Three solutions for a fractional elliptic problems with critical and supercritical growth

Acta Mathematica Scientia 2016,36B(6):1819–1831 http://actams.wipm.ac.cn

THREE SOLUTIONS FOR A FRACTIONAL ELLIPTIC PROBLEMS WITH CRITICAL AND SUPERCRITICAL GROWTH∗

Ü7I)

Jinguo ZHANG (

School of Mathematics, Jiangxi Normal University, Nanchang 330022, China E-mail : [email protected]

4¡S)

Xiaochun LIU (

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail : [email protected] Abstract In this paper, we deal with the existence and multiplicity of solutions to the fractional elliptic problems involving critical and supercritical Sobolev exponent via variational arguments. By means of the truncation combining with the Moser iteration, we prove that our problem has at least three solutions. Key words

fractional elliptic equation; variational methods; three solutions; Moser iteration

2010 MR Subject Classification

1

35J60; 47J30

Introduction

In this paper, we consider the existence and multiplicity of solutions for the fractional elliptic problem ( (−∆)s u = λf (x, u) + µ|u|p−2 u in Ω, (1.1) u=0 on ∂Ω, where Ω ⊂ RN , N > 2s, is a smooth bounded domain and 0 < s < 1. Here (−∆)s stands for the fractional Laplacian, p ≥ 2∗s = N2N −2s , µ and λ are nonnegative constants and f : Ω × R → R is a Car´atheodory function. In a bounded domain Ω ⊂ RN , we define the operator (−∆)s as follows, see [13] for details. Let {λk , ϕk }∞ k=1 be the eigenvalues and corresponding eigenfunctions of the Laplacian operator −∆ in Ω with zero Dirichlet boundary values on ∂Ω normalized by kϕk kL2 (Ω) = 1, i.e., −∆ϕk = λk ϕk For any u ∈ L2 (Ω), we may write ∞ X u= uk ϕk , k=1

in Ω;

ϕk = 0 on ∂Ω.

where uk =

Z

uk ϕk dx.

Ω

∗ Received February 28, 2015; revised April 28, 2016. Supported by NSFC (11371282, 11201196) and Natural Science Foundation of Jiangxi (20142BAB211002).

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With this spectral decomposition, the fractional powers of the Dirichlet Laplacian (−∆)s can be defined for u ∈ C0∞ (Ω) by ∞ X (−∆)s u = λsk uk ϕk . k=1

By density, the operator (−∆)s can be etended to the Hilbert Space   ∞ ∞ X X s 2 s 2 H (Ω) = u = uk ϕk ∈ L (Ω) : λk uk < ∞ , k=1

(1.2)

k=1

which is equipped with the norm kukHs (Ω) =

X ∞

λsk u2k

k=1

 21

.

The theory of Hilbert scales presented in the classical book by Lions and Magenes [11] shows that s

[H01 (Ω), L2 (Ω)]θ = Dom(−∆) 2 , where θ = 1 − s. This implies the following characterization of the space Hs (Ω),  1   H s (Ω), if s ∈ (0 , ),   2   1 1 s 2 H (Ω) = H00 (Ω), if s = ,  2    1  s  H (Ω), if s ∈ ( , 1). 0 2

One of the main difficulties in the study of problem (1.1) is that the fractional Laplacian is a nonlocal operator. To localize it, Caffarelli and Silvestre [1] developed a local interpretation of the fractional Laplacian in RN by considering a Dirichlet to Neumann type operator in the domain {(x, y) ∈ RN +1 : y > 0}. A similar extension, in a bounded domain with zero Dirichlet boundary condition, was establish by Cabr´e and Tan in [2] , Tan [3] and by Br¨andle, Colorado, 1 de Pablo and S´anchez [4]. For any u ∈ Hs (Ω), the solution w ∈ H0,L (CΩ ) of   div(y 1−2s ∇w) = 0 in CΩ = Ω × (0, ∞),   (1.3) w=0 on ∂L CΩ = ∂Ω × (0, ∞),    w=u on Ω × {0}

is called the s-harmonic extension w = Es (u) of u, and it belongs to the space   Z 2 1−2s 1−2s 2 1 ) : w = 0 on ∂L CΩ , y |∇w| dxdy < ∞ . H0,L (CΩ ) = w ∈ L (CΩ , y CΩ

It is proved that −ks lim y 1−2s y→0+

∂w = (−∆)s u, ∂y

1 where ks = 21−2s Γ(1 − s)/Γ(s). Here H0,L (CΩ ) is a Hilbert space endowed with the norm

1 (C ) = kwkH0,L Ω



ks

Z

CΩ

y

1−2s

2

|∇w| dxdy

 12

.

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Therefore, the nonlocal problem (1.1) can be reformulated to the following local problem  − div(y 1−2s ∇w) = 0 in CΩ ,     w=0 on ∂L CΩ , (1.4)    1−2s ∂w p−2  y = λf (x, w(x, 0)) + µ|w(x, 0)| w(x, 0) in Ω × {0}, ∂ν

∂ where ∂ν is the outward normal derivative of ∂CΩ . We then turn to study the equivalent 1 problem (1.4). Define on H0,L (CΩ ), the functional Z Z Z 1 µ 1−2s 2 p I(w) = ks y |∇w| dxdy − |w(x, 0)| dx − λ F (x, w(x, 0))dx, 2 p Ω CΩ Ω Ru where F (x, u) = 0 f (x, t)dt. Critical points of I(w) are weak solutions of (1.4). In this paper, we study the existence and multiplicity of solutions for the problem with critical and supercritical growth. In our problem, the first difficulty lies in that the fractional Laplacian operator (−∆)s is nonlocal, and this makes some calculations difficult. To overcome this difficulty, we do not work on the space Hs (Ω) directly, and we transform the nonlocal problem into a local problem by the extension introduced by Caffarelli and Silvestre in [1]. The second difficulty lies in which problem (1.4) is a supercritical problem. We can not use directly the variational techniques because the corresponding energy functional is not well-defined on 1 Hilbert space H0,L (CΩ ). To overcome this difficulty, one usually uses the truncation and the Moser iteration. This spirt has been widely applied in the supercritical Laplacian equation in the past decades, see [5–10] and references therein. The aim of this paper is to study problem (1.4) when p ≥ 2∗s . In order to state our main results, we formulate the following assumptions (f1 ) lim f (x,t) = 0 uniformly in x ∈ Ω; |t|

(f2 ) (f3 )

|t|→+∞ lim f (x,t) |t|→0 |t|

sup u∈Hs (Ω)

Set 1 θ := inf 2

R



Ω

= 0 uniformly in x ∈ Ω; F (x, u)dx > 0, and for every M > 0, f (x, u) ∈ L∞ (Ω) for each |u| ≤ M .

ks

y 1−2s |∇w|2 dxdy 1 R CΩ : w ∈ H0,L (CΩ ) and F (x, w(x, 0))dx Ω R

Z

Ω

 F (x, w(x, 0))dx > 0 .

The main results are as follows.

Theorem 1.1 Assume that (f1 )–(f3 ) hold. Then there exists a δ > 0 such that for any µ ∈ [0 , δ], there are a compact interval [a , b] ⊂ ( 1θ , +∞) and a constant γ > 0 such that 1 1 problem (1.4) has at least three solutions in H0,L (CΩ ) for each λ ∈ [a , b], whose H0,L (CΩ )norms are less than γ. For the general problem ( (−∆)s u = λf (x, u) + µg(x, u) u=0

in Ω, on ∂Ω,

where Ω ⊂ RN is a bounded smooth domain, and (g) |g(x, u)| ≤ C(1 + |u|p−1 ), where p ≥ 2∗s = N2N −2s , C > 0. If f satisfies conditions (f1 )–(f3 ), we also have similar result.

(1.5)

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Theorem 1.2 Assume that (f1 )–(f3 ) and (g) hold. Then there exists a, δ > 0 such that for any µ ∈ [0 , δ], there are a compact interval [a , b] ⊂ ( θ1 , +∞) and a constant γ > 0 such that 1 1 problem (1.5) has at least three solutions in H0,L (CΩ ) for each λ ∈ [a , b], whose H0,L (CΩ )-norms are less than γ. The paper is organized as follows. In Section 2, we introduce a variational setting of the problem and present some preliminary results. In Section 3, some properties of the fractional operator are discussed, and we apply the truncation and the Moser iteration to obtain the proof of Theorems 1.1 and 1.2. For convenience we fix some notations. Lp (Ω) (1 < p ≤ ∞) denotes the usual Lebesgue ¯ denotes the space of continuous real functions in Ω ¯ vanishing space with norm k · kLp ; C0 (Ω) on the boundary ∂Ω; C or Ci (i = 1, 2, · · · , ) denote different positive constants.

2

Preliminaries and Functional Setting

Let us recall some function spaces, for details the reader to [12, 13]. For 0 < s < 1, we introduce the so-called Gagliardo-Slobodeckii seminorm Z Z |u(x) − u(y)|2 [u]2H s (Ω) = dxdy. N +2s Ω Ω |x − y|

The Sobolev space H s (Ω) of order is defined by n H s (Ω) = u ∈ L2 (Ω) :

which, equipped with the norm

o [u]H s (Ω) < ∞

  12 kukH s (Ω) = kuk2L2 (Ω) + [u]2H s (Ω)

is a Hilbert space. Let H0s (Ω) be the closure of Cc∞ (Ω) with respect to the norm k · kH s (Ω) , i.e., H s (Ω)

H0s (Ω) = C0∞ (Ω)

.

If the boundary of Ω is smooth, the space H s (Ω) can be defined as interpolation spaces of index θ = 1 − 2s for pair [H 1 (Ω) , L2 (Ω)]θ . Analogously, for s ∈ [0 , 1]\{ 21 }, the spaces H0s (Ω) are defined as interpolation spaces of index θ = 1 − 2s for pair [H01 (Ω) , L2 (Ω)]θ , that is, 1 H0s (Ω) = [H01 (Ω) , L2 (Ω)]θ , θ 6= . 2 1

2 The space H00 (Ω) = [H01 (Ω) , L2 (Ω)] 12 ,2 is the so-called Lions-Magenes space, which can be characterized as   Z 2 1 1 u (x) 2 2 dx < ∞ H00 (Ω) = u ∈ H (Ω) : Ω d(x)

and d(x) = dist(x , ∂Ω) for all x ∈ Ω. It was known from [11] that for 0 < s ≤ 12 , H0s (Ω) = H s (Ω); for 21 < s < 1, H0s (Ω) $ H s (Ω). Moreover, from [11] (see [13]), it is known that  1   H s (Ω), if s ∈ (0 , ),   2   1 1 s 2 H (Ω) = H00 (Ω), if s = ,  2    1  s  H (Ω), if s ∈ ( , 1). 0 2 Furthermore, we recall a result in [14].

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1 Lemma 2.1 There exists a trace operator from H0,L (CΩ ) in to H0s (Ω). Furthermore, the space Hs (Ω) given by (1.2) is characterized by n o 1 Hs (Ω) = u = trw : w ∈ H0,L (CΩ ) .

1 Lemma 2.1 was proved in [14]. In its proof, we see in fact that the mapping tr : H0,L (CΩ ) ֒→ s H (Ω) is continuous, and this operator has its image contained in H0 (Ω). Next, we have the Sobolev embedding theorem. s

Lemma 2.2 Given s > 0 and 1p > 1 so that p1 ≥ 12 − Ns , the inclusion map i : H s (Ω) → Lp (Ω) is well defined and bounded. If the above inequality is strict, then the inclusion is compact. By Lemma 2.1 and Lemma 2.2, we now that there exists a continuous linear mapping from to Lp (Ω) if 2 ≤ p ≤ N2N −2s . Then we will list following lemma.

1 H0,L (CΩ )

1 Lemma 2.3 For every 2 ≤ r ≤ 2∗s and w ∈ H0,L (CΩ ), it holds

Z

Ω

 r2 Z ≤C |w(x, 0)|r dx

y 1−2s |∇w|2 dxdy,

CΩ

where C > 0 depends on r, s, N and Ω. Theorems 1.1 and 1.2 will be proved in an idea from a recent result on the existence of at least three critical points by Ricceri [15, 16]. For the readers convenience, we state it as follows. If X is a real Banach space, we denote by X the class of all function φ : X → R possessing the following property: if {un } ⊂ X is a sequence converging weakly to u ∈ X and lim inf φ(un ) ≤ φ(u), then {un } has a subsequence converging strongly to u. n→∞

Theorem 2.4 Let X be a separable and reflexive real Banach space and I ⊆ R be an interval. A C 1 functional Φ : X → R a sequentially weakly lower semi-continuous, bounded on each bounded subset of X, and belonging to X . The derivative of Φ admits a continuous inverse on X ∗ . The functional J : X → R is a C 1 functional with compact derivative. Assume that the functional Φ − λJ is coercive for each λ ∈ I, and it has a strict local but not global minimum, say u ˆλ . Then there exists a number γ > 0 for each compact interval [a, b] ⊆ I for which sup (Φ(ˆ uλ ) − λJ(ˆ uλ )) < +∞, such that the following property holds: there exists λ∈[a,b]

δ0 > 0, for every λ ∈ [a, b] and every C 1 functional Ψ : X → R with compact derivative, such that the equation Φ′ (u) = λJ ′ (u) + µΨ′ (u) has at least three solutions whose norm are less than γ for each µ ∈ [0 , δ0 ].

3

Proof of Main Results Let Φ(u) =

1 ks 2

Z

y 1−2s |∇w|2 dxdy and J(w) =

CΩ

Z

Ω

Obviously, condition (f3 ) implies J(w) > 0. Φ(w) Φ(w)66=0

θ = sup

F (x, w(x, 0))dx.

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Lemma 3.1 Let f satisfy (f1 )–(f3 ). Then for every λ ∈ (0 , ∞), the functional Φ − λJ 1 is sequentially weakly lower continuous and coercive on H0,L (CΩ ), and has a global minimizer 1 wλ ∈ H0,L (CΩ ). Proof By (f1 ) and (f3 ), for any ε > 0, there exist M0 > 0 and C1 > 0 such that for all 1 w ∈ H0,L (CΩ ), we have |f (x, w(x, 0))| ≤ ε|w(x, 0)|, |f (x, w(x, 0))| ≤ C1 ,

∀ |w(x, 0)| ≥ M0 ,

∀ |w(x, 0)| ≤ M0 + 1,

x ∈ Ω, x ∈ Ω,

which implies that |f (x, w(x, 0))| ≤ C1 + ε|w(x, 0)| and

(3.1)

Z ε |F (x, w(x, 0))| = f (x, w(x, 0))dx ≤ C1 |w(x, 0)| + |w(x, 0)|2 . 2 Ω

Then

Z Z 1 ks y 1−2s |∇w|2 dxdy − λ F (x, w(x, 0))dx 2 CΩ Ω Z Z 1 ε ≥ y 1−2s |∇w|2 dxdy − λ (C1 |w(x, 0)| + |w(x, 0)|2 )dx 2 CΩ 2 Ω Z Z 1 ε 2 = kwkH 1 (CΩ ) − λ |w(x, 0)|2 dx − λC1 |w(x, 0)|dx 0,L 2 2 Ω Ω 1 εC2  1 (C ) , ≥ −λ kwk2H 1 (CΩ ) − λC1 C3 kwkH0,L Ω 0,L 2 2

Φ(w) − λJ(w) =

where the constants C2 > 0, C3 > 0. Let ε > 0 small enough such that we have 1 (C ) → ∞. Φ(w) − λJ(w) → +∞ as kwkH0,L Ω

1 2

− λ εC2 2 > 0, and then

Hence Φ − λJ is coerciveness. 1 Moreover, from the embedding H0,L (CΩ ) ֒→ Lr (Ω) is continuous and (3.1), J is weakly continuous. Obviously, Z 1 1 Φ(u) = ks y 1−2s |∇w|2 dxdy = kwk2H 1 (CΩ ) 0,L 2 2 CΩ

1 is weakly lower semi-continuous on H0,L (CΩ ). We can deduce that Φ − λJ is a sequentially weakly lower semi-continuous, that is, Φ − λJ ∈ X . Therefore, Φ − λJ has a global minimizer 1 wλ ∈ H0,L (CΩ ). The proof is completed. 

Next, we will show that Φ − λJ has a strictly local, but not global minimizer for some λ, when f satisfies (f1 )–(f3 ). Lemma 3.2 Let f satisfy (f1 )–(f3 ). Then (i) 0 is a strictly local minimizer of the functional Φ − λJ for λ ∈ (0 , +∞). (ii) wλ 6= 0, i.e., 0 is not the global minimizer wλ for λ ∈ ( 1θ , +∞), where wλ is given by Lemma 3.1. Proof

First, we prove that J(w) = 0, →0 Φ(w) (CΩ )

lim

kwkH 1

0,L

1 ∀w ∈ H0,L (CΩ ).

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In fact, by (f2 ), for any ε > 0, there exists a δ > 0, such that |w(x, 0)| < δ and |f (x, w(x, 0))| ≤ ε|w(x, 0)|.

(3.2)

Considering inequality (3.2), (f1 ) and (f3 ), there exists r ∈ (1 , 2∗s − 1) such that |f (x, w(x, 0))| ≤ ε|w(x, 0)| + |w(x, 0)|r ,

1 ∀ w ∈ H0,L (CΩ ).

(3.3)

Then from Sobolev embedding theorem, there exist C4 , C5 > 0, such that |J(w)| ≤ ε C4 kwk2H 1

+ C5 kwkr+1 H1

0,L (CΩ )

0,L (CΩ )

.

This implies lim

kwkH 1

0,L

Next, we will prove (i) and (ii). (i) For λ ∈ (0 , +∞), since

(CΩ

→0 )

J(w) = 0. Φ(w)

J(w) →0 Φ(w)

lim

kwkH 1

(C ) 0,L Ω

=0<

1 λ

and Φ(w) > 0 for each w 6= 0 in

some neighborhood U of 0, there exists a neighborhood V ⊆ U of 0 such that Φ(w) − λJ(w) > 0,

∀ w ∈ V \ {0}.

Hence 0 is a strictly local minimizer of Φ − λJ. 1 (ii) For λ ∈ ( θ1 , +∞), from the definition of θ, there exists w ˆ ∈ H0,L (CΩ ) such that J(w) ˆ 1 Φ(w) ˆ > 0, J(w) ˆ > 0 and Φ(w) > . So we have ˆ λ Φ(w) ˆ − λJ(w) ˆ < 0 = Φ(0) − λJ(0). It yields that 0 is not a global minimizer of Φ − λJ. This completes the proof.



Let K > 0 be a real number whose value will be fixed later. Define the truncation of |w| w with p > 2∗s by ( p−2 |w| w, if 0 ≤ |w| ≤ K, gK (w) = K p−q |w|q−2 w, if |w| > K, p−2

where q ∈ (2 , 2∗s ). Then gK (w) satisfies |gK (w)| ≤ K p−q |w|q−1 for K large enough. Then, we study the truncated problem  1−2s ∇w) = 0 in CΩ ,   − div(y   w=0 on ∂L CΩ ,     y 1−2s ∂w = λf (x, w) + µgK (w) in Ω × {0}. ∂ν 1 We say that w ∈ H0,L (CΩ ) is a weak solution of problem (3.4) if Z Z Z ks y 1−2s ∇w · ∇ϕdxdy = λ f (x, w(x, 0))ϕ(x, 0)dx + µ gK (w(x, 0))ϕ(x, 0)dx CΩ

Ω

1 holds for every ϕ ∈ H0,L (CΩ ). Let

Ψ(w) =

Ω

Z

Ω

GK (w(x, 0))dx,

(3.4)

(3.5)

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where GK (w) =

Z

Vol.36 Ser.B

w

gK (t)dt.

0

So from |gK (w)| ≤ K p−q |w|q−1 , 2 < q < 2∗s , we get that gK (w) is a super-linear function 1 with subcritical growth. Then Ψ(w) has a compact derivative in H0,L (CΩ ). Moreover, for 1 each compact interval [a, b] ⊂ ( θ , +∞) and λ ∈ [a, b], we have that J(w) has also a compact 1 derivative in H0,L (CΩ ). Therefore, it is easy to see that the functional E(w) = Φ(w) − λJ(w) − µΨ(w) is C 1 and its derivative is given by Z Z hE ′ (w), ϕi = ks y 1−2s ∇w∇ϕdxdy − λ f (x, w(x, 0))ϕ(x, 0)dx CΩ Ω Z −µ gK (w(x, 0))ϕ(x, 0)dx Ω

1 H0,L (CΩ )

1 for all w ∈ and ϕ ∈ H0,L (CΩ ). By Lemma 3.1 and Lemma 3.2, we know that all hypotheses of Theorem 2.4 are satisfied. So there exists γ > 0 with the following property: for every λ ∈ [a, b] ⊂ ( 1θ , +∞), there exists δ0 > 0, such that for µ ∈ [0, δ0 ], problem (3.4) has at least three solutions w0 , w1 and w2 in 1 H0,L (CΩ ) and 1 (C ) ≤ γ, kwk kH0,L Ω

k = 0, 1, 2,

where γ depends on λ, but does not depend on µ or K. If these three solutions satisfy |wk | ≤ K

a.e. (x, y) ∈ Ω × {0} for k = 0, 1, 2,

(3.6)

then in the view of the definition gK , we have gK (x, w) = µ|w|p−2 w, and therefore wk , k = 0, 1, 2, are also solutions of the original problem (1.4). This implies that problem (1.1) has at least three solutions uk (x) = trwk (k = 0, 1, 2). Thus, in order to prove Theorem 1.1, it suffices to show that exists δ0 > 0 such that the solutions obtained by Theorem 2.4 satisfy inequality (3.6) for µ ∈ [0, δ0 ]. Proof of Theorem 1.1 Our aim is to show that exits δ0 > 0 such that for µ ∈ [0, δ0 ], the solution wk , k = 0, 1, 2, satisfy inequality (3.6). For simplicity, we will denote w := wk , k = 0, 1, 2. Set w+ = max{w , 0}, w− = − min{w , 0}. Then |w| = w+ + w− . We can argue with the positive and negative part of w separately. We first deal with w+ . For each L > 0, we define the following function ( w+ , if w+ ≤ L, wL = L, if w+ > L. For β > 1 to be determined later, we choose in (3.5) that 2(β−1)

ϕ = wL

w+

and 2(β−1)

∇ϕ = wL

2(β−1)−1

∇w+ + 2(β − 1)wL

w+ ∇wL ,

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Then we obtain Z y 1−2s ∇w∇ϕdxdy ZCΩ 2(β−1) = y 1−2s (∇(w+ − w− ))∇(wL w+ )dxdy CΩ Z 2(β−1) 2(β−1)−1 = y 1−2s (∇w+ − ∇w− )(wL ∇w+ + 2(β − 1)wL w+ ∇wL )dxdy CΩ Z   2(β−1) 2(β−1)−1 + 2(β − 1)wL w+ ∇wL ∇w+ dxdy = y 1−2s |∇w+ |2 wL C Z Ω Z 2(β−1) 2(β−1)−1 = y 1−2s wL |∇w+ |2 dxdy + 2(β − 1) y 1−2s wL w+ ∇wL ∇w+ dxdy. (3.7) CΩ

CΩ

From the definition of wL , we have Z 2(β−1)−1 2(β − 1) y 1−2s wL w+ ∇wL ∇w+ dxdy CΩ Z 2(β−1)−1 = 2(β − 1) y 1−2s wL w+ ∇wL ∇w+ dxdy {w+
≥ 0.

(3.8)

Set hK (x, w) = λf (x, w) + µgK (x, w),

1 ∀w ∈ H0,L (CΩ ).

From (3.3) and |gK (x, w)| ≤ K p−q |w|q−1 , we can choose a constant C6 > 0 such that |hK (x, w)| ≤ C6 |w| + µK p−q |w|q−1 .

(3.9)

We deduce from (3.5), (3.7), (3.8) and (3.9) for β > 1 that Z 2(β−1) y 1−2s wL |∇w+ |2 dxdy CΩ Z = hK (x, w(x, 0))ϕdx ZΩ ≤ |hK (x, w(x, 0))ϕ|dx Ω Z   2(β−1) ≤ C6 |w(x, 0)| + µK p−q |w(x, 0)|q−1 wL (x, 0)w+ (x, 0)dx Ω Z   2(β−1) = C6 (w+ (x, 0) + w− (x, 0)) + µK p−q (w+ (x, 0) + w− (x, 0))q−1 wL (x, 0)w+ (x, 0)dx ZΩ   2(β−1) 2(β−1) q 2 = C6 w+ (x, 0)wL (x, 0) + µK p−q w+ wL (x, 0) dx. (3.10) Ω

β−1 Let w ˆL = w+ wL , we have

β−1 β−2 ∇w ˆL = wL ∇w+ + (β − 1)w+ wL ∇wL .

By the Sobolev embedding theorem, we obtain Z  22∗ ∗ |w ˆL (x, 0)|2s dx s Ω

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Z

Z

Vol.36 Ser.B

y 1−2s |∇w ˆL |2 dxdy CΩ

β−1 β−2 y 1−2s |wL ∇w+ + (β − 1)w+ wL ∇wL |2 dxdy Z Z  β−2 β−1 ≤ 2S y 1−2s |(β − 1)w+ wL ∇wL |2 dxdy + y 1−2s |wL ∇w+ |2 dxdy CΩ CΩ Z Z β−1 = 2S y 1−2s (β − 1)2 |wL |2(β−1) |∇w+ |2 dxdy + y 1−2s |wL ∇w+ |2 dxdy) CΩ CΩ Z Z   2(β−1) 1−2s 2(β−1) 2 2 y wL |∇w+ | dxdy + y 1−2s wL |∇w+ |2 dxdy ≤ 2S (β − 1) CΩ CΩ  Z 2(β−1) = 2S (β − 1)2 + 1 y 1−2s wL |∇w+ |2 dxdy CΩ Z  β−1 1  2(β−1) 2 2 = 2S β ( ) + 2 y 1−2s wL |∇w+ |2 dxdy, (3.11) β β CΩ

=S

CΩ

where S > 0 is the best Sobolev embedding constant. Since β > 1, we have

1 β2

2 < 1 and ( β−1 β ) < 1. From (3.10) and (3.11), we get

Z  β−1 1  2(β−1) 2S β 2 ( )2 + 2 y 1−2s wL |∇w+ |2 dxdy β β CΩ Z 2 1−2s 2(β−1) < 4 Sβ y wL |∇w+ |2 dxdy ZCΩ  2(β−1) 2(β−1) q 2 2 ≤ 4 Sβ C6 w+ (x, 0)wL (x, 0) + µK p−q w+ (x, 0)wL (x, 0) dx.

(3.12)

Ω

∗

1 1 (C ) ≤ γ, we have Moreover, by the Sobolev embedding H0,L (CΩ ) ֒→ L2s (Ω) and kw+ kH0,L Ω

Z

Ω

Z  22∗ ∗ |w+ (x, 0)|2s dx s ≤ S

y 1−2s |∇w+ |2 dxdy ≤ S γ.

(3.13)

CΩ

2 2∗

2(β−1)

2(β−1)

q q 2 −2 q−2 2 2 2 s . Since w+ wL = w+ w ˆL w+ = w+ Let t = 2∗ −q+2 w ˆL and w ˆL = w+ wL , we s can use the H¨older’s inequality, (3.11), (3.12) and (3.13) to conclude that, whenever w ˆL (·, 0) ∈ Lt (Ω), it holds

 22∗ ∗ |w ˆL (x, 0)|2s dx s Ω Z   2(β−1) 2(β−1) q 2 2 (x, 0)wL (x, 0) dx, ≤ 4S β C w+ (x, 0)wL (x, 0) + µK p−q w+ Ω Z   Z q−2 2 p−q 2 2 (x, 0)dx , w ˆL (x, 0)dx + µK w+ (x, 0)wˆL = 4S β C Ω Ω  Z  2t Z  q−2 Z  2t  q−2  2∗ s 2 t p−q 2∗ t 2∗ s s ≤ 4S β |Ω| w ˆL (x, 0)dx + µK |w+ (x, 0)| dx w ˆL (x, 0)dx Ω Ω Ω  q−2  q−2 ≤ 4S β 2 |Ω| 2∗s + µK p−q (S γ) 2 kw ˆL (x, 0)k2Lt . Z

2∗

2∗ −q

Set β := ts = 1 + s2 > 1. By the definition of wL , we have wL ≤ w+ , and then we conclude that w ˆL (·, 0) ∈ Lt (Ω), whenever (w+ (·, 0))β ∈ Lt (Ω). If it is the case, it follows from

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J.G. Zhang & X.C. Liu: THREE SOLUTIONS FOR A FRACTIONAL ELLIPTIC PROBLEMS

1829

the above inequality that  22∗ Z Z  22∗ ∗ 2∗ (β−1) 2∗ |w ˆL |2s (x, 0)dx s = wLs (x, 0)w+s (x, 0)dx s Ω

Ω

 q−2  Z  2t q−2 β−1 ≤ 4Sβ 2 |Ω| 2∗s + µK p−q (S γ) 2 |wL (x, 0) w+ (x, 0)|t dx . Ω

By Fatou’s Lemma on the variable L, we get Z  22β Z  2β ∗β tβ 2∗ β w+s (x, 0)dx s ≤ 4 Sβ 2 Cµ,K |w+ (x, 0)|tβ dx , Ω

Ω

i.e.,

Z

Ω

1 Z  tβ1  2∗1β   2β 1 2∗ β ββ |w+ (x, 0)|tβ dx , w+s (x, 0)dx s ≤ 4S Cµ,K

(3.14)

Ω

q−2

q−2

where Cµ,K = |Ω| 2∗s + µK p−q (S γ) 2 . ∗ 2∗ Since β = ts > 1 and w+ (·, 0) ∈ L2s (Ω), inequality (3.14) holds. Therefore, from β 2 t = β2∗s , we have that inequality (3.14) also holds with β replaced by β 2 . Hence we get Z  ∗1 2   12 Z  12 1 2 2 β 2β tβ 2∗ β 2 w+s (x, 0)dx s ≤ 4S Cµ,K (β 2 ) β2 |w+ (x, 0)|tβ dx Ω Ω   12 Z  2∗1β 1 ∗ 2β (β 2 ) β2 = 4S Cµ,K |w+ (x, 0)|2s β dx s Ω 1   12   2β Z  tβ1 1 1 2β ≤ 4S Cµ,K (β 2 ) β2 4S Cµ,K ββ |w+ (x, 0)|tβ dx Ω   21 ( 12 + β1 ) 2 1  Z  tβ1 β + = 4S Cµ,K β β2 β |w+ (x, 0)|tβ dx . Ω

2∗s ,

By iterating this process and β t = we obtain Z   21 ( β1m +···+ 12 + β1 ) m Z  2∗ 1βm  21∗ ∗ m ∗ β 2 β +···+ β22 + β1 w+s (x, 0)dx s ≤ 4S Cµ,K β βm |w+ (x, 0)|2s dx s . Ω

Ω

(3.15)

Taking the limit as m → ∞ in (3.15), we have 1

kw+ kL∞ ≤ (4S Cµ,K )θ1 β θ2 kw+ kL2∗s ≤ (4S Cµ,K )θ1 β θ2 (S γ) 2 , where θ1 =

1 2

∞ P

m=1

1 βm ,

θ2 =

∞ P

m=1

m βm

and β > 1.

Next, we will find some suitable value of K and µ such that the inequality 1 K (4 S Cµ,K )θ1 β θ2 (S γ) 2 ≤ 2 holds. From (3.16), we get   θ1 q−2 1 q−2 1 K p−q 2∗ Cµ,K = |Ω| s + µK . (S γ) 2 ≤ 1 θ 4S 2(γS) 2 β 2 Then we can choose K to satisfy the inequality   θ1 q−2 1 1 K − |Ω| 2∗s > 0, 1 4 S 2(Sγ) 2 β θ2 and fix µ0 such that ′

0 < µ0 < µ :=

1 K p−q (Sγ)

q−2 2



1 4S



K 1

2(Sγ) 2 β θ2

 θ1

1

− |Ω|

q−2 2∗ s

 .

(3.16)

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ACTA MATHEMATICA SCIENTIA

Vol.36 Ser.B

Thus we obtain (3.16) for µ ∈ [0, µ0 ], i.e., K for µ ∈ [0, µ0 ]. 2 Similarly, we can also have the estimate for the w− , i.e., kw+ kL∞ ≤

(3.17)

K for µ ∈ [0, µ0 ]. (3.18) 2 Now, let δ = min{δ0 , µ0 }. For each µ ∈ [0, δ], from (3.17), (3.18) and |w| = w+ + w− , we have kwkL∞ ≤ K for µ ∈ [0, µ0 ], kw− kL∞ ≤

which implies that kwk kL∞ ≤ K for k = 0, 1, 2 and µ ∈ [0, δ]. Therefore, we obtain inequality (3.6). The proof is completed. Proof of Theorem 1.2 ( gK (x, s) =



In fact, the truncation of gK (x, s) can be given by g(x, s)

if |s| ≤ K,

min{g(x, s), C0 (1 + K

p−q

q−2

|s|

s)}

if |s| > K,

where q ∈ (2 , 2∗s ). Then gK satisfies |gK (x, s)| ≤ C0 (1 + K p−q |s|q−2 ),

∀s ∈ R.

1 Let hK (x, w) = λf (x, w) + µgK (x, w), ∀w ∈ H0,L (CΩ ). The truncated problems associated to hK is the following  − div(y 1−2s ∇w) = 0 in CΩ ,     w=0 on ∂L CΩ , (3.19)   ∂w   y 1−2s = hK (x, w) on Ω × {0}. ∂ν Similar as in the proof of Theorem 1.1, by Theorem 2.4 we can prove that there exists δ > 0 such that the solutions w for the truncated problems (3.19) satisfy kwkL∞ ≤ K for µ ∈ [0, δ]; and in view of the definition gK , we have

hK (x, w) = λf (x, w) + µg(x, w). Therefore wk , k = 0, 1, 2, are also solutions of problem (3.19). This implies that problem (1.5) has at least three solutions uk (x) = trwk (k = 0, 1, 2).  References [1] Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differ Equ, 2007, 32: 1245–1260 [2] Cabr´ e X, Tan J. Positive solutions for nonlinear problems involving the square root of the Laplacian. Adv Math, 2010, 224: 2052–2093 [3] Tan J. The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc Var, 2011, 42: 21–41 [4] Barrios B, Colorado E, de Pablo A, S´ anchez U. On some critical problems for the fractional Laplacian operator. J Differ Equ, 2012, 252: 6133–6162 [5] Cohabrowski J, Yang J. Existence theorems for elliptic equations involving supercritical Sobolev exponent. Adv Differ Equ, 1997, 2: 231–256

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