We consider the fractional elliptic problem with singular nonlinearity (−∆)su=a(x)u−q+λb(x)up in Ω, u >0 in Ω, u= 0 in ∂Ω

Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 145, pp. 1–23.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

FRACTIONAL ELLIPTIC EQUATIONS WITH SIGN-CHANGING AND SINGULAR NONLINEARITY

SARIKA GOYAL, KONIJETI SREENADH

Abstract. In this article, we study the fractional Laplacian equation with singular nonlinearity

(−∆)^su=a(x)u^−q+λb(x)u^p in Ω, u >0 in Ω, u= 0 in∂Ω,

where Ω is a bounded domain inRⁿ with smooth boundary∂Ω,n >2s,s∈ (0,1),λ >0. Using variational methods, we show existence and multiplicity of positive solutions.

1. Introduction

Let Ω⊂Rⁿbe a bounded domain with smooth boundary,n >2sands∈(0,1).

We consider the fractional elliptic problem with singular nonlinearity (−∆)^su=a(x)u^−q+λb(x)u^p in Ω,

u >0 in Ω, u= 0 in ∂Ω. (1.1) We use the following assumptions onaandb:

(A1) a: Ω⊂Rⁿ→Rsuch that 0< a∈L

2∗ s 2∗

s−1+q(Ω).

(A2) b : Ω⊂Rⁿ →Ris a sign-changing function such thatb⁺ 6≡0 andb(x)∈ L

2∗ s 2∗

s−1−p(Ω).

Here λ > 0 is a parameter, 0 < q < 1 < p < 2^∗_s−1, with 2^∗_s = _n−2s²ⁿ , known as fractional critical Sobolev exponent and where (−∆)^s is the fractional Laplacian operator in Ω with zero Dirichlet boundary values on∂Ω.

To define the fractional Laplacian operator (−∆)^s in Ω, let {λ_k, φ_k} be the eigenvalues and the corresponding eigenfunctions of −∆ in Ω with zero Dirichlet boundary values on∂Ω

(−∆)^sφ_k=λ_kφ_k in Ω, φ_k = 0 on∂Ω.

normalized bykφkkL²(Ω)= 1. Then one can define the fractional Laplacian (−∆)^s fors∈(0,1) by

(−∆)^su=

∞

X

k=1

λ^s_kckφk,

2010Mathematics Subject Classification. 35A15, 35J75, 35R11.

Key words and phrases. Non-local operator; singular nonlinearity; Nehari manifold;

sign-changing weight function.

c

2016 Texas State University.

Submitted January, 21, 2016. Published June 14, 2016.

1

which clearly maps H₀^s(Ω) :=

u=

∞

X

k=1

ckφk ∈L²(Ω) :kukH^s₀(Ω)=X^∞

k=1

λ^s_kc²_k^1/2

<∞ intoL²(Ω). MoreoverH₀^s(Ω) is a Hilbert space endowed with an inner product

∞

X

k=1

c_kφ_k,

∞

X

k=1

d_kφ_k

H^s₀(Ω)=

∞

X

k=1

λ^s_kc_kd_kφ_k, if

∞

X

k=1

c_kφ_k,

∞

X

k=1

d_kφ_k ∈H₀^s(Ω).

Definition 1.1. A function u∈ H₀^s(Ω) such that u(x)> 0 in Ω is a solution of (1.1) such that for every functionv∈H₀^s(Ω), it holds

Z

Ω

(−∆)^s/2u(−∆)^s/2vdx= Z

Ω

a(x)u^−qvdx+λ Z

Ω

b(x)u^pvdx.

Associated with (1.1), we consider the energy functional for u∈H₀^s(Ω),u > 0 in Ω such that

Iλ(u) = Z

Ω

|(−∆)^s/2u|²dx− 1 1−q

Z

Ω

a(x)|u|^1−qdx− λ p+ 1

Z

Ω

b(x)|u|^p+1dx.

The fractional power of Laplacian is the infinitesimal generator of L´evy stable diffusion process and arise in anomalous diffusions in plasma, population dynamics, geophysical fluid dynamics, flames propagation, chemical reactions in liquids and American options in finance. For more details, we refer to [3, 14] and reference therein.

Recently the study of existence, multiplicity of solutions for fractional elliptic equations attracted a lot of interest by many researchers. Among the works dealing with fractional elliptic equations we cite [6, 9, 21, 22, 23, 24, 25, 26] and references therein, with no attempt to provide a complete list. Caffarelli and Silvestre [8]

gave a new formulation of fractional Laplacian through Dirichlet-Neumann maps.

This formulation transforms problems involving the fractional Laplacian into a local problem which allows one to use the variational methods.

On the other hand, there are some works where multiplicity results are shown using the structure of associated Nehari manifold. In [15, 16] authors studied subcritical problems and in [28] the authors obtained the existence of multiplicity for critical growth nonlinearity. In the case of the square root of Laplacian, the multiplicity results for sublinear and superlinear type of nonlinearity with sign- changing weight functions are studied in [7, 27].

In the local setting, s= 1, the paper by Crandall, Robinowitz and Tartar [12]

is the starting point on semilinear problem with singular nonlinearity. There is a large body of literature on singular problems, see [1, 2, 11, 12, 17, 18, 19, 20] and reference therein. Recently, Chen and Chen in [10] studied the following problem with singular nonlinearity

−∆u− λ

|x|² =h(x)u^−q+µW(x)u^p in Ω\ {0}, u >0 in Ω\ {0}, u= 0 on∂Ω, where 0 ∈ Ω⊂ Rⁿ(n ≥ 3) is a bounded smooth domain with smooth boundary, 0 < λ < ⁽ⁿ⁻²⁾₄ ² and 0 < q < 1 < p < ⁿ⁺²_n−2. Also h, W both are continuous functions in Ω with h >0 and W is sign-changing. By variational methods, they showed that there exists T_λ such that forµ ∈(0, T_λ) the above problem has two positive solutions.

In case of the fractional Laplacian, Fang [13] proved the existence of a solution of the singular problem

(−∆)^sw=w^−p, u >0 in Ω, u= 0 inRⁿ\Ω,

with 0< p < 1, using the method of sub and super solution. Recently, Barrios, Peral and et al [4] extend the result of [13]. They studied the existence result for the singular problem

(−∆)^su=λf(x)

u^γ +M u^p, u >0 in Ω, u= 0 inRⁿ\Ω,

where Ω is a bounded smooth domain of Rⁿ, n > 2s, 0 < s < 1,γ > 0, λ > 0, p >1 andf ∈L^m(Ω), m≥1 is a nonnegative function. For M = 0, they proved the existence of solution for every γ >0 and λ > 0. ForM = 1 andf ≡1, they showed that there exist Λ such that it has a solution for every 0 < λ < Λ, and have no solution forλ >Λ. Here the authors first studied the uniform estimates of solutions{un}of the regularized problems

(−∆)^su=λ f(x)

(u+¹_n)^γ +u^p, u >0 in Ω, u= 0 inRⁿ\Ω. (1.2) Then they obtained the solutions by taking limit in the regularized problem (1.2).

As far as we know, there is no work related to fractional Laplacian for singular nonlinearity and sign-changing weight functions. So, in this paper, we study the multiplicity results for problem (1.1) for 0< q <1< p <2^∗_s−1 andλ >0. This work is motivated by the work of Chen and Chen in [10]. Due to the singularity of problem, it is not easy to deal the problem (1.1) as the associated functional is not differentiable even in sense of Gˆateaux and the strong maximum principle is not applicable to show the positivity of solutions. Moreover one can not directly extend all the results from Laplacian case to fractional Laplacian, due to the non- local behavior of the operator and the bounded support of the test function is not preserved. To overcome these difficulties, we first use the Cafferelli and Silvestre [9] approach to convert the problem (1.1) into the local problem. Then we use the variational technique to study the local problem as in [10]. In this paper, the proofs of some Lemmas follow the similar lines as in [10] but for completeness, we give the details.

The article is organized as follows: In section 2 we present some preliminaries on extension problem and necessary weighted trace inequalities required for variational settings. We also state our main results. In section 3, we study the decomposition of Nehari manifold and local charts using the fibering maps. In Section 4, we show the existence of a nontrivial solutions and show how these solutions arise out of nature of Nehari manifold.

We will use the following notation throughout this paper: The same symbol k · kdenotes the norms in the three spaces L

2∗ s 2∗

s−1−q(Ω),L

2∗ s 2∗

s−1−p(Ω), andH_0,L^s (CΩ) defined by (2.1). AlsoS:=ksS(s, n) whereS(s, n) is the best constant of Sobolev embedding (see (2.2)).

2. Preliminaries and main results

In this section we give some definitions and functional settings. At the end of this section, we state our main results. To state our main result, we introduce some

notation and basic preliminaries results. Denote the upper half-space inRⁿ⁺¹ by Rⁿ⁺¹+ :={z= (x, y) = (x₁, x₂, . . . , x_n, y)∈Rⁿ⁺¹|y >0},

the half cylinder standing on a bounded smooth domain Ω ⊂ Rⁿ by CΩ := Ω× (0,∞)⊂Rⁿ⁺¹+ and its lateral boundary is denoted by∂LCΩ=∂Ω×[0,∞). Define the function spaceH_0,L^s (CΩ) as the completion ofC_0,L^∞(CΩ) ={w∈C^∞(CΩ) :w= 0 on∂_LCΩ}under the norm

kwk_H^s

0,L(CΩ)= ks

Z

CΩ

y^1−2s|∇w(x, y)|²dx dy^1/2

, (2.1)

where ks := ^21−2s_Γ(s)^Γ(1−s) is a normalization constant. Then it is a Hilbert space endowed with the inner product

hw, vi_H^s

0,L(CΩ)=ks

Z

Ω×{0}

y^1−2s∇w∇v dx dy.

If Ω is a smooth bounded domain then it is verified that (see [8, Proposition 2.1], [6, Proposition 2.1], [26, section 2])

H₀^s(Ω) :={u= tr|_Ω×{0}w:w∈H_0,L^s (C_Ω)}

and there exists a constantC >0 such that

kw(·,0)kH₀^s(Ω)≤CkwkH_0,L^s (CΩ) for allw∈H_0,L^s (CΩ).

Now we define the extension operator and fractional Laplacian for functions in H₀^s(Ω).

Definition 2.1. Given a function u∈H₀^s(Ω), we define its s-harmonic extension w=Es(u) to the cylinderCΩ as a solution of the problem

div(y^1−2s∇w) = 0 in CΩ, w= 0 on∂_LCΩ, w=u on Ω× {0}.

Definition 2.2. For any regular function u ∈ H₀^s(Ω), the fractional Laplacian (−∆)^sacting onuis defined by

(−∆)^su(x) =−k_s lim

y→0⁺y^1−2s∂w

∂y(x, y) for all (x, y)∈ C_Ω.

From [5] and [9], the mapE_s(·) is an isometry between H₀^s(Ω) andH_0,L^s (CΩ).

Furthermore, we have (1)

k(−∆)^suk_H−s(Ω)=kuk_H^s

0(Ω)=kEs(u)k_H^s

0,L(CΩ), whereH^−s(Ω) denotes the dual space ofH₀^s(Ω);

(2) For any w ∈ H_0,L^s (CΩ), there exists a constant C independent of w such that

ktrΩwk_Lr(Ω)≤Ckwk_H^s

0,L(CΩ)

holds for everyr∈[2,_n−2s²ⁿ ]. Moreover, H_0,L^s (CΩ) is compactly embedded intoL^r(Ω) forr∈[2,_n−2s²ⁿ ).

Lemma 2.3. For every 1≤r≤ _n−2s²ⁿ and every w∈H_0,L^s (CΩ), it holds Z

Ω×{0}

|w(x,0)|^rdx^2/r

≤Ck_s Z

CΩ

y^1−2s|∇w(x, y)|²dx dy, where the constant C depends onr,s,nand|Ω|.

Lemma 2.4. For every w∈H^s(Rⁿ⁺¹+ ), it holds S(s, n)Z

Rⁿ

|u(x)|^n−2s2n dx^n−2s_n

≤ Z

Rⁿ⁺¹₊

y^1−2s|∇w(x, y)|²dx dy, (2.2) whereu= tr_Ωw. The constant S(s, n)is known as the best constant and takes the value

S(s, n) =2π^sΓ(^2−2s₂ )Γ(^n+2s₂ )(Γ(ⁿ₂))^2sn Γ(s)Γ(^n−2s₂ )(Γ(n))^2sn .

Now we can transform the nonlocal problem (1.1) into the local problem

−div(y^1−2s∇w) = 0 inC_Ω:= Ω×(0,∞), w= 0 on∂LCΩ, w >0 on Ω× {0},

∂w

∂v^2s =a(x)w^−q+λb(x)w^p on Ω× {0},

(2.3)

where _∂v^∂w2s :=−kslim_y→0+y^{1−2s ∂w}_∂y(x, y), for allx∈Ω.

Definition 2.5. A weak solution of (2.3) is a function w ∈H_0,L^s (CΩ), w >0 in Ω× {0}such that for every v∈H_0,L^s (CΩ),

ks

Z

CΩ

y^1−2s∇w∇v dx dy

= Z

Ω×{0}

a(x)(w^−qv)(x,0)dx+λ Z

Ω×{0}

b(x)(w^pv)(x,0)dx.

Ifwsatisfies (2.3), thenu= trΩw=w(x,0)∈H₀^s(Ω) is a weak solution to problem (1.1).

Letc= 1−2s, then the associated functionalJλ:H_0,L^s (CΩ)→Rto the problem (2.3) is

J_λ(w) =ks

2 Z

CΩ

y^c|∇w|²dx dy− 1 1−q

Z

Ω×{0}

a(x)|w|^1−qdx

− λ p+ 1

Z

Ω×{0}

b(x)|w(x,0)|^p+1dx.

Now forw∈H_0,L^s (CΩ), we define the fiber mapφw:R⁺ →Ras φw(t) =Jλ(tw) =t²

2kwk²− 1 1−q

Z

Ω×{0}

a(x)|tw|^1−qdx

−λt^p+1 p+ 1

Z

Ω×{0}

b(x)|w(x,0)|^p+1dx.

Then

φ⁰_w(t) =tkwk²−t^−q Z

Ω×{0}

a(x)|w(x,0)|^1−qdx−λt^p Z

Ω×{0}

b(x)|w(x,0)|^p+1dx,

φ⁰⁰_w(t) =kwk²+qt^−q−1 Z

Ω×{0}

a(x)|w(x,0)|^1−qdx

−pλt^p−1 Z

Ω×{0}

b(x)|w(x,0)|^p+1dx.

It is easy to see that the energy functionalJλ is not bounded below on the space H_0,L^s (CΩ). But we will show that it is bounded below on an appropriate subset of H_0,L^s (CΩ) and a minimizer on subsets of this set gives rise to solutions of (2.3). In order to obtain the existence results, we define

N_λ: ={w∈H_0,L^s (C_Ω) :hJ_λ⁰(w), wi= 0}

=n

w∈H_0,L^s (CΩ) :kwk²= Z

Ω×{0}

a(x)|w(x,0)|^1−qdx +λ

Z

Ω×{0}

b(x)|w(x,0)|^p+1dxo .

Note thatw∈ Nλ ifwis a solution of (2.3). Also one can easily see thattw∈ Nλ

if and only if φ⁰_w(t) = 0 and in particular, w ∈ Nλ if and only if φ⁰_w(1) = 0. In order to obtain our result, we decomposeNλ withN_λ^±,N_λ⁰as follows:

N_λ^±={w∈ Nλ:φ⁰⁰_w(1)≷0}

=

w∈ N_λ: (1 +q)kwk²≷λ(p+q) Z

Ω×{0}

b(x)|w(x,0)|^p+1dx , N_λ⁰={w∈ N_λ:φ⁰⁰_w(1) = 0}

=

w∈ Nλ: (1 +q)kwk²=λ(p+q) Z

Ω×{0}

b(x)|w(x,0)|^p+1dx . Inspired by [9] and [10], we show that how variational methods can be used to established some existence and multiplicity results. Our results are as follows:

Theorem 2.6. Suppose that λ∈(0,Λ), where Λ := 1 +q

p+q p−1

p+q ^p−1_1+q 1

kbk S^p+q

kak^p−1

1/(1+q)

.

Then problem (2.3) has at least two solutions w0 ∈ N_λ⁺, W0 ∈ N_λ⁻ with kW0k>

kw0k. Moreover, u0(x) = w0(·,0) ∈ H₀^s(Ω) and U0(x) = W0(·,0) ∈ H₀^s(Ω) are positive solutions of the problem (1.1).

Next, we obtain the blow up behavior of the solutionW∈ N_λ⁻ of problem (2.3) withp= 1 +as →0⁺.

Theorem 2.7. letW∈ N_λ⁻ be the solution of problem (2.3)withp= 1 +, where λ∈(0,Λ), then

kWk> C

Λ λ

1/

, where

C=

1 + 1 +q

1/(1+q)

kak^1/(q+1) 1

√ S

^1−q_1+q

→ ∞ as→0⁺. Namely,W blows up faster than exponentially with respect to .

We remark that ifwis a positive solution of the problem (−∆)^su=a(x)u^−q+λb(x)u^p in Ω

u >0 in Ω, u= 0 on∂Ω.

Then one can easily see thatv=λ^1/(p−1)uis a positive solution of the problem (−∆)^sv=λ^1+qp−1a(x)v^−q+b(x)v^p in Ω

v >0 in Ω, v= 0 in∂Ω. (2.4) That is, the problem (2.4) has two positive solution forλ∈(0,Λ^p−1).

3. Fibering map analysis

In this section, we show thatN_λ^± is nonempty andN_λ⁰ ={0}. Moreover,Jλ is bounded below and coercive.

Lemma 3.1. Let λ∈(0,Λ). Then for eachw∈H_0,L^s (CΩ)with Z

Ω×{0}

a(x)|w(x,0)|^1−qdx >0, we have the following:

(i) R

Ω×{0}b(x)|w(x,0)|^p+1dx ≤ 0, then there exists a unique t1 < tmax such that t1w∈ N_λ⁺ andJλ(t1w) = inft>0Jλ(tw);

(ii) R

Ω×{0}b(x)|w(x,0)|^p+1dx > 0, then there exists a unique t₁ and t₂ with 0 < t1 < tmax < t2 such that t1w ∈ N_λ⁺, t2w ∈ N_λ⁻ and Jλ(t1w) = inf_0≤t≤tmaxJ_λ(tw),J_λ(t₂w) = sup_t≥t

1J_λ(tw).

Proof. Fort >0, we define ψw(t) =t^1−pkwk²−t^−p−q

Z

Ω×{0}

a(x)|w(x,0)|^1−qdx−λ Z

Ω×{0}

b(x)|w(x,0)|^p+1dx.

One can easily see thatψw(t)→ −∞ast→0⁺. Now ψ_w⁰ (t) = (1−p)t^−pkwk²+ (p+q)t^−p−q−1

Z

Ω×{0}

a(x)|w(x,0)|^1−qdx.

ψ_w⁰⁰(t) =−p(1−p)t^−p−1kwk²

−(p+q)(p+q+ 1)t^−p−q−2 Z

Ω×{0}

a(x)|w(x,0)|^1−qdx.

Then ψ_w⁰ (t) = 0 if and only if t = tmax := [_(p+q)R ^(p−1)kwk2

Ω×{0}a(x)|w(x,0)|^1−qdx]^−1/(1+q). Also

ψ_w⁰⁰(t_max) =p(p−1)h (p−1)kwk² (p+q)R

Ω×{0}a(x)|w(x,0)|^1−qdx i^p+1_q+1

kwk²

−(p+q)(p+q+ 1)h (p−1)kwk² (p+q)R

Ω×{0}a(x)|w(x,0)|^1−qdx i^p+q+2_q+1

× Z

Ω×{0}

a(x)|w(x,0)|^1−qdx

=−kwk²(p−1)(1 +q)h (p−1)kwk² (p+q)R

Ω×{0}a(x)|w(x,0)|^1−qdx i^p+1_q+1

<0.

Thusψwachieves its maximum att=tmax. Now using the H¨older’s inequality and Sobolev inequality (2.2), we obtain

Z

Ω×{0}

a(x)|w(x,0)|^1−qdx

≤hZ

Ω×{0}

|a(x)|

2∗ s 2∗

s−1+qi

2∗ s+q−1

2∗ s hZ

Ω×{0}

|w(x,0)|^2∗sdxi^1−q₂∗ s

≤ kakkwk

√ S

1−q

.

(3.1)

Z

Ω×{0}

b(x)|w(x,0)|^p+1dx

≤hZ

Ω×{0}

|b(x)|

2∗ s 2∗

s−1−pi

2∗ s−p−1

2∗ s hZ

Ω×{0}

|w(x,0)|^2∗sdxi^p+1₂∗ s

≤ kbkkwk

√S ^p+1

.

(3.2)

Using (3.1) and (3.2), we obtain ψw(tmax)

=1 +q p+q

p−1 p+q

^p−1_1+q kwk^2(p+q)1+q [R

Ω×{0}a(x)|w(x,0)|^1−qdx]^p−11+q

−λ Z

Ω×{0}

b(x)|w(x,0)|^p+1dx

≥h1 +q p+q

p−1 p+q

^p−1_1+q(√ S)^(1−q)

kak

^p−1_1+q

−λkbk 1

√S p+1i

kwk^p+1

≡Eλkwk^p+1,

(3.3) where

Eλ:=h1 +q p+q

p−1 p+q

^p−1_1+q(√ S)^(1−q)

kak

^p−1_1+q

−λkbk 1

√S p+1i

. Then we see thatEλ= 0 if and only ifλ= Λ, where

Λ := 1 +q p+q

p−1 p+q

^p−1_1+q 1 kbk

S^p+q kak^p−1

1/(1+q)

.

Thus for λ ∈ (0,Λ), we have E_λ > 0, and therefore it follows from (3.3) that ψw(tmax)>0.

(i) IfR

Ω×{0}b(x)|w(x,0)|^p+1dx≥0, then ψw(t)→ −λ

Z

Ω×{0}

b(x)|w(x,0)|^p+1dx <0

as t → ∞. Consequently, ψ_w(t) has exactly two points 0 < t₁ < t_max < t₂ such that

ψ_w(t₁) = 0 =ψ_w(t₂) andψ⁰_w(t₁)>0> ψ_w⁰ (t₂).

From the definition ofN_λ^±, we gett1w∈ N_λ⁺andt2w∈ N_λ⁻. Nowφ⁰_w(t) =t^pψw(t).

Thus φ⁰_w(t)< 0 in (0, t₁), φ⁰_w(t) >0 in (t₁, t₂) and φ⁰_w(t) <0 in (t₂,∞). Hence J_λ(t₁w) = inf_0≤t≤tmaxJ_λ(tw),J_λ(t₂w) = sup_t≥t

1J_λ(tw).

(ii) IfR

Ω×{0}b(x)|w(x,0)|^p+1dx <0 and ψw(t)→ −λ

Z

Ω×{0}

b(x)|w(x,0)|^p+1dx >0

ast→ ∞. Consequently,ψ_w(t) has exactly one point 0< t₁< t_max such that ψw(t1) = 0 and ψ⁰_w(t1)>0.

Now φ⁰_w(t) = t^pψ_w(t). Then φ⁰_w(t) < 0 in (0, t₁), φ⁰_w(t) > 0 in (t₁,∞). Thus J_λ(t₁w) = inf_t≥0J_λ(tw). Hence, it follows thatt₁w∈ N_λ⁺. Corollary 3.2. Suppose thatλ∈(0,Λ), then N_λ^±6=∅.

Proof. From (A1) and (A2), we can choosew∈H_0,L^s (CΩ)\ {0}such that Z

Ω×{0}

a(x)|w(x,0)|^1−qdx >0 and Z

Ω×{0}

b(x)|w(x,0)|^p+1dx >0.

Then by (ii) of Lemma 3.1, there exists a unique t1 and t2 such thatt1w ∈ N_λ⁺,

t2w∈ N_λ⁻. In conclusion,N_λ^± 6=∅.

Lemma 3.3. Forλ∈(0,Λ), we have N_λ⁰={0}.

Proof. We prove this by contradiction. Assume that there exists 0 6≡ w ∈ N_λ⁰. Then it follows fromw∈ N_λ⁰ that

(1 +q)kwk²=λ(p+q) Z

Ω×{0}

b(x)|w(x,0)|^p+1dx and consequently

0 =kwk²− Z

Ω×{0}

a(x)|w(x,0)|^1−qdx−λ Z

Ω×{0}

b(x)|w(x,0)|^p+1dx

= p−1 p+qkwk²−

Z

Ω×{0}

a(x)|w(x,0)|^1−qdx.

Therefore, asλ∈(0,Λ) andw6≡0, we use similar arguments as those in (3.3) to obain

0< Eλkwk^p+1

≤ 1 +q p+q

p−1 p+q

^p−1_1+q kwk^2(p+q)1+q R

Ω×{0}a(x)|w(x,0)|^1−qdx^p−1_1+q

−λ Z

Ω×{0}

b(x)|w(x,0)|^p+1dx

= 1 +q p+q

p−1 p+q

^p−1_1+q kwk^2(p+q)1+q

p−1

p+qkwk^2p−1_1+q

−1 +q

p+qkwk²= 0,

a contradiction. Hencew≡0. That is,N_λ⁰={0}.

We note that Λ is also related to a gap structure inNλ:

Lemma 3.4. Suppose that λ∈(0,Λ), then there exist a gap structure inNλ: kWk> A_λ> A₀>kwk for allw∈ N_λ⁺, W ∈ N_λ⁻,

where

Aλ=h 1 +q λ(p+q)kbk(√

S)^p+1i1/(p−1)

and A0=hp+q p−1kak 1

√S

1−qi1/(q+1)

. Proof. Ifw∈ N_λ⁺⊂ Nλ, then

0<(1 +q)kwk²−λ(p+q) Z

Ω×{0}

b(x)|w(x,0)|^p+1dx

= (1 +q)kwk²−(p+q)h kwk²−

Z

Ω×{0}

a(x)|w(x,0)|^1−qdxi

= (1−p)kwk²+ (p+q) Z

Ω×{0}

a(x)|w(x,0)|^1−qdx.

Hence it follows from (3.1) that (p−1)kwk²<(p+q)

Z

Ω×{0}

a(x)|w(x,0)|^1−qdx≤(p+q)kakkwk

√S 1−q

which yields

kwk<hp+q

p−1kak 1

√ S

^1−qi^1/(q+1)

≡A₀. IfW ∈ N_λ⁻, then it follows from (3.2) that

(1 +q)kWk²< λ(p+q) Z

Ω×{0}

b(x)|W(x,0)|^p+1dx≤λ(p+q)kbkkWk

√ S

^p+1

which yields

kWk>h 1 +q λ(p+q)kbk(√

S)^p+1i1/(p−1)

≡A_λ. Now we show thatA_λ=A₀if and only if λ= Λ.

λ= Λ = 1 +q p+q

p−1 p+q

^p−1_1+q 1 kbk

S^p+q kak^p−1

^1/(1+q)

if and only if A_λ=λ^−1/(p−1)1 +q

p+q

^1/(p−1) 1 kbk

^1/(p−1)√ S^p+1p−1

=p+q p−1

^1/(1+q)

kak^1/(q+1)√

S−_(1+q)(p−1)^2(p+q) +^p+1_p−1

=h (p+q)kak (p−1)(√

S)^1−q

i1/(q+1)

≡A₀. Thus for allλ∈(0,Λ), we can conclude that

kWk> Aλ> A0>kwk for allw∈ N_λ⁺, W ∈ N_λ⁻.

This completes the proof.

Lemma 3.5. Suppose thatλ∈(0,Λ), thenN_λ⁻is a closed set inH_0,L^s (C_Ω)-topology.

Proof. Let{Wk} be a sequence inN_λ⁻ withWk→W0inH_0,L^s (CΩ). Then we have kW₀k²= lim

k→∞kW_kk²

= lim

k→∞

hZ

Ω×{0}

a(x)|Wk(x,0)|^1−qdx+λ Z

Ω×{0}

b(x)|Wk(x,0)|^p+1dxi

= Z

Ω×{0}

a(x)|W₀(x,0)|^1−qdx+λ Z

Ω×{0}

b(x)|W₀(x,0)|^p+1dx and

(1 +q)kW₀k²−λ(p+q) Z

Ω×{0}

b(x)|W₀(x,0)|^p+1dx

= lim

k→∞

h

(1 +q)kWkk²−λ(p+q) Z

Ω×{0}

b(x)|Wk(x,0)|^p+1dxi

≤0.

That is,W₀∈ N_λ⁻∩ N_λ⁰. Since{W_k} ⊂ N_λ⁻, from Lemma 3.4 we have kW₀k= lim

k→∞kW_kk ≥A₀>0,

which imply, W0 6= 0. It follows from Lemma 3.1, that W0 6∈ N_λ⁰ for any λ ∈ (0,Λ). Thus W0 ∈ N_λ⁻. Hence, N_λ⁻ is a closed set inH_0,L^s (CΩ)-topology for any

λ∈(0,Λ).

Lemma 3.6. Let w ∈ N_λ^±. Then for any φ ∈ C_0,L^∞ (CΩ), there exists a number >0 and a continuous functionf :B(0) :={v∈H_0,L^s (C_Ω) :kvk< } →R⁺ such that

f(v)>0, f(0) = 1, f(v)(w+vφ)∈ N_λ^± for allv∈B(0).

Proof. We give the proof only for the casew∈ N_λ⁺, the caseN_λ⁻may be preceded exactly. For anyφ∈C_0,L^∞(CΩ), we defineF :H_0,L^s (CΩ)×R⁺→Ras follows:

F(v, r) =r^1+qkw+vφk²− Z

Ω×{0}

a(x)|(w+vφ)(x,0)|^1−q

−λr^p+q Z

Ω×{0}

b(x)|(w+vφ)(x,0)|^p+1. Sincew∈ N_λ⁺(⊂ Nλ), we have

F(0,1) =kwk²− Z

Ω×{0}

a(x)|w(x,0)|^1−qdx−λ Z

Ω×{0}

b(x)|w(x,0)|^p+1dx= 0, and

∂F

∂r(0,1) = (1 +q)kwk²−λ(p+q) Z

Ω×{0}

b(x)|w(x,0)|^p+1dx >0.

Applying the implicit function theorem at (0,1), we have that there exists ¯ > 0 such that for kvk < ¯, v ∈ H_0,L^s (CΩ), the equation F(v, r) = 0 has a unique continuous solution r=f(v)>0. It follows from F(0,1) = 0 that f(0) = 1 and fromF(v, f(v)) = 0 forkvk<¯,v∈H_0,L^s (CΩ) that

0 =f^1+q(v)kw+vφk²− Z

Ω×{0}

a(x)|(w+vφ)(x,0)|^1−q

−λf^p+q(v) Z

Ω×{0}

b(x)|(w+vφ)(x,0)|^p+1

=

kf(v)(w+vφ)k²− Z

Ω×{0}

a(x)|f(v)(w+vφ)(x,0)|^1−q

−λ Z

Ω×{0}

b(x)|f(v)(w+vφ)(x,0)|^p+1

/f^1−q(v);

that is,

f(v)(w+vφ)∈ Nλ for allv∈H_0,L^s (CΩ),kvk<˜. Since ^∂F_∂r(0,1)>0 and

∂F

∂r(v, f(v))

= (1 +q)f^q(v)kw+vφk²−λ(p+q)f^p+q−1(v) Z

Ω×{0}

b(x)|(w+vφ)(x,0)|^p+1dx

=

(1 +q)kf(v)(w+vφ)k²−λ(p+q)R

Ω×{0}b(x)|f(v)(w+vφ)(x,0)|^p+1dx

f^2−q(v) ,

we can take >0 possibly smaller ( <¯) such that for anyv∈H_0,L^s (C_Ω),kvk< , (1 +q)kf(v)(w+vφ)k²−λ(p+q)

Z

Ω×{0}

b(x)|f(v)(w+vφ)(x,0)|^p+1dx >0;

that is,

f(v)(w+vφ)∈ N_λ⁺ for allv∈B(0).

This completes the proof.

Lemma 3.7. Jλ is bounded below and coercive onNλ. Proof. Forw∈ Nλ, from (3.1), we obtain

Jλ(w) =1 2 − 1

p+ 1

kwk²− 1

1−q − 1 p+ 1

Z

Ω×{0}

a(x)|w(x,0)|^1−qdx

≥1 2 − 1

p+ 1

kwk²− 1

1−q − 1 p+ 1

kakkwk

√ S

1−q

.

(3.4)

Now consider the functionρ:R⁺→Ras ρ(t) =αt²−βt^1−q, where α,β are both positive constants. One can easily show that ρ is convex(ρ⁰⁰(t)>0 for all t >0) with ρ(t) → 0 as t → 0 and ρ(t) → ∞ as t → ∞. ρ achieves its minimum at t_min= [^β(1−q)_2α ]^1/(1+q) and

ρ(t_min) =αβ(1−q) 2α

_1+q²

−ββ(1−q) 2α

^1−q_1+q

=−1 +q

2 β^1+q2 1−q 2α

^1−q_1+q . Applying ρ(t) with α = ¹₂ −_p+1¹

, β = _1−q¹ − _p+1¹ kak ^√1

S

1−q

and t = kwk, w∈ Nλ, we obtain from (3.4) that

kwk→∞lim Jλ(w)≥ lim

t→∞ρ(t) =∞.

ThusJλ is coercive onNλ. Moreover, it follows from (3.4) that

Jλ(w)≥ρ(t)≥ρ(tmin) (a constant), (3.5) i.e.,

J_λ(w)≥ −1 +q

2 β^1+q2 1−q 2α

^1−q_1+q

=− 1 +q (1−q)(p+ 1)

(p+q)kak 2(√

S)^1−q

_1+q² 1 p−1

^1−q_1+q .

ThusJλ is bounded below onNλ. 4. Existence of solutions in N_λ^±

Now from Lemma 3.5,N_λ⁺∪N_λ⁰andN_λ⁻are two closed sets inH_0,L^s (CΩ) provided λ∈(0,Λ). Consequently, the Ekeland variational principle can be applied to the problem of finding the infimum of J_λ on bothN_λ⁺∪ N_λ⁰ andN_λ⁻. First, consider {wk} ⊂ N_λ⁺∪ N_λ⁰ with the following properties:

J_λ(w_k)< inf

w∈N_λ⁺∪N_λ⁰

J_λ(w) +1

k (4.1)

J_λ(w)≥J_λ(w_k)−1

kkw−w_kk, ∀w∈ N_λ⁺∪ N_λ⁰. (4.2) FromJλ(|w|) =Jλ(w), we may assume that wk ≥0 onCΩ.

Lemma 4.1. Show that the sequence {wk} is bounded in Nλ. Moreover, there exists06=w0∈H_0,L^s (CΩ)such that wk * w0 weakly inH_0,L^s (CΩ).

Proof. By equations (3.5) and (4.1), we have at²−bt^1−q =ρ(t)≤J_λ(w)< inf

w∈N_λ⁺∪N_λ⁰

J_λ(w) +1 k ≤C₅,

for sufficiently large k and a suitable positive constant. Hence putting t =wk in the above equation, we obtain{wk} is bounded.

Let{wk} is bounded in H_0,L^s (CΩ). Then, there exists a subsequence of{wk}k, still denoted by{wk}k andw0∈H_0,L^s (CΩ) such thatwk * w0weakly inH_0,L^s (CΩ), w_k(·,0)→w₀(·,0) strongly inL^p(Ω) for 1≤p <2^∗_s andw_k(·,0)→w₀(·,0) a.e. in Ω.

For anyw∈ N_λ⁺, we have from 0< q <1< pthat Jλ(w) =1

2 − 1 1−q

kwk²+ 1

1−q − 1 p+ 1

λ

Z

Ω×{0}

b(x)|w(x,0)|^p+1dx

<1 2 − 1

1−q

kwk²+ 1

1−q − 1 p+ 1

1 +q p+qkwk²

= 1 p+ 1−1

2 1 +q

1−qkwk²<0, which means that inf_N+

λ Jλ <0. Now for λ ∈ (0,Λ), we know from Lemma 3.1, thatN_λ⁰={0}. Together, these imply thatw_k∈ N_λ⁺ forklarge and

inf

w∈N_λ⁺∪N_λ⁰

J_λ(w)≤ inf

w∈N_λ⁺

J_λ(w)<0.

Therefore, by weak lower semi-continuity of norm, Jλ(w0)≤lim inf

k→∞ Jλ(wk) = inf

N_λ⁺∪N_λ⁰

Jλ<0,

that is,w₀≥0,w₀6≡0.

Lemma 4.2. Supposewk∈ N_λ⁺ such thatwk* w0 weakly inH_0,L^s (CΩ). Then for λ∈(0,Λ),

(1 +q) Z

Ω×{0}

a(x)w^1−q₀ (x,0)dx−λ(p−1) Z

Ω×{0}

b(x)w₀^p+1(x,0)dx >0. (4.3)

Moreover, there exists a constantC2>0 such that (1 +q)kwkk²−λ(p+q)

Z

Ω×{0}

b(x)w_k^p+1(x,0)dx≥C₂>0. (4.4) Proof. For{w_k} ⊂ N_λ⁺(⊂ N_λ), since

(1 +q) Z

Ω×{0}

a(x)w₀^1−q(x,0)dx−λ(p−1) Z

Ω×{0}

b(x)w₀^p+1(x,0)dx

= lim

k→∞

h (1 +q)

Z

Ω×{0}

a(x)w^1−q_k (x,0)dx−λ(p−1) Z

Ω×{0}

b(x)w^p+1_k (x,0)dxi

= lim

k→∞

h

(1 +q)kwkk²−λ(p+q) Z

Ω×{0}

b(x)w^p+1_k (x,0)dxi

≥0, we can argue by a contradiction and assume that

(1 +q) Z

Ω×{0}

a(x)w^1−q₀ (x,0)dx−λ(p−1) Z

Ω×{0}

b(x)w₀^p+1(x,0)dx= 0. (4.5) Sincew_k∈ Nλ, from the weak lower semi continuity of norm and (4.5) we have

0 = lim

k→∞

hkwkk²− Z

Ω×{0}

a(x)w^1−q_k (x,0)dx−λ Z

Ω×{0}

b(x)w^p+1_k (x,0)dxi

≥ kw0k²− Z

Ω×{0}

a(x)w^1−q₀ (x,0)dx−λ Z

Ω×{0}

b(x)w^p+1₀ (x,0)dx

=

(kw0k²−λ^p+q_1+qR

Ω×{0}b(x)w^p+1₀ (x,0)dx kw0k²−^p+q_p−1R

Ω×{0}a(x)w^1−q₀ (x,0)dx.

Thus for anyλ∈(0,Λ) andw₀6≡0, by similar arguments as those in (3.3) we have that

0< E_λkw₀k^p+1

≤ 1 +q p+q

p−1 p+q

^p−1_1+q kw0k^2(p+q)1+q R

Ω×{0}a(x)w^1−q₀ (x,0)dx^p−1_1+q

−λ Z

Ω×{0}

b(x)w₀^p+1(x,0)dx

= 1 +q p+q

p−1 p+q

^p−1_1+q kw0k^2(p+q)1+q

p−1

p+qkw0k^2p−1_1+q

−1 +q

p+qkw0k²= 0, which is clearly impossible. Now by (4.3), we have that

(1 +q) Z

Ω×{0}

a(x)w^1−q_k (x,0)dx−λ(p−1) Z

Ω×{0}

b(x)w^p+1_k (x,0)dx≥C₂ for sufficiently largekand a suitable positive constantC₂. This, together with the

fact thatw_k∈ Nλ we obtain equation (4.4).

Fixφ ∈C_0,L^∞(CΩ) withφ≥0. Then we apply Lemma 3.6 with w=w_k ∈ N_λ⁺ (k large enough such that ^(1−q)C_k ¹ < C₂), we obtain a sequence of functions f_k : Bk(0)→Rsuch thatfk(0) = 1 andfk(w)(wk+wφ)∈ N_λ⁺ for allw∈B_k(0). It follows fromwk ∈ Nλ andfk(w)(wk+wφ)∈ Nλ that

kw_kk²− Z

Ω×{0}

a(x)w_k^1−q(x,0)dx−λ Z

Ω×{0}

b(x)w_k^p+1(x,0)dx= 0 (4.6)

and

f_k²(w)kwk+wφk²−f_k^1−q(w) Z

Ω×{0}

a(x)(wk+wφ)^1−q(x,0)dx

−λf_k^p+1(w) Z

Ω×{0}

b(x)(wk+wφ)^p+1(x,0)dx= 0.

(4.7)

Choose 0 < ρ < _k and w = ρv with kvk < 1. Then we find f_k(w) such that f_k(0) = 1 andf_k(w)(w_k+wφ)∈ N_λ⁺ for allw∈B_ρ(0).

Lemma 4.3. Forλ∈(0,Λ)we have|hf_k⁰(0), vi|is finite for every0≤v∈H_0,L^s (CΩ) withkvk ≤1.

Proof. By (4.6) and (4.7) we have

0 = [f_k²(w)−1]kwk+wφk²+kwk+wφk²− kwkk²

−[f_k^1−q(w)−1]

Z

Ω×{0}

a(x)(wk+wφ)^1−q(x,0)dx

− Z

Ω×{0}

a(x)[((wk+wφ)^1−q−w^1−q_k )(x,0)]dx

−λ[f_k^p+1(w)−1]

Z

Ω×{0}

b(x)(w_k+wφ)^p+1(x,0)dx

−λ Z

Ω×{0}

b(x)[((wk+wφ)^p+1−w^p+1_k )(x,0)]dx

≤[f_k²(ρv)−1]kwk+ρvφk²+kwk+ρvφk²− kwkk²

−[f_k^1−q(ρv)−1]

Z

Ω×{0}

a(x)(w_k+ρvφ)^1−q(x,0)

−λ[f_k^p+1(ρv)−1]

Z

Ω×{0}

b(x)(wk+ρvφ)^p+1(x,0)

−λ Z

Ω×{0}

b(x)[((w_k+ρvφ)^p+1−w_k^p+1)(x,0)]

Dividing byρ >0 and passing to the limitρ→0, we derive that 0≤2hf_k⁰(0), vikwkk²+ 2ks

Z

CΩ

y^c∇wk∇(vφ)dx dy

−(1−q)hf_k⁰(0), vi Z

Ω×{0}

a(x)w_k^1−q(x,0)dx

−λ(p+ 1)hf_k⁰(0), vi Z

Ω×{0}

b(x)w^p+1_k (x,0)dx

−λ(p+ 1) Z

Ω×{0}

b(x)(w^p_kvφ)(x,0)dx

=hf_k⁰(0), vih

2kwkk²−(1−q) Z

Ω×{0}

a(x)w_k^1−q(x,0)dx

−λ(p+ 1) Z

Ω×{0}

b(x)w^p+1_k (x,0)dxi + 2k_s

Z

CΩ

y^c∇w_k∇(vφ)dx dy−λ(p+ 1) Z

Ω×{0}

b(x)(w_k^pvφ)(x,0)dx

=hf_k⁰(0), vih

(1 +q)kwkk²

−λ(p+q) Z

Ω×{0}

b(x)w^p+1_k (x,0)dxi + 2k_s

Z

CΩ

y^c∇wk∇(vφ)dx dy−λ(p+ 1) Z

Ω×{0}

b(x)(w_k^pvφ)(x,0)dx. (4.8) From this inequality and (4.4) we know thathf_k⁰(0), vi 6=−∞. Now we show that hf_k⁰(0), vi 6= +∞. Arguing by contradiction, we assume thathf_k⁰(0), vi= +∞. Now we note that

|fk(ρv)−1|kwkk+fk(ρv)kρvφk ≥ k[fk(ρv)−1]wk+ρvfk(ρv)φk

=kf_k(ρv)(w_k+ρvφ)−w_kk (4.9) andfk(ρv)> fk(0) = 1 for sufficiently largek.

From the definition of derivative hf_k⁰(0), vi, applying equation (4.2) with w = f_k(ρv)(w_k+ρvφ)∈ N_λ⁺, we clearly have

[fk(ρv)−1]kwkk

k +fk(ρv)kρvφk k

≥ 1

kkfk(ρv)(wk+ρvφ)−wkk

≥J_λ(w_k)−J_λ(f_k(ρv)(w_k+ρvφ))

=1 2 − 1

1−q

kwkk²+λ 1

1−q − 1 p+ 1

Z

Ω×{0}

b(x)w^p+1_k (x,0)dx + 1

1−q −1 2

f_k²(ρv)kwk+ρvφk²

− λ(p+q)

(1−q)(p+ 1)f_k^p+1(ρv) Z

Ω×{0}

b(x)(w_k+ρvφ)^p+1(x,0)dx

= 1 1−q−1

2

(kw_k+ρvφk²− kw_kk²) + 1 1−q−1

2

[f_k²(ρv)−1]kw_k+ρvφk²

−λ 1

1−q− 1 p+ 1

f_k^p+1(ρv) Z

Ω×{0}

b(x)[((wk+ρvφ)^p+1−w_k^p+1)(x,0)]dx

−λ 1

1−q− 1 p+ 1

[f_k^p+1(ρv)−1]

Z

Ω×{0}

b(x)w^p+1_k (x,0)dx.

Dividing byρ >0 and passing to the limit asρ→0, we can obtain hf_k⁰(0), vikwkk

k +kvφk k

≥1 +q 1−q

ks

Z

CΩ

y^c∇wk∇(vφ)dx dy+1 +q 1−q

hf_k⁰(0), vikwkk²

−λp+q 1−q

hf_k⁰(0), vi Z

Ω×{0}

b(x)w^p+1_k (x,0)dx

−λp+q 1−q

Z

Ω×{0}

b(x)(w^p_kvφ)(x,0)dx

= hf_k⁰(0), vi 1−q

h

(1 +q)kw_kk²−λ(p+q) Z

Ω×{0}

b(x)w^p+1_k (x,0)dxi

+1 +q 1−q

ks

Z

CΩ

y^c∇wk∇(vφ)dx dy−λp+q 1−q

Z

Ω×{0}

b(x)(w_k^pvφ)(x,0)dx.

That is, kvφk

k ≥ hf_k⁰(0), vi 1−q

h

(1 +q)kw_kk²−λ(p+q) Z

Ω×{0}

b(x)w^p+1_k (x,0)dx

−(1−q)kwkk k

i

+1 +q 1−q

ks

Z

CΩ

y^c∇wk∇(vφ)dx dy

−λp+q 1−q

Z

Ω×{0}

b(x)(w^p_kvφ)(x,0)dx,

(4.10)

which is impossible becausehf_k⁰(0), vi= +∞and (1+q)kwkk²−λ(p+q)

Z

Ω×{0}

b(x)w^p+1_k (x,0)dx−(1−q)kwkk

k ≥C2−(1−q)C₁ k >0.

In conclusion, |hf_k⁰(0), vi| < +∞. Furthermore, (4.4) with kw_kk ≤ C₁ and two inequalities (4.8) and (4.10) also imply that

|hf_k⁰(0), vi| ≤C3

forksufficiently large and a suitable constantC₃. Lemma 4.4. For each 0 ≤φ ∈ C_0,L^∞ (CΩ) and for every 0 ≤ v ∈ H_0,L^s (CΩ) with kvk ≤1, we have a(x)w₀^−qvφ∈L¹(Ω)and

k_s Z

CΩ

y^c∇w0∇(vφ)dx dy− Z

Ω×{0}

a(x)(w^−q₀ vφ)(x,0)dx

−λ Z

Ω×{0}

b(x)(w₀^pvφ)(x,0)dx≥0.

(4.11)

Proof. Applying (4.9) and (4.2) again, we obtain [f_k(ρv)−1]kwkk

k +f_k(ρv)kρvφk k

≥Jλ(wk)−Jλ(fk(ρv)(wk+ρvφ))

=−f_k²(ρv)−1

2 kw_kk²−f_k²(ρv)

2 (kw_k+ρvφk²− kw_kk²) +f_k^1−q(ρv)−1

1−q Z

Ω×{0}

a(x)(wk+ρvφ)^1−q(x,0)

+ 1

1−q Z

Ω×{0}

a(x)[((w_k+ρvφ)^1−q−w_k^1−q)(x,0)]

+λf_k^p+1(ρv)−1 p+ 1

Z

Ω×{0}

b(x)(wk+ρvφ)^p+1(x,0)

+ λ

p+ 1 Z

Ω×{0}

b(x)[((w_k+ρvφ)^p+1−w^p+1_k )(x,0)].

Dividing byρ >0 and passing to the limitρ→0⁺, we obtain

|hf_k⁰(0), vi|kwkk

k +kvφk k