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

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

where Ω is a bounded domain inR^{n} 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⊂R^{n}be a bounded domain with smooth boundary,n >2sands∈(0,1).

We consider the fractional elliptic problem with singular nonlinearity
(−∆)^{s}u=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⊂R^{n}→Rsuch that 0< a∈L

2∗ s 2∗

s−1+q(Ω).

(A2) b : Ω⊂R^{n} →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−2s}^{2n} , 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

(−∆)^{s}φ_{k}=λ_{k}φ_{k} in Ω, φ_{k} = 0 on∂Ω.

normalized bykφkkL^{2}(Ω)= 1. Then one can define the fractional Laplacian (−∆)^{s}
fors∈(0,1) by

(−∆)^{s}u=

∞

X

k=1

λ^{s}_{k}ckφk,

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
H_{0}^{s}(Ω) :=

u=

∞

X

k=1

ckφk ∈L^{2}(Ω) :kukH^{s}_{0}(Ω)=X^{∞}

k=1

λ^{s}_{k}c^{2}_{k}^{1/2}

<∞
intoL^{2}(Ω). MoreoverH_{0}^{s}(Ω) is a Hilbert space endowed with an inner product

∞

X

k=1

c_{k}φ_{k},

∞

X

k=1

d_{k}φ_{k}

H^{s}_{0}(Ω)=

∞

X

k=1

λ^{s}_{k}c_{k}d_{k}φ_{k}, if

∞

X

k=1

c_{k}φ_{k},

∞

X

k=1

d_{k}φ_{k} ∈H_{0}^{s}(Ω).

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

Z

Ω

(−∆)^{s/2}u(−∆)^{s/2}vdx=
Z

Ω

a(x)u^{−q}vdx+λ
Z

Ω

b(x)u^{p}vdx.

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

Iλ(u) = Z

Ω

|(−∆)^{s/2}u|^{2}dx− 1
1−q

Z

Ω

a(x)|u|^{1−q}dx− λ
p+ 1

Z

Ω

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

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⊂ R^{n}(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

(−∆)^{s}w=w^{−p}, u >0 in Ω, u= 0 inR^{n}\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

(−∆)^{s}u=λf(x)

u^{γ} +M u^{p}, u >0 in Ω, u= 0 inR^{n}\Ω,

where Ω is a bounded smooth domain of R^{n}, 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

(−∆)^{s}u=λ f(x)

(u+^{1}_{n})^{γ} +u^{p}, u >0 in Ω, u= 0 inR^{n}\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 inR^{n+1} by
R^{n+1}+ :={z= (x, y) = (x_{1}, x_{2}, . . . , x_{n}, y)∈R^{n+1}|y >0},

the half cylinder standing on a bounded smooth domain Ω ⊂ R^{n} by CΩ := Ω×
(0,∞)⊂R^{n+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∂_{L}CΩ}under the norm

kwk_{H}^{s}

0,L(CΩ)= ks

Z

CΩ

y^{1−2s}|∇w(x, y)|^{2}dx dy^{1/2}

, (2.1)

where ks := ^{2}^{1−2s}_{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])

H_{0}^{s}(Ω) :={u= tr|_{Ω×{0}}w:w∈H_{0,L}^{s} (C_{Ω})}

and there exists a constantC >0 such that

kw(·,0)kH_{0}^{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
H_{0}^{s}(Ω).

Definition 2.1. Given a function u∈H_{0}^{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∂_{L}CΩ,
w=u on Ω× {0}.

Definition 2.2. For any regular function u ∈ H_{0}^{s}(Ω), the fractional Laplacian
(−∆)^{s}acting onuis defined by

(−∆)^{s}u(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 H_{0}^{s}(Ω) andH_{0,L}^{s} (CΩ).

Furthermore, we have (1)

k(−∆)^{s}uk_{H}−s(Ω)=kuk_{H}^{s}

0(Ω)=kEs(u)k_{H}^{s}

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

(2) For any w ∈ H_{0,L}^{s} (CΩ), there exists a constant C independent of w such
that

ktrΩwk_{L}r(Ω)≤Ckwk_{H}^{s}

0,L(CΩ)

holds for everyr∈[2,_{n−2s}^{2n} ]. Moreover, H_{0,L}^{s} (CΩ) is compactly embedded
intoL^{r}(Ω) forr∈[2,_{n−2s}^{2n} ).

Lemma 2.3. For every 1≤r≤ _{n−2s}^{2n} and every w∈H_{0,L}^{s} (CΩ), it holds
Z

Ω×{0}

|w(x,0)|^{r}dx^{2/r}

≤Ck_{s}
Z

CΩ

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

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

R^{n}

|u(x)|^{n−2s}^{2n} dx^{n−2s}_{n}

≤ Z

R^{n+1}_{+}

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

S(s, n) =2π^{s}Γ(^{2−2s}_{2} )Γ(^{n+2s}_{2} )(Γ(^{n}_{2}))^{2s}^{n}
Γ(s)Γ(^{n−2s}_{2} )(Γ(n))^{2s}^{n} .

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}^{∂w}2s :=−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^{−q}v)(x,0)dx+λ
Z

Ω×{0}

b(x)(w^{p}v)(x,0)dx.

Ifwsatisfies (2.3), thenu= trΩw=w(x,0)∈H_{0}^{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|^{2}dx dy− 1
1−q

Z

Ω×{0}

a(x)|w|^{1−q}dx

− λ p+ 1

Z

Ω×{0}

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

Now forw∈H_{0,L}^{s} (CΩ), we define the fiber mapφw:R^{+} →Ras
φw(t) =Jλ(tw) =t^{2}

2kwk^{2}− 1
1−q

Z

Ω×{0}

a(x)|tw|^{1−q}dx

−λt^{p+1}
p+ 1

Z

Ω×{0}

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

Then

φ^{0}_{w}(t) =tkwk^{2}−t^{−q}
Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx−λt^{p}
Z

Ω×{0}

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

φ^{00}_{w}(t) =kwk^{2}+qt^{−q−1}
Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx

−pλt^{p−1}
Z

Ω×{0}

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

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_{λ}^{0}(w), wi= 0}

=n

w∈H_{0,L}^{s} (CΩ) :kwk^{2}=
Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx
+λ

Z

Ω×{0}

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

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

if and only if φ^{0}_{w}(t) = 0 and in particular, w ∈ Nλ if and only if φ^{0}_{w}(1) = 0. In
order to obtain our result, we decomposeNλ withN_{λ}^{±},N_{λ}^{0}as follows:

N_{λ}^{±}={w∈ Nλ:φ^{00}_{w}(1)≷0}

=

w∈ N_{λ}: (1 +q)kwk^{2}≷λ(p+q)
Z

Ω×{0}

b(x)|w(x,0)|^{p+1}dx ,
N_{λ}^{0}={w∈ N_{λ}:φ^{00}_{w}(1) = 0}

=

w∈ Nλ: (1 +q)kwk^{2}=λ(p+q)
Z

Ω×{0}

b(x)|w(x,0)|^{p+1}dx .
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) ∈ H_{0}^{s}(Ω) and U0(x) = W0(·,0) ∈ H_{0}^{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
(−∆)^{s}u=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
(−∆)^{s}v=λ^{1+q}^{p−1}a(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_{λ}^{0} ={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−q}dx >0,
we have the following:

(i) R

Ω×{0}b(x)|w(x,0)|^{p+1}dx ≤ 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+1}dx > 0, then there exists a unique t_{1} and t_{2} with
0 < t1 < tmax < t2 such that t1w ∈ N_{λ}^{+}, t2w ∈ N_{λ}^{−} and Jλ(t1w) =
inf_{0≤t≤t}_{max}J_{λ}(tw),J_{λ}(t_{2}w) = sup_{t≥t}

1J_{λ}(tw).

Proof. Fort >0, we define
ψw(t) =t^{1−p}kwk^{2}−t^{−p−q}

Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx−λ
Z

Ω×{0}

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

One can easily see thatψw(t)→ −∞ast→0^{+}. Now
ψ_{w}^{0} (t) = (1−p)t^{−p}kwk^{2}+ (p+q)t^{−p−q−1}

Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx.

ψ_{w}^{00}(t) =−p(1−p)t^{−p−1}kwk^{2}

−(p+q)(p+q+ 1)t^{−p−q−2}
Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx.

Then ψ_{w}^{0} (t) = 0 if and only if t = tmax := [_{(p+q)}R ^{(p−1)kwk}^{2}

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

ψ_{w}^{00}(t_{max}) =p(p−1)h (p−1)kwk^{2}
(p+q)R

Ω×{0}a(x)|w(x,0)|^{1−q}dx
i^{p+1}_{q+1}

kwk^{2}

−(p+q)(p+q+ 1)h (p−1)kwk^{2}
(p+q)R

Ω×{0}a(x)|w(x,0)|^{1−q}dx
i^{p+q+2}_{q+1}

× Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx

=−kwk^{2}(p−1)(1 +q)h (p−1)kwk^{2}
(p+q)R

Ω×{0}a(x)|w(x,0)|^{1−q}dx
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−q}dx

≤hZ

Ω×{0}

|a(x)|

2∗ s 2∗

s−1+qi

2∗ s+q−1

2∗ s hZ

Ω×{0}

|w(x,0)|^{2}^{∗}^{s}dxi^{1−q}_{2}∗
s

≤ kakkwk

√ S

1−q

.

(3.1)

Z

Ω×{0}

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

≤hZ

Ω×{0}

|b(x)|

2∗ s 2∗

s−1−pi

2∗ s−p−1

2∗ s hZ

Ω×{0}

|w(x,0)|^{2}^{∗}^{s}dxi^{p+1}_{2}∗
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−q}dx]^{p−1}^{1+q}

−λ Z

Ω×{0}

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

≥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+1}dx≥0, then
ψw(t)→ −λ

Z

Ω×{0}

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

as t → ∞. Consequently, ψ_{w}(t) has exactly two points 0 < t_{1} < t_{max} < t_{2} such
that

ψ_{w}(t_{1}) = 0 =ψ_{w}(t_{2}) andψ^{0}_{w}(t_{1})>0> ψ_{w}^{0} (t_{2}).

From the definition ofN_{λ}^{±}, we gett1w∈ N_{λ}^{+}andt2w∈ N_{λ}^{−}. Nowφ^{0}_{w}(t) =t^{p}ψw(t).

Thus φ^{0}_{w}(t)< 0 in (0, t_{1}), φ^{0}_{w}(t) >0 in (t_{1}, t_{2}) and φ^{0}_{w}(t) <0 in (t_{2},∞). Hence
J_{λ}(t_{1}w) = inf_{0≤t≤t}_{max}J_{λ}(tw),J_{λ}(t_{2}w) = sup_{t≥t}

1J_{λ}(tw).

(ii) IfR

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

Z

Ω×{0}

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

ast→ ∞. Consequently,ψ_{w}(t) has exactly one point 0< t_{1}< t_{max} such that
ψw(t1) = 0 and ψ^{0}_{w}(t1)>0.

Now φ^{0}_{w}(t) = t^{p}ψ_{w}(t). Then φ^{0}_{w}(t) < 0 in (0, t_{1}), φ^{0}_{w}(t) > 0 in (t_{1},∞). Thus
J_{λ}(t_{1}w) = inf_{t≥0}J_{λ}(tw). Hence, it follows thatt_{1}w∈ 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−q}dx >0 and
Z

Ω×{0}

b(x)|w(x,0)|^{p+1}dx >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_{λ}^{0}={0}.

Proof. We prove this by contradiction. Assume that there exists 0 6≡ w ∈ N_{λ}^{0}.
Then it follows fromw∈ N_{λ}^{0} that

(1 +q)kwk^{2}=λ(p+q)
Z

Ω×{0}

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

0 =kwk^{2}−
Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx−λ
Z

Ω×{0}

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

= p−1
p+qkwk^{2}−

Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx.

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−q}dx^{p−1}_{1+q}

−λ Z

Ω×{0}

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

= 1 +q p+q

p−1 p+q

^{p−1}_{1+q} kwk^{2(p+q)}^{1+q}

p−1

p+qkwk^{2}^{p−1}_{1+q}

−1 +q

p+qkwk^{2}= 0,

a contradiction. Hencew≡0. That is,N_{λ}^{0}={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}> A_{0}>kwk for allw∈ N_{λ}^{+}, W ∈ N_{λ}^{−},

where

Aλ=h 1 +q λ(p+q)kbk(√

S)^{p+1}i1/(p−1)

and A0=hp+q p−1kak 1

√S

1−qi1/(q+1)

.
Proof. Ifw∈ N_{λ}^{+}⊂ Nλ, then

0<(1 +q)kwk^{2}−λ(p+q)
Z

Ω×{0}

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

= (1 +q)kwk^{2}−(p+q)h
kwk^{2}−

Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dxi

= (1−p)kwk^{2}+ (p+q)
Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx.

Hence it follows from (3.1) that
(p−1)kwk^{2}<(p+q)

Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx≤(p+q)kakkwk

√S 1−q

which yields

kwk<hp+q

p−1kak 1

√ S

^{1−q}i^{1/(q+1)}

≡A_{0}.
IfW ∈ N_{λ}^{−}, then it follows from (3.2) that

(1 +q)kWk^{2}< λ(p+q)
Z

Ω×{0}

b(x)|W(x,0)|^{p+1}dx≤λ(p+q)kbkkWk

√ S

^{p+1}

which yields

kWk>h 1 +q λ(p+q)kbk(√

S)^{p+1}i1/(p−1)

≡A_{λ}.
Now we show thatA_{λ}=A_{0}if 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+1}p−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)

≡A_{0}.
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
kW_{0}k^{2}= lim

k→∞kW_{k}k^{2}

= lim

k→∞

hZ

Ω×{0}

a(x)|Wk(x,0)|^{1−q}dx+λ
Z

Ω×{0}

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

= Z

Ω×{0}

a(x)|W_{0}(x,0)|^{1−q}dx+λ
Z

Ω×{0}

b(x)|W_{0}(x,0)|^{p+1}dx
and

(1 +q)kW_{0}k^{2}−λ(p+q)
Z

Ω×{0}

b(x)|W_{0}(x,0)|^{p+1}dx

= lim

k→∞

h

(1 +q)kWkk^{2}−λ(p+q)
Z

Ω×{0}

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

≤0.

That is,W_{0}∈ N_{λ}^{−}∩ N_{λ}^{0}. Since{W_{k}} ⊂ N_{λ}^{−}, from Lemma 3.4 we have
kW_{0}k= lim

k→∞kW_{k}k ≥A_{0}>0,

which imply, W0 6= 0. It follows from Lemma 3.1, that W0 6∈ N_{λ}^{0} 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+q}kw+vφk^{2}−
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) =kwk^{2}−
Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx−λ
Z

Ω×{0}

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

∂F

∂r(0,1) = (1 +q)kwk^{2}−λ(p+q)
Z

Ω×{0}

b(x)|w(x,0)|^{p+1}dx >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+vφk^{2}−
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)k^{2}−
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+vφk^{2}−λ(p+q)f^{p+q−1}(v)
Z

Ω×{0}

b(x)|(w+vφ)(x,0)|^{p+1}dx

=

(1 +q)kf(v)(w+vφ)k^{2}−λ(p+q)R

Ω×{0}b(x)|f(v)(w+vφ)(x,0)|^{p+1}dx

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)k^{2}−λ(p+q)

Z

Ω×{0}

b(x)|f(v)(w+vφ)(x,0)|^{p+1}dx >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

kwk^{2}− 1

1−q − 1 p+ 1

Z

Ω×{0}

a(x)|w(x,0)|^{1−q}dx

≥1 2 − 1

p+ 1

kwk^{2}− 1

1−q − 1 p+ 1

kakkwk

√ S

1−q

.

(3.4)

Now consider the functionρ:R^{+}→Ras ρ(t) =αt^{2}−βt^{1−q}, where α,β are both
positive constants. One can easily show that ρ is convex(ρ^{00}(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+q}^{2}

−ββ(1−q) 2α

^{1−q}_{1+q}

=−1 +q

2 β^{1+q}^{2} 1−q
2α

^{1−q}_{1+q}
.
Applying ρ(t) with α = ^{1}_{2} −_{p+1}^{1}

, β = _{1−q}^{1} − _{p+1}^{1}
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+q}^{2} 1−q
2α

^{1−q}_{1+q}

=− 1 +q (1−q)(p+ 1)

(p+q)kak 2(√

S)^{1−q}

_{1+q}^{2} 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_{λ}^{0}andN_{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_{λ}^{0} andN_{λ}^{−}. First, consider
{wk} ⊂ N_{λ}^{+}∪ N_{λ}^{0} with the following properties:

J_{λ}(w_{k})< inf

w∈N_{λ}^{+}∪N_{λ}^{0}

J_{λ}(w) +1

k (4.1)

J_{λ}(w)≥J_{λ}(w_{k})−1

kkw−w_{k}k, ∀w∈ N_{λ}^{+}∪ N_{λ}^{0}. (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
at^{2}−bt^{1−q} =ρ(t)≤J_{λ}(w)< inf

w∈N_{λ}^{+}∪N_{λ}^{0}

J_{λ}(w) +1
k ≤C_{5},

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)→w_{0}(·,0) strongly inL^{p}(Ω) for 1≤p <2^{∗}_{s} andw_{k}(·,0)→w_{0}(·,0) a.e. in
Ω.

For anyw∈ N_{λ}^{+}, we have from 0< q <1< pthat
Jλ(w) =1

2 − 1 1−q

kwk^{2}+ 1

1−q − 1 p+ 1

λ

Z

Ω×{0}

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

<1 2 − 1

1−q

kwk^{2}+ 1

1−q − 1 p+ 1

1 +q
p+qkwk^{2}

= 1 p+ 1−1

2 1 +q

1−qkwk^{2}<0,
which means that inf_{N}+

λ Jλ <0. Now for λ ∈ (0,Λ), we know from Lemma 3.1,
thatN_{λ}^{0}={0}. Together, these imply thatw_{k}∈ N_{λ}^{+} forklarge and

inf

w∈N_{λ}^{+}∪N_{λ}^{0}

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_{λ}^{0}

Jλ<0,

that is,w_{0}≥0,w_{0}6≡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−q}_{0} (x,0)dx−λ(p−1)
Z

Ω×{0}

b(x)w_{0}^{p+1}(x,0)dx >0. (4.3)

Moreover, there exists a constantC2>0 such that
(1 +q)kwkk^{2}−λ(p+q)

Z

Ω×{0}

b(x)w_{k}^{p+1}(x,0)dx≥C_{2}>0. (4.4)
Proof. For{w_{k}} ⊂ N_{λ}^{+}(⊂ N_{λ}), since

(1 +q) Z

Ω×{0}

a(x)w_{0}^{1−q}(x,0)dx−λ(p−1)
Z

Ω×{0}

b(x)w_{0}^{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)kwkk^{2}−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−q}_{0} (x,0)dx−λ(p−1)
Z

Ω×{0}

b(x)w_{0}^{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→∞

hkwkk^{2}−
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

≥ kw0k^{2}−
Z

Ω×{0}

a(x)w^{1−q}_{0} (x,0)dx−λ
Z

Ω×{0}

b(x)w^{p+1}_{0} (x,0)dx

=

(kw0k^{2}−λ^{p+q}_{1+q}R

Ω×{0}b(x)w^{p+1}_{0} (x,0)dx
kw0k^{2}−^{p+q}_{p−1}R

Ω×{0}a(x)w^{1−q}_{0} (x,0)dx.

Thus for anyλ∈(0,Λ) andw_{0}6≡0, by similar arguments as those in (3.3) we have
that

0< E_{λ}kw_{0}k^{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−q}_{0} (x,0)dx^{p−1}_{1+q}

−λ Z

Ω×{0}

b(x)w_{0}^{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^{2}^{p−1}_{1+q}

−1 +q

p+qkw0k^{2}= 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≥C_{2}
for sufficiently largekand a suitable positive constantC_{2}. 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} < C_{2}), 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_{k}k^{2}−
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_{k}^{2}(w)kwk+wφk^{2}−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_{k}^{0}(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_{k}^{2}(w)−1]kwk+wφk^{2}+kwk+wφk^{2}− kwkk^{2}

−[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_{k}^{2}(ρv)−1]kwk+ρvφk^{2}+kwk+ρvφk^{2}− kwkk^{2}

−[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_{k}^{0}(0), vikwkk^{2}+ 2ks

Z

CΩ

y^{c}∇wk∇(vφ)dx dy

−(1−q)hf_{k}^{0}(0), vi
Z

Ω×{0}

a(x)w_{k}^{1−q}(x,0)dx

−λ(p+ 1)hf_{k}^{0}(0), vi
Z

Ω×{0}

b(x)w^{p+1}_{k} (x,0)dx

−λ(p+ 1) Z

Ω×{0}

b(x)(w^{p}_{k}vφ)(x,0)dx

=hf_{k}^{0}(0), vih

2kwkk^{2}−(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}^{p}vφ)(x,0)dx

=hf_{k}^{0}(0), vih

(1 +q)kwkk^{2}

−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}^{p}vφ)(x,0)dx. (4.8)
From this inequality and (4.4) we know thathf_{k}^{0}(0), vi 6=−∞. Now we show that
hf_{k}^{0}(0), vi 6= +∞. Arguing by contradiction, we assume thathf_{k}^{0}(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_{k}k (4.9)
andfk(ρv)> fk(0) = 1 for sufficiently largek.

From the definition of derivative hf_{k}^{0}(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

kwkk^{2}+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_{k}^{2}(ρv)kwk+ρvφk^{2}

− 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}+ρvφk^{2}− kw_{k}k^{2}) + 1
1−q−1

2

[f_{k}^{2}(ρv)−1]kw_{k}+ρvφk^{2}

−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_{k}^{0}(0), vikwkk

k +kvφk k

≥1 +q 1−q

ks

Z

CΩ

y^{c}∇wk∇(vφ)dx dy+1 +q
1−q

hf_{k}^{0}(0), vikwkk^{2}

−λp+q 1−q

hf_{k}^{0}(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}_{k}vφ)(x,0)dx

= hf_{k}^{0}(0), vi
1−q

h

(1 +q)kw_{k}k^{2}−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}^{p}vφ)(x,0)dx.

That is, kvφk

k ≥ hf_{k}^{0}(0), vi
1−q

h

(1 +q)kw_{k}k^{2}−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}_{k}vφ)(x,0)dx,

(4.10)

which is impossible becausehf_{k}^{0}(0), vi= +∞and
(1+q)kwkk^{2}−λ(p+q)

Z

Ω×{0}

b(x)w^{p+1}_{k} (x,0)dx−(1−q)kwkk

k ≥C2−(1−q)C_{1}
k >0.

In conclusion, |hf_{k}^{0}(0), vi| < +∞. Furthermore, (4.4) with kw_{k}k ≤ C_{1} and two
inequalities (4.8) and (4.10) also imply that

|hf_{k}^{0}(0), vi| ≤C3

forksufficiently large and a suitable constantC_{3}.
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)w_{0}^{−q}vφ∈L^{1}(Ω)and

k_{s}
Z

CΩ

y^{c}∇w0∇(vφ)dx dy−
Z

Ω×{0}

a(x)(w^{−q}_{0} vφ)(x,0)dx

−λ Z

Ω×{0}

b(x)(w_{0}^{p}vφ)(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_{k}^{2}(ρv)−1

2 kw_{k}k^{2}−f_{k}^{2}(ρv)

2 (kw_{k}+ρvφk^{2}− kw_{k}k^{2})
+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_{k}^{0}(0), vi|kwkk

k +kvφk k