lemma 2.5 gives a positive number λ0, independent of ε and such that, for λ=λ0, problem (1.6) has a weak solution wε. From this result, and from some properties of the function wε, in Lemma 2.11 we show that, for λ≥λ0 and for any ε>0, there exists a weak solution uε of problem (1.6), with the following properties:
a) cdsΩ≤uε≤c′dsΩ for some positive constants c and c′ independent of ε,
b) uε≤¯u, where ¯u is the solution of the problem (−Δ)s¯u=λh in Ω, ¯u=0 in Rn∖Ω,
c) uε≥ψ for any ψ∈Xs0(Ω) such that (−Δ)sψ=−g(.,ψ+ε)+λh in Ω.
In section 3 we prove Theorems 1.2 and 1.3. To prove Theorem 1.2, we consider a decreasing sequence {εj}j∈N such that limj→∞εj=0, and we show that, for λ≥λ0, the sequence of functions {uεj}j∈N given by Lemma 2.11 converges, in Xs0(Ω), to a weak solution u of problem (1.5) which has the properties required by the theorem. An adaptation of some of the arguments of [10] gives that, if problem (1.5) has a weak solution in E, then it has a maximal (in the sense stated in the theorem) weak solution in E and that if for some λ=λ′ (1.5) has a weak solution in E, then it has a weak solution in E for any λ≥λ′. Finally, the assertion iii) of Theorem 1.2 is proved with the same argument given in [10].
We fix, from now on, h∈L∞(Ω) such that 0≤h≢0 in Ω. We assume also from now on (except in Lemma 3.2) that g:Ω×(0,∞)→[0,∞) satisfies the assumptions g1)-g5 of Theorem 1.2.
In the next remark we collect some general facts concerning the operator (−Δ)s.
Remark 2.1. ⅰ) (see e.g., [27], Proposition 4.1 and Corollary 4.2) The following comparison principle holds: If u,v∈Xs0(Ω) and (−Δ)su≥(−Δ)sv in Ω then u≥v in Ω. In particular, the following maximum principle holds: If v∈Xs0(Ω), (−Δ)sv≥0 in Ω and v≥0 in Rn∖Ω, then v≥0 in Ω.
ⅱ) (see e.g., [27], Lemma 7.3) If f:Ω→R is a nonnegative and not identically zero measurable function in f∈(Xs0(Ω))′, then the weak solution u of problem (1.4) satisfies, for some positive constant c,
u≥cdsΩ in Ω. | (2.1) |
ⅲ) (see e.g., [28], Proposition 1.1) If f∈L∞(Ω) then the weak solution u of problem (1.4) belongs to Cs(Rn). In particular, there exists a positive constant c such that
|u|≤cdsΩ in Ω. | (2.2) |
ⅳ) (Poincaré inequality, see [11], Theorem 6.5) Let s∈(0,1) and let 2∗s:=2nn−2s. Then there exists a positive constant C=C(n,s) such that, for any measurable and compactly supported function f:Rn→R,
‖f‖L2∗s(Rn)≤C∫Rn×Rn(f(x)−f(y))2|x−y|n+spdxdy. |
ⅴ) If v∈L(2∗s)′(Ω) then v∈(Xs0(Ω))′, and ‖v‖(Xs0(Ω))′≤C‖v‖(2∗s)′, with C as in i). Indeed, for φ∈Xs0(Ω), from the Hölder inequality and iii), ∫Ω|vφ|≤‖v‖(2∗s)′‖φ‖2∗s≤C‖v‖(2∗s)′‖φ‖Xs0(Ω).
ⅵ) (Hardy inequality, see [25], Theorem 2.1) There exists a positive constant c such that, for any φ∈Xs0(Ω),
‖d−sΩφ‖2≤c‖φ‖Xs0(Ω). | (2.3) |
Remark 2.2. Let z∗∈Hs(Rn) be the solution of the problem
{(−Δ)sz∗=τ1h in Ωz∗=0 in Rn∖Ω, | (2.4) |
with τ1 chosen such that ‖z∗‖L∞(Rn)=1. Since h∈L∞(Rn), Remark 2.1 ⅲ) gives z∗∈C(Rn) (see also [12], Theorem 1.2). Thus, since supp(z∗)⊂¯Ω and z∗∈C(¯Ω), we have z∗∈L∞(Rn). Moreover, by Remark 2.1 ⅱ), there exists a positive constant c∗ such that
z∗≥c∗dsΩ in Ω. | (2.5) |
Remark 2.3. There exist positive numbers M0 and M1 such that
12c∗M1≥‖d−sΩ((−Δ)s)−1(g(.,12c∗M1dsΩ))‖∞,M1<M0,12c∗M1≥‖d−sΩ((−Δ)s)−1(g(.,M0))‖L∞(Ω). | (2.6) |
Indeed, by g4), limσ→∞‖(σdsΩ)−1((−Δ)s)−1(dsΩg(.,σdsΩ))‖∞=0 and so the first one of the above inequalities hold for M1 large enough. Fix such a M1. Since, from g4), limσ→∞‖d−sΩ((−Δ)s)−1(g(.,σ))‖L∞(Ω)=0, the remaining inequalities of (2.6) hold for M0 large enough.
Lemma 2.4. Let ε>0 and let z∗, τ1 and c∗ be as in Remark 2.2. Let M0 and M1 be as in Remark 2.3. Let z:=M1z∗ and let w0,ε:Rn→R be the constant function w0,ε=M0. Then there exist sequences {wj,ε}j∈N and {ζj,ε}j∈N in Xs0(Ω)∩L∞(Ω) such that, for all j∈N:
i) wj−1,ε≥wj,ε≥0 in Rn,
ii) wj,ε≥12c∗M1dsΩ in Ω,
iii) wj,ε is a weak solution of the problem
{(−Δ)swj,ε=−g(.,wj−1,ε+ε)+τ1M1hinΩ,wj,ε=0inRn∖Ω. | (2.7) |
iv) wj,ε=z−ζj,ε in Rn and ζj,ε is a weak solution of the problem
{(−Δ)sζj,ε=g(.,wj−1,ε+ε)inΩ,ζj,ε=0inRn∖Ω. | (2.8) |
v) ‖wj,ε‖Xs0(Ω)≤c for some positive constant c independent of j and ε.
Proof. The sequences {wj,ε}j∈N and {ζj,ε}j∈N with the properties i)−v) will be constructed inductively. Let ζ1,ε∈X10(Ω) be the solution of the problem
{(−Δ)sζ1,ε=g(.,w0,ε+ε) in Ωζ1,ε=0 in Rn∖Ω |
(thus iv) holds for j=1). From g1) and g2) we have 0≤g(.,w0,ε+ε)≤g(.,ε)∈L∞(Ω). Thus g(.,w0,ε+ε)∈L∞(Ω). Then, by Remark 2.1 iii), ζ1,ε∈C(Rn). Therefore, since supp(ζ1,ε)⊂¯Ω, we have ζ1,ε∈L∞(Ω). By g1), g(.,M0)∈L∞(Ω) and so g(.,M0)∈(X10(Ω))′. Let u0:=((−Δ)s)−1(g(.,M0)). Then, by g1) and g3), d−sΩu0∈L∞(Ω). We have, in weak sense,
{(−Δ)s(ζ1,ε−u0)=g(.,w0,ε+ε)−g(.,M0)≤0 in Ωζ1,ε−u0=0 in Rn∖Ω. |
Then, by the maximum principle of Remark 2.1 i),
0≤ζ1,ε≤u0≤‖d−sΩu0‖L∞(Ω)dsΩ in Ω. | (2.9) |
Let z:=M1z∗. By Remark 2.2, z∈Hs(Rn)∩C(¯Ω) and
z≥c∗M1dsΩ in Ω. | (2.10) |
Also, z≤M1 in Ω, and z is a weak solution of the problem
{(−Δ)sz=τ1M1h in Ω,z=0 in Rn∖Ω. |
Let w1,ε:=z−ζ1,ε. Then w1,ε∈Hs(Rn) and w1,ε=0 in Rn∖Ω. Thus w1,ε∈Xs0(Ω). Also w1,ε∈L∞(Ω). Since ζ1,ε≥0 in Ω, we have
w0,ε−w1,ε=M0−z+ζ1,ε≥M0−z≥M0−M1>0 in Ω. |
Then w1,ε≤w0,ε in Ω. Thus i) holds for j=1. Now, in weak sense,
{(−Δ)sw1,ε=(−Δ)s(z−ζ1,ε)=τ1M1h−(−Δ)s(ζ1,ε)=−g(.,w0,ε+ε)+τ1M1h in Ω,w1,ε=0 in Rn∖Ω, |
and so iii) holds for j=1. Also, from (2.9), (2.10), and taking into account that (2.6),
w1,ε=z−ζ1,ε≥c∗M1dsΩ−‖d−sΩ((−Δ)s)−1(g(.,M0))‖L∞(Ω)dsΩ≥12c∗M1dsΩ in Ω. |
and then w1,ε≥12c∗M1dsΩ in Ω. Thus ii) holds for j=1.
Suppose constructed, for k≥1, functions w1,ε,..., wk,ε and ζ1,ε,...,ζk,ε, belonging to Xs0(Ω)∩L∞(Ω), and with the properties i)-iv). Let ζk+1,ε∈Xs0(Ω) be the solution of the problem
{(−Δ)sζk+1,ε=g(.,wk,ε+ε) in Ω,ζk+1,ε=0 on Rn∖Ω. | (2.11) |
(and so iv) holds for j=k+1) and let wk+1,ε:=z−ζk+1,ε. Then wk+1,ε∈Hs(Rn) and wk+1,ε=0 in Rn∖Ω. Thus wk+1,ε∈Xs0(Ω). Also,
wk,ε−wk+1,ε=ζk+1,ε−ζk,ε in Rn | (2.12) |
and
{(−Δ)s(ζk+1,ε−ζk,ε)=g(.,wk,ε+ε)−g(.,wk−1,ε+ε)≥0 in Ω,ζk+1,ε−ζk,ε=0 in Rn∖Ω, |
the last inequality because, by g1), s→g(.,s) is nonincreasing and (by our inductive hypothesis) wk,ε≤wk−1,ε in Ω. Then, by the maximum principle, ζk+1,ε−ζk,ε≥0 in Rn. Therefore, by (2.12), wk,ε≥wk+1,ε in Rn,and then i) holds for j=k+1. Also,
{(−Δ)swk+1,ε=(−Δ)sz−(−Δ)sζk+1,ε=−g(.,wk,ε+ε)+τ1M1h in Ω,wk+1,ε=0 in Rn∖Ω. |
Then iii) holds for j=k+1. By g4), g(.,12c∗M1dsΩ)∈(Xs0(Ω))′. Let u1:=((−Δ)s)−1(g(.,12c∗M1dsΩ))∈Xs0(Ω). By the inductive hypothesis we have wk,ε≥12c∗M1dsΩ in Ω. Now,
{(−Δ)s(ζk+1,ε−u1)=g(.,wk,ε+ε)−g(.,12c∗M1dsΩ)≤0 in Ω,ζk+1,ε−u1=0 on Rn∖Ω, |
then the comparison principle gives ζk+1,ε≤u1. Thus, in Ω,
wk+1,ε=z−ζk+1,ε≥c∗M1dsΩ−u1=c∗M1dsΩ−((−Δ)s)−1(g(.,12c∗M1dsΩ))≥c∗M1dsΩ−‖d−sΩ((−Δ)s)−1(g(.,12c∗M1dsΩ))‖∞dsΩ≥12c∗M1dsΩ, |
the last inequality by (2.6). Thus ii) holds for j=k+1. This complete the inductive construction of the sequences {wj,ε}j∈N and {ζj,ε}j∈N with the properties i)−iv).
To see v), we take ζj,ε as a test function in (2.8). Using ii), the Hölder inequality, the Poincaré inequality of Remark 2.1 iv), we get, for any j∈N,
‖ζj,ε‖2Xs0(Ω)=∫Ωg(.,wj−1,ε+ε)ζj,ε≤∫Ωg(.,12c∗M1dsΩ)ζj,ε=∫ΩdsΩg(.,12c∗M1dsΩ)d−sΩζj,ε≤‖dsΩg(.,12c∗M1dsΩ)‖2‖d−sΩζj,ε‖2≤c‖ζj,ε‖Xs0(Ω). |
where c is a positive constant c independent of j and ε, and where, in the last inequality, we have used g5). Then ‖ζj,ε‖Xs0(Ω) has an upper bound independent of j and ε. Since wj,ε=z−ζj,ε, the same assertion holds for wj,ε.
Lemma 2.5. Let ε>0 and let τ1 and c∗ be as in Remark 2.2. Let M0 and M1 be as in Remark 2.3 and let {wj,ε}j∈N and {ζj,ε}j∈N be as in Lemma 2.4. Let wε:=limj→∞wj,ε and let ζε:=limj→∞ζj,ε. Then
i) wε and ζε belong to Hs(Rn)∩C(¯Ω),
ii) 12c∗M1dsΩ≤wε≤M0 in Ω, and there exists a positive constant c independent of ε such that wε≤cdsΩ in Ω.
iii) wε satisfies, in weak sense,
{−Δwε=−g(.,wε+ε)+τ1M1hinΩ,wε=0inRn∖Ω. | (2.13) |
iv) ζε satisfies, in weak sense,
{(−Δ)sζε=g(.,wε+ε)inΩ,ζε=0inRn∖Ω. | (2.14) |
Proof. Let z∗ be as in Remark 2.2, and let z:=M1z∗. Let M0 and M1 be as in Remark 2.3. By Lemma 2.4, {wj,ε}j∈N is a nonincreasing sequence of nonnegative functions in Rn, and so there exists wε=limj→∞wj,ε. Since ζj,ε=z−wj−1,ε, there exists also ζε=limj→∞ζj,ε. Again by Lemma 2.4 we have, for any j∈N, 0≤wj,ε=z−ζj,ε≤z∈L∞(Ω). Thus, by the Lebesgue dominated convergence theorem,
{wj,ε}j∈N converges to wε in Lp(Ω) for any p∈[1,∞), | (2.15) |
and so {g(.,wj,ε+ε)}j∈N converges to g(.,wε+ε) in Lp(Ω) for any p∈[1,∞). We claim that
ζε∈Xs0(Ω) and {ζj,ε}j∈N converges in Xs0(Ω) to ζε. | (2.16) |
Indeed, for j, k∈N, from (2.8),
{(−Δ)s(ζj,ε−ζk,ε)=g(.,wj−1,ε+ε)−g(.,wk−1,ε+ε) in Ω,ζj,ε−ζk,ε=0 in Rn∖Ω. | (2.17) |
We take ζj,ε−ζk,ε as a test function in (2.17) to obtain
‖ζj,ε−ζk,ε‖2Xs0(Ω)=∫Ω(ζj,ε−ζk,ε)(g(.,wj−1,ε+ε)−g(.,wk−1,ε+ε))≤‖ζj,ε−ζk,ε‖2∗s‖g(.,wj−1,ε+ε)−g(.,wk−1,ε+ε)‖(2∗s)′ |
where 2∗s:=2nn−2s. Then
‖ζj,ε−ζk,ε‖Xs0(Ω)≤c‖g(.,wj−1,ε+ε)−g(.,wk−1,ε+ε)‖(2∗s)′. |
where c is a constant independent of j and k. Since {g(.,wj−1,ε+ε)}j∈N converges to g(.,wε+ε) in L(2∗s)′(Ω), we get
limj,k→∞‖ζj,ε−ζk,ε‖Xs0(Ω)=0, |
and thus {ζj,ε}j∈N converges in Xs0(Ω). Since {ζj,ε}j∈N converges to ζε in pointwise sense, (2.16) follows. Also, wj,ε=z−ζj,ε, and then {wj,ε}j∈N converges to wε,ρ in Xs0(Ω). Thus
wε∈Xs0(Ω) and {wj,ε}j∈N converges in Xs0(Ω) to wε. | (2.18) |
To prove (2.14) observe that, from (2.8), we have, for any φ∈Xs0(Ω) and j∈N,
∫Rn×Rn(ζε,j(x)−ζε,j(y))(φ(x)−φ(y))|x−y|n+2sdxdy=∫Ωg(.,wε,j−1+ε)φ. | (2.19) |
Taking limj→∞ in (2.19) and using (2.16) and (2.15), we obtain (2.14). From (2.14) and since, by g1) and g2), g(.,wε+ε)∈L∞(Ω), Remark 2.1 iii) gives that, in addition, ζε∈C(¯Ω) (and so, since wε=z−ζε, then also wε∈C(¯Ω)).
Let us see that (2.13) holds. Let φ∈Xs0(Ω). From (2.7), we have, for any j∈N,
∫Rn×Rn(wj,ε(x)−wj,ε(y))(φ(x)−φ(y))|x−y|n+2sdxdy=∫Ω∖¯Bρ(y)(−g(.,wj−1,ε+ε)+τ1M1h)φ. | (2.20) |
Since φ∈Xs0(Ω) and {wj,ε}j∈N converges to wε in Xs0(Ω) we have
limj→∞∫Rn×Rn(wj,ε(x)−wj,ε(y))(φ(x)−φ(y))|x−y|n+2sdxdy=∫Rn×Rn(wε(x)−wε(y))(φ(x)−φ(y))|x−y|n+2sdxdy. | (2.21) |
Also, wε(x)=limj→∞wj,ε(x) for any x∈Ω, and
|g(.,wj−1,ε+ε)φ|≤g(.,ε)|φ|∈L1(Ω), |
and clearly |τ1M1hφ|∈L1(Ω). Then, by the Lebesgue dominated convergence theorem,
limj→∞∫Ω(−g(.,wj−1,ε+ε)+τ1M1h)φ=∫Ω(−g(.,wε+ε)+τ1M1h)φ. | (2.22) |
Now (2.13) follows from (2.20), (2.21) and (2.22). Finally, by Lemma 2.4 we have, for all j∈N, 12c∗M1dsΩ≤wj,ε in Ω and so the same inequality hold with wj,ε replaced by wε. Also, since wj,ε≤z0 in Ω we have wj,ε≤cdsΩ with c independent of j and ε.
Remark 2.6. Let G:Ω×Ω→R∪{∞} be the Green function for (−Δ)s in Ω, with homogeneous Dirichlet boundary condition on Rn∖Ω. Then, for f∈C(¯Ω) the solution u of problem (1.4) is given by u(x)=∫ΩG(x,y)f(y)dy for x∈Ω and by u(x)=0 for x∈Rn∖Ω. Let us recall the following estimates from [4]:
ⅰ) (see [4], Theorems 1.1 and 1.2) There exist positive constants c and c′, depending only on Ω and s, such that for x;y∈Ω,
G(x,y)≤cdΩ(x)s|x−y|n−s, | (2.23) |
G(x,y)≤cdΩ(x)sdΩ(y)s1|x−y|n−2s, | (2.24) |
G(x,y)≤cdΩ(x)sdΩ(y)s|x−y|n | (2.25) |
G(x,y)≥c′1|x−y|n−2s if |x−y|≤max{dΩ(x)2,dΩ(y)2} | (2.26) |
G(x,y)≥c′dΩ(x)sdΩ(y)s|x−y|n if |x−y|>max{dΩ(x)2,dΩ(y)2} | (2.27) |
ⅱ) From ⅰ) it follows that there exists a positive constant c′′, depending only on Ω and s, such that for x;y∈Ω,
G(x;y)≥c′′dΩ(x)sdΩ(y)s. | (2.28) |
Indeed:
If |x−y|>max{dΩ(x)2,dΩ(y)2} then, from (2.27), G(x;y)≥c′dΩ(x)sdΩ(y)s|x−y|n and so G(x;y)≥c′dΩ(x)sdΩ(y)s(diam(Ω))n.
If |x−y|≤max{dΩ(x)2,dΩ(y)2} then either |x−y|≤dΩ(x)2 or |x−y|≤dΩ(y)2. If |x−y|≤dΩ(x)2 consider z∈∂Ω such that dΩ(y)=|z−y|. then dΩ(y)=|z−y|≥|x−z|−|x−y|≥dΩ(x)−|x−y|≥12dΩ(x). Then dΩ(y)≥12dΩ(x)≥|x−y|. Thus, since also |x−y|≤dΩ(x)2, we have |x−y|≤1√2(dΩ(x)dΩ(y))12, and so, from (2.26), G(x,y)≥c′1|x−y|n−2s≥c′1(1√2(dΩ(x)dΩ(y))12)n−2s=2n2−sc′dsΩ(x)dsΩ(y)(dΩ(x)dΩ(y))n2≥2n2−sc′(diam(Ω))ndsΩ(x)dsΩ(y). If |x−y|≤dΩ(y)2, by interchanging the roles of x and y in the above argument, the same conclusion is reached.
ⅲ) If 0<β<s, then
G(x,y)≤cdΩ(x)sdΩ(y)β|x−y|n−s+β. | (2.29) |
Indeed, If dΩ(y)≥|x−y| then, from (2.23),
G(x,y)≤cdΩ(x)s|x−y|n−s=cdΩ(x)sdΩ(y)β|x−y|n−sdΩ(y)β≤cdΩ(x)sdΩ(y)β|x−y|n−s+β, |
and if dΩ(y)≤|x−y| then, from (2.27),
G(x,y)≤cdΩ(x)sdΩ(y)s|x−y|n=cdΩ(x)sdΩ(y)βdΩ(y)s−β|x−y|n−s+β|x−y|s−β≤cdΩ(x)sdΩ(y)β|x−y|n−s+β, |
ⅳ) If f\in C\left(\overline{\Omega}\right) then the unique solution u\in X_{0}^{s}\left(\Omega\right) of problem (1.4) is given by u\left(x\right) : = \int_{\Omega}G\left(x, y\right) f\left(y\right) dy for x\in\Omega, and u\left(x\right) : = 0 for x\in\mathbb{R}^{n}\setminus\Omega.
Lemma 2.7. Let a\in L^{\infty}\left(\Omega\right) and let \beta\in\left[0, s\right). Then ad_{\Omega}^{-\beta}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime} and the weak solution u\in X_{0}^{s}\left(\Omega\right) of the problem
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u = ad_{\Omega}^{-\beta}\; in \;\Omega, \\ u = 0\; in \;\mathbb{R}^{n}\setminus\Omega \end{array} \right. | (2.30) |
satisfies d_{\Omega}^{-s}u\in L^{\infty}\left(\Omega\right).
Proof. Let \varphi\in X_{0}^{s}\left(\Omega\right). By the Hölder and Hardy inequalities we have \int_{\Omega}\left\vert ad_{\Omega}^{-\beta} \varphi\right\vert = \int_{\Omega}\left\vert ad_{\Omega}^{s-\beta}d_{\Omega }^{-s}\varphi\right\vert \leq\left\Vert a\right\Vert _{\infty}\left\Vert d_{\Omega}^{s-\beta}\right\Vert _{2}\left\Vert d_{\Omega}^{-s}\varphi \right\Vert _{2}\leq c\left\Vert \varphi\right\Vert _{X_{0}^{s}\left(\Omega\right) } with c a positive constant independent of \varphi. Thus ad_{\Omega}^{-\beta}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}. Let u\in X_{0}^{s}\left(\Omega\right) be the unique weak solution (given by the Riesz Theorem) of problem (2.30) and consider a decreasing sequence \left\{ \varepsilon_{j}\right\} _{j\in N} in \left(0, 1\right) such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0. For j\in\mathbb{N}, let u_{\varepsilon_{j}}\in X_{0}^{s}\left(\Omega\right) be the weak solution of the problem
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{\varepsilon_{j}} = a\left( d_{\Omega }+\varepsilon_{j}\right) ^{-\beta}\text{ in }\Omega, \\ u_{\varepsilon_{j}} = 0\text{ in }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. | (2.31) |
Thus u_{\varepsilon_{j}} = \int_{\Omega}G\left(., y\right) a\left(y\right) \left(d_{\Omega}\left(y\right) +\varepsilon_{j}\right) ^{-\beta}dy in \Omega, where G is the Green function for \left(-\Delta\right) ^{s} in \Omega, with homogeneous Dirichlet boundary condition on \mathbb{R} ^{n}\setminus\Omega. Since \beta < s we have \int_{\Omega}\frac {1}{\left\vert x-y\right\vert ^{n-s+\beta}}dy < \infty. Thus, recalling (2.29), there exists a positive constant c such that, for any j\in\mathbb{N} and \left(x, y\right) \in\Omega\times\Omega,
\begin{align*} 0 & \leq G\left( x, y\right) a\left( y\right) \left( d_{\Omega}\left( y\right) +\varepsilon_{j}\right) ^{-\beta}\leq c\frac{d_{\Omega}^{s}\left( x\right) d_{\Omega}^{\beta}\left( y\right) }{\left\vert x-y\right\vert ^{n-s+\beta}}\left( d_{\Omega}\left( y\right) +\varepsilon_{j}\right) ^{-\beta}\\ & \leq cd_{\Omega}^{s}\left( x\right) \frac{1}{\left\vert x-y\right\vert ^{n-s+\beta}}\in L^{1}\left( \Omega, dy\right) . \end{align*} |
Since also \lim_{j\rightarrow\infty}G\left(x, y\right) a\left(y\right) \left(d_{\Omega}\left(y\right) +\varepsilon_{j}\right) ^{-\beta } = G\left(x, y\right) a\left(y\right) d_{\Omega}^{-\beta}\left(y\right) for a.e. y\in\Omega, by the Lebesgue dominated convergence theorem, \left\{ u_{\varepsilon_{j}}\left(x\right) \right\} _{j\in\mathbb{N}} converges to \int_{\Omega}G\left(x, y\right) a\left(y\right) d_{\Omega }^{-\beta}\left(y\right) dy for any x\in\Omega. Let u\left(x\right) : = \lim_{j\rightarrow\infty}u_{\varepsilon_{j}}\left(x\right). Thus u\left(x\right) = \int_{\Omega}G\left(x, y\right) a\left(y\right) d_{\Omega}^{-\beta}\left(y\right) dy. Again from (2.29), u\leq cd_{\Omega}^{s} a.e. in \Omega, with c constant c independent of x. Now we take u_{\varepsilon_{j}} as a test function in (2.31) to obtain that
\begin{align*} \int_{\Omega\times\Omega}\frac{\left( u_{\varepsilon_{j}}\left( x\right) -u_{\varepsilon_{j}}\left( y\right) \right) ^{2}}{\left\vert x-y\right\vert ^{n+2s}} & = \int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\left( u_{\varepsilon_{j}}\left( x\right) -u_{\varepsilon_{j}}\left( y\right) \right) ^{2}}{\left\vert x-y\right\vert ^{n+2s}}\\ & = \int_{\Omega}u_{\varepsilon_{j}}\left( y\right) \left( d_{\Omega }\left( y\right) +\varepsilon_{j}\right) ^{-\beta}dy\\ & \leq c\int_{\Omega}d_{\Omega}^{s}\left( y\right) \left( d_{\Omega }\left( y\right) +\varepsilon\right) j^{-\beta}dy\leq c^{\prime} \int_{\Omega}d_{\Omega}^{s-\beta}\left( y\right) dy = c^{\prime\prime}, \end{align*} |
with c and c^{\prime} constants independent of j. For j\in\mathbb{N}, let U_{\varepsilon_{j}} and U be the functions, defined on \mathbb{R} ^{n}\times\mathbb{R}^{n}, by
U_{\varepsilon_{j}}\left( x, y\right) : = u_{\varepsilon_{j}}\left( x\right) -u_{\varepsilon_{j}}\left( y\right) , \text{ }U\left( x, y\right) : = u\left( x\right) -u\left( y\right) . |
Then \left\{ U_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} is bounded in \mathcal{H} = L^{2}\left(\mathbb{R}^{n}\times\mathbb{R}^{n}, \frac {1}{\left\vert x-y\right\vert ^{n+2s}}dxdy\right). Thus, after pass to a subsequence if necessary, we can assume that \left\{ U_{\varepsilon_{j} }\right\} _{j\in\mathbb{N}} is weakly convergent in \mathcal{H} to some V\in\mathcal{H}. Since \left\{ U_{\varepsilon_{j}}\right\} _{j\in \mathbb{N}} converges pointwise to U on \mathbb{R}^{n}\times\mathbb{R} ^{n}, we conclude that U\in\mathcal{H} and that \left\{ U_{\varepsilon _{j}}\right\} _{j\in\mathbb{N}} converges weakly to U in \mathcal{H}. Thus u\in X_{0}^{s}\left(\Omega\right) and, for any \varphi\in X_{0}^{s}\left(\Omega\right),
\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\left( u\left( x\right) -u\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy\\ = \lim\limits_{j\rightarrow\infty}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}} \frac{\left( u_{\varepsilon_{j}}\left( x\right) -u_{\varepsilon_{j}}\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy\\ = \lim\limits_{j\rightarrow\infty}\int_{\Omega}a\left( d_{\Omega}+\varepsilon _{j}\right) ^{-\beta}\varphi = \int_{\Omega}ad_{\Omega}^{-\beta}\varphi, |
Then u is the weak solution of (2.30). Finally, since for all j, u_{\varepsilon_{j}}\leq c^{\prime}d_{\Omega}^{s} a.e. in \Omega, we have u\leq c^{\prime}d_{\Omega}^{s} a.e. in \Omega.
Lemma 2.8. Let \lambda>0 and let \varepsilon \geq0. Suppose that \left\{ u_{j}\right\} _{j\in\mathbb{N}}\subset X_{0}^{s}\left(\Omega\right) is a nonincreasing sequence with the following properties i) and ii):
i) There exist positive constants c and c^{\prime} such that cd_{\Omega}^{s}\leq u_{j}\leq c^{\prime}d_{\Omega }^{s} a.e. in \Omega for any j\in\mathbb{N}.
ii) for any j\in\mathbb{N}, u_{j} is a weak solution of the problem
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{j} = -g\left( ., u_{j}+\varepsilon\right) +\lambda h\; in\;\Omega, \\ u_{j} = 0\; in \;\mathbb{R}^{n}\setminus\Omega, \\ u_{j}>.0\; in \;\Omega \end{array} \right. | (2.32) |
Then \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges in X_{0} ^{s}\left(\Omega\right) to a weak solution u of the problem
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u = -g\left( ., u+\varepsilon\right) +\lambda h\; in \;\Omega, \\ u_{j} = 0\; in \;\mathbb{R}^{n}\setminus\Omega, \\ u>0\; in \;\Omega, \end{array} \right. | (2.33) |
which satisfies cd_{\Omega}^{s}\leq u\leq c^{\prime}d_{\Omega}^{s} a.e. in \Omega. Moreover, the same conclusions holds if, instead of ii), we assume the following ii'):
ii') for any j\geq2, u_{j} is a weak solution of the problem
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{j} = -g\left( ., u_{j-1}+\varepsilon\right) +\lambda h\; in\;\Omega, \\ u_{j} = 0\; in \;\mathbb{R}^{n}\setminus\Omega, \\ u_{j}>0\; in \;\Omega. \end{array} \right. |
Proof. Assume i) and ii). For x\in\mathbb{R}^{n}, let u\left(x\right) : = \lim_{j\rightarrow\infty}u_{j}\left(x\right). For j, k\in\mathbb{N} we have, in weak sense,
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\left( u_{j}-u_{k}\right) = g\left( ., u_{k}+\varepsilon\right) -g\left( u_{j}+\varepsilon\right) \text{ in }\Omega, \\ u_{j}-u_{k} = 0\text{ in }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. | (2.34) |
We take u_{j}-u_{k} as a test function in (2.34) to get
\begin{align} \left\Vert u_{j}-u_{k}\right\Vert _{X_{0}^{s}\left( \Omega\right) }^{2} & = \int_{\Omega}\left( g\left( ., u_{k}+\varepsilon\right) -g\left( ., u_{j}+\varepsilon\right) \right) \left( u_{j}-u_{k}\right) \label{inecuacion domingo dos new}\\ & = \int_{\Omega}d_{\Omega}^{s}\left( g\left( ., u_{k}+\varepsilon\right) -g\left( ., u_{j}+\varepsilon\right) \right) d_{\Omega}^{-s}\left( u_{j}-u_{k}\right) \nonumber\\ & \leq\left\Vert d_{\Omega}^{-s}\left( \overline{u}_{j}-\overline{u} _{k}\right) \right\Vert _{2}\left\Vert d_{\Omega}^{s}\left( g\left( ., \overline{u}_{k}+\varepsilon\right) -g\left( ., \overline{u}_{j} +\varepsilon\right) \right) \right\Vert _{2}.\nonumber \end{align} | (2.35) |
By the Hardy inequality, \left\Vert d_{\Omega}^{-s}\left(u_{j} -u_{k}\right) \right\Vert _{2}\leq c^{\prime\prime}\left\Vert u_{j} -u_{k}\right\Vert _{X_{0}^{s}\left(\Omega\right) } where c^{\prime\prime } is a constant independent of j and k. Thus, from (2.35),
\left\Vert u_{j}-u_{k}\right\Vert _{X_{0}^{s}\left( \Omega\right) }\leq c^{\prime\prime}\left\Vert d_{\Omega}^{s}\left( g\left( ., u_{k} +\varepsilon\right) -g\left( ., u_{j}+\varepsilon\right) \right) \right\Vert _{2}. | (2.36) |
Now, \lim_{j, k\rightarrow\infty}\left\vert d_{\Omega}^{s}\left(g\left(., u_{k}+\varepsilon\right) -g\left(., u_{j}+\varepsilon\right) \right) \right\vert ^{2} = 0 a.e. in \Omega. Also, since u_{l}\geq cd_{\Omega} ^{s} a.e. in \Omega for any l\in\mathbb{N}, and taking into account g5) and g2),
\left\vert d_{\Omega}^{s}\left( g\left( ., u_{k}+\varepsilon\right) -g\left( ., u_{j}+\varepsilon\right) \right) \right\vert ^{2}\leq c^{\prime }\left( d_{\Omega}^{s}g\left( ., cd_{\Omega}^{s}\right) \right) ^{2}\in L^{1}\left( \Omega\right) , |
where c^{\prime} is a constant independent of j and k. Then, by the Lebesgue dominated convergence theorem \lim_{j, k\rightarrow\infty}\left\Vert d_{\Omega}^{s}\left(g\left(., u_{k}+\varepsilon\right) -g\left(., u_{j}+\varepsilon\right) \right) \right\Vert _{2} = 0. Therefore, from (2.36), \lim_{j, k\rightarrow\infty}\left\Vert u_{j}-u_{k}\right\Vert _{X_{0}^{s}\left(\Omega\right) } = 0 and so \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges in X_{0}^{s}\left(\Omega\right) to some u^{\ast}\in X_{0}^{s}\left(\Omega\right). Then, by the Poincaré inequality of Remark 2.1 iv), \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges to u^{\ast} in L^{2_{s}^{\ast}}\left(\Omega\right), and thus there exists a subsequence \left\{ u_{j_{k}}\right\} _{k\in\mathbb{N}} that converges to u^{\ast} a.e. in \Omega. Since \left\{ u_{j_{k}}\right\} _{k\in\mathbb{N}} converges pointwise to u_{\varepsilon}, we conclude that u^{\ast} = u. Then \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges to u_{\varepsilon} in X_{0}^{s}\left(\Omega\right). Now, for \varphi\in X_{0}^{s}\left(\Omega\right) and j\in\mathbb{N},
\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\left( u_{j}\left( x\right) -u_{j}\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s} }dxdy = \int_{\Omega}\left( -g\left( ., u_{j}+\varepsilon\right) +\lambda h\right) \varphi. | (2.37) |
Since \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges to u in X_{0}^{s}\left(\Omega\right), we have
\lim\limits_{j\rightarrow\infty}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}} \frac{\left( u_{j}\left( x\right) -u_{j}\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy\label{ecuacion domingo cinco}\\ = \int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\left( u\left( x\right) -u\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy.\nonumber | (2.38) |
On the other hand, \left\vert \left(-g\left(., u_{j}+\varepsilon\right) +\lambda h\right) \varphi\right\vert \leq\left(g\left(., cd_{\Omega }\right) +\lambda\left\Vert h\right\Vert _{\infty}\right) \left\vert \varphi\right\vert \in L^{1}\left(\Omega\right) (with c as in i)). Also, \left\{ \left(-g\left(u_{j}+\varepsilon\right) +\lambda h\right) \varphi\right\} _{j\in\mathbb{N}} converges to \left(-g\left(u+\varepsilon\right) +\lambda h\right) \varphi a.e. in \Omega. Then, by the Lebesgue dominated convergence theorem,
\lim\limits_{j\rightarrow\infty}\int_{\Omega}\left( -g\left( ., u_{j}+\varepsilon \right) +\lambda h\right) \varphi = \int_{\Omega}\left( -g\left( ., u+\varepsilon\right) +\lambda h\right) \varphi. | (2.39) |
From (2.37), (2.38) and (2.39) we get, for any \varphi\in X_{0}^{s}\left(\Omega\right),
\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\left( u\left( x\right) -u\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy = \int_{\Omega }\left( -g\left( ., u+\varepsilon\right) +\lambda h\right) \varphi. |
and so u is a weak solution of problem (2.33) which clearly satisfies cd_{\Omega}^{s}\leq u\leq c^{\prime}d_{\Omega}^{s} a.e. in \Omega. If instead of ii) we assume ii'), the proof is the same. Only replace, for j\geq2, k\geq2 and in each appearance, g\left(., u_{j}\right) and g\left(., u_{k}\right) by g\left(., u_{j-1}\right) and g\left(., u_{k-1}\right) respectively.
Lemma 2.9. Let \lambda>0, and let \overline{u} be the weak solution of
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\overline{u} = \lambda h\; in \;\Omega, \\ \overline{u} = 0\; in \;\mathbb{R}^{n}\setminus\Omega. \end{array} \right. | (2.40) |
Assume that, for each \varepsilon>0, we have a function \widetilde {v}_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) satisfying, in weak sense,
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\widetilde{v}_{\varepsilon}\leq-g\left( ., \widetilde{v}_{\varepsilon}+\varepsilon\right) +\lambda h\;in \; \Omega, \\ \widetilde{v}_{\varepsilon} = 0\; in \;\mathbb{R}^{n}\setminus\Omega. \end{array} \right. | (2.41) |
and such that \widetilde{v}_{\varepsilon}\geq cd_{\Omega}^{s} a.e. in \Omega, where c is a positive constant independent of \varepsilon. Then for any \varepsilon>0 there exists a sequence \left\{ u_{j}\right\} _{j\in\mathbb{N}}\subset X_{0}^{s}\left(\Omega\right) such that:
i) u_{1} = \overline{u} and u_{j}\leq u_{j-1} for any j\geq2.
ii) \widetilde{v}_{\varepsilon}\leq u_{j}\leq\overline{u} for all j\in N.
iii) For any j\geq2, u_{j} satisfies, in weak sense,
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{j} = -g\left( ., u_{j-1}+\varepsilon\right) +\lambda h\; in \;\Omega, \\ u_{j} = 0\; in \;\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
iv) There exist positive constants c and c^{\prime}independent of \varepsilon and j such that, for all j, cd_{\Omega}^{s}\leq u_{j}\leq c^{\prime}d_{\Omega}^{s} a.e. in \Omega.
Proof. By Remark 2.1 iii), there exists a positive constant c^{\prime} such that \overline{u}\leq c^{\prime}d_{\Omega}^{s} in \Omega. We construct inductively a sequence \left\{ u_{j}\right\} _{j\in\mathbb{N}} satisfying the assertions i)-iii) of the lemma: Let u_{1}: = \overline{u}. Thus, in weak sense, \left(-\Delta\right) ^{s}u_{1} = \lambda h\geq-g\left(., \widetilde{v}_{\varepsilon}+\varepsilon \right) +\lambda h in \Omega. Thus \left(-\Delta\right) ^{s}\left(u_{1}-\widetilde{v}_{\varepsilon}\right) \geq0 in \Omega. Then, by the maximum principle in Remark 2.1 i), u_{1} \geq\widetilde{v}_{\varepsilon} in \Omega, and so u_{1}\geq cd_{\Omega }^{s} in \Omega. Then, for some positive constant c^{\prime\prime}, \left\vert -g\left(., u_{1}+\varepsilon\right) +\lambda h\right\vert \leq c^{\prime\prime}\left(1+g\left(., cd_{\Omega}^{s}\right) \right) in \Omega and, by g1), g\left(., cd_{\Omega}^{s}\right) \in L^{\infty}\left(\Omega\right). Thus there exists a weak solution u_{2}\in X_{0}^{s}\left(\Omega\right) to the problem
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{2} = -g\left( ., u_{1}+\varepsilon\right) +\lambda h\text{ in }\Omega, \\ u_{2} = 0\text{ in }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
Since, in weak sense, \left(-\Delta\right) ^{s}u_{2}\leq\lambda h = \left(-\Delta\right) ^{s}u_{1} in \Omega, the maximum principle in Remark 2.1 i) gives u_{2}\leq u_{1} in \Omega. Since u_{1}\geq\widetilde{v}_{\varepsilon} in \Omega we have, in weak sense, \left(-\Delta\right) ^{s}u_{2} = -g\left(., u_{1}+\varepsilon \right) +\lambda h\geq-g\left(., \widetilde{v}_{\varepsilon}+\varepsilon \right) +\lambda h in \Omega. Also, \left(-\Delta\right) ^{s}\widetilde{v}_{\varepsilon}\leq-g\left(., \widetilde{v}_{\varepsilon }+\varepsilon\right) +\lambda h in \Omega and so, by the maximum principle, u_{2}\geq\widetilde{v}_{\varepsilon} in \Omega. Then i)-iii) hold for j = 1.
Supposed constructed u_{1}, ..., u_{k} such that i)-iii) hold for 1\leq j\leq k. Then, for some positive constant c^{\prime\prime\prime}, \left\vert -g\left(., u_{k}+\varepsilon\right) +\lambda h\right\vert \leq c^{\prime\prime}\left(1+g\left(., cd_{\Omega}^{s}\right) \right) in \Omega and so, as before, there exists a weak solution u_{k+1}\in X_{0}^{s}\left(\Omega\right) to the problem
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{k+1} = -g\left( ., u_{k}+\varepsilon\right) +\lambda h\text{ in }\Omega, \\ u_{k+1} = 0\text{ in }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
By our inductive hypothesis, u_{k}\geq\widetilde{v}_{\varepsilon} in \Omega. Then, in weak sense, \left(-\Delta\right) ^{s}u_{k+1} = -g\left(., u_{k}+\varepsilon\right) +\lambda h\geq-g\left(., \widetilde {v}_{\varepsilon}+\varepsilon\right) +\lambda h\geq\left(-\Delta\right) ^{s}\widetilde{v}_{\varepsilon} in \Omega and thus, by the maximum principle, u_{k+1}\geq\widetilde{v}_{\varepsilon} in \Omega. If k = 2 we have u_{k}\leq u_{k-1} in \Omega. If k>2, by the inductive hypothesis we have, in weak sense, \left(-\Delta\right) ^{s}u_{k} = -g\left(., u_{k-1}+\varepsilon\right) +\lambda h\leq-g\left(., u_{k-2}+\varepsilon \right) +\lambda h in \Omega. Also, \left(-\Delta\right) ^{s} u_{k} = -g\left(., u_{k-1}+\varepsilon\right) +\lambda h in \Omega. Thus, by the maximum principle, u_{k+1}\leq u_{k} in \Omega. Again by the inductive hypothesis u_{k}\leq\overline{u} in \Omega and then, since u_{k+1}\leq u_{k} in \Omega, we get u_{k+1}\leq\overline{u} in \Omega.
Since for all j, v_{\varepsilon}\leq u_{j}\leq\overline{u} in \Omega, iv) follows from the facts that \overline{u}\leq c^{\prime}d_{\Omega}^{s} in \Omega, and that \widetilde{v}_{\varepsilon }\geq cd_{\Omega}^{s} in \Omega, with c and c^{\prime} positive constants independent of \varepsilon and j.
Lemma 2.10. Let \lambda>0. Assume that we have, for each \varepsilon>0, a function \widetilde{v}_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) satisfying, in weak sense, (2.41), and such that \widetilde{v}_{\varepsilon}\geq cd_{\Omega}^{s} a.e. in \Omega, with c a positive constant independent of \varepsilon. Then for any \varepsilon>0 there exists a weak solution u_{\varepsilon} of the problem
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{\varepsilon} = -g\left( ., u_{\varepsilon }+\varepsilon\right) +\lambda h\;in \;\Omega, \\ u_{\varepsilon} = 0\; in \;\mathbb{R}^{n}\setminus\Omega, \\ u_{\varepsilon}>0\;in \;\Omega. \end{array} \right. | (2.42) |
such that:
i) u_{\varepsilon}\geq\widetilde{v}_{\varepsilon} and there exist positive constants c and c^{\prime} independent of \varepsilon such that cd_{\Omega}^{s}\leq u_{\varepsilon}\leq c^{\prime }d_{\Omega}^{s} in \Omega,
ii) If \underline{u}_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) and \left(-\Delta\right) ^{s} \underline{u}_{\varepsilon}\leq-g\left(., \underline{u}_{\varepsilon }+\varepsilon\right) +\lambda h in \Omega, then \underline{u} _{\varepsilon}\leq u_{\varepsilon} in \Omega,
iii) If 0 < \varepsilon < \eta then u_{\varepsilon}\leq u_{\eta}.
Proof. Let \left\{ u_{j}\right\} _{j\in\mathbb{N}} be as given by Lemma 2.9. For x\in\Omega, let u_{\varepsilon}\left(x\right) : = \lim_{j\rightarrow\infty}u_{j}\left(x\right). By Lemma 2.8, \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges to u_{\varepsilon} in X_{0} ^{s}\left(\Omega\right) and u_{\varepsilon} is a weak solution to (2.42). From Lemma 2.9 iv) we have u_{\varepsilon}\geq\widetilde{v}_{\varepsilon} in \Omega and cd_{\Omega }^{s}\leq u_{\varepsilon}\leq c^{\prime}d_{\Omega}^{s} in \Omega, for some positive constants c and c^{\prime} independent of \varepsilon. Then i) holds. If \underline{u}_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) and \left(-\Delta\right) ^{s}\underline{u}_{\varepsilon }\leq-g\left(., \underline{u}_{\varepsilon}+\varepsilon\right) +\lambda h in \Omega, then \left(-\Delta\right) ^{s}\underline{u}_{\varepsilon} \leq\lambda h = \left(-\Delta\right) ^{s}u_{1} in \Omega, and so \underline{u}_{\varepsilon}\leq u_{1}. Thus \left(-\Delta\right) ^{s}\underline{u}_{\varepsilon}\leq-g\left(., \underline{u}_{\varepsilon }+\varepsilon\right) +\lambda h\leq-g\left(., u_{1}+\varepsilon\right) +\lambda h = \left(-\Delta\right) ^{s}u_{2}, then \underline{u} _{\varepsilon}\leq u_{2} and, iterating this procedure, we obtain that \underline{u}_{\varepsilon}\leq u_{j} for all j. Then \underline {u}_{\varepsilon}\leq u_{\varepsilon}. Thus ii) holds. Finally, iii) is immediate from ii).
Lemma 2.11. Let \varepsilon>0 and let \tau_{1} and M_{1} be as in Remarks 2.2, and 2.3 respectively. Let w_{\varepsilon} be as in Lemma 2.5. Then, for \lambda\geq\tau_{1} M_{1}, there exists a weak solution u_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) of problem (2.42) such that
i) u_{\varepsilon}\geq w_{\varepsilon} and there exist positive constants c and c^{\prime}, both independent of \varepsilon, such that cd_{\Omega }^{s}\leq u_{\varepsilon}\leq c^{\prime}d_{\Omega}^{s} in \Omega,
ii) If \underline{u}_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) and \left(-\Delta\right) ^{s}\underline{u}_{\varepsilon}\leq-g\left(., \underline{u}_{\varepsilon}+\varepsilon\right) +\lambda h in \Omega, then \underline{u}_{\varepsilon}\leq u_{\varepsilon} in \Omega,
iii) If 0 < \varepsilon_{1} < \varepsilon_{2} then u_{\varepsilon_{1}}\leq u_{\varepsilon_{2}}.
Proof. Let \lambda\geq\tau_{1}M_{1} and let w_{\varepsilon} be as in Lemma 2.5. We have, in weak sense,
\left\{ \begin{array} [c]{c} -\Delta w_{\varepsilon} = -g\left( ., w_{\varepsilon}+\varepsilon\right) +\tau_{1}M_{1}h\text{ in }\Omega, \\ w_{\varepsilon} = 0\text{ on }\mathbb{R}^{n}\setminus\Omega, \end{array} \right. |
Also, -g\left(., w_{\varepsilon}+\varepsilon\right) +\tau_{1}M_{1} h\leq-g\left(., w_{\varepsilon}+\varepsilon\right) +\lambda h in \Omega, and cd_{\Omega}^{s}\leq w_{\varepsilon}\leq c^{\prime}d_{\Omega}^{s} in \Omega, with c and c^{\prime} positive constants independent of \varepsilon. Then the lemma follows from Lemma 2.10.
Lemma 3.1. Let \lambda>0. If problem (1.5) has a weak solution in \mathcal{E}, then it has a weak solution u\in\mathcal{E} satisfying u\geq\psi a.e. in \Omega for any \psi \in\mathcal{E} such that -\Delta\psi\leq-g\left(., \psi\right) +\lambda h in \Omega.
Proof. Let u^{\ast}\in\mathcal{E} be a weak solution of (1.5), and let \overline{u} be as in (2.40). By the comparison principle u^{\ast}\leq \overline{u} in \Omega. We construct inductively a sequence \left\{ u_{j}\right\} _{j\in\mathbb{N}}\subset\mathcal{E} with the following properties: u_{1} = \overline{u} and
i) u^{\ast}\leq u_{j}\leq\overline{u} for all j\in\mathbb{N}
ii) g\left(., u_{j}\right) \in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime} for all j\in\mathbb{N}.
iii) u_{j}\leq u_{j-1} for all j\geq2.
iv) For all j\geq2
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{j} = -g\left( ., u_{j-1}\right) +\lambda h\text{ in }\Omega, \\ u_{j} = 0\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
To do it, define u_{1} = :\overline{u}. Thus u_{1}\in\mathcal{E}. By the comparison principle, u^{\ast}\leq\overline{u}, i.e., u^{\ast}\leq u_{1}. By Remark 2.1there exist positive constants c and c^{\prime} such that cd_{\Omega}^{s}\leq\overline{u}\leq c^{\prime }d_{\Omega}^{s} in \Omega. Thus \left\vert -g\left(., \overline {u}\right) +\lambda h\right\vert \leq g\left(., cd_{\Omega}^{s}\right) +\lambda\left\Vert h\right\Vert _{\infty} and so -g\left(., u_{1}\right) +\lambda h\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}. Thus i) and ii) hold for j = 1. Define u_{2} as the weak solution of
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{2} = -g\left( ., u_{1}\right) +\lambda h\text{ in }\Omega, \\ u_{2} = 0\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
Then, in weak sense,
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{2}\leq\left( -\Delta\right) ^{s}u_{1}\text{ in }\Omega, \\ u_{2} = 0 = u_{1}\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
and so u_{2}\leq u_{1} = \overline{u} a.e. in \Omega. Since u_{1}\geq u^{\ast}, we have, in weak sense,
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{2} = -g\left( ., u_{1}\right) +\lambda h\geq-g\left( ., u^{\ast}\right) +\lambda h = \left( -\Delta\right) ^{s}u^{\ast}\text{ in }\Omega, \\ u_{2} = 0 = u^{\ast}\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
and then u_{2}\geq u^{\ast} a.e. in \Omega. Thus u^{\ast}\leq u_{2} \leq\overline{u}. In particular this gives u_{2}\in\mathcal{E}. Let c^{\prime\prime}>0 such that u^{\ast}\geq c^{\prime\prime}d_{\Omega} in \Omega. Now, \left\vert -g\left(., u_{2}\right) +\lambda h\right\vert \leq g\left(., u_{2}\right) +\lambda h\leq g\left(., u^{\ast}\right) +\lambda h\leq g\left(., c^{\prime\prime}d_{\Omega}^{s}\right) +\lambda\left\Vert h\right\Vert _{\infty} and so -g\left(., u_{2}\right) +\lambda h\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}. Thus i)-iv) hold for j = 2. Suppose constructed, for 2\leq j\leq k, functions u_{j}\in\mathcal{E} with the properties i)-iv). Define u_{k+1} by
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{k+1} = -g\left( ., u_{k}\right) +\lambda h\text{ in }\Omega, \\ u_{k+1} = 0\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
Thus, by the comparison principle, u_{k+1}\leq\overline{u}. Also, by the inductive hypothesis, u_{k}\geq u^{\ast}, then
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{k+1} = -g\left( ., u_{k}\right) +\lambda h\geq-g\left( ., u^{\ast}\right) +\lambda h\text{ in }\Omega, \\ u_{k+1} = 0 = u^{\ast}\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
and so u_{k+1}\geq u^{\ast}. Then u^{\ast}\leq u_{k+1}\leq\overline{u}. In particular u_{k+1}\in\mathcal{E}. Again by the inductive hypothesis, u_{k}\leq u_{k-1}. Then
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{k+1} = -g\left( ., u_{k}\right) +\lambda h\leq-g\left( ., u_{k-1}\right) +\lambda h = \left( -\Delta\right) ^{s} u_{k}\text{ in }\Omega, \\ u_{k+1} = 0 = u_{k}\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
and so u_{k+1}\leq u_{k}. Also, \left\vert -g\left(., u_{k+1}\right) +\lambda h\right\vert \leq g\left(., u_{k+1}\right) +\lambda h\leq g\left(., u^{\ast}\right) +\lambda h\leq g\left(., c^{\prime\prime}d_{\Omega} ^{s}\right) +\lambda\left\Vert h\right\Vert _{\infty} and so -g\left(., u_{2}\right) +\lambda h\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}. Thus i)-iv) hold for j = k+1, which completes the inductive construction of the sequence \left\{ u_{j}\right\} _{j\in \mathbb{N}}. For x\in\mathbb{R}^{n} let u\left(x\right) : = \lim _{j\rightarrow\infty}u_{j}\left(x\right). By i) we have c^{\prime\prime}d_{\Omega}^{s}\leq u_{j}\leq c^{\prime}d_{\Omega}^{s} in \Omega for all j\in\mathbb{N}, and so c^{\prime\prime}d_{\Omega}^{s}\leq u\leq c^{\prime}d_{\Omega}^{s} in \Omega. By Lemma 2.8 \left\{ u_{j}\right\} _{j\in \mathbb{N}} converges in X_{0}^{s}\left(\Omega\right) to some weak solution u^{\ast\ast}\in X_{0}^{s}\left(\Omega\right) of problem (1.5). Thus, by the Poincaré inequality, \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges to u^{\ast\ast} in L^{2_{s}^{\ast}}\left(\Omega\right), which implies u = u^{\ast\ast}. Thus u\in X_{0}^{s}\left(\Omega\right) and u is a weak solution of problem (1.5). Since c^{\prime\prime}d_{\Omega}^{s}\leq u_{j}\leq c^{\prime}d_{\Omega}^{s} in \Omega for all j, we have c^{\prime\prime}d_{\Omega}^{s}\leq u\leq c^{\prime}d_{\Omega}^{s} in \Omega. Thus u\in\mathcal{E}. Let \psi\in\mathcal{E} such that -\Delta\psi\leq-g\left(., \psi\right) +\lambda h in \Omega. By the comparison principle, \psi\leq u a.e. in \Omega. An easy induction shows that \psi\leq u_{j} for all j. Indeed, by the comparison principle, \psi\leq\overline{u} = u_{1}. Then
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\psi\leq-g\left( ., \psi\right) +\lambda h\leq-g\left( ., u_{1}\right) +\lambda h = \left( -\Delta\right) ^{s} u_{2}\text{ in }\Omega, \\ \psi = 0 = u_{2}\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
Thus, again by the comparison principle, \psi\leq u_{2}. Suppose that k\geq2 and \psi\leq u_{k}. Then, in weak sense,
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\psi\leq-g\left( ., \psi\right) +\lambda h\leq-g\left( ., u_{k}\right) +\lambda h = \left( -\Delta\right) ^{s} u_{k+1}\text{ in }\Omega, \\ \psi = 0 = u_{k+1}\text{ on }\mathbb{R}^{n}\setminus\Omega, \end{array} \right. |
which gives \psi\leq u_{k+1}. Thus \psi\leq u_{j} for all j, and so \psi\leq u.
Proof of Theorem 1. Let \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N}}\subset\left(0, \infty\right) be a decreasing sequence such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0. For \lambda\geq \tau_{1}M_{1} and j\in\mathbb{N}, let u_{\varepsilon_{j}} be the weak solution of problem (2.42), given by Lemma 2.11, taking there \varepsilon = \varepsilon_{j}. Then \left\{ u_{\varepsilon_{j}}\right\} _{j\in \mathbb{N}} is a nonincreasing sequence in X_{0}^{s}\left(\Omega\right) and there exist positive constants c and c^{\prime} such that cd_{\Omega }^{s}\leq u_{j}\leq c^{\prime}d_{\Omega}^{s} in \Omega for all j\in\mathbb{N}. Therefore, by Lemma 2.8, \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} converges in X_{0}^{s}\left(\Omega\right) to some weak solution u\in X_{0} ^{s}\left(\Omega\right) of problem (1.5). Let
\mathcal{T}: = \left\{ \lambda>0:\text{ problem (1.5) has a weak solution }u\in\mathcal{E}\right\} . |
Thus \lambda\in\mathcal{T} whenever \lambda\geq\tau_{1}M_{1}. Consider now an arbitrary \lambda\in\mathcal{T}, and let \lambda^{\prime}>\lambda. Let u\in\mathcal{E} be a weak solution of the problem
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u = -g\left( ., u\right) +\lambda h\text{ in }\Omega, \\ u = 0\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
Let \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N}}\subset\left(0, \infty\right) be a decreasing sequence such that \lim_{j\rightarrow \infty}\varepsilon_{j} = 0. We have, in weak sense,
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u = -g\left( ., u\right) +\lambda h\leq-g\left( ., u+\varepsilon_{j}\right) +\lambda^{\prime}h\text{ in }\Omega, \\ u = 0\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. |
Then, by Lemma 2.9, used with \varepsilon = \varepsilon_{j}, \widetilde{v}_{\varepsilon_{j}} = u, and with \lambda replaced by \lambda^{\prime}, there exists a nonincreasing sequence \left\{ \widetilde{u}_{\varepsilon_{j}}\right\} _{j\in\mathbb{N} }\subset X_{0}^{s}\left(\Omega\right) such that
\left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\widetilde{u}_{\varepsilon_{j}} = -g\left( ., \widetilde{u}_{\varepsilon_{j}}+\varepsilon_{j}\right) +\lambda^{\prime }h\text{ in }\Omega, \\ \widetilde{u}_{\varepsilon_{j}} = 0\text{ in }\mathbb{R}^{n}\setminus\Omega, \end{array} \right. |
satisfying that \widetilde{u}_{\varepsilon_{j}}\geq u for all j\in \mathbb{N}, and cd_{\Omega}^{s}\leq\widetilde{u}_{\varepsilon_{j}}\leq c^{\prime}d_{\Omega}^{s} in \Omega for some positive constants c and c^{\prime} independent of j. Let \widetilde{u}: = \lim_{j\rightarrow\infty }\widetilde{u}_{\varepsilon_{j}}. Proceeding as in the first part of the proof, we get that \widetilde{u}\in\mathcal{E} and that \widetilde{u} is a weak solution of problem (1.5). Then \lambda^{\prime} \in\mathcal{T} whenever \lambda^{\prime}>\lambda for some \lambda \in\mathcal{T}. Thus there exists \lambda^{\ast}\geq0 such that \left(\lambda^{\ast}, \infty\right) \subset\mathcal{T}\subset\left[\lambda^{\ast }, \infty\right).
By Lemma 3.1, for any \lambda\in\mathcal{T} there exists a weak solution u\in\mathcal{E} of problem (1.5) such that u\geq\psi a.e. in \Omega for any \psi\in\mathcal{E} such that \left(-\Delta\right) ^{s}\psi \leq-g\left(., \psi\right) +\lambda h in \Omega.
Suppose now that g\left(., s\right) \geq bs^{-\beta} a.e. in \Omega for any s\in\left(0, \infty\right) for some b\in L^{\infty}\left(\Omega\right) such that 0\leq b\not \equiv 0 in \Omega. Then there exist a constant \eta_{0}>0 and a measurable set \Omega_{0}\subset\Omega such that \left\vert \Omega_{0}\right\vert >0 and b\geq\eta_{0} in \Omega_{0}. Let \lambda_{1} be the principal eigenvalue for \left(-\Delta\right) ^{s} in \Omega with Dirichlet boundary condition \varphi_{1} = 0 on \mathbb{R}^{n}\setminus\Omega, and let \varphi_{1}\in X_{0}^{s}\left(\Omega\right) be an associated positive principal eigenfunction. Then
\lambda_{1}\int_{\Omega}\varphi\varphi_{1} = \int_{\mathbb{R}^{n}\times \mathbb{R}^{n}}\frac{\left( \varphi\left( x\right) -\varphi\left( y\right) \right) \left( \varphi_{1}\left( x\right) -\varphi_{1}\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy |
and \varphi_{1}>0 a.e. in \Omega (see e.g., [25], Theorem 3.1). Let \lambda\in\mathcal{T} and let u\in\mathcal{E} be a weak solution of (1.5). Thus
\begin{align*} \lambda_{1}\int_{\Omega}u\varphi_{1} & = \int_{\mathbb{R}^{n}\times \mathbb{R}^{n}}\frac{\left( u\left( x\right) -u\left( y\right) \right) \left( \varphi_{1}\left( x\right) -\varphi_{1}\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy\\ & = \int_{\Omega}\left( -\varphi_{1}g\left( ., u\right) +\lambda h\varphi_{1}\right) \leq\int_{\Omega}\left( -bu^{-\beta}\varphi_{1}+\lambda h\varphi_{1}\right) \end{align*} |
and so
\lambda\int_{\Omega}h\varphi_{1}\geq\int_{\Omega_{0}}\left( \lambda _{1}u+bu^{-\beta}\right) \varphi_{1}\geq\inf\limits_{s>0}\left( \lambda_{1} s+\eta_{0}s^{-\beta}\right) \int_{\Omega_{0}}\varphi_{1} |
thus \lambda\geq\inf_{s>0}\left(\lambda_{1}s+\eta_{0}s^{-\beta}\right) \left(\int_{\Omega}h\varphi_{1}\right) ^{-1}\int_{\Omega_{0}}\varphi_{1} for any \lambda\in\mathcal{T}. Then \lambda^{\ast}>0.
Lemma 3.2. Let g:\Omega\times\left(0, \infty\right) \rightarrow\left[0, \infty\right) be a Carathéodory function. Assume that s\rightarrow g\left(x, s\right) is nonincreasing for a.e. x\in\Omega, and that, for some a\in L^{\infty}\left(\Omega\right) and \beta\in\left[0, s\right), g\left(., s\right) \leq as^{-\beta} a.e. in \Omega for any s\in\left(0, \infty\right). Then g satisfies the conditions g1)-g5) of Theorem 1.2.
Proof. Clearly g satisfies g1) and g2). Let \sigma>0. By Lemma 2.7, 0\leq g\left(., \sigma d_{\Omega} ^{s}\right) \leq a\sigma^{-\beta}d_{\Omega}^{-s\beta}\in\left(X_{0} ^{s}\left(\Omega\right) \right) ^{\prime} and so g\left(., \sigma d_{\Omega}^{s}\right) \in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}. Also, from the comparison principle, 0\leq\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{s}g\left(., \sigma ad_{\Omega}^{s}\right) \right) \leq\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(\sigma^{-\beta}ad_{\Omega}^{s-\beta}\right) in \Omega, and, since ad_{\Omega}^{s-\beta}\in L^{\infty}\left(\Omega\right), by Remark 2.1 iii), there exists a positive constant c such that \left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(\sigma ad_{\Omega}^{s-\beta}\right) \leq cd_{\Omega}^{s} in \Omega. Thus d_{\Omega}^{-s}\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{s}g\left(., \sigma d_{\Omega}^{s}\right) \right) \in L^{\infty}\left(\Omega\right). Then g satisfies g3). In particular, d_{\Omega}^{-s}\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{s}g\left(., d_{\Omega }^{s}\right) \right) \in L^{\infty}\left(\Omega\right). Since, for \sigma\geq1,
0\leq\left( \sigma d_{\Omega}^{s}\right) ^{-1}\left( \left( -\Delta \right) ^{s}\right) ^{-1}\left( d_{\Omega}^{s}g\left( ., \sigma d_{\Omega }^{s}\right) \right) \leq\sigma^{-1}d_{\Omega}^{-s}\left( \left( -\Delta\right) ^{s}\right) ^{-1}\left( d_{\Omega}^{s}g\left( ., d_{\Omega }^{s}\right) \right) , |
we get \lim_{\sigma\rightarrow\infty}\left\Vert \left(\sigma d_{\Omega} ^{s}\right) ^{-1}\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{s}g\left(., \sigma d_{\Omega}^{s}\right) \right) \right\Vert _{\infty} = 0. Also, by the comparison principle,
0\leq d_{\Omega}^{-s}\left( \left( -\Delta\right) ^{s}\right) ^{-1}\left( g\left( ., \sigma\right) \right) \leq\sigma^{-\beta}d_{\Omega}^{-s}\left( \left( -\Delta\right) ^{s}\right) ^{-1}\left( a\right) , |
and, by Remark 2.1 iii), d_{\Omega} ^{-s}\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(a\right) \in L^{\infty}\left(\Omega\right). Thus
\lim\limits_{\sigma\rightarrow\infty}\left\Vert d_{\Omega}^{-s}\left( \left( -\Delta\right) ^{s}\right) ^{-1}\left( g\left( ., \sigma\right) \right) \right\Vert _{L^{\infty}\left( \Omega\right) } = 0. |
Then g4) holds. Finally, 0\leq d_{\Omega}^{s}g\left(., \sigma d_{\Omega}^{s}\right) \leq\sigma^{-\beta}d_{\Omega}^{s-\beta}a and so g5) holds.
Proof of Theorem 1.3. The theorem follows from Lemma 3.2 and Theorem 1.2.
The author declare no conflicts of interest in this paper.
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