Multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

Multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent

Tiexiang Li

^a

, Tsung-fang Wu

^b

^,∗

aDepartment of Mathematics, Southeast University, Nanjing 211189, PR China

bDepartment of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

a r t i c l e i n f o a b s t r a c t

Article history:

Received 30 September 2009 Available online 16 March 2010 Submitted by J. Wei

Keywords:

Ljusternik–Schnirelmann category Multiple positive solutions Critical Sobolev exponent Nehari manifold

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the Ljusternik–

Schnirelmann category to prove that the existence of multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.

©2010 Elsevier Inc. All rights reserved.

1. Introduction

In this paper, we study the multiplicity of positive solutions for the following semilinear elliptic problem:

−

^u

=

^fλ

(

x

)|

^u

|

^q−2u

+

^g

(

x

)|

^u

|

^{p−2u in}

Ω,

u

=

0 in

∂Ω,

^(Eλ)

where 1

<

q

<

2

<

p^2∗

=

_N^2N₋₂ ^(N3),

Ω ⊂

R^N is a bounded domain with smooth boundary, the parameter

λ >

0 and the weight functions f_λ

= λ

f₊

+

^f−(f_±

= ±

^max

{

^f

,

0

}

^),g are continuous on

Ω

which satisfy the following condition:

(

Q

)

there exist a non-empty closed setM

= {

^x

∈ Ω |

^g

(

x

) =

^max_x∈Ω^g

(

x

) ≡

¹

}

^and

ρ >

N

−

2 such that M

⊂

x

∈ Ω

f

(

x

) >

0

and

g

(

z

) −

g

(

x

) =

O

|

x

−

z

|

^ρ

asx

→

zand uniformly inz

∈

M

.

Remark 1.1.LetMr

= {

^x

∈

R^N

|

^dist

(

x

,

M

) <

r

}

^forr

>

0. Then by the condition

(

Q

)

, we may assume that there exist two positive constantsC0andr0such that

f

(

x

),

g

(

x

) >

0 for allx

∈

^Mr0

⊂ Ω

*

Corresponding author.

E-mail addresses:txli@seu.edu.cn(T. Li),tfwu@nuk.edu.tw(T.-f. Wu).

0022-247X/$ – see front matter ©2010 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmaa.2010.03.022

and

g

(

z

) −

^g

(

x

)

^C0

|

^x

−

^z

|

^{ρ for allx}

∈

^BN

(

z

,

r0

)

and for allz

∈

^M

.

That the existence and multiplicity of positive solutions of Eq.

(

E_λ

)

may be influenced by the concave and convex nonlinearities. These issues have been the focus of a great deal of research in recent years. When the weight functions fλ

≡ λ

and g

≡

1, Ambrosetti, Brezis and Cerami [1] has proved that there exists

λ

0

>

0 such that Eq.

(

Eλ

)

admits at least two positive solutions for

λ ∈ (

0

, λ

0

)

, a positive solution for

λ = λ

0 and no positive solution for

λ > λ

0. Actually, Adimurthy, Pacella and Yadava [2], Ouyang and Shi [20] and Tang [22] proved that there exists

λ

0

>

0 such that Eq.

(

E_λ

)

in unit ball B^N

(

0

;

¹

)

has exactly two positive solutions for all

λ ∈ (

0

, λ

0

)

, has exactly one positive solution for

λ = λ

0 and no positive solution for all

λ > λ

0. When the weight functions f

,

g are sign-changing, Brown and Wu [6,7], de Figueiredo, Gossez and Ubilla [14], Hsu and Lin [15] and Wu [25,26] they proved that Eq.

(

E_λ

)

has at least two positive solutions when

λ

is sufficiently small. Furthermore, if p

=

^{2∗ and f}

,

g

∈

^C

(Ω)

satisfy the following conditions:

(

D1

)

B^N

(

2

δ

0

) \

^BN

(δ

0

) ⊂ Ω

for some

δ

0

>

0, where B^N

(

r

) = {

^x

∈

R^N

| |

^x

| <

r

}

^;

(

D2

)

f

=

^f+

−

^f−, where f^±

: Ω →

Rare continuous functions and there exists a domain B^N

(

2

δ

0

) \

^BN

(δ

0

) ⊂ Θ ⊂ Ω

of classC¹such that for allx

∈ Θ

, f₋

(

x

) =

^{0, f}+

(

x

) >

0 and for allx

∈ Ω\Θ

^{, f}−

(

x

)

^{0, f}+

(

x

) =

^0;

(

D3

)

theN-ball B^N

(

2

δ

0

) ⊂ Ω

such that for allx

∈

^BN

(δ

0

)

, 0

<

g

(

x

) <

1 and for allx

∈ Ω\

^BN

(

2

δ

0

)

, 0^g

(

x

) <

1;

(

D4

)

g

(

x

) =

^{0 for allx}

∈

^BN

(

2

δ

0

) \

^BN

(δ

0

)

,

then Wu [27] proved that Eq.

(

E_λ

)

has at least three positive solutions when

λ

is sufficiently small.

The main purpose of this paper is to consider the relation between the number of solutions and the topology of the global maxima set of weight function g. Our main result improves a recent multiplicity result of [15,25,27]. The following theorem is our main result.

Theorem 1.1.Suppose that p

=

^2∗. Then for each

δ <

r₀there exists

Λ

δ

>

0such that for

λ < Λ

δ, Eq.

(

E_λ

)

has at least cat_Mδ

(

M

) +

¹ positive solutions.

Hereaftercatis the Ljusternik–Schnirelmann category (see e.g. [17]).

In the following sections, we proceed to proof Theorem 1.1. We use the variational methods to find positive solutions of Eq.

(

E_λ

)

. Associated with Eq.

(

E_λ

)

, we consider the energy functional J_λinH₀¹

(Ω)

,

J_λ

(

u

) =

¹

2

^u

²_H1

−

¹ q

Ω

f_λ

|

^u

|

^qdx

−

¹ 2^∗

Ω

g

|

^u

|

^2∗dx

where

^u

H¹

= (

_Ω

|∇

^u

|

^2dx

)

^1/2 is the standard norm in H₀¹

(Ω)

. It is well known that the solutions of Eq.

(

E_λ

)

are the critical points of the energy functional J_λinH₀¹

(Ω)

.

This paper is organized as follows. In Section 2, we give some notations and preliminaries. In Section 3, we discuss some concentration behavior. In Section 4, we prove Theorem 1.1.

2. Notations and preliminaries

Throughout this paper, we denote by S the best Sobolev constant for the embedding ofD^1,2

(

R^N

)

intoL^2∗

(

R^N

)

which is given by

S

=

inf

u∈^D1,2(R^N)\{⁰}

R^N

|∇

^u

|

^2dx

(

_RN

|

u

|

^2∗dx

)

^2/2∗

>

0

.

(2.1)

It is well known that Sis independent of

Ω ⊂

R^N in the sense that if S

(Ω) =

^inf

u∈H¹₀(Ω)\{0}

Ω

|∇

^u

|

^2dx

(

_Ω

|

^u

|

^2∗dx

)

^2/2∗

>

0

,

then S

(Ω) =

^S

(

R^N

) =

S, and the function u_ε

(

x

) = ε

^(N−2)/2

( ε

²

+ |

^x

|

²

)

^(N−2)/2

, ε >

0 andx

∈ R

^N

,

is an extremal function for the minimum problem

(

2

.

1

)

. Moreover, for each

ε >

0, v_ε

(

x

) = [

N

(

N

−

2

) ε

²

]

^(N−2)/4

( ε

²

+ |

x

|

²

)

^{(N−2)/2 (2.2)}

which is a positive solution of critical problem:

−

^u

= |

^u

|

^{2∗−2u in}

R

^{N (2.3)}

with _RN

|∇

^vε

|

^2dx

=

_R^N

|

^vε

|

^2∗dx

=

^SN2.

We define the Palais–Smale (simply by (PS)) sequences, (PS)-values, and (PS)-conditions in H¹₀

(Ω)

for J_λas follows.

Definition 2.1.

(i) For

β ∈

R, a sequence

{

^un

}

^{is a}

(

PS

)

_β-sequence in H¹₀

(Ω)

for J_λif J_λ

(

un

) = β +

^o

(

1

)

; and J_λ

(

un

) =

^o

(

1

)

; strongly in H⁻¹

(Ω)

asn

→ ∞

^.

(ii) J_λsatisfies the

(

PS

)

β-condition inH₀¹

(Ω)

if every

(

PS

)

β-sequence in H₀¹

(Ω)

for J_λcontains a convergent subsequence.

As the energy functional Jλ is not bounded below on H¹₀

(Ω)

, it is useful to consider the functional on the Nehari manifold

N_λ

=

u

∈

^H10

(Ω) \{

⁰

}

^Jλ

(

u

),

u

=

⁰

.

Thus,u

∈

^Nλif and only if

u

²_H1

−

Ω

fλ

|

u

|

^qdx

−

Ω

g

|

u

|

^2∗dx

=

0

.

Moreover, we have the following results.

Lemma 2.2.The energy functional Jλis coercive and bounded below onNλ.

Proof. Ifu

∈

^Nλ, then by the Young and Sobolev inequalities J_λ

(

u

) =

¹

N

^u

²_H1

−

1

q

−

¹ 2^∗

Ω

f_λ

|

^u

|

^{qdx (2.4)}

¹

N

^u

²_H1

− λ

²

∗

−

^q

q2^∗

^f+

_Lq∗S^−q2

^u

^q_H1 (2.5)

¹

N

^u

²_H1

−

¹

N

^u

²_H1

−

^D

λ

^2/(2−q)

= −

^D0

λ

^2/(2−q)

,

(2.6)

whereq^∗

=

₂∗²−^∗q andD is a positive constant depending onq

,

N

,

S and

^f+

L^q∗. Thus, J_λis coercive and bounded below onNλ. 2

Define

ψ

_λ

(

u

) =

J_λ

(

u

),

u

=

^u

²_H1

−

Ω

f_λ

|

^u

|

^qdx

−

Ω

g

|

^u

|

^2∗dx

.

Then foru

∈

^Nλ,

ψ

_λ

(

u

),

u

=

2

u

²_H1

−

q

Ω

fλ

|

u

|

^qdx

−

2^∗

Ω

g

|

u

|

^2∗dx

= −

⁴

N

−

²

^u

²_H1

−

q

−

^2∗

Ω

f_λ

|

^u

|

^{qdx (2.7)}

= (

2

−

^q

)

^u

²_H1

−

2^∗

−

^q

Ω

g

|

^u

|

^2∗dx

.

(2.8)

Similarly to the method used in [26], we splitN_λinto three parts:

N⁺_λ

=

u

∈

^Nλ

ψ

_λ

(

u

),

u

>

0

;

^N0_λ

=

u

∈

^Nλ

ψ

_λ

(

u

),

u

=

⁰

;

^N−_λ

=

u

∈

^Nλ

ψ

_λ

(

u

),

u

<

0

.

We now derive some basic properties ofN⁺_λ,N⁰_λandN⁻_λ.

Lemma 2.3.Suppose that u0is a local minimizer for JλonNλand that u0

∈ /

^N0λ. Then J_λ

(

u0

) =

^{0in H−1}

(Ω)

. Furthermore, if u0is a non-trivial nonnegative function in

Ω

, then u₀is a positive solution of Eq.

(

E_λ

)

.

Proof. Similarly the argument in [8, Theorem 2.3] (or see [3]), we have J_λ

(

u0

) =

^{0 in H−1}

(Ω)

. This implies u0 is a weak solution of Eq.

(

E_λ

)

. Now, ifu₀ is a non-trivial nonnegative function in

Ω

, then by [13, Lemma 2.1], we haveu₀

∈

^L∞

(Ω)

. Moreover, we can apply the Harnack inequality due to [23] in order to getu0 is positive in

Ω

. 2

Lemma 2.4.For each

λ >

0we have the following:

(i) For any u

∈

^N+λ, we have _Ω fλ

|

^u

|

^qdx

>

0.

(ii) For any u

∈

^N0_λ^{, we have} _Ω ^fλ

|

^u

|

^qdx

>

0and _Ωg

|

^u

|

^2∗dx

>

0.

(iii) For any u

∈

^N−_λ^{, we have} _Ω^g

|

^u

|

^2∗dx

>

0.

Proof. The result now follows immediately from

(

2

.

7

)

and

(

2

.

8

)

. 2

Let

Λ

1

=

^S

N(2−q) 4 +q

2

^f+

Lq∗

(

₂²_∗⁻₋^q_q

)

^{(N−2)(2−q)4}

(

²₂^∗_∗⁻₋²_q

)

. Then by an argument similar to that the proof Lemma 2.1 in [26], we have N⁰_λ

= ∅

^{for all}

λ ∈ (

0

, Λ

1

)

. Thus, we can writeN_λ

=

^N+_λ

∪

^N−_λ ^{and define}

α

_λ⁺

₌

inf

u∈^N+_λ J_λ

(

u

)

and

α

_λ⁻

₌

inf

u∈^N−_λ J_λ

(

u

).

Moreover, we have the following result, whose proof can be found in [27, Theorem 3.1].

Theorem 2.5.We have the following:

(i)

α

⁺_λ

<

0for all

λ ∈ (

0

, Λ

1

)

.

(ii) If

λ < Λ

2

=

^q₂

Λ

1, then

α

⁻_λ

>

c0for some c0

>

0.

In particular,

α

⁺_λ

₌

infu∈^Nλ Jλ

(

u

)

for all

λ ∈ (

0

, Λ

2

)

.

For each u

∈

^H1₀

(Ω) \{

⁰

}

^with _Ω^g

|

^u

|

^2∗dx

>

0, we write

t_max

=

(

2

−

^q

)

^u

²_H1

(

2^∗

−

q

)

_Ωg

|

u

|

^2∗dx

^N−₄²

>

0

.

Then we have the following lemma.

Lemma 2.6.For each u

∈

^H10

(Ω) \{

⁰

}

we have the following:

(i) If _Ω fλ

|

^u

|

^qdx0, then there is a unique t⁻

=

^t−

(

u

) >

tmaxsuch that t⁻u

∈

^N−_λ ^{and h}uis increasing on

(

0

,

t⁻

)

and decreasing on

(

t⁻

, ∞ )

. Moreover,

J_λ

t⁻u

=

^sup

t⁰ J_λ

(

tu

).

(2.9)

(ii) If _Ω f_λ

|

^u

|

^qdx

>

0, then there are unique0

<

t⁺

=

^t+

(

u

) <

t_max

<

t⁻ such that t⁺u

∈

^N+_λ^{, t−u}

∈

^N−_λ^{, h}u is decreasing on

(

0

,

t⁺

)

, increasing on

(

t⁺

,

t⁻

)

and decreasing on

(

t⁻

, ∞)

. Moreover,

J_λ

t⁺u

=

^inf

0^ttmax

J_λ

(

tu

) ;

^Jλ

t⁻u

=

^sup

tt⁺

J_λ

(

tu

).

(2.10)

Proof. The proof is almost the same as the in [26, Lemma 2.4]. 2 Forc

>

0, we define

J₀^c

(

u

) =

¹

2

^u

²_H1

−

^c 2^∗

Ω

g

|

^u

|

^2∗dx

;

^Nc₀

=

u

∈

^H1₀

(Ω)\{

⁰

}

^Jc₀

(

u

),

u

=

⁰

.

Note thatN^c₀

=

^N0

=

^N−₀ ^forc

=

1 and for eachu

∈

^N−_λ there is a uniquet_u

>

0 such thatt_uu

∈

^N0. Moreover, we have the following result, whose proof can be found in [25, Lemma 5.2].

Lemma 2.7.Let q^∗

=

₂∗²−^∗q. Then for each u

∈

^N−λ we have the following:

(i) There is a unique t^c

(

u

) >

0such that t^c

(

u

)

u

∈

^Nc₀^and

sup

t⁰ J^c₀

(

tu

) =

^J₀^c

t^c

(

u

)

u

=

¹ N

^u

²_H^∗1

c _Ωg

|

^u

|

^2∗dx

^N−₂²

.

(ii)

J_λ

(

u

) (

1

− λ)

^N2 J₀

(

t_uu

) − λ(

2

−

^q

)

2q

^f+

_Lq∗S^−2q

₂₋²_q

and

J_λ

(

u

) (

1

+ λ)

^N2 J0

(

tuu

) + λ(

2

−

^q

)

2q

^f+

L^q∗S^−2q

₂₋²_q

.

3. Concentration behavior

First, we consider the following critical problem:

−

u

= |

^u

|

^{2∗−2u in}

Ω,

u

∈

H¹₀

(Ω).

⁽

_E0)

Associated with Eq.

(

E₀

)

, we consider the energy functional J^∞ inH¹₀

(Ω)

, J^∞

(

u

) =

¹

2

^u

²_H1

−

¹ 2^∗

Ω

|

^u

|

^2∗dx

.

It is well known that inf

u∈N^∞(R^N)

J^∞

(

u

) =

^inf

u∈^N∞(Ω)J^∞

(

u

) =

¹

NS^N2 for all domain

Ω ⊂ R

^N

,

where N^∞

R

^N

=

u

∈

D^1,2

R

^N

\{

0

}

J^∞

(

u

),

u

=

0

and

N^∞

(Ω) =

u

∈

^H1₀

(Ω)\{

⁰

} (

u

),

u

=

⁰

are the Nehari manifolds. Actually, infu∈^N∞(Ω)J^∞

(

u

)

is never attained on a domain

Ω

R^N. Following the method of [4]

and Remark 1.1, let

η _∈

C₀^∞

(

R^N

)

be a radially symmetric function such that 0

η

1,

|∇ η _|

C and

η (

x

) =

1

, |

^x

|

^r0

/

2

,

0

, |

^x

|

^r0

.

For anyz

∈

M, let

w_ε,z

(

x

) = η (

x

−

z

)

v_ε

(

x

−

z

),

(3.1)

wherevε

(

x

)

as in

(

2

.

2

)

. Then, by an argument similar to that the proof Lemma 4.2 in [16] (or see Struwe [21]), we have

w_ε,z

²_H1

=

S^N2

+

O

ε

^N−2

uniformly inz

∈

M

.

(3.2)

Moreover, we have the following results.

Lemma 3.1.We have uinf∈^N0

J0

(

u

) =

^inf

u∈^N∞(Ω)J^∞

(

u

) =

¹ NS^N2

.

Furthermore, Eq.

(

E0

)

does not admit any positive solution u0such that J0

(

u0

) =

_N^1SN2.

Proof. Letwε,zbe as in

(

3

.

1

)

and define g

:

R^N

→

R^by g

(

x

) =

g

(

x

),

ifx

∈ Ω,

0

,

ifx

∈ Ω

^c

,

as an extension of g. Then, by Lemma 2.6, there is a uniquet0

(

wε,z

) >

0 such thatt0

(

wε,z

)

wε,z

∈

^N0for all

ε >

0, that is

t₀

(

w_ε,z

)

w_ε,z

²

H¹

=

Ω

g

t₀

(

w_ε,z

)

w_ε,z

^2∗dx

and so

t0

(

w_ε,z

)

4/(N−²)

=

^Ωg

|

^wε,z

|

^2∗dx

^wε,z

²_H1

.

(3.3)

By the definition ofvε, we get that

Ω

g

|

^wε,z

|

^2∗dx

=

B^N(z,2r0)

g

(

x

) η (

x

−

^z

)

v_ε

(

x

−

^z

)

^2∗dx

=

R^N

[

^N

(

N

−

²

) ε

²

_]

^N/2g

(

x

+

^z

) η

^2∗

(

x

) ( ε

²

_{+ |}

x

|

²

)

^{N dx}

.

Thus, by the condition

(

Q

)

,

0

¹

[

^N

(

N

−

²

) ε

²

_]

^N/2

R^N

|

v_ε

|

^2∗dx

−

Ω

g

|

w_ε,z

|

^2∗dx

=

R^N\^BN(0,r₀/2)

[

¹

−

^g

(

x

+

^z

) η

^2∗

(

x

)]

( ε

²

+ |

^x

|

²

)

^{N dx}

+

B^N(0,r₀/2)

[

¹

−

^g

(

x

+

^z

)]

( ε

²

+ |

^x

|

²

)

^{N dx}

R^N\^BN(0,r₀/2) 1

|

x

|

^2Ndx

+

C0

B^N(0,r₀/2)

|

x

|

^ρ

( ε

²

+ |

x

|

²

)

^Ndx

^N

ω

N

∞

r0/2

r^−(N+1)dr

+

^C0N

ω

N

ε

²

r0/2

0

r^ρ−N+1dr

= ω

N

r₀ 2

₋N

+

^C0N

ω

N

ε

²

( ρ − (

N

−

²

))

r₀

2

ρ−(N−2)

=

^C1

+

^C2

ε

^{2 for allz}

^∈

^M

^,

^(3.4)

where

ω

N is the volume of the unit ball B^N

(

0

,

1

)

inR^{N. Then} εlim→⁰

Ω

g

|

^wε,z

|

^2∗dx

=

^SN2 uniformly inz

∈

^M

.

(3.5)

From the condition

(

Q

)

and the same argument of

(

3

.

5

)

, we have εlim→0

Ω

f₋

|

^wε,z

|

^2∗dx

=

0 uniformly inz

∈

^M

.

(3.6)

By

(

3

.

2

)

,

(

3

.

5

)

and

(

3

.

6

)

,

εlim→⁰t₀

(

w_ε,z

) =

^{1 and lim}

ε→⁰

^wε,z

²_H1

=

^SN2 uniformly inz

∈

^M

.

Thus, uinf∈^N0

J₀

(

u

)

^J0

t₀

(

w_ε,z

)

w_ε,z

→

¹

NS^N2 as

ε →

⁰

and so inf

u∈N0

J₀

(

u

)

^inf

u∈N^∞(Ω)J^∞

(

u

) =

¹ NS^N2

.

Letu

∈

^N0. Then, by Lemma 2.6(i), J0

(

u

) =

^supt⁰ J0

(

tu

)

. Moreover, there is a uniquetu

>

0 such thattuu

∈

^N∞

(Ω)

. Thus, J₀

(

u

)

^J0

(

t_uu

)

^J∞

(

t_uu

)

¹

NS^N2

.

This implies infu∈^N0 J0

(

u

)

¹_N^SN2. Therefore, uinf∈N0

J0

(

u

) =

inf

u∈N^∞(Ω)J^∞

(

u

) =

¹ NS^N2

.

Next, we will show that Eq.

(

E₀

)

does not admit any solution u₀ such that J₀

(

u₀

) =

^infu∈N0 J₀

(

u

)

. Suppose the contrary.

Then we can assume that there existsu₀

∈

^N0 such that J₀

(

u₀

) =

^infu∈N0 J₀

(

u

)

. Since J₀

(

u₀

) =

^J0

(|

^u0

|)

^and

|

^u0

| ∈

^N0, by Lemma 2.3, we may assume thatu0 is a positive solution of Eq.

(

E0

)

. Moreover, by Lemma 2.6(i), J0

(

u0

) =

^supt⁰ J0

(

tu0

)

. Thus, there is a uniquetu0

>

0 such thattu0u0

∈

^N∞

(Ω)

and so

1

NS^N2

=

^inf

u∈^N0

J₀

(

u

) =

^J0

(

u₀

)

^J0

(

t_u0u₀

)

=

J^∞

(

tu₀u0

) +

^t2

∗ u₀

2^∗

Ω

(

1

−

g

)|

u0

|

^2∗dx

¹

NS^N2

+

^t

2^∗ u0

2^∗

Ω

(

1

−

^g

) |

^u0

|

^2∗dx

.

This implies _Ω

(

1

−

^g

) |

^u0

|

^2∗dx

=

0, which is a contradiction. This completes the proof. 2 Lemma 3.2.Suppose that

{

^un

}

is a minimizing sequence for J0inN0. Then

(i) _Ω f₋

|

^un

|

^qdx

=

^o

(

1

)

; (ii) _Ω

(

1

−

^g

)|

^un

|

^2∗dx

=

^o

(

1

)

. Furthermore,

{

^un

}

^{is a}

(

PS

)

₁

NS^N2-sequence for J^∞in H₀¹

(Ω)

.

Proof. For eachn, there is a uniquetn

>

0 such thattnun

∈

^N∞

(Ω)

, that is t_n²

Ω

|∇

^un

|

^2dx

=

^t2n^∗

Ω

|

^un

|

^2∗dx

.

Then, by Lemma 2.6(i),

J0

(

un

)

^J0

(

tnun

) =

^J∞

(

tnun

) −

^t

q n

q

Ω

f₋

|

^un

|

^qdx

+

^t2

∗ n

2^∗

Ω

(

1

−

^g

) |

^un

|

^2∗dx

¹

NS^N2

−

^t

q n

q

Ω

f₋

|

^un

|

^qdx

+

^t2

∗ n

2^∗

Ω

(

1

−

^g

) |

^un

|

^2∗dx

.

Since J0

(

un

) =

¹_N^SN2

+

^o

(

1

)

from Lemma 3.1, we have t_n^q

q

Ω

f₋

|

^un

|

^qdx

=

^o

(

1

)

and t_n^2∗

2^∗

Ω

(

1

−

^g

)|

^un

|

^2∗dx

=

^o

(

1

).

We will show that there exists c₀

>

0 such thatt_n

>

c₀ for alln. Suppose the contrary. Then we may assume t_n

→

^{0 as} n

→ ∞

^{. Since}

J0

(

un

) =

¹

NS^N2

+

^o

(

1

)

and

J0

(

un

) =

¹

N

^un

²_H1

+

^o

(

1

),

by Lemma 2.2,

^un

is uniformly bounded and so

^tnun

H¹

→

^{0 or J∞}

(

tnun

) →

0 and this contradicts J^∞

(

tnun

)

_N^1SN2

>

0.

Thus,

Ω

f₋

|

^un

|

^qdx

=

^o

(

1

)

and

Ω

(

1

−

^g

) |

^un

|

^2∗dx

=

^o

(

1

),

this implies

Ω

|∇

^un

|

^2dx

=

Ω

|

^un

|

^2∗dx

+

^o

(

1

)

and

J^∞

(

un

) =

¹

NS^N2

+

^o

(

1

).

Similar to the method used in [24, Lemma 7], we have

{

^un

}

^{is a}

(

PS

)

₁

NS^N2-sequence for J^∞in H¹₀

(Ω)

. 2 For the positive numbersd, consider the filtration of the Nehari manifoldN0 as follows:

N₀

(

d

) =

u

∈

^N0

^J0

(

u

)

¹ NS^N2

+

^d

.

Let

Φ :

^H1₀

(Ω) →

R^N be a barycenter map defined by

Φ(

u

) =

^Ωx

|

^u

|

^2∗dx

Ω

|

^u

|

^2∗dx

.

(3.7)

Such a map has been constructed in Bartsch and Weth [5], Coron [10], Cerami and Passaseo [11] and Clapp and Weth [12], etc. Then we have the following result.

Lemma 3.3.For each positive number

δ <

r0, there exists dδ

>

0such that

Φ(

u

) ∈

^Mδ for all u

∈

^N0

(

d_δ

).

Proof. Suppose the contrary. Then we can assume that there exist a sequence

{

^un

} ∈

^N0 and

δ

0

<

r0 such that J0

(

un

) =

1

NS^N2

+

^o

(

1

)

and

Φ(

un

) / ∈

^Mδ0 for alln

.

(3.8)

Then, by Lemma 3.2, we have

{

^un

}

^{is a}

(

PS

)

₁

NS^N2-sequence for J^∞ inH¹₀

(Ω)

. It follows from Lemma 2.2 that there exist a subsequence

{

^un

}

^andu0

∈

^H1₀

(Ω)

such thatun

u0weakly inH¹₀

(Ω)

. Since

Ω

is a bounded domain, by the concentration- compactness principle (see Lions [18,19] or Struwe [21, Theorem 3.1]), there exist two sequences

{

^xn

} ⊂ Ω

,

{

^Rn

} ⊂

R^+, x0

∈ Ω

and a positive solutionv0

∈

^D1,2

(R

^N

)

of critical problem

(

2

.

3

)

with J^∞

(

v0

) =

_N^{1SN2 such thatx}n

→

^x0 andRn

→ ∞

asn

→ ∞

^and

u_n

(

x

) −

^Rn^N−22v₀

R_n

(

x

−

^xn

)

D^1,2(R^N)

→

^{0 asn}

→ ∞.

^(3.9)