J. Korean Math. Soc. 57 (2020), No. 6, pp. 1451–1470

https://doi.org/10.4134/JKMS.j190693

pISSN: 0304-9914 / eISSN: 2234-3008

EXISTENCE, MULTIPLICITY AND REGULARITY

OF SOLUTIONS FOR THE FRACTIONAL

p-LAPLACIAN EQUATION

Yun-Ho Kim

Abstract. We are concerned with the following elliptic equations:(−∆)spu = λf(x, u) in Ω,

u = 0 on RN\Ω,

where λ are real parameters, (−∆)sp is the fractional p-Laplacian oper-ator, 0 < s < 1 < p < +∞, sp < N , and f : Ω × R → R satisfies a

Caratheodory condition. By applying abstract critical point results, we

establish an estimate of the positive interval of the parameters λ for whichour problem admits at least one or two nontrivial weak solutions when the

nonlinearity f has the subcritical growth condition. In addition, under

adequate conditions, we establish an apriori estimate in L∞(Ω) of anypossible weak solution by applying the bootstrap argument.

1. Introduction

In the present paper, we consider the existence of nontrivial weak solutionsto the nonlinear elliptic equations involving the fractional p-Laplacian of theform

(Pλ)

(−∆)spu = λf(x, u) in Ω,

u = 0 on RN\Ω,

where the fractional p-Laplacian operator (−∆)sp is defined by

(−∆)spu(x) = 2 limε0

∫RN\Bε(x)

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+psdy, x ∈ RN .

Here, λ are real parameters, 0 < s < 1 < p < +∞, sp < N , Bε(x) := y ∈RN : |x− y| < ε and f : Ω× R→ R satisfies a Caratheodory condition.

In the last years the study of fractional and nonlocal problems of elliptic typehas received a tremendous popularity because the interest in such operators has

Received October 15, 2019; Accepted February 13, 2020.

2010 Mathematics Subject Classification. 35R11, 35A15, 35J60, 49R05.Key words and phrases. Fractional p-Laplacian, weak solution, critical points, variational

method.

c©2020 Korean Mathematical Society

1451

1452 Y.-H. KIM

consistently increased within the framework of the mathematical theory to con-crete some phenomena such as social sciences, fractional quantum mechanics,materials science, continuum mechanics, phase transition phenomena, imageprocess, game theory and Levy processes; see [6, 8, 15, 18, 21, 27, 28] and thereferences therein. Especially, in terms of fractional quantum mechanics, thenonlinear fractional Schrodinger equation was originally suggested by Laskin in[21,22] as an extension of the Feynman path integral, from the Brownian-like tothe Levy-like quantum mechanical paths. Fractional operators are closely re-lated to financial mathematics, because Levy processes with jumps revealed asmore adequate models of stock pricing in comparison with the Brownian onesused in the celebrated Black–Scholes option pricing model. In these directions,many researchers have been extensively studied the fractional Laplacian typeproblems in various way; see [5, 7, 8, 14, 15, 19, 23, 31–33, 36] and the referencestherein. Especially, a mountain pass theorem and applications to Dirichletproblems involving non-local integro-differential operators of fractional Lapla-cian type are given in [33]; see also [32]. Iannizzotto et al. [19] have investigatedthe existence and multiplicity results for the fractional p-Laplacian type prob-lems. One of key ingredients for obtaining these results is the Ambrosetti andRabinowitz condition ((AR)-condition for short) in [2];

(AR) There exist positive constants C1 and ζ such that ζ > p and

0 < ζF (x, t) ≤ f(x, t)t for x ∈ Ω and |t| ≥ C1,

where F (x, t) =∫ t

0f(x, s) ds, and Ω is a bounded domain in RN .

As is known, the (AR)-condition is significant to ascertain the Palais-Smalecondition of an energy functional which plays a momentous role in employingcritical point theory originally introduced by the paper [2]. However this con-dition is very restrictive and gets rid of many nonlinearities. In this regard,Miyagaki and Souto [29] established the existence of a nontrivial solution forthe superlinear problems without the (AR)-condition. Inspired by this paper,the existence and multiplicity of solutions for the p-Laplacian equation in abounded domain Ω ⊂ RN were obtained by Liu-Li [25] (see also [24]) under thefollowing assumption:

(LL) There exists C∗ > 0 such that

F(x, t) ≤ F(x, τ) + C∗

for each x ∈ Ω, 0 < t < τ or τ < t < 0, where F(x, t) = f(x, t)t− pF (x, t).

Under this condition, Wei and Su [34] showed that the fractional Laplacianproblem possesses infinitely many weak solutions. Recently, the authors in [4]investigated the existence of at least one nontrivial weak solutions of ellipticequations with variable exponent by utilizing an abstract nonsmooth criticalpoint result provided in [13] (see [9]). In particular, they obtained this existenceresult without assuming the condition (LL) as well as the (AR)-condition.

FRACTIONAL p-LAPLACIAN EQUATION 1453

In this respect, the first aim of this paper is to concretely provide an esti-mate of the positive interval of the parameters λ for which the problem (Pλ)possesses at least one nontrivial weak solution when the nonlinear term f fulfilsthe subcritical growth condition. In addition, under adequate conditions, weestablish an apriori estimate in L∞(Ω) of any possible weak solution apply-ing the bootstrap argument. As compared with the local case, the value of(−∆)spu(x) at any point x ∈ Ω relies not only on the values of u on the whole

Ω, but actually on the whole space RN . Hence more complicated analysis thanthe papers [4, 11, 12] has to be carefully carried out when we investigate theaccurate interval for the parameters for which problem (Pλ) possesses at leastone nontrivial weak solution. As far as we are aware, there were no such ex-istence results for fractional p-Laplacian problems in this situation althoughour result is motivated by the paper [4]. Also it is worth noticing that weobtain the existence of at least one nontrivial weak solution for our problemwithout using the facts that the energy functional related to (Pλ) fulfils thePalais-Smale condition and the mountain pass geometry that is crucial to takeadvantage of the mountain pass theorem in [2].

The other aim of this paper is to consider the existence and uniform estimatesof two distinct solutions for our problem. To do this, we utilize an abstractcritical point result in [3] which is a variant of the work of G. Bonanno [10].Recently, G. Bonanno and A. Chinnı [12] investigated the existence of at leasttwo distinct weak solutions to the p(x)-Laplacian problems by applying criticalpoint theorem in [10]. Contrast to our first main result, (AR)-condition isneeded to ensure the existence of two distinct weak solutions for nonlinearelliptic equations; see [12]. In that sense, we show that the problem (Pλ) hastwo distinct weak solutions provided that f fulfils a weaker condition than the(AR)-condition.

This paper is structured as follows. First we recall briefly some basic resultsfor the fractional Sobolev spaces. Next, under certain conditions on the non-linear term f , we obtain the existence of at least one or two nontrivial weaksolutions for problem (Pλ) whenever the parameter λ belongs to a positiveinterval.

2. Preliminaries and main results

In this section, we introduce the definitions and some underlying propertiesof the fractional Sobolev spaces. We mention the reader to [1,17,18] for furtherreferences and for some of the proofs of results. From this, we establish the ex-istence of a nontrivial weak solution for the problem (Pλ) when the nonlinearityf has the subcritical growth condition.

Let s ∈ (0, 1) and p ∈ (1,+∞). The fractional Sobolev space W s,p(RN ) isdefined as

W s,p(RN ) :=

u ∈ Lp(RN ) :

∫RN

∫RN

|u(x)− u(y)|p

|x− y|N+psdxdy < +∞

1454 Y.-H. KIM

endowed with the norm

||u||W s,p(RN ) :=

(||u||p

Lp(RN )+ |u|p

W s,p(RN )

) 1p

,

where

||u||pLp(RN )

:=

∫RN|u|p dx and |u|p

W s,p(RN ):=

∫RN

∫RN

|u(x)− u(y)|p

|x− y|N+psdxdy.

Let s ∈ (0, 1) and 1 < p < +∞. Then W s,p(RN ) is a separable and reflex-ive Banach space, and also the space C∞0 (RN ) is dense in W s,p(RN ), that isW s,p

0 (RN ) = W s,p(RN ) (see e.g. [1, 17]).We consider the problem (Pλ) in the closed linear subspace defined by

Xps (Ω) =

u ∈W s,p(RN ) : u(x) = 0 a.e. in RN\Ω

with the norm

||u||Xps (Ω) =

(|u|p

W s,p(RN )+ ‖u‖pLp(Ω)

) 1p

.

Then Xps (Ω) is a uniformly convex Banach space; see Lemma 2.4 in [35]. Now

we list some basic results which will be needed to obtain our main result.

Lemma 2.1 ([31]). Let Ω be an bounded open set in RN , s ∈ (0, 1) and p ∈[1,+∞). Then

||u||pLp(Ω) ≤sp |Ω|

spN

2ωspN +1

N

|u|pW s,p(RN )

for any u ∈ W s,p(RN ). Here |Ω| means the Lebesgue measure of Ω, ωN denotes

the volume of the N -dimensional unit ball and we denoted by W s,p(RN ) thespace of all u ∈ Xp

s (Ω) such that u ∈ W s,p(RN ), where u is the extension byzero of u.

Lemma 2.2 ([16]). Let Ω ⊂ RN be a bounded open set with Lipschitz boundary,s ∈ (0, 1) and p ∈ (1,+∞). Then we have the following continuous embeddings:

W s,p(Ω) → Lq(Ω) for all q ∈ [1, p∗s], if sp < N ;

W s,p(Ω) → Lq(Ω) for every q ∈ [1,∞), if sp = N ;

W s,p(Ω) → C0,λb (Ω) for all λ < s−N/p, if sp > N,

where p∗s is the fractional critical Sobolev exponent, that is

p∗s :=

NpN−sp if sp < N,

+∞ if sp ≥ N.

In particular, the space W s,p(Ω) is compactly embedded in Lq(Ω) for any q ∈[p, p∗s).

FRACTIONAL p-LAPLACIAN EQUATION 1455

Lemma 2.3 ([26]). Let s ∈ (0, 1) and p ∈ (1,+∞) be such that sp < N . Then,for all u ∈W s,p(RN ), there holds

||u||pLp∗s (RN )

≤ Cp∗s |u|pW s,p(RN )

,

where

Cp∗s =(N + 2p)3ppp+22(N+1)(N+2)s(1− s)

Npp∗s |SN−1| psN +1(N − sp)p−1

.

Here∣∣SN−1

∣∣ denotes the surface area of the (N − 1)-dimensional unit sphere.

Thanks to the above lemma, it is possible to establish the estimate of apositive constant denoted by Cq which is crucial to obtain the positive intervalof the parameters λ for which the problem (Pλ) possesses at least one or twonontrivial weak solutions.

Remark 2.4. Recall that for each 1 < hs < N , putting h∗s = hN/(N−hs), fromLemma 2.2 one has Xh

s (Ω) → Lh∗s (Ω) with continuous embedding. Precisely,

according to Lemma 2.3, for each u ∈ Xhs (Ω), it results

(2.1) ||u||Lh∗s (Ω) ≤ C1h

h∗s|u|W s,h(RN ) .

Let q ∈ [1, h∗s] be fixed. Set ` = h∗s/q and `′ = h∗s/(h∗s−q). Since |u|q ∈ L

h∗sq (Ω),

the Holder inequality implies that∫Ω

|u(x)|qdx = ||uq||L1(Ω) ≤ ||uq||L`(Ω)||1||L`′ (Ω)

=(∫

Ω

|u(x)|h∗sdx) qh∗s |Ω| 1

`′ = ||u||qLh∗s (Ω)|Ω|

h∗s−qh∗s

and so

||u||Lq(Ω) = (||uq||L`(Ω))1q ≤ ||u||Lh∗s (Ω)|Ω|

h∗s−qh∗sq .

By (2.1) one has

(2.2) ||u||Lq(Ω) ≤ ||u||Lh∗s (Ω)|Ω|h∗s−qh∗sq ≤ C

1h

h∗s|Ω|

h∗s−qh∗sq |u|W s,h(RN ) .

Now, let 1 < q ≤ p∗s . By applying (2.2) for h = p, it follows from Lemma 2.1that

(2.3) ||u||Lq(Ω) ≤ C1p

p∗s|Ω|

p∗s−qp∗sq |u|W s,p(RN )

for each u ∈ Xps (Ω). Hence we will denote the positive constant Cq for which

one has

(2.4) ||u||Lq(Ω) ≤ Cq |u|W s,p(RN ) ,

where

Cq ≤ C1p

p∗s|Ω|

p∗s−qp∗sq .

1456 Y.-H. KIM

Definition 2.5. Let (X, || · ||X) be a real Banach space If Φ,Ψ : X → R aretwo continuously Gateaux differentiable functionals and r ∈ R, we say thatJ = Φ − Ψ satisfies the Cerami condition cut off upper at r ((C)[r]-conditionfor short) if any sequence unn∈N in X such that

(C1) J(un)n∈N is bounded;(C2) ||J ′(un)||X∗(1 + ||un||X)→ 0 as n→∞;(C3) 0 < Φ(un) < r for all n ∈ N;

has a convergent subsequence.

Now we recall the key lemma to get our main result. This assertion can befound in [20]; see [13] for the case of the Palais-Smale condition.

Lemma 2.6. Let X be a real Banach space and let Φ,Ψ : X → R be twocontinuously Gateaux differentiable functionals such that

infu∈X

Φ(u) = Φ(0) = Ψ(0) = 0.

Assume that there are a positive constant µ and an element u in X, with0 < Φ(u) < µ, such that

(2.5)supΦ(u)≤µ Ψ(u)

µ<

Ψ(u)

Φ(u)

holds and for each λ ∈ Λµ :=(

Φ(u)Ψ(u) ,

µsupΦ(u)≤µ Ψ(u)

), the functional Iλ :=

Φ−λΨ satisfies the (C)[µ]-condition. Then, for each λ ∈ Λµ, the functional Iλhas a nontrivial point xλ in Φ−1((0, µ)) such that Iλ(xλ) ≤ Iλ(x) for all x inΦ−1((0, µ)) and I ′λ(xλ) = 0.

The following assertion is crucial to ensure that the problem (Pλ) has atleast two distinct weak solutions.

Lemma 2.7 ([3]). Let Φ,Ψ : X → R be two continuously Gateaux differentiablefunctionals such that Gateaux derivative of Ψ is compact and

infu∈X

Φ(u) = Φ(0) = Ψ(0) = 0.

Assume that there are a constant µ > 0 and an u in X, with 0 < Φ(u) < µ,such that

supΦ(u)≤µ Ψ(u)

µ<

Ψ(u)

Φ(u)

holds and for every λ ∈ Λµ :=(

Φ(u)Ψ(u) ,

µsupΦ(u)≤µ Ψ(u)

), the functional Iλ :=

Φ − λΨ satisfies the (C)-condition and it is unbounded from below. Then, foreach λ ∈ Λµ the functional Iλ has two distinct critical points u0 and u1 suchthat u0 is a nontrivial local minimum satisfying Φ(u0) < µ and

Iλ(u1) = infψ∈Υ

maxt∈[0,1]

Iλ(ψ(t)),

FRACTIONAL p-LAPLACIAN EQUATION 1457

where Υ is the family of paths γ : [0, 1] → (X, || · ||X) with ψ(0) = u0 andψ(1) = z, and z is such that Iλ(z) ≤ Iλ(u0).

Definition 2.8. Let 0 < s < 1 < p < +∞. We say that u ∈ Xps (Ω) is a weak

solution of the problem (Pλ) if

(2.6)

∫RN

∫RN

|u(x)− u(y)|p−2(u(x)− u(y))(v(x)− v(y))

|x− y|N+psdxdy

= λ

∫Ω

f(x, u)v dx

for all v ∈ Xps (Ω).

Let us define a functional Φs,p : Xps (Ω)→ R by

Φs,p(u) =1

p

∫RN

∫RN

|u(x)− u(y)|p

|x− y|N+psdxdy.

Then Φs,p is well defined on Xps (Ω), Φs,p ∈ C1(Xp

s (Ω),R) and its Frechetderivative is given by

〈Φ′s,p(u), v〉 =

∫RN

∫RN

|u(x)− u(y)|p−2(u(x)− u(y))(v(x)− v(y))

|x− y|N+psdxdy

for any v ∈ Xps (Ω) where 〈·, ·〉 denotes the pairing of Xp

s (Ω) and its dual(Xp

s (Ω))∗; see [31].

Lemma 2.9 ([31]). Let 0 < s < 1 < p < +∞. The functional Φ′s,p : Xps (Ω)→

(Xps (Ω))∗ is of type (S+), i.e., if un u in Xp

s (Ω) and

lim supn→∞

⟨Φ′s,p(un)− Φ′s,p(u), un − u

⟩≤ 0,

then un → u in Xps (Ω) as n→∞.

We assume that for 1 < q < p∗s and x ∈ Ω,

(F1) f : Ω × R → R satisfies the Caratheodory condition and there existnonnegative functions ρ ∈ L∞(Ω) and σ ∈ L∞(Ω) such that

|f(x, t)| ≤ ρ(x) + σ(x) |t|q−1

for all (x, t) ∈ Ω× R.

We denote with F the function defined by

F (x, ξ) :=

∫ ξ

0

f(x, t) dt for (x, ξ) ∈ Ω× R.

Define the functional Ψ : Xps (Ω)→ R by

Ψ(u) =

∫Ω

F (x, u) dx.

Next we define a functional Iλ : Xps (Ω)→ R by

Iλ(u) = Φs,p(u)− λΨ(u).

1458 Y.-H. KIM

Theorem 2.10. Suppose that (F1) holds and the following condition is verified:

(F2) lim supt→0+

∫ΩF (x, t) dx

tp= +∞.

Then, put

λ∗ =1

Cp(p)1p |Ω|

1p′ ||ρ||L∞(Ω) + q−1(p)

qpCqq ||σ||L∞(Ω)

for each λ ∈ (0, λ∗), the problem (Pλ) has at least one nontrivial weak solution.Furthermore, if q in (F1) satisfies p ≤ q < p∗s, then any weak solution of (Pλ)belongs to the space L∞(Ω).

Proof. It is clear that the functional Φs,p is in C1(Xps (Ω),R). In addition, from

the condition (F1) and Lemma 2.2, the functional Ψ is in C1(Xps (Ω),R) and

has compact derivative.We prove the existence of a nontrivial solution for λ ∈ (0, λ∗). Fix λ ∈

(0, λ∗), and choose µ = 1 to satisfy the condition (2.5) of Lemma 2.6. From(F2), there exists

(2.7) 0 < ξλ < min

1,

(p

2ω2Nd

N−spM

) 1p

such that

(2.8)p dspess infx∈Ω F (x, ξλ)

2N+1ξpλωNM>

1

λ,

where ωN is given in Lemma 2.1 and

M :=22p+N−sp−1

(p− sp)(N − sp+ p)+

2

2N−spsp(N + p− sp)+

1

sp(N − sp).

First of all, we show that Iλ satisfies the (C)[µ]-condition. Let µ be a fixedpositive number and let un be a Cerami sequence in Xp

s (Ω) satisfying (C1),(C2), and (C3). Since Φs,p(un) < µ, it follows from Lemma 2.1 that

||un||pXps (Ω)≤∫RN

∫RN

|un(x)− un(y)|p

|x− y|N+spdxdy +

1

p

∫Ω

|un(x)|p dx

≤ pΦs,p(un) + ||un||pLp(Ω) ≤ pµ

(1 +

s |Ω|spN

2ωspN +1

N

).

Thus, un is a bounded sequence in Xps (Ω) and we may suppose that un u

as n → ∞ for some u ∈ Xps (Ω). Then taking into account Lemma 2.2 we

have un → u as n → ∞ in Lp(Ω). From (C2), there is a sequence εnn∈N in(0,+∞), εn → 0+ such that

〈Φ′(un), u− un〉+ 〈(−λΨ)′(un), u− un〉 ≥−εn||v − un||Xps (Ω)

1 + ||un||Xps (Ω)

FRACTIONAL p-LAPLACIAN EQUATION 1459

for each n ∈ N . Since εn → 0+, we deduce

lim supn→+∞

〈Φ′(un), un − u〉 ≤ lim supn→+∞

〈(−λΨ)′(un), un − u〉.

Invoking the compactness of (−λΨ)′, we obtain

lim supn→+∞

〈Φ′(un), un − u〉 ≤ 0.

Since Φ′s,p is of type (S+), we assert that un → u in Xps (Ω) as n→∞.

Next, to apply Lemma 2.6 with Φ = Φs,p, we provide that there is an elementu ∈ Xp

s (Ω) satisfying Φs,p(u) < 1 and the relation (2.5). Define

u(x) =

0 if x ∈ RN \BN (x0, d),

ξλ if x ∈ BN (x0,d2 ),

2ξλd (d− |x− x0|) if x ∈ BN (x0, d) \BN (x0,

d2 ).

Then it is clear that 0 ≤ u(x) ≤ ξλ for all x ∈ RN , and so u ∈ Xps (Ω). Put

Bd = BN (x0, d). Then, it follows that

Φs,p(u) =1

p

∫RN

∫RN

|u(x)− u(y)|p

|x− y|N+spdxdy

=1

p

∫Bd\B d

2

∫Bd\B d

2

|u(x)− u(y)|p

|x− y|N+spdxdy

+2

p

∫Bd\B d

2

∫RN\Bd

|u(x)− u(y)|p

|x− y|N+spdxdy

+2

p

∫B d

2

∫Bd\B d

2

|u(x)− u(y)|p

|x− y|N+spdxdy

+2

p

∫RN\Bd

∫B d

2

|u(x)− u(y)|p

|x− y|N+spdxdy

=:1

p(I1 + 2I2 + 2I3 + 2I4).

Next we estimate I1–I4, by the direct calculation, respectively:

• Estimate of I1: For any positive constant ε small enough,

I1 =

∫Bd\B d

2

∫Bd\B d

2

|u(x)− u(y)|p

|x− y|N+spdxdy

≤2pξpλdp

∫Bd\B d

2

∫Bd\B d

2

|x− y|p

|x− y|N+spdxdy

≤2pξpλωNdp

∫Bd\B d

2

∫ d+|y−x0|

ε

rp−sp−1 drdy

1460 Y.-H. KIM

≤2pξpλωNdp

∫Bd\B d

2

(d+ |y − x0|)p−sp

p− spdy

=2pξpλω

2N

(p− sp)dp

∫ 2d

32d

rp+N−sp−1 dr

=2pξpλω

2Nd

N−sp

(p− sp)(p+N − sp)(2p+N−sp −

(3

2

)p+N−sp).

• Estimate of I2:

I2 =

∫Bd\B d

2

∫RN\Bd

|u(x)− u(y)|p

|x− y|N+spdxdy

≤2pξpλdp

∫Bd\B d

2

∫RN\Bd

|d− |y − x0||p

|x− y|N+spdxdy

=2pξpλωNdp

∫Bd\B d

2

∫ ∞d−|y−x0|

|d− |y − x0||p

rsp+1drdy

=2pξpλωNdpsp

∫Bd\B d

2

|d− |y − x0| |p−sp dy

=2pξpλω

2N

dpsp

∫ d2

0

rN+p−sp−1 dr

=ξpλd

N−spω2N

2N−spsp(N + p− sp).

• Estimate of I3:

I3 =

∫B d

2

∫Bd\B d

2

|u(x)− u(y)|p

|x− y|N+spdxdy

=2pξpλdp

∫B d

2

∫Bd\B d

2

| − d2 + |x− x0||p

|x− y|N+spdxdy

=2pξpλdp

∫Bd\B d

2

∫B d

2

| − d2 + |x− x0||p

|x− y|N+spdydx

=2pξpλωNdp

∫Bd\B d

2

| − d

2+ |x− x0| |p

∫ |x−x0|+ d2

|x−x0|− d2

1

rsp+1drdx

≤2pξpλωNdpsp

∫Bd\B d

2

| − d

2+ |x− x0| |p−sp dx

FRACTIONAL p-LAPLACIAN EQUATION 1461

=2pξpλω

2N

dpsp

∫ d2

0

tN+p−sp−1 dt

=dN−spξpλω

2N

2N−spsp(N + p− sp).

• Estimate of I4:

I4 =

∫B d

2

∫RN\Bd

|u(x)− u(y)|p

|x− y|N+spdxdy = ξpλ

∫B d

2

∫RN\Bd

1

|x− y|N+spdxdy

= ξpλωN

∫B d

2

∫ ∞d−|y−x0|

r−sp−1 drdy

= ξpλωN

∫B d

2

1

sp(d− |y − x0|)spdy

=ξpλω

2N

sp

∫ d

d2

rN−sp−1 dr

=ξpλω

2Nd

N−sp

sp(N − sp)(1− 1

2N−sp)

≤ξpλω

2Nd

N−sp

sp(N − sp).

Hence it follows from the relation (2.7) that

Φs,p(u) ≤2ξpλω

2Nd

N−spMp

≤ 1.

Owing to the relation (2.8), we infer that

Ψ(u) ≥∫B d

2

F (x, u) dx

≥ ess infx∈Ω

F (x, ξλ)

(ωNd

N

2N

)and thus

(2.9)Ψ(u)

Φs,p(u)≥ p dspess infx∈Ω F (x, ξλ)

2N+1ξpλωNM>

1

λ.

Since |u|W s,p(RN ) = Φs,p(u) ≤ p1p , the condition (F1), Holder’s inequality and

Remark 2.4 imply that, for each u ∈ Φ−1s,p((−∞, 1]), we get

Ψ(u) =

∫Ω

F (x, u) dx

≤∫

Ω

|ρ(x)| |u(x)| dx+

∫Ω

|σ(x)|q|u(x)|q dx

1462 Y.-H. KIM

≤ ||ρ||L∞(Ω) |Ω|1p′ ||u||Lp(Ω) + q−1||σ||L∞(Ω)||u||qLq(Ω)

≤ Cpp1p |Ω|

1p′ ||ρ||L∞(Ω) + q−1p

qpCqq ||σ||L∞(Ω),

and hence

(2.10)

supu∈Φ−1

s,p((−∞,1])

Ψ(u) ≤ Cpp1p |Ω|

1p′ ||ρ||L∞(Ω) + q−1p

qpCqq ||σ||L∞(Ω)

=1

λ∗<

1

λ.

Due to the inequalities (2.9) and (2.10), we have

supu∈Φ−1

s,p((−∞,1])

Ψ(u) <1

λ<

Ψ(u)

Φs,p(u).

Since λ ∈ (Φs,p(u)

Ψ(u) , 1supΦs,p(u)≤1 Ψ(u) ), Lemma 2.6 with µ = 1 and Φ = Φs,p

guarantees that the problem (Pλ) has at least one nontrivial weak solution foreach λ ∈ (0, λ∗).

For the case that p ≤ q < p∗s, we will show that a weak solution uλ,0 of theproblem (Pλ) belongs to the space L∞(Ω) for any λ ∈ (0, λ∗). Suppose thatuλ,0 is non-negative. For K > 0, we define

%K(x) = minuλ,0(x),K

and choose v = %mp+1K (m ≥ 0) as a test function in (2.6). Then, v ∈W (RN )∩

L∞(RN ), and it follows from (2.6) that

∫RN

∫RN

|uλ,0(x)− uλ,0(y)|p−2(uλ,0(x)− uλ,0(y))(%mp+1K (x)− %mp+1

K (y))

|x− y|N+psdxdy

(2.11)

= λ

∫Ω

f(x, uλ,0)%mp+1K dx.

The left-hand side of (2.11) can be estimated as follows:

∫RN

∫RN

|uλ,0(x)− uλ,0(y)|p−2(uλ,0(x)− uλ,0(y))(vmp+1K (x)− vmp+1

K (y))

|x− y|N+psdxdy

(2.12)

≥∫RN

∫RN

|uλ,0(x)− uλ,0(y)|p−1∣∣∣%mp+1K (x)− %mp+1

K (y)∣∣∣

|x− y|N+psdxdy

≥ C5

∫RN

∫RN

|%m+1K (x)− %m+1

K (y)|p

|x− y|N+psdxdy

≥ C6||%m+1K ||p

W (RN )

≥ C7

(∫Ω

|%K |(m+1)p∗s dx

) pp∗s

FRACTIONAL p-LAPLACIAN EQUATION 1463

for some positive constants C5, C6, C7.

Now, two cases arise, i.e., p < q < p∗s or q = p.

Case I: For p < q < p∗s, due to the assumption (F1) and the Holder in-equality, we get the following estimation for the right-hand side of (2.11) asfollows:

λ

∫Ω

f(x, uλ,0)%mp+1K dx

(2.13)

≤ λ

∫Ω

ρ(x)|uλ,0|mp+1 + σ(x)|uλ,0|mp+p dx

≤ λ

∫Ω

ρ(x)(|uλ,0|mp+p + |uλ,0|m+1) dx

+ λ||σ||L∞(Ω) |Ω|1τ′1

(∫Ω

|uλ,0|(m+1)pτ ′1 |uλ,0|(q−p)τ′1 dx

) 1τ′1

≤ λ||ρ||L∞(Ω)

∫Ω

|uλ,0|(m+1)p dx+ λ||ρ||L∞(Ω) |Ω|1p′

(∫Ω

|uλ,0|(m+1)p dx

) 1p

+ λ||σ||L∞(Ω) |Ω|1τ′1

(∫Ω

|uλ,0|(m+1)ζ dx

) pζ(∫

RN|uλ,0|

(q−p)τ ′1ζ

ζ−pτ′1 dx

) ζ−pτ′1ζτ′1

,

where τ1 =p∗sp∗s−q

, and ζ =pp∗sτ

′1

p∗s−(q−p)τ ′1. Obviously ζ ≤ p∗s, 1 < ζ

pτ ′1, and

(q−p)τ ′1ζζ−pτ ′1

= p∗s, and hence (2.13) yields

λ

∫Ω

f(x, uλ,0)%mp+1K dx

(2.14)

≤ λ||ρ||L∞(RN )

∫Ω

|uλ,0|(m+1)p dx+ λ||ρ||L∞(Ω) |Ω|1p′

(∫Ω

|uλ,0|(m+1)p dx

) 1p

+ λ||σ||L∞(Ω) |Ω|1τ′1

(∫Ω

|uλ,0|p∗s dx

) ζ−pτ′1ζτ′1

(∫RN|uλ,0|(m+1)ζ dx

) pζ

.

It follows from (2.11), (2.12), (2.14), and the Sobolev inequality that there existpositive constants C7, C8 and C9 (independent of K and m > 0) such that

||%K ||(m+1)p

L(m+1)p∗s (Ω)≤ C7||uλ,0||(m+1)p

L(m+1)p(Ω)+C8||uλ,0||m+1

L(m+1)p(Ω)+C9||uλ,0||(m+1)p

L(m+1)ζ(Ω).

Hence, for any positive constant K there are positive constants C10 and C11

such that

||%K ||L(m+1)p∗s (Ω) ≤

C1

(m+1)p

10 ||uλ,0||L(m+1)t(Ω) if ||uλ,0||L(m+1)p(Ω) ≥ 1,

C1

(m+1)p

11 ||uλ,0||L(m+1)ζ(Ω) if ||uλ,0||L(m+1)p(Ω) < 1,

1464 Y.-H. KIM

where t is either p or ζ. This estimation is a starting point for a bootstraptechnique. Since uλ,0(x) = lim

K→∞%K(x) for almost every x ∈ Ω, obvious modi-

fications of the proof of Proposition 1 in [20] yields that ||uλ,0||L∞(Ω) ≤ C14 forsome positive constant C14.

Case II: For q = p, as in the relation (2.13), the right-hand side of (2.11)can be estimated by using the assumption (F2) and the Holder inequality:

λ

∫Ω

f(x, uλ,0)%mp+1K dx

≤ λ||ρ||L∞(Ω)||uλ,0||(m+1)p

L(m+1)p(Ω)+ λ||ρ||L∞(Ω) |Ω|

1p′ ||uλ,0||m+1

L(m+1)p(Ω)

+ λ

∫Ω

σ(x)|uλ,0|(m+1)p dx

≤ λ(||ρ||L∞(Ω) + ||σ||L∞(Ω))||uλ,0||(m+1)p

L(m+1)p(Ω)+ λ||ρ||L∞(Ω) |Ω|

1p′ ||uλ,0||m+1

L(m+1)p(Ω).

This together with (2.11) and (2.12) yields that

||%K ||(m+1)p

L(m+1)p∗s (Ω)≤ C15||uλ,0||(m+1)p

L(m+1)p(Ω)+ C16||uλ,0||m+1

L(m+1)p(Ω)

for any positive constant K and for some positive constants C15 and C16. Usingthe bootstrap argument (similarly as in the proof of Proposition 1 in [20]), weconclude that uλ,0 belongs to L∞(Ω).

Similarly we can handle the case uλ,0(x) < 0 for almost all x ∈ Ω if wereplace uλ,0 by −uλ,0 in the above arguments. Consequently, we have an aprioriestimate in L∞(Ω) of any possible weak solution of (Pλ).

Theorem 2.11. Let f be satisfied (F1)–(F2). Assume furthermore that thefollowing conditions are verified:

(F3) lim|t|→∞f(x,t)

|t|p−1 =∞ uniformly for almost all x ∈ Ω.

(F4) There exist a constant ν ≥ 1 and a positive constant C∗ such that

νF(x, t) ≥ F(x, st)− C∗for (x, t) ∈ Ω× R and s ∈ [0, 1], where F(x, t) = f(x, t)t− pF (x, t).

Furthermore, if q in (F1) satisfies p < q < p∗s and λ∗ is given in Theorem 2.10,then, for each λ ∈ (0, λ∗], the problem (Pλ) has at least two nontrivial weaksolutions which belong to L∞(Ω).

Proof. With the analogous arguments as Theorem 1.6 in [30], the functional Iλsatisfies the (C)-condition. By the condition (F3), for any C > 0, there existsa constant δ > 0 such that

(2.15) F (x, t) ≥ C |t|p

for |t| > δ and for almost all x ∈ RN . Take v ∈ Xps (Ω) \ 0. Then the relation

(2.15) implies that

Iλ(tv) = Φs,p(tv)− λΨ(tv) ≤ tp(

1

p||v||p

Xps (Ω)− λC

∫Ω

|v(x)|p dx)

FRACTIONAL p-LAPLACIAN EQUATION 1465

for large enough t > 1. If C is sufficiently large, then we know that Iλ(tv)→ −∞as t → ∞ and thus Iλ is unbounded from below. If we follow the basic linesof the proof in Theorem 2.10, Lemma 2.7 with µ = 1 implies the existenceof at least two distinct nontrivial weak solutions to the problem (Pλ) for eachλ ∈ (0, λ∗).

To finish the proof, we will prove that (Pλ∗) has at least two nontrivialsolutions. The proof is quite close to that of [4, Theorem 3.1] (see also [3]).From Lemma 2.6 in the case µ = 1, for each λ ∈ (0, λ∗), we obtain the ex-istence of a nontrivial solution uλ for the problem (Pλ) in Φ−1

s,p((0, 1)) such

that Iλ(uλ) ≤ Iλ(v) for all v ∈ Φ−1s,p((0, 1)). From this, we drive a sequence

vn ⊂ Φ−1s,p((0, 1)) such that ||vn||Xps (Ω) → 0 as n→∞ and

Iλ(uλ) ≤ Iλ(vn).

Combining this with the continuity of Iλ, we deduce

(2.16) Iλ(uλ) ≤ 0 for all λ ∈ (0, λ∗).

Fix λ0 ∈ (0, λ∗). Choose a sequence λn such that 0 < λ0 < λn < λ∗

and λn → λ∗ as n → ∞. Then there is a corresponding sequence uλn inΦ−1s,p((0, 1)) such that uλn is a minimal point for the problem (Pλn). From

Definition 2.8, we arrive at

(2.17) 〈Φ′s,p(uλn), v〉 = λn

∫Ω

f(x, uλn)v dx

for any v ∈ Xps (Ω). Since uλn is bounded, we may suppose that uλn uλ∗

in Xps (Ω) and uλn → uλ∗ in Lq(Ω) as n → ∞ for any 1 < q < p∗s, due to

Lemma 2.2. It follows from (2.17) with v = uλn − u∗ that we get

lim supn→∞

〈Φ′s,p(uλn), uλn − uλ∗〉 = lim supn→∞

λn

∫Ω

f(x, uλn)(uλn − uλ∗) dx = 0.

Since Φ is of type (S+) by Lemma 2.9, we conclude that uλn → uλ∗ in Xps (Ω)

and so Iλ∗(uλ∗) ≤ Iλ∗(v) for all v ∈ Φ−1s,p((0, 1)), because λn → λ∗ as n → ∞.

Letting n→∞ in (2.17) gives

〈Φ′s,p(uλ∗), v〉 = λ∗∫

Ω

f(x, uλ∗)v dx

for all v ∈ Xps (Ω). Hence uλ∗ is a critical point for Iλ∗ and so is a weak solution

for (Pλ∗). Also since uλn → uλ∗ in Xps (Ω) and λn → λ∗ as n → ∞, one has

Iλ∗(uλ∗) ≤ Iλ∗(v) for all v ∈ Φ−1s,p((0, 1)).

Now, we claim that uλ∗ 6= 0. Note that every uλ is a minimal point forIλ and Iλ(uλ) ≤ Iλ(v) for all v ∈ Φ−1

s,p((0, 1)) and all λ ∈ (0, λ∗). Since

uλ ∈ Φ−1s,p((0, 1)) for every λ ∈ (0, λ∗), we see that

Iλn(uλn) ≤ Iλn(uλ0) and Iλ0(uλ0) ≤ Iλ0(uλn) for all n ∈ N.According to λ0 < λn for all n ∈ N, these inequality imply that

Ψ(uλ0) ≤ Ψ(uλn) for all n ∈ N,

1466 Y.-H. KIM

and thus

(2.18) Ψ(uλ0) ≤ Ψ(uλ∗).

In fact, if we let uλ∗ = 0 in (2.18), then we get Ψ(uλ0) ≤ 0 and so Iλ0(uλ0) > 0which contradicts (2.16). Since Iλ∗ is unbounded from below and uλ∗ is anontrivial local minimum, it is not strictly global. This together with Lemma2.7 guarantees that the given problem has a nontrivial weak solution whichis different from uλ∗ . As in the analogous arguments in Theorem 2.10, weconclude that any possible weak solution of (Pλ) belongs to the space L∞(Ω).

3. Appendix

In this section, our main results continue to hold when (Pλ) is replaced bythe equations driven by a non-local integro-differential operator of elliptic typeas follows:

(PK)

−LKu = λf(x, u) in Ω,

u = 0 on RN\Ω.

Here LK is a non-local operator LK defined pointwise as

LKu(x) = 2

∫RN|u(x)− u(y)|p−2(u(x)− u(y))K(x− y)dy for all x ∈ RN ,

where K : RN \0 → (0,+∞) is a kernel function with the following properties

(K1) mK ∈ L1(RN ), where m(x) = min|x|p, 1;(K2) there exist positive constants θ0, θ1 such that θ0 ≤ K(x)|x|N+ps ≤ θ1

for almost all x ∈ RN \ 0;(K3) K(x) = K(−x) for all x ∈ RN \ 0.

By the condition (K1), the function

(x, y) 7→ (u(x)− u(y))K(x− y)1p ∈ Lp(R2N )

for all u ∈ C∞0 (RN ). Let us denote with W s,pK (RN ) the completion of C∞0 (RN )

with respect to the norm

||u||W s,pK (RN ) :=

(||u||p

Lp(RN )+ |u|p

W s,pK (RN )

) 1p

,

where

|u|pW s,pK (RN )

:=

∫RN

∫RN|u(x)− u(y)|pK(x− y) dxdy.

Lemma 3.1 ([35]). Let K : RN \ 0 → (0,∞) be a function satisfying theconditions (K1)–(K3). Then if u ∈W s,p

K (RN ), then u ∈W s,p(RN ). Moreover

||u||W s,p(RN ) ≤ max1, θ−1p

0 ||u||W s,pK (RN );

From Lemmas 2.1 and 3.1, we can obtain the following assertion immediately.

FRACTIONAL p-LAPLACIAN EQUATION 1467

Lemma 3.2 ([35]). Let K : RN \ 0 → (0,∞) satisfy the conditions (K1)–(K3). Then there exists a positive constant C0 = C0(N, p, s) such that for anyu ∈W s,p

K (RN ) and 1 ≤ q ≤ p∗s

||u||pLq(Ω) ≤ C0∫RN

∫RN

|u(x)− u(y)|p

|x− y|N+psdxdy

≤ C0θ0

∫RN

∫RN|u(x)− u(y)|pK(x− y) dxdy;

In this section, the basic space is the closed linear subspace defined as

XK(Ω) =u ∈W s,p

K (RN ) : u(x) = 0 a.e. in RN\Ω

with the norm

||u||XK(Ω) :=

(|u|p

W s,pK (RN )

+ ||u||pLp(Ω)

) 1p

.

Then XK(Ω) is a uniformly convex Banach space; see Lemma 2.4 in [35].

Definition 3.3. Let 0 < s < 1 < p < +∞ with ps < N and conditions (K1)-(K3) are satisfied. We say that u ∈ XK(Ω) is a weak solution of the problem(PK) if ∫

RN

∫RN|u(x)− u(y)|p−2(u(x)− u(y))(v(x)− v(y))K(x− y) dxdy

= λ

∫Ω

f(x, u)v dx

for all v ∈ XK(Ω).

Let us define a functional Φp,K : XK(Ω)→ R by

Φp,K(u) =1

p

∫RN

∫RN|u(x)− u(y)|pK(x− y) dxdy.

Then the functional Φp,K is well defined on XK(Ω), Φp,K ∈ C1(X0,R) and itsFrechet derivative is given by

〈Φ′p,K(u), v〉

=

∫RN

∫RN|u(x)− u(y)|p−2(u(x)− u(y))(v(x)− v(y))K(x− y) dxdy

for any v ∈ XK(Ω); see [31].With the help of Lemma 3.2, if we replace Xp

s (Ω) and Φs,p by XK(Ω) andΦp,K , respectively, then the proofs of the following consequences are almostidentical to those of Theorems 2.10 and 2.11. Hence we skip their proofs.

Theorem 3.4. Suppose that conditions (K1)–(K3) are satisfied. Let (F1)–(F2) hold. Then there exists λ∗ > 0 such that the problem (PK) has at leastone nontrivial weak solution for each λ ∈ (0, λ∗). Furthermore, if q in (F1)satisfies p ≤ q < p∗s, then any weak solution of (PK) belongs to the spaceL∞(Ω).

1468 Y.-H. KIM

Theorem 3.5. Suppose that conditions (K1)–(K3) are satisfied. Assume that(F1)–(F4) hold. Furthermore, if q in (F1) satisfies p < q < p∗s and λ∗ is givenin Theorem 3.4, then, for each λ ∈ (0, λ∗], the problem (PK) has at least twonontrivial weak solutions which belong to L∞(Ω).

Acknowledgements. This research was supported by a 2017 Research Grantfrom Sangmyung University.

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Yun-Ho Kim

Department of Mathematics Education

Sangmyung UniversitySeoul 03016, Korea

Email address: kyh1213@smu.ac.kr