# Multiplicity of Weak Positive Solutions for Fractionalp & q Laplacian Problem with Singular Nonlinearity

Authors: Dandan Yang; Chuanzhi Bai
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we prove the existence and multiplicity of positive solutions for a class of fractionalp & q Laplacian problem with singular nonlinearity. Our approach relies on the variational method, some analysis techniques, and the method of Nehari manifold.


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## Body

## 1. Introduction

In this paper, we consider the following fractionalp & q Laplacian problem(1)−Δpsu+−Δqsu=axu−γ+λbxuβ,x∈Ω,u>0,x∈Ω,u=0,x∈∂Ω,whereΩ⊂RNN≥3 is an open bounded domain with smooth boundary ∂Ω, N>2s0<s<1, 1≤q<p<N/s, 0<γ<1<β<ps∗−1, λ is a positive parameter. The weight functions a:Ω→R is in Lps∗/ps∗+γ−1 with ax>0 for almost every x∈Ω, b:Ω→R is bounded with bx>0 for almost every x∈Ω, and ps∗=pN/N−ps denotes the critical Sobolev exponent. −Δrs, with r∈p,q, is the fractional r-Laplacian operator defined for any u∈Cc∞RN by(2)−Δrsux=2limε↘0∫Bεxcux−uyr−2ux−uyx−yN+srdy,x∈RN.Recently, Wang and Zhang [1] investigated the following singular elliptic boundary value problem involving the fractional Laplacian(3)−Δsu=λuβ+axu−γ,x∈Ω,u>0,x∈Ω,u=0,x∈∂Ω,the existence and multiplicity of weak positive solutions of (3) have been obtained in [1] by using the variational method, Nehari Maniford method, and Fibering Map analysis.In [2], Crandall et al. firstly studied the semilinear problem with singular nonlinearity, since then, the local setting (s=1) for (3) and some other versions of the problem have been extensively studied during the past decades, see for example [3–10] and the references therein.In recent years, the fractional Laplacian problems have been extensively investigated. For more details, we cite the reader to [11–15]. There are many different definitions of weak solutions for the fractional Laplacian equation (3). In [16], Fang say that u∈H0sΩ is a weak solution of (3) with λ=0 and a=1 if the identity(4)∫Ω−Δs/2u−Δs/2φdx=∫Ωu−γφdx,∀φ∈H0sΩholds. In virtue of the method of sub-supersolution, the author gives the sufficient conditions for the existence and uniqueness of positive solution.Very recently, great attention has been devoted to the study of fractionalp-Laplacian problems, see for instance [17–20]. However, in literature, there are only a few papers [21–23] dealing with fractional p & q problems. Motivated by the works [1, 23, 24], in this paper, we investigate the existence and multiplicity of solutions for the fractional p & q Laplacian problem (1) and extend the main results of Wang and Zhang [1].This article is organized as follows. In Section2, we give some notations and preliminaries. Section 3 is devoted to proving that problem (1) has at least two positive solutions for λ sufficiently small.


## 2. Preliminaries

For anys∈0,1, 1<p<∞, we define(5)Ws,pRN=uu:RN→Rismeasurable,uΩ∈LpΩ,and∫Qux−uypx−yN+spdxdy<∞,whereQ=R2N\CΩ×CΩ with CΩ=RN\Ω.The spaceW0s,pΩ=u∈Ws,pRN:u=0inRN\Ω endowed with the norm(6)u=us,p=∫Qux−uypx−yN+psdxdy1/p.From [23], we know that W0s,pΩ⊂W0s,qΩq≤p, which allows us to study (1) in W0s,pΩ.Throughout this section, we denote the best Sobolev constantSp for the embedding of W0s,pΩ into Lps∗Ω, defined as(7)Sp=infu∈W0s,pΩ\0∫R2Nux−uyp/x−yN+spdxdy∫RNups∗dxp/ps∗>0.By a weak solutionu of (1), we mean u∈W0s,pΩ satisfies (1) weakly, that is, we are looking for a function u∈W0s,pΩ, ux>0 a.e. in Ω such that(8)∫Qux−uyp−2ux−uyϕx−ϕyx−yN+psdxdy+∫Qux−uyq−2ux−uyϕx−ϕyx−yN+qsdxdy=∫Ωaxu−γϕ+λbxuβϕdx,∀ϕ∈W0s,pΩ.To study weak solutions to (1), we consider the following energy functional(9)Jλu=1pus,pp+1qus,qq−11−γ∫Ωaxu1−γdx−λ1+β∫Ωbxu1+βdx.The following Lemmas will be useful in the study of our problem (1).Lemma 1 [23].
Ifq≤p, then W0s,pΩ⊂W0s,qΩ with continuous embedding.Lemma 2 [23].
IfUnn∈ℕ is a sequence weakly convergent to some u in W0s,pΩ, then(10)un−us,pp=uns,pp−us,pp+o1.The Nehari manifoldNλΩ is closed linked to the behavior of functions of the form hλ:t→Jλtu for t>0 that named fibering maps [25]. If u∈W0s,pΩ, we have (11)hλ′t=tppus,pp+tqqus,qq−t1−γ1−γ∫Ωaxu1−γdx−λt1+β1+β∫Ωbxu1+βdx,(12)
h
λ
t
=
t
p
−
1
u
s
,
p
p
+
t
q
−
1
u
s
,
q
q
−
t
−
γ
∫
Ω
a
x
u
1
−
γ
d
x
−
λ
t
β
∫
Ω
b
x
u
1
+
β
d
x
,and(13)hλ″t=p−1tp−2us,pp+q−1tq−2us,qq+γt−γ−1∫Ωaxu1−γdx−λβtβ−1∫Ωbxu1+βdx.Obviously,(14)thλ′t=tpus,pp+tqus,qq−t1−γ∫Ωaxu1−γdx−λt1+β∫Ωbxu1+βdx=Jλ′tu,tu,which implies that foru∈W0s,pΩ\{0} and t>0, hλ′t=0 if and only if tu∈NλΩ, i.e., positive critical points of hλ′ correspond to points on the Nehari manifold. In particular, hλ′1=0 if and only if u∈NλΩ. Hence, we define(15)Nλ+Ω=u∈NλΩ:hλ″1>0=u∈NλΩ:p+γ−1us,pp+q+γ−1us,qq>λβ+γ∫Ωbxu1+βdx,Nλ0Ω=u∈NλΩ:hλ″1=0=u∈NλΩ:p+γ−1us,pp+q+γ−1us,qq=λβ+γ∫Ωbxu1+βdx,Nλ−Ω=u∈NλΩ:hλ″1<0=u∈NλΩ:p+γ−1us,pp+q+γ−1us,qq<λβ+γ∫Ωbxu1+βdx.Throughout this paper, we make the following assumptions:(H1)a∈Lp/p−1Ω with ax>0 for all almost every x∈Ω.(H2)bx>0 is bounded for all almost every x∈Ω.By (H2), we know that there existsb¯>0 such that 0<bx≤b¯ for all almost every x∈Ω.Forps∗>p, one has that ps∗/ps∗−1+γ<p/p−1+γ<p/p−1. Thus, there exists a constant C such that(16)aps∗/ps∗−1+γ≤Cap/p−1.Lemma 3.
Assume that condition (H1) holds withp<1+β. Then there exists M>0 such that ‖u‖≤M, for each u∈Nλ+Ω.Proof.
Ifu∈Nλ+Ω, by the definition of NλΩ and Nλ+Ω, we get that(17)p+γ−1us,pp+q+γ−1us,qq>β+γus,pp+us,qq−∫Ωaxu1−γdx.
By (H1), (16), and the Ho¨lder inequality and fractional Sobolev inequalities (7), we obtain(18)∫Ωaxu1−γdx≤∫Ωaxps∗/ps∗−1+γdxps∗−1+γ/ps∗∫Ωups∗dx1−γ/ps∗=aps∗/ps∗−1+γups∗1−γ≤aps∗/ps∗−1+γSp−1−γ/pu1−γ≤Cap/p−1Sp−1−γ/pu1−γ.
It follows from (17) and (18) that(19)β+1−pup<β+1−pus,pp+β+1−qus,qq<β+γ∫Ωaxu1−γdx≤β+γCSp−1−γ/papp−1u1−γ.
Thus, one has(20)u≤M≔β+γβ+1−pCSp−1−γ/pap/p−11/p+γ−1,∀u∈Nλ+Ω.
The proof is complete.Lemma 4.
If (H1) holds withp<1+β, then the functional Jλ is coercive and bounded below on NλΩ.Proof.
By using the definition ofNλΩ and (16), we have(21)Jλu=1p−11+βus,pp+1q−11+βus,qq−11−γ−11+β∫Ωaxu1−γdx≥1p−11+βup−11−γ−11+βCSp−1−γ/pap/p−1u1−γ.
This implies thatJλΩ is coercive and bounded from below on NλΩ. The proof is complete.Lemma 5.
Assume that conditions (H1) and (H2) hold withp<1+β. Then the minimal value mλ=infu∈Nλ+ΩJλu<0.Proof.
For1<β<ps∗−1, applying (H2), the Ho¨lder inequality and fractional Sobolev inequalities (7), we have(22)∫Ωbxu1+βdx≤b¯∫Ωdxps∗−1−β/ps∗∫Ωups∗dx1+β/ps∗=b¯Ωps∗−1−β/ps∗ups∗1+β≤b¯Ωps∗−1−β/ps∗Sp−1+β/pu1+β.
Using inequality (18) and (22), we obtain(23)Jλu≥1pus,pp−11−γ∫Ωaxu1−γdx−λ1+β∫Ωbxu1+βdx≥1pup−C1−γSp−1−γ/pap/p−1u1−γ−λ1+βb¯Sp−1+β/pΩps∗/ps∗−β−1u1+β.
It follows from0<γ<1≤q<p<1+β<ps∗ that there exist ρ>0, ε>0, and λ1>0 small enough such that for any fixed λ∈0,λ1(24)Jλu≥ε2>0,∀u∈∂Bρ,
whereBρ=u∈Nλ+Ω:u≤ρ. On the other hand, for any fixed v∈Nλ+Ω, since 0<γ<1≤q<p<1+β, we can derive that(25)Jλtv=tppvs,pp+tqqvs,qq−t1−γ1−γ∫Ωaxv1−γdx−λ1+βt1+β∫Ωbxv1+βdx=−At1−γ+otqt→0,
whereA=1/1−γ∫Ωax|v|1−γdx. Thus, Jλtv<0 for all v≠0, provided t>0 is sufficiently small. Hence, mλ=infu∈Nλ+ΩJλu<0. This completes the proof of Lemma 5.Lemma 6.
For eachλ∈0,λ1, there exists uλ∈Nλ+Ω such that Jλuλ=mλ=infNλΩJλ.Proof.
Let{un}⊂Bρ be a minimizing sequence such that Jλun→mλ as n→∞. From Lemma 3 we know that the sequence {un} is bounded in W0s,pΩ. Hence, we can obtain that there exists a sub-sequence of {un} (still denoted by {un}) such that un⇀uλ weakly in W0s,pΩ, strongly in LrΩ (1≤r<ps∗) and pointwise a.e. in Ω. By using Ho¨lder inequality, we derive that as n→∞,(26)∫Ωaxun1−γdx≤∫Ωaxuλ1−γdx+∫Ωaxun−uλ1−γdx≤∫Ωaxuλ1−γdx+∫Ωaxp/p−1dxp−1/p∫Ωun−uλpdx1−γ/p=∫Ωaxuλ1−γdx+app−1un−uλp1−γ=∫Ωaxuλ1−γdx+o1,
and(27)∫Ωaxuλ1−γdx≤∫Ωaxun1−γdx+∫Ωaxun−uλ1−γdx≤∫Ωaxun1−γdx+ap/p−1un−uλp1−γ=∫Ωaxun1−γdx+o1.
Thus, we obtain that(28)∫Ωaxun1−γdx=∫Ωaxuλ1−γdx+o1.
Sincebx>0 is bounded in Ω, we have that bx1/1+βunx is also bounded in W0s,pΩ. If r∈1,∞ and {gn}n∈N⊂LrRk is a bounded sequence such that gn→g a.e. in Rk, then we have from the Brezis–Lieb Lemma that(29)gn−gLrRkr=gnLrRkr−gLrRkr+o1.
Taking(30)gn=bx1/1+βunx,g=bx1/1+βux,r=1+βandk=N
in (29), we get(31)∫Ωbxunx−ux1+βdx=∫Ωbxunx1+βdx−∫Ωbxux1+βdx+o1.
Since(32)0≤∫Ωbxunx−ux1+βdx≤bun−u1+β1+β→0,n→∞,
we can deduce that asn→∞(33)∫Ωbxunx1+βdx=∫Ωbxux1+βdx+o1.
By Lemmas1 and 2, we have(34)un−us,qq=uns,qq−us,qq+o1,
and(35)un−us,pp=uns,pp−us,pp+o1.
Frommλ<0, we have un≤r0<r for some positive constant r independent of n. Thus, by (35) and uλ∈Br we obtain un−uλ∈Br if n is sufficiently large. Combining above arguments with (28), (33), (34), and (35), we can obtain(36)mλ=Jλun+o1=Jλuλ+1pun−uλs,pp+1qun−uλs,qq+o1≥Jλuλ+o1≥mλ+o1n→∞,
which yields that(37)0≥Jλuλ−mλ+o1≥o1.
Passing to the limit asn→∞, we get Jλuλ=mλ.
In the following, we will show thatuλ∈Nλ+Ω. It suffices to prove that un→uλ strongly in W0s,pΩ. By Jλuλ=mλ and(38)mλ=Jλuλ+1pun−uλs,pp+1qun−uλs,qq+o1,
we have(39)0=1pun−uλs,pp+1qun−uλs,qq+o1≥1pun−uλp+o1,
which implies thatun−uλp→0 as n→∞, i.e., un→uλ strongly in W0s,pΩ. Thus, uλ∈Nλ+Ω is a minimizer of Jλ in Nλ+Ω. The proof is complete.


## 3. Main Result

Similar to the proof of Lemma 3.7 in [1], we can easily obtain the following Lemma.Lemma 7.
For anyu∈Nλ+Ω, then there exists ε>0 and a continuous function f=fω>0, ω∈W0s,pΩ with ω<ε satisfying that f0=1, fωu+ω∈Nλ+Ω.Lemma 8.
For each givenϕ∈W0s,pΩ, ϕ≥0, there exists T>0 such that Jλuλ+tϕ≥Jλuλ for all t∈0,T and uλ∈Nλ+Ω.Proof.
Let(40)ηt=p−1uλ+tϕs,pp+q−1uλ+tϕs,qq+γ∫Ωaxuλ+tϕ1−γdx−λβ∫Ωbxuλ+tϕ1+βdx,∀t>0.
By using the continuity ofη, η0>0 and uλ∈Nλ+Ω, we can infer that there exists T>0, such that ηt>0 for all t∈0,T. On the other hand, applying Lemma 7 we get that for any t>0 there exists t′>0 such that t′uλ+tϕ∈Nλ+Ω. Hence, t′→1 as t→0 and for each t∈0,T we obtain(41)Jλuλ+tϕ≥Jλt′uλ+tϕ≥infNλ+ΩJλ=Jλuλ.
The proof of this Lemma is completed.Lemma 9.
The minimizeruλ∈Nλ+Ω is a weak solution of problem (1). Moreover, uλx>0 for each x∈Ω.Proof.
Firstly, we prove thatuλ∈Nλ+Ω is a weak solution of (1). From Lemma 8, we derive that for any ϕ∈W0s,pΩ, ϕ≥0 and t∈0,T.(42)0≤Jλuλ+tϕ−Jλuλt=1pt∫Quλx−uλy+tϕx−ϕyp−uλx−uλypx−yN+qsdxdy+1qt∫Quλx−uλy+tϕx−ϕyq−uλx−uλyqx−yN+qsdxdy−11−γ∫Ωaxuλ+tϕ1−γ−uλ1−γtdx−λ1+β∫Ωbxuλ+tϕ1+β−uλ1+βtdx.
Lettingt→0+, we infer that(43)11−γliminft→0+∫Ωaxuλ+tϕ−uλ1−γtdx≤∫Quλx−uλyp−2uλx−uλyϕx−ϕyx−yn+psdxdy+∫Quλx−uλyq−2uλx−uλyϕx−ϕyx−yn+qsdxdy−λ∫Ωbxuλβϕdx.
From Fatou’s Lemma, we have(44)∫Ωaxuλ−γϕdx≤11−γliminft→0+∫Ωaxuλ+tϕ1−γ−uλ1−γtdx.
Together with (42) with (43), it yields(45)∫Quλx−uλyp−2uλx−uλyϕx−ϕyx−yn+psdxdy+∫Quλx−uλyq−2uλx−uλyϕx−ϕyx−yn+qsdxdy−λ∫Ωbxuλβϕdx−∫Ωaxuλ−γϕdx≥0.
For any givenψ∈W0s,pΩ, taking(46)ϕ=uλ+tψ+∈W0s,pΩ,ϕ≥0
into (44), we obtain(47)0≤∫Q′uλx−uλyp−2uλx−uλyuλ+tψx−uλ+tψyx−yn+psdxdy+∫Q′uλx−uλyq−2uλx−uλyuλ+tψx−uλ+tψyx−yn+qsdxdy−λ∫xuλ+tψ≥0bxuλβuλ+tψdx−∫xuλ+tψ≥0axuλ−γuλ+tψdx=uλp+uλq−λ∫Ωbxuλβ+1dx−∫Ωaxuλ1−γdx+t∫Quλx−uλyp−2uλx−uλyψx−ψyx−yn+psdxdy+∫Quλx−uλyq−2uλx−uλyψx−ψyx−yn+qsdxdy−λ∫Ωbxuλβψdx−∫Ωaxuλ−γψdx−∫Q″uλx−uλyp−2uλx−uλyuλ+tψx−uλ+tψyx−yn+psdxdy−∫Q″uλx−uλyq−2uλx−uλyuλ+tψx−uλ+tψyx−yn+qsdxdy−λ∫xuλ+tψ<0bxuλβuλ+tψdx−∫xuλ+tψ<0axuλ−γuλ+tψdx≤t∫Quλx−uλyp−2uλx−uλyψx−ψyx−yn+psdxdy+∫Quλx−uλyq−2uλx−uλyψx−ψyx−yn+qsdxdy−λ∫Ωbxuλβψdx−∫Ωaxuλ−γψdx−t∫Q″uλx−uλyp−2uλx−uλyψx−ψyx−yn+psdxdy−t∫Q″uλx−uλyq−2uλx−uλyψx−ψyx−yn+qsdxdy,
whereQ′=R2n\CΩ′×CΩ′, CΩ′=Rn\{x|uλ+tψ≥0} and Q″=R2n\CΩ″×CΩ″, CΩ″=Rn\{x|uλ+tψ<0}. Since the measure of the set {x|uλ+tψ<0} tends to 0 as t→0+, one has(48)∫Q″uλx−uλyp−2uλx−uλyψx−ψyx−yn+psdxdy→0,t→0+,
(49)
∫
Q
″
u
λ
x
−
u
λ
y
q
−
2
u
λ
x
−
u
λ
y
ψ
x
−
ψ
y
x
−
y
n
+
q
s
d
x
d
y
→
0
,
t
→
0
+
.
Hence, dividing byt>0 we infer that(50)∫Quλx−uλyp−2uλx−uλyψx−ψyx−yn+psdxdy+∫Quλx−uλyq−2uλx−uλyψx−ψyx−yn+qsdxdy−λ∫Ωbxuλβψdx−∫Ωaxuλ−γψdx≥0,∀ψ∈W0s,pΩ.
Sinceψ∈W0s,pΩ is arbitrary, replacing ψ by −ψ in above inequality, one obtains(51)∫Quλx−uλyp−2uλx−uλyψx−ψyx−yn+psdxdy+∫Quλx−uλyq−2uλx−uλyψx−ψyx−yn+qsdxdy−λ∫Ωbxuλβψdx−∫Ωaxuλ−γψdx≤0,∀ψ∈W0s,pΩ.
Thus(52)∫Quλx−uλyp−2uλx−uλyψx−ψyx−yn+psdxdy+∫Quλx−uλyq−2uλx−uλyψx−ψyx−yn+qsdxdy−λ∫Ωbxuλβψdx−∫Ωaxuλ−γψdx=0,∀ψ∈W0s,pΩ.
Hence,uλ is a weak solution of (1).
Secondly, we prove thatuλ>0 for almost each x∈Ω. Since Jλuλ=c<0, we get uλ≥0 and uλ≠0. Let e1∈W0s,pΩ be the first eigenfunction of the operator −Δps with e1>0 and ‖e1‖=1. Taking ψ=e1 in (50), one gets  (53)∫Ωaxuλ−γe1dx≤∫Quλx−uλyp−1x−yn+psdxdyp−1/p·∫Qe1x−e1ypx−yn+psdxdy1/p+∫Quλx−uλyq−1x−yn+qsdxdyq−1/q·∫Qe1x−e1yqx−yn+qsdxdy1/q+λb¯∫Ωuλps∗dxβ/ps∗·∫Ωe1ps∗/ps∗−βdxps∗−β/ps∗=uλp−1/pe11/p+uλq−1/qe11/q+λb¯uλps∗β∫Ωe1ps∗/ps∗−βdxps∗−β/ps∗≤uλp−1/pe11/p+uλq−1/qe11/q+λb¯Sp−β/puλβ∫Ωdxps∗−β−1/ps∗−β·∫Ωe1ps∗dx1/ps∗−βps∗−β/ps∗=uλp−1/pe11/p+uλq−1/qe11/q+λb¯Sp−β/pΩps∗−β−1/ps∗uλβe1ps∗≤uλp−1/pe11/p+uλq−1/qe11/q+λb¯Sp−1+β/pΩps∗−β−1/ps∗uλβe1<∞,
which implies thatuλ>0 for almost every x∈Ω. This completes the proof of Lemma 9.Lemma 10.
Assume that (H1) and (H2) hold withp<1+β. Then there exists λ2>0 such that for any λ∈0,λ2, one has Nλ0Ω={0}. Moreover, Nλ−Ω is closed in W0s,pΩfor all λ∈0,λ2.Proof.
If not, there is au0∈Nλ0Ω with u0≠0. Then by the definition of Nλ0Ω, we get(54)p+γ−1u0s,pp≤p+γ−1u0s,pp+q+γ−1u0s,qq=λβ+γ∫Ωbxu01+βdx,
which implies that(55)u0p=u0s,pp≤λβ+γp+γ−1∫Ωbxu01+βdx≤λbβ+γp+γ−1Ωps∗−1−p/ps∗Sp−1+β/pu01+β,
and so(56)u0≥p+γ−1λbβ+γΩp+1−ps∗/ps∗Sp1+β/p1/1+β−p.
On the other hand, from the definition ofNλ0Ω, we can obtain(57)β+1−pu0s,pp+β+1−qu0s,qq=β+γ∫Ωaxu01−γdx,u0∈Nλ0Ω.
By using the Hölder inequalities and (7), we have by (57) that(58)β+1−pu0s,pp≤β+1−pu0s,pp+β+1−qu0s,qq=β+γ∫Ωaxu01−γdx≤β+γaps∗/ps∗−1+γSp−1−γ/pu01−γ,
which implies that(59)u0≤β+γβ+1−paps∗/ps∗−1+γSp−1−γ/p1/p+γ−1.
Ifλ>0 is sufficiently small, however, (55) contradicts to (58). Hence, we conclude that there exists λ2>0 such that Nλ0Ω={0} for λ∈0,λ2.
Let{un}⊂Nλ−Ω be a sequence satisfying un→u∗ in the W0s,pΩ. By using the Sobolev inequalities and continuous compact embedding, we have un→u∗ in L1+βΩ and u∗∈Nλ−Ω∪Nλ0Ω. From the definition of Nλ−Ω, we obtain that(60)b¯λβ+γΩps∗−1−β/ps∗u∗ps∗1+β≥b¯λβ+γu∗1+β1+β≥λβ+γ∫Ωbxu∗1+βdx>p+γ−1u∗s,pp+q+γ−1u∗s,qq≥p+γ−1u∗p≥p+γ−1Spu∗ps∗p
that is,(61)u∗ps∗≥p+γ−1b¯λβ+γSpΩ1+β−ps∗/ps∗1/1+β−p>0
which yieldsu∗≠0 i.e., u∗∈Nλ−Ω. The proof of this Lemma is completed.Lemma 11.
Assume that (H1) and (H2) hold withp<1+β. Then there exists λ3>0, such that Jλu≥0, for each u∈Nλ−Ω while λ∈0,λ3.Proof.
If not, there is av∈Nλ−Ω such that Jλv<0. Then(62)1pvs,pp+vs,qq−11−γ∫Ωaxv1−γdx−λ1+β∫Ωbxv1+βdx≤1pvs,pp+1qvs,qq−11−γ∫Ωaxv1−γdx−λ1+β∫Ωbxv1+βdx<0.
By (62) and the definition of Nλ−Ω, it follows that(63)λ1p−11+β∫Ωbxv1+βdx−11−γ−1p∫Ωaxv1−γdx<0.
Combining with (18) and (22), we get(64)λb¯1p−11+βΩps∗−1−β/ps∗vps∗1+β<11−γ−1paps∗/ps∗−1+γvps∗1−γ,
which implies that(65)vps∗<1+βp+γ−1λb¯1−γβ+1−pΩ1+β−ps∗/ps∗aps∗/ps∗−1+γ1/β+γ.
With the help of inequalities (61) and (65), for any u∈Nλ−Ω, we get that(66)p+γ−1b¯λβ+γSpΩ1+β−ps∗/ps∗1/1+β−p≤ups∗<1+βp+γ−1λb¯1−γβ+1−pΩ1+β−ps∗/ps∗aps∗/ps∗−1+γ1/β+γ.
Direct calculations show that(67)p+γ−1b¯β+γSpΩ1+β−ps∗/ps∗1/1+β−p1+βp+γ−1b¯1−γβ+1−pΩ1+β−ps∗/ps∗aps∗/ps∗−1+γ1/β+γ<λp+γ−1/1+β−pβ+γ,
which contradicts the factλp+γ−1/1+β−pβ+γ→0 as λ→0+. We complete the proof of Lemma 11.Lemma 12.
Assume that (H1) and (H2) hold withp<1+β. Then there exists λ4>0 small enough such that for each λ∈0,λ4, there exists vλ∈Nλ−Ω satisfying Jλvλ=m¯λ=infu∈Nλ−ΩJλu. Furthermore, vλ is a weak positive solution of problem (1).Proof.
From Lemma4, Jλ is coercive on NλΩ, and it is also true for Nλ−Ω. If sequence {vn}⊂Nλ−Ω satisfying Jλvn→m¯λ=infu∈Nλ−ΩJλu as n→∞. By using the coercive of Jλ, we can get {vn} is bounded in Nλ−Ω. Hence, we may assume that vn⇀vλ weakly as n→∞ in Nλ−Ω. Since Nλ−Ω is completed in W0s,pΩ (Lemma 10), it follows from the same arguments as in proving the existence of minimizer uλ (Lemma 6) and the compactness of the embedding W0s,pΩ→L1+βΩ (β<ps∗−1) we get vλ∈Nλ−Ω is the minimizer of Jλ. Furtherover, similar to the proof of weak positive solution uλ as in Lemma 9, one can prove that vλ∈W0s,pΩ is also a weak positive solution for problem (1.1). The proof is complete.Theorem 13.
Let0<γ<1≤q<p<1+β<ps∗. Assume that (H1) and (H2) hold. Then there exists a positive number λ∗ such that for each λ∈0,λ∗ problem (1.1) possesses at least two weak positive solution uλ,vλ∈W0s,pΩ.Proof.
Letλ∗=min{λ1,λ2,λ3,λ4}. Obviously, Lemma 3 and Lemma 12 are true for all λ∈0,λ∗. Thus, it follows from Lemma 9 and Lemma 12 that uλ and vλ are the weak positive solutions of problem (1.1). The proof is complete.


## 4. Conclusions

In this paper, the existence and multiplicity of positive solutions for a class of fractionalp & q Laplacian problem with singular nonlinearity have been investigated. It is worthy to point out that few studies have been done on this issue. By means of the variational method, Nehari manifold method and some analysis techniques, the sufficient conditions of existence and multiplicity of positive solutions to this problem have been presented in Theorem 13. Our results generalize the main conclusions of Wang and Zhang in [1].



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