Volume 7, Number 2, March 2008 pp. 383–405

ON SEMILINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS

AND SIGN-CHANGING WEIGHT FUNCTION

Tsung-fang Wu Department of Applied Mathematics

National University of Kaohsiung, Kaohsiung 811, Taiwan (Communicated by Manuel del Pino)

Abstract. In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result to prove that the semilinear elliptic equation with a sign-changing weight function has at least two positive solutions.

1. Introduction. In this paper, we consider the multiplicity results of positive solutions of the following semilinear elliptic equation:

½

*−∆u = λf (x) |u|q−2u + |u|*2*∗−2u in Ω,*
*u ∈ H*1

0*(Ω) ,*

*(Eλ,f*)
where Ω is a bounded domain in R*N _{(N ≥ 3) , 1 < q < 2 < 2}∗*

_{=}

*2N*

*N −2, λ > 0 and*
*f : Ω → R is a continuous function with f*+_{(x) = max {f (x) , 0} 6≡ 0 in Ω (f is}

*possibly sign-changing on Ω). Associated with the equation (Eλ,f) , we consider the*
*energy functional Jλ* *in H*01(Ω)
*Jλ(u) =*
1
2
Z
Ω
*|∇u|*2*dx −λ*
*q*
Z
Ω
*f (x) |u|qdx −* 1
2*∗*
Z
Ω
*|u|*2*∗dx.*

*It is well known that the solutions of equation (Eλ,f*) are the critical points of the
*energy functional Jλ* *in H*1

0(Ω)(see Rabinowitz [15]).

*It is known that the number of positive solutions of equation (Eλ,f*) is affected
by the concave and convex nonlinearities. This issue has been the focus of a great
*deal of research in recent years. If the weight function f (x) ≡ 1, the authors*
Ambrosetti–Brezis–Cerami [3] have investigated the following equation

½

*−∆u = λ |u|q−2u + |u|p−2u in Ω,*
*u ∈ H*1

0*(Ω) ,*

*(Eλ,1)*
*where 1 < q < 2 < p ≤ 2∗ _{. They found that there exists λ}*

0 *> 0 such that the*

*equation (Eλ,1) admits at least two positive solutions for λ ∈ (0, λ*0*) , has a positive*

*solution for λ = λ*0 *and no positive solution exists for λ > λ*0*. Moreover, they*

*proved that when the domain Ω is starshaped and p = 2∗*_{, one has the following}
*result: given a sequence {λn} ⊂ R*+ _{with λn}_{& 0 as n → ∞, then there exists a}

*2000 Mathematics Subject Classification. Primary 35J65; Secondary 35J20.*

*Key words and phrases. Critical Sobolev exponent, Nehari manifold, concave-convex*

nonlinearities.

Partially supported by the National Science Council of Republic of China.

*sequence {uλn} of positive solutions of equation (Eλn,f) such that kuλnk∞* *→ ∞*
*as n → ∞. Actually, Adimurthi–Pacella–Yadava [1], Damascelli–Grossi–Pacella [7],*
Ouyang–Shi [14] and Tang [18] proved that if 1 < q < 2 < p < 2*∗*_{then there exists a}
*λ*0*> 0 such that the equation (Eλ,1) in unit ball BN*(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 exists for all λ > λ*0*.*

For more general results, Ambrosetti–Azorezo–Peral [2], de Figueiredo-Grossez-Ubilla [10,11] and Wu [21] considered the following general subcritical problem:

½

*−∆u = λa (x) |u|q−2u + b (x) |u|p−2u in Ω,*
*0 ≤ u ∈ H*1

0*(Ω) ,*

*(Eλ,a,b)*
*where 1 < q < 2 < p ≤ 2∗* _{and a, b satisfy some integrability conditions or change}*sign in Ω. They proved that the equation (Eλ,a,b) has at least two positive solutions*
*if λ is sufficiently small.*

The main purpose of this paper is to generalize the partial results of Ambrosetti–
Brezis–Cerami [3], using the decomposition of the Nehari manifold as λ varies. First,
*we prove that the equation (Eλ,f) has at least two positive solutions for λ sufficiently*
small.

Theorem 1.1. *There exists λ*0 *> 0 such that for λ ∈ (0, λ*0*) , the equation (Eλ,f*)

*has at least two positive solutions.*

As for the asymptotic behavior of the solutions obtained in Theorem1.1*as λ → 0,*
we have the following results.

Theorem 1.2. *Assume that a sequence {λn} ⊂ R satisfies λn> 0 and*
*λn→ 0 as n → ∞.*

*Then there exists a subsequence {λn} and two sequences {u(j)n* *(x)} (j = 1, 2) of*
*positive solutions of equation (Eλn,f) such that*

*(i) ku*(1)*n* *kH*1*→ 0 as n → ∞;*

*(ii) there exist two sequences {xn} ⊂ Ω, {Rn} ⊂ R*+ *and a positive solution v*0 *∈*

*D1,2*¡_{R}*N*¢ _{of critical problem}*−∆u = |u|*2*∗−2u in RN _{,}*

*such that*

*Rn→ ∞ as n → ∞*

*and*

_{°}°

*°u*(2)

*n*

*(x) − R*

*N −2*2

*n*

*v*0

*(Rn(x − xn*)) ° ° °

*D1,2*

_{(R}

*N*

_{)}

*→ 0 as n → ∞.*

This paper is organized as follows. In Section 2, we give some notation and
*preliminaries. In Section 3, we establish the existence of a local minimum for Jλ*on
Mλ*(Ω). In Section 4, we prove that the equation (Eλ,f*) has at least two positive
*solutions for λ sufficiently small. In Section 5, we prove Theorem*1.2.

2. Notation and preliminaries. We define the Palais–Smale (PS) sequences,
*(PS)–values, and (PS)–conditions in H*1

0*(Ω) for Jλ* as follows:

Definition 2.1. *(i) For β ∈ R, a sequence {un} is a (PS)β–sequence in H*01(Ω) for

*Jλ* *if Jλ(un) = β + o(1) and Jλ0(un) = o(1) strongly in H−1(Ω) as n → ∞;*
*(ii) β ∈ R is a (PS)–value in H*1

0*(Ω) for Jλ* if there exists a (PS)β–sequence in

*H*1

*(iii) Jλ* satisfies the (PS)*β–condition in H*01(Ω) if every (PS)*β–sequence in H*01(Ω)

*for Jλ* contains a convergent subsequence.

*Throughout this section, we denote by S the best Sobolev constant for the *
*imbed-ding of H*1
0*(Ω) into L*2
*∗*
(Ω), i.e.,
*S =* inf
*u∈H*1
0*(Ω)\{0}*
R
Ω*|∇u|*
2
*dx*
³R
Ω*|u|*
2*∗*
*dx*
´*2/2∗* *> 0.*
*In particular, kuk _{L}2∗*

*≤ S−*1 2

*kuk*

*H*1 *for all u ∈ H*_{0}1*(Ω) \ {0} . Now, we consider the*

*Nehari minimization problem: for λ ≥ 0,*

*αλ(Ω) = inf {Jλ(u) | u ∈ Mλ(Ω)} ,*
where Mλ(Ω) =©*u ∈ H*1

0*(Ω) \ {0} | hJλ0* *(u) , ui = 0*
ª

*. Define*
*ψλ(u) = hJλ0* *(u) , ui = kuk*2*H*1*− λ*

Z
Ω
*f (x) |u|qdx −*
Z
Ω
*|u|*2*∗dx.*
*Then for u ∈ Mλ(Ω) ,*
*hψ0*
*λ(u) , ui = 2 kuk*
2
*H*1*− λq*
Z
Ω
*f (x) |u|qdx − 2∗*
Z
Ω
*|u|*2*∗dx.*

Similarly to the method used in Tarantello[17], we split Mλ(Ω) into three parts:
M+_{λ}(Ω) = {u ∈ Mλ(Ω) | hψλ0*(u) , ui > 0} ,*

M0

*λ(Ω) = {u ∈ Mλ(Ω) | hψλ0* *(u) , ui = 0} ,*
M*− _{λ}(Ω) = {u ∈ Mλ(Ω) | hψλ0*

*(u) , ui < 0} .*Then, we have the following results.

Lemma 2.2. *There exists λ*1*> 0 such that for each λ ∈ (0, λ*1*) we have M*0*λ*(Ω) =
*∅.*

*Proof.* We consider the following two cases.

*Case (I): u ∈ Mλ*(Ω) and R_{Ω}*f (x) |u|qdx = 0. We have*
*kuk*2* _{H}*1

*−*Z Ω

*|u|*2

*∗dx = 0.*Thus,

*hψ0λ(u) , ui = 2 kuk*2

*H*1

*− 2∗*Z Ω

*|u|*2

*∗dx = (2 − 2∗) kuk*2

*1*

_{H}*< 0*

*and so u /∈ M*0

*λ(Ω) .*

*Case (II): u ∈ Mλ*(Ω) and R_{Ω}*f (x) |u|qdx 6= 0.*
Suppose that M0

*λ(Ω) 6= ∅ for all λ > 0. If u ∈ M*0*λ*(Ω), then we have
*0 = hψ0*
*λ(u) , ui = 2 kuk*
2
*H*1*− λq*
Z
Ω
*f (x) |u|qdx − 2∗*
Z
Ω
*|u|*2*∗dx*
*= (2 − q) kuk*2* _{H}*1

*− (2∗− q)*Z Ω

*|u|*2

*∗dx.*Thus,

*kuk*2

*1= 2*

_{H}*∗*

_{− q}*2 − q*Z Ω

*|u|*2

*∗dx*(2)

and
*λ*
Z
Ω
*f (x) |u|qdx = kuk*2* _{H}*1

*−*Z Ω

*|u|*2

*∗dx =*2

*∗*

_{− 2}*2 − q*Z Ω

*|u|*2

*∗dx.*(3) Moreover, µ 2

*∗*2

_{− 2}*∗*¶

_{− q}*kuk*2

*1*

_{H}*= kuk*2

*1*

_{H}*−*Z Ω

*|u|*2

*∗dx = λ*Z Ω

*f (x) |u|qdx*

*≤ λ kf k*

_{L}q∗S−*q*2

*kukq*

*H*1

*, where q∗*= 2

*∗*2

*∗*This implies

_{− q}.*kuk*1

_{H}*≤*·

*λ*µ 2

*∗*2

_{− q}*∗*¶

_{− 2}*kf k*

_{L}q∗S−*q*2 ¸ 1

*2−q*

*.*(4)

*Let Iλ*: Mλ*(Ω) → R be given by*

*Iλ(u) = K (2∗, q)*
* kuk*
*2N +4*
*N −2*
*H*1
R
Ω*|u|*
2*∗*
*dx*
*N −2*
4
*− λ*
Z
Ω
*f (x) |u|qdx,*
*where K (2∗ _{, q) =}*³

*2−q*2

*∗*´

_{−q}*N +2*4 ³

_{2}

*∗*

_{−2}*2−q*´

*. Then Iλ(u) = 0 for all u ∈ M*0

*λ(Ω) . Indeed,*
by (2) and (3),
*Iλ(u) = K (2∗, q)*
* kuk*
*2N +4*
*N −2*
*H*1
R
Ω*|u|*
2*∗*
*dx*
*N −2*
4
*− λ*
Z
Ω
*f (x) |u|qdx*
=
µ
*2 − q*
2*∗ _{− q}*
¶

*N +2*4 µ

_{2}

*∗*

_{− 2}*2 − q*¶ ³ 2

*∗*

_{−q}*2−q*´

*N +2*

*N −2*³R Ω

*|u|*2

*∗*

*dx*´

*N +2*

*N −2*R Ω

*|u|*2

*∗*

*dx*

*N −2*4

*−*2

*∗*

_{− 2}*2 − q*Z Ω

*|u|*2

*∗dx*

*= 0.*(5)

However, by (4) , the H¨older and Sobolev inequalities,

*Iλ(u) = K (2∗, q)*
* kuk*
*2N +4*
*N −2*
*H*1
R
Ω*|u|*
2*∗*
*dx*
*N −2*
4
*− λ*
Z
Ω
*f (x) |u|qdx*
*≥ K (2∗, q)*
* kuk*
*2N +4*
*N −2*
*H*1
R
Ω*|u|*
2*∗*
*dx*
*N −2*
4
*− λ kf k _{L}q∗kuk*

*q*

*L2∗*

*≥ kukq*Ã

_{L}2∗*K (2∗*

_{, q)}*S*

*q+N*2

*kukq−1*1

_{H}*− λ kf k*!

_{L}q∗*≥ kukq*(

_{L}2∗*K (2∗*2

_{, q) S}q+N*λ*

*1−q*

*2−q*·µ 2

*∗*2

_{− q}*∗*¶

_{− 2}*kf k*

_{L}q∗S−*q*2 ¸

*1−q*

*2−q*

*−λ kf k*

_{L}q∗} .*This implies that there exists λ*1*> 0 such that for λ ∈ (0, λ*1*) , we have Iλ(u) > 0*
*for all u ∈ M*0

*λ*(Ω), this contradicts (5). Thus, we can conclude that M0*λ(Ω) = ∅*
*for all λ ∈ (0, λ*1*).*

Lemma 2.3. *If u ∈ M*+* _{λ}(Ω) , then* R

_{Ω}

*f (x) |u|qdx > 0.*

*Proof.*Since

*kuk*2

*1*

_{H}*− λ*Z Ω

*f (x) |u|qdx −*Z Ω

*|u|*2

*∗dx = 0*and

*kuk*2

*1*

_{H}*>*2

*∗*

_{− q}*2 − q*Z Ω

*|u|*2

*∗dx.*We have

*λ*Z Ω

*f (x) |u|qdx = kuk*2

*1*

_{H}*−*Z Ω

*|u|*2

*∗dx >*2

*∗*

_{− 2}*2 − q*Z Ω

*|u|*2

*∗dx > 0.*This completes the proof.

By Lemma2.2, for λ ∈ (0, λ1) we write Mλ(Ω) = M+*λ* *(Ω) ∪ M−λ* (Ω) and define
*α*+
*λ* (Ω) = inf
*u∈M*+
*λ*(Ω)
*Jλ(u) ; α− _{λ}* (Ω) = inf

*u∈M−*

*λ*(Ω)

*Jλ(u) .*

The following lemma shows that the minimizers on Mλ(Ω) are “usually” critical
*points for Jλ.*

Lemma 2.4. *For λ ∈ (0, λ*1*). If u*0 *is a local minimizer for Jλ* *on Mλ(Ω) , then*
*J0*

*λ(u*0*) = 0 in H−1(Ω) .*

*Proof.* *If u*0 *is a local minimizer for Jλ* on Mλ*(Ω) , then u*0 is a solution of the

optimization problem

*minimize Jλ(u) subject to ψλ(u) = 0.*

*Hence, by the theory of Lagrange multipliers, there exists θ ∈ R such that*
*J0*
*λ(u*0*) = θψ0λ(u*0*) in H−1(Ω) .*
This implies
*hJ0*
*λ(u*0*) , u*0*i _{H}*1

*= θ hψ*

_{λ}0*(u*0

*) , u*0

*i*1

_{H}*.*(6)

*Since u*0

*∈ Mλ(Ω) , that is ku*0

*k*2

*1*

_{H}*− λ*R Ω

*f (x) |u*0

*|*

*q*R Ω

_{dx −}*|u*0

*|*2

*∗*

*dx = 0. We have*

*hψ0*

*λ(u*0

*) , u*0

*i*1

_{H}*= (2 − q) ku*0

*k*2

*1*

_{H}*− (2∗− q)*Z Ω

*|u*0

*|*2

*∗*

*dx.*

*Moreover, hψ0*

*λ(u*0*) , u*0*i _{H}*1

*6= 0 and so by (*6) , θ = 0. This completes the proof.

*For each u ∈ H*1
0*(Ω) \ {0} , we write*
*t*max=
Ã
*(2 − q) kuk*2* _{H}*1
(2

*∗*R Ω

_{− q)}*|u|*2

*∗*

*dx*! 1

*2∗−2*

*> 0.*Then, we have the following lemma.

Lemma 2.5. *Let q∗*_{=} 2*∗*
2*∗ _{−q}*

*and λ*2= ³ 2

*∗*2

_{−2}*∗*´ ³

_{−q}*2−q*2

*∗*´

_{−q}*2−q*

*2∗−2*

*S2∗−q2∗−2kf k−1*

*Lq∗. Then for*

*each u ∈ H*1 0

*(Ω) \ {0} and λ ∈ (0, λ*2

*), we have*

*(i) there is a unique t− _{= t}−_{(u) > t}*

max*> 0 such that t−u ∈ M− _{λ}*

*(Ω) and Jλ(t−u) =*

maxt≥tmax*Jλ(tu) ;*

*(ii) t− _{(u) is a continuous function for nonzero u;}*

*(iii) M−*(Ω) = n

_{λ}*u ∈ H*1 0

*(Ω) \ {0} |*

*kuk*1

*³*

_{H1}t−*u*

*kuk*´ = 1 o ;

_{H1}*(iv) if* R_{Ω}*f (x) |u|qdx > 0, then there is a unique 0 < t*+* _{= t}*+

_{(u) < t}max *such that*

*t*+* _{u ∈ M}*+

*λ* *(Ω) and Jλ(t*+*u) = min0≤t≤t−J _{λ}(tu) .*

*Proof.* *(i) Fix u ∈ H*1

0*(Ω) \ {0}. Let*
*s (t) = t2−q _{kuk}*2

*H*1

*− t*2

*∗*Z Ω

_{−q}*|u|*2

*∗dx for t ≥ 0.*

*We have s(0) = 0, s(t) → −∞ as t → ∞, s (t) is concave and achieves its maximum*
*at t*max*. Moreover,*
*s(t*max) =
Ã
*(2 − q) kuk*2* _{H}*1
(2

*∗*R Ω

_{−q)}*|u|*2

*∗*

*dx*!

*2−q*

*2∗−2*

*kuk*2

*1*

_{H}*−*Ã

*(2 − q) kuk*2

*1 (2*

_{H}*∗*R Ω

_{− q)}*|u|*2

*∗*

*dx*!

*2∗−q*

*2∗−2*Z Ω

*|u|*2

*∗dx*

*= kukq*1 µ

_{H}*2 − q*2

*∗*¶

_{− q}*2−q*

*2∗−2*

*−*µ

*2 − q*2

*∗*¶

_{− q}*2∗−q*

*2∗−2* Ã

*kuk*2

*1 R Ω*

_{H}∗*|u|*2

*∗*

*dx*!

*2−q*

*2∗−2*

*≥ kukq*1 µ 2

_{H}*∗*2

_{− 2}*∗*¶ µ

_{− q}*2 − q*2

*∗*¶

_{− q}*2−q*

*2∗−2*

*SN (2−q)*4

*,*or

*s (t*max

*) ≥ kukqH*1 µ 2

*∗*2

_{− 2}*∗*¶ µ

_{− q}*2 − q*2

*∗*¶

_{− q}*2−q*

*2∗−2*

*SN (2−q)*4

*.*(7)

*Case (I) :*R_{Ω}*f (x) |u|qdx ≤ 0.*
*Then there is a unique t− _{> t}*

max*such that s (t−*) =

R
Ω*f (x) |u|*
*q*
*dx and s0 _{(t}−_{) < 0.}*
Now,

*(2 − q)*°

*°t−*°

_{u}_{°}2

*H*1

*− (2∗− q)*Z Ω ¯

*¯t−*¯

_{u}_{¯}2

*∗*= ¡

_{dx}*t−*¢

*1+q*·

*(2 − q)*¡

*t−*¢

*1−q*2

_{kuk}*H*1

*− (2∗− q)*¡

*t−*¢2

*∗−q−1*Z Ω ¯

*¯t−*¯

_{u}_{¯}2

*∗*¸ = ¡

_{dx}*t−*¢

*1+q*¡

_{s}0*¢*

_{t}−*and *

_{< 0}*J0*

*λ*¡

*t−*¢

_{u}*®*

_{, t}−_{u}_{=}¡

*¢2*

_{t}−*2*

_{kuk}*H*1

*−*¡

*t−*¢

*q*Z Ω

_{λ}*f (x) |u|qdx −*¡

*t−*¢2

*∗*Z Ω

*|u|*2

*∗dx*= ¡

*t−*¢q ·

*s*¡

*t−*¢

*− λ*Z Ω

*f (x) |u|qdx*¸

*= 0.*

*Thus, t−*

_{u ∈ M}−*λ(Ω) . Moreover, for t > t*max*, we have*
*(2 − q) ktuk*2* _{H}*1

*− (2∗− q)*Z Ω

*|tu|*2

*∗dx < 0,*

*d*2

*dt*2

*Jλ(tu) < 0*

and
*d*
*dtJλ(tu) = t kuk*
2
*H*1*− tq−1λ*
Z
Ω
*f (x) |u|qdx − t*2*∗ _{−1}*Z
Ω

*|u|*2

*∗dx = 0 for t = t−*

_{.}*This implies Jλ(t−*

_{u) = max}*t≥t*max*Jλ(tu) .*

*Case (II) :*R_{Ω}*f (x) |u|qdx > 0.*
By (7) and
*s (0) = 0 < λ*
Z
Ω
*f (x) |u|qdx ≤ λ kf k _{L}q∗S*

*−q*2

*kukq*

*H*1

*< kukq*1 µ 2

_{H}*∗*2

_{− 2}*∗*¶ µ

_{− q}*2 − q*2

*∗*¶

_{− q}*2−q*

*2∗−2*

*SN (2−q)*4

*≤ s (t*

_{max}

*) for λ ∈ (0, λ*

_{2}

*),*

*there are unique t*+ _{and t}−* _{such that 0 < t}*+

_{< t}max*< t−,*
*s*¡*t*+¢* _{= λ}*
Z
Ω

*f (x) |u|qdx = s*¡

*t−*¢ and

*s0*¡

*+¢*

_{t}*¡*

_{> 0 > s}0*¢*

_{t}−

_{.}*We have t±*

_{u ∈ M}±*λ(Ω) , Jλ(t−u) ≥ Jλ(tu) ≥ Jλ(t*+*u) for each t ∈ [t*+*, t−*] and
*Jλ(t*+*u) ≤ Jλ(tu) for each t ∈ [0, t*+*] . Thus,*

*Jλ*
¡
*t−u*¢= max
*t≥t*max
*Jλ(tu) and Jλ*
¡
*t*+*u*¢= min
*0≤t≤t−Jλ(tu) .*

*(ii) By the uniqueness of t− _{(u) and the external property of t}−_{(u) , we see that}*

*t−*

_{(u) is a continuous function of u 6= 0.}*(iii) For u ∈ M−*

*λ* *(Ω), let v =* *kuku _{H1}. By part (i), there is a unique t−(v) > 0 such*

*that t−*

_{(v) v ∈ M}−*λ*

*(Ω) or t−*³

*u*

*kukH1*´ 1

*kukH1u ∈ M*

*−*

*λ*

*(Ω) . Since u ∈ M−λ*

*(Ω) , we*

*have t−*³

*u*

*kuk*´ 1

_{H1}*kuk _{H1}*

*= 1, and this implies*M

*−*

*λ*

*(Ω) ⊂*½

*u ∈ H*1 0

*(Ω) \ {0} |*1

*kuk*1

_{H}*t−*µ

*u*

*kuk*1 ¶ = 1 ¾

_{H}*.*

*Conversely, let u ∈ H*1

0*(Ω) \ {0} such that* *kuk*1* _{H1}t−*
³

*u*

*kuk*´

_{H1}*= 1. Then*

*t−*µ

*u*

*kuk*1 ¶

_{H}*u*

*kuk*1

_{H}*∈ M−*

_{λ}*(Ω) .*Thus, M

*−*(Ω) = ½

_{λ}*u ∈ H*1 0

*(Ω) \ {0} |*1

*kuk*1

_{H}*t−*µ

*u*

*kuk*1 ¶ = 1 ¾

_{H}*.*

*(iv) By case (II) of part (i).*

3. Existence of a local minimum. First, we establish the existence of positive solutions for the equation:

½

*−∆u = λf (x) |u|q−2u in Ω,*
*0 ≤ u ∈ H*1

0*(Ω) .*

(8) Associated with equation (8) , we consider the energy functional

*Kλ(u) =*
1
2
Z
Ω
*|∇u|*2*dx −λ*
*q*
Z
Ω
*f (x) |u|qdx*

and the minimization problem

*βλ(Ω) = inf {Kλ(u) | u ∈ Nλ(Ω)} ,*
where Nλ(Ω) =©*u ∈ H*1

0*(Ω) \ {0} | hKλ0* *(u) , ui = 0*
ª

*. Then we have the following*
result.

Theorem 3.1. *For each λ > 0, the equation (*8) there exists a positive solution vλ
*such that Iλ(vλ) = βλ(Ω) < 0.*

*Proof.* *First, we need to show that Iλ* is bounded below on Nλ*(Ω) and βλ(Ω) < 0.*
*Then for u ∈ Nλ(Ω) ,*
*kuk*2* _{H}*1

*= λ*Z Ω

*f (x) |u|qdx ≤ λ kf k*

_{L}q∗S−*q*2

*kukq*

*H*1

*.*

*where q∗*

_{=}2

*∗*2

*∗*

_{−q}. This implies*kuk*1

_{H}*≤*³

*λ kf k*

_{L}q∗S−*q*2 ´ 1

*2−q*

*.*(9) Hence

*Kλ(u) =*µ 1 2

*−*1

*q*¶

*kuk*2

*1*

_{H}*≤*µ 1 2

*−*1

*q*¶ ³

*λ kf k*

_{L}q∗S−*q*2 ´ 1

*2−q*

*for all u ∈ Nλ(Ω) and βλ(Ω) < 0. Let {vn} be a minimizing sequence for Iλ* on
Nλ*(Ω) , then by (9) and the compact imbedding theorem, there exist a subsequence*
*{vn} and vλin H*01(Ω) such that

*vn* vλ* *weakly in H*01(Ω)

and

*vn→ vλ* *strongly in Lq(Ω).* (10)

First, we claim thatR_{Ω}*f (x) |vλ|qdx > 0. If not, by (*10) we can conclude that
Z
Ω
*f (x) |vλ|qdx = 0 and*
Z
Ω
*f (x) |vn|qdx → 0 as n → ∞.*
Thus, _{Z}
Ω
*|∇vn|*2*dx = o (1)*
and
*Kλ(vn) =* 1
2
Z
Ω
*|∇vn|*2*dx −λ*
*q*
Z
Ω
*f (x) |vn|qdx = 0 as n → ∞,*

*this contradicts Kλ(vn) → βλ(Ω) < 0 as n → ∞. Thus,* R_{Ω}*f (x) |vλ|qdx > 0.*
*In particular, vλ* *6≡ 0. Now, we prove that vn* *→ vλ* *strongly in H*01*(Ω). Suppose*

*otherwise, then kvλk _{H}*1

*< lim inf*

*n→∞* *kvnkH*1 and so
*kvλk*2*H*1*− λ*
Z
Ω
*f (x) |vλ|qdx < lim inf _{n→∞}*
µ

*kvnk*2

*H*1

*− λ*Z Ω

*f (x) |vn|qdx*¶

*= 0.*ByR

_{Ω}

*f (x) |vλ|qdx > 0, there is a unique t*0

*6= 1 such that t*0

*vλ∈ Nλ(Ω) . Thus,*

*t*0*vn* t*0*vλ* *weakly in H*01*(Ω).*

Moreover,

*which is a contradiction. Hence vn* *→ vλ* *strongly in H*01*(Ω). This implies vλ* *∈*
Nλ(Ω) and

*Kλ(vn) → Kλ(vλ) = βλ(Ω) as n → ∞.*

*Since Kλ(vλ) = Kλ(|vλ|) and |vλ| ∈ Nλ(Ω) , without loss of generality, we may*
*assume that vλ* is nonnegative. Similar to the argument of proof in Lemma 2.4,
*vλ* is a nonnegative solution of equation (8) . Now, by Dr´abek–Kufner–Nicolosi [8,
*Lemma 2.1] we have vλ∈ L∞ _{(Ω) . Then we can apply the Harnack inequality due}*
to Trudinger [19] (or see Gilbarg–Trudinger [12]) in order to get that vλ is positive

*in Ω.*

Lemma 3.2. *(i) Let vλ* *be a positive solution of equation (*8) as in Theorem *3.1.*
*Then there exists tλ> 0 such that αλ(Ω) <* * _{N}*2

*t*2

*λβλ(Ω) < 0;*

*(ii) Jλ* *is coercive and bounded below on Mλ(Ω) for all λ ∈ (0, _{N (2−q)+2q}*4

*].*

*Proof.*

*(i) Let vλ∈ H*1

0(Ω) be a positive solution of equation (8) as in Theorem3.1

such that
*Kλ(vλ) = βλ(Ω) .*
Then _{Z}
Ω
*f (x) |vλ|qdx = kvλk*2* _{H}*1=
µ

*2q*

*q − 2*¶

*βλ(Ω) > 0.*

*Set tλ= t*+

_{(vλ) > 0 as defined by Lemma}_{2.5. Hence tλ}

_{v}*λ∈ M*+*λ*(Ω) and
*Jλ(tλvλ) =*
µ
1
2*−*
1
*q*
¶
*t*2*λkvλk*2*H*1+
µ
1
*q* *−*
1
2*∗*
¶
*t*2*λ∗*
Z
Ω
*|vλ|*2
*∗*
*dx*
*< −*1
*N*
*2 − q*
*q* *t*
2
*λkvλk*2* _{H}*1
= 2

*Nt*2

*λβλ(Ω) .*This yields,

*αλ(Ω) <*2

*Nt*2

*λβλ(Ω) < 0.*(11)

*(ii) For u ∈ Mλ(Ω) , we have kuk*2* _{H}*1

*= λ*

R

Ω*f (x) |u|*

*q*

*+ dx*R_{Ω}*|u|*2*∗dx. Then by the*
H¨older inequality and the Young inequality,

*Jλ(u) =* 1
*N* *kuk*
2
*H*1*− λ*
µ
1
*q* *−*
1
2*∗*
¶ Z
Ω
*f (x) |u|qdx*
*≥* 1
*N* *kuk*
2
*H*1*− λ*
µ
1
*q* *−*
1
2*∗*
¶
*kf k _{L}q∗S*

*−q*2

*kukq*

*H*1

*≥*· 1

*N*

*− λ*µ 2

*∗*

_{− q}*2 × 2∗*¶¸

*kuk*2

*1*

_{H}*− λ*µ

*2 − q*

*2q∗*¶ ³

_{q}*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*= 1

*N*·

*1 − λN (2 − q) + 2q*4 ¸

*kuk*2

*1*

_{H}*− λ*µ

*2 − q*

*2q∗*¶ ³

_{q}*kf k*

_{L}q∗S*q*2 ´ 2

*2−q*

*.*

*Thus, Jλ*is coercive on Mλ(Ω) and

*Jλ(u) ≥ −λ*
µ
*2 − q*
*2q∗ _{q}*
¶ ³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*(12)

*for all λ ∈ (0,*4

_{N (2−q)+2q}*].*

Lemma 3.3. *For each u ∈ Mλ(Ω) , there exist ² > 0 and differentiable function*
*ξ : B (0; ²) ⊂ H*1

0*(Ω) → R*+ *such that ξ (0) = 1, the function ξ (v) (u − v) ∈ Mλ*(Ω)
*and*
*hξ0(0) , vi =*2
R
Ω*∇u∇vdx − qλ*
R
Ω*f (x) |u|*
*q−2*
*uvdx − 2∗*R
Ω*|u|*
2*∗ _{−2}*

*uvdx*

*(2 − q)*R

_{Ω}

*|∇u|*2

*dx − (2∗*R Ω

_{− q)}*|u|*2

*∗*

*dx*(13)

*for all v ∈ H*1 0

*(Ω) .*

*Proof.* *For u ∈ Mλ(Ω), define a function F : R × H*1

0*(Ω) → R given by*
*Fu(ξ, w) = hJλ0(ξ (u − w)) , ξ (u − w)i*
*= ξ*2
Z
Ω
*|∇ (u − w)|*2*dx − ξqλ*
Z
Ω
*f (x) |u − w|qdx − ξ*2*∗*
Z
Ω
*|u − w|*2*∗dx.*
*Then Fu(1, 0) = hJ0*
*λ(u) , ui = 0 and*
*d*
*dtFu(1, 0) = 2*
Z
Ω
*|∇u|*2*dx − qλ*
Z
Ω
*f (x) |u|qdx − 2∗*
Z
Ω
*|u|*2*∗dx*
*= (2 − q)*
Z
Ω
*|∇u|*2*dx − (2∗− q)*
Z
Ω
*|u|*2*∗dx 6= 0.*

*According to the implicit function theorem, there exists ² > 0 and a differentiable*
*function ξ : B (0; ²) ⊂ H*1¡_{R}*N*¢_{→ R such that ξ (0) = 1,}

*hξ0 _{(0) , vi =}* 2
R
Ω

*∇u∇vdx − qλ*R Ω

*f (x) |u|*

*q−2*

*uvdx − 2∗*R Ω

*|u|*2

*∗*

_{−2}*uvdx*

*(2 − q)*R

_{Ω}

*|∇u|*2

*dx − (2∗*R Ω

_{− q)}*|u|*2

*∗*

*dx*and

*Fu(ξ (v) , v) = 0 for all v ∈ B (0; ²)*which is equivalent to

*hJ0*

*λ(ξ (v) (u − v)) , ξ (v) (u − v)i = 0 for all v ∈ B (0; ²) ,*
*that is, ξ (v) (u − v) ∈ Mλ(Ω) .*

Proposition 1. *Let λ*0= min

n

*λ*1*, λ*2*, _{N (2−q)+2q}*4

o

*, then for λ ∈ (0, λ*0*) there exists*

*a minimizing sequence {un} ⊂ Mλ(Ω) such that*

*Jλ(un) = αλ(Ω) + o (1) and Jλ0* *(un) = o (1) in H−1(Ω) .*
*i.e. {un} is a (PS)αλ*(Ω)*–sequence in H*

1

0*(Ω) for Jλ.*

*Proof.* By Lemma 3.2*(ii) and the Ekeland variational principle [9], there exists a*
*minimizing sequence {un} ⊂ Mλ*(Ω) such that

*Jλ(un) < αλ*(Ω) + 1

*n* (14)

and

*Jλ(un) < Jλ(w) +* 1

*nkw − unkH*1 *for each w ∈ Mλ(Ω) .* (15)

*By taking n large, from Lemma*3.2*(i) we have*
*Jλ(un) =* 1
*N* *kunk*
2
*H*1*−*
µ
1
*q−*
1
2*∗*
¶
*λ*
Z
Ω
*f (x) |un|qdx* (16)
*< αλ*(Ω) + 1
*n*
*<* 2
*Nt*
2
*λβλ(Ω) .*

This implies
*λ*
Z
Ω
*f (x) |un|qdx >* *−2*
*∗ _{q}*
2

*∗*2

_{− q}*Nt*2

*λβλ(Ω) > 0.*(17)

*Consequently un6= 0 and putting (*16), (17) and the H¨older inequality together, we
obtain
·
*−2∗ _{q}*
(2

*∗*2

_{− q) N}*λt*2

*λβλ(Ω) S*

*q*2

*kf k−1*

*Lq∗*¸1

*q*

*< kunkH*1

*<*·

*λN (2∗*2

_{− q)}*∗*

_{q}*kf kLq∗S*

*−q*2 ¸ 1

*2−q*

*.*(18) Now, we will show that

*kJλ0* *(un)kH−1* *→ 0 as n → ∞.*

Applying Lemma3.3*with un* *to obtain the functions ξn: B (0; ²n) → R*+ _{for some}

*²n* *> 0, such that ξn(w) (un− w) ∈ Mλ(Ω) . Choose 0 < ρ < ²n. Let u ∈ H*01(Ω)

*with u 6≡ 0 and let wρ*=_{kuk}ρu

*H1. We set ηρ= ξn(wρ) (un− wρ) . Since ηρ∈ Mλ(Ω) ,*

we deduce from (15) that

*Jλ(ηρ) − Jλ(un) ≥ −*1

*nkηρ− unkH*1

and by the mean value theorem, we have
*hJ0*
*λ(un) , ηρ− uni + o*
¡
*kηρ− unk _{H}*1
¢

*≥ −*1

*nkηρ− unkH*1

*.*Thus,

*hJ0*

*λ(un) , −wρi + (ξn(wρ) − 1) hJλ0* *(un) , (un− wρ)i* (19)
*≥ −*1
*nkηρ− unkH*1*+ o*
¡
*kηρ− unk _{H}*1
¢

*.*

*From ξn(wρ) (un− wρ) ∈ Mλ*(Ω) and (19) that

*−ρ*
¿
*J0*
*λ(un) ,*
*u*
*kuk _{H}*1
À

*+ (ξn(wρ) − 1) hJ0*

*λ(un) − Jλ0* *(ηρ) , (un− wρ)i*
*≥ −*1
*nkηρ− unkH*1*+ o*
¡
*kηρ− unkH*1
¢
*.*
Thus,
¿
*Jλ0* *(un) ,*
*u*
*kuk _{H}*1
À

*≤*

*kηρ− unkH*1

*nρ*+

*o*¡

*kηρ− unkH*1 ¢

*ρ*+

*(ξn(wρ) − 1)*

*ρ*

*hJ*

*0*

*λ(un) − Jλ0* *(ηρ) , (un− wρ)i .* (20)
Since
*kηρ− unk _{H}*1

*≤ ρ |ξn(wρ)| + |ξn(wρ) − 1| kunkH*1 and lim

*ρ→0*

*|ξn(wρ) − 1|*

*ρ*

*≤ kξ*

*0*

*n(0)k .*

*If we let ρ → 0 in (20) for a fixed n, then by (18) we can find a constant C > 0,*
*independent of ρ, such that*

¿
*J0*
*λ(un) ,*
*u*
*kuk _{H}*1
À

*≤*

*C*

*n*

*(1 + kξ*

*0*

*n(0)k) .*

*We are done once we show that kξ0*

*n(0)k is uniformly bounded in n. By (13) , (18)*
and the H¨older inequality, we have

*hξn0* *(0) , vi ≤*
*b kvk _{H}*1
¯
¯

*¯(2 − q)*R

_{Ω}

*|∇un|*2

*dx − (2∗− q)*R Ω

*|un|*2

*∗*

*dx*¯ ¯ ¯

*for some b > 0.*We only need to show that

¯
¯
¯
*¯(2 − q)*
Z
Ω
*|∇un|*2*dx − (2∗− q)*
Z
Ω
*|un|*2
*∗*
*dx*
¯
¯
¯
*¯ > c* (21)

*for some c > 0 and n large. We argue by way of contradiction. Assume that there*
*exists a subsequence {un} , we have*

*(2 − q)*
Z
Ω
*|∇un|*2*dx − (2∗− q)*
Z
Ω
*|un|*2
*∗*
*dx = o (1) .* (22)
Combining (22) with (18) , we can find a suitable constant d > 0 such that

Z

Ω

*|un|*2

*∗*

*dx ≥ d for n sufficiently large.* (23)
In addition, (22) , and the fact that un*∈ Mλ*(Ω) also give

*λ*
Z
Ω
*f (x) |un|qdx = kunk*2*H*1*−*
Z
Ω
*|un|*2
*∗*
*dx =* 2
*∗ _{− 2}*

*2 − q*Z Ω

*|un|*2

*∗*

*dx + o (1)*and

*kunkH*1

*≤*·

*λ*µ 2

*∗*2

_{− 2}*∗*¶

_{− q}*kf k*

_{L}q∗S−*q*2 ¸ 1

*2−q*

*+ o (1) .*(24) This implies

*Iλ(un) = K (2∗, q)*

* kunk*

*2N +4*

*N −2*

*H*1 R Ω

*|un|*2

*∗*

*dx*

*N −2*4

*− λ*Z Ω

*f (x) |un|qdx*= µ

*2 − q*2

*∗*¶

_{− q}*N +2*4 µ

_{2}

*∗*

_{− 2}*2 − q*¶ ³ 2

*∗*

_{−q}*2−q*´

*N +2*

*N −2*³R Ω

*|un|*2

*∗*

*dx*´

*N +2*

*N −2*R Ω

*|un|*2

*∗*

*dx*

*N −2*4

*−*2

*∗− 2*

*2 − q*Z Ω

*|un|*2

*∗*

*dx + o (1)*

*= o (1) .*(25)

However, by (23) , (24) and λ ∈ (0, λ0)
*Iλ(un) = K (2∗, q)*
* kunk*
*2N +4*
*N −2*
*H*1
R
Ω*|un|*
2*∗*
*dx*
*N −2*
4
*− λ*
Z
Ω
*f (x) |un|qdx*
*≥ K (2∗ _{, q)}*

* kunk*

*2N +4*

*N −2*

*H*1 R Ω

*|un|*2

*∗*

*dx*

*N −2*4

*− λ kf k*

_{L}q∗kunkq_{L}2∗*≥ kunkq*

_{L}2∗*K (2∗, q) Sq+N*2

*kunk*

*2N +4*

*N −2*

*H*1

*kunk*2

*∗*

_{(q+1)−2q}*H*1

*N −2*4

*− λ kf k*

_{L}q∗*≥ kunkq*(

_{L}2∗*K (2∗, q) Sq+N*2

*λ*

*1−q*

*2−q*·µ 2

*∗*2

_{− q}*∗*¶

_{− 2}*kf k*

_{L}q∗S−*q*2 ¸

*1−q*

*2−q*

*−λ kf k*

_{L}q∗}*> d*0 *for some d*0*> 0 and n sufficiently large.*

This contradicts (25). We get
¿
*J0*
*λ(un) ,*
*u*
*kuk _{H}*1
À

*≤C*

*n.*

*This shows that {un} is a (PS)αλ(Ω)–sequence for Jλ.*

*Final, we establish the existence of a local minimum for Jλ*on Mλ(Ω).

Theorem 3.4. *Let λ*0*> 0 be as in Proposition1, then for λ ∈ (0, λ*0*), the functional*

*Jλ* *has a minimizer u*+0 *in M*+*λ(Ω) and it satisfies*
*(i) Jλ*¡*u*+0

¢

*= α*+_{λ}*(Ω) = αλ*(Ω) ;
*(ii) u*+

0 *is a nonnegative (nontrivial) solution of equation (Eλ,f*) ;
*(iii) Jλ*¡*u*+_{0}¢*→ 0 as λ → 0.*

*Proof.* *Let {un} ⊂ Mλ*(Ω) is a (PS)*αλ(Ω)–sequence for Jλ, then by Lemma*3.2and

*the compact imbedding theorem, there exist a subsequence {un} and u*+_{0} *∈ H*1
0(Ω),

*a solution of equation (Eλ,f*) such that

*un* u*+0 *weakly in H*01(Ω)

and

*un→ u*+0 *strongly in Lq(Ω).* (26)

First, we claim that R_{Ω}*f (x)*¯*¯u*+
0

¯
¯*q*

*dx 6= 0. If not, by Lemma* 2.3 and (26) we can

conclude that _{Z}
Ω
*f (x)*¯*¯u*+_{0}¯¯*qdx = 0*
and _{Z}
Ω
*f (x) |un|qdx → 0 as n → ∞.*
Thus, _{Z}
Ω
*|∇un|*2*dx =*
Z
Ω
*|un|*2
*∗*
*dx + o (1)*

and
1
*N*
Z
Ω
*|un|*2
*∗*
*dx =* 1
2
Z
Ω
*|∇un|*2*dx −*
*λ*
*q*
Z
Ω
*f (x) |un|qdx −*
1
2*∗*
Z
Ω
*|un|*2
*∗*
*dx + o (1)*
*= αλ(Ω) + o (1) ,*

*this contradicts αλ(Ω) < 0. Thus,* R_{Ω}*f (x)*¯*¯u*+
0

¯
¯*q*

*dx 6= 0. In particular, u*+
0 is a

*nontrivial solution of equation (Eλ,f) . Moreover,*
*αλ(Ω) ≤ Jλ*
¡
*u*+_{0}¢= 1
*N*
Z
Ω
¯
*¯∇u*+
0
¯
¯2
*dx −*
µ
1
*q−*
1
2*∗*
¶
*λ*
Z
Ω
*f (x)*¯*¯u*+_{0}¯¯*qdx*
*≤* lim
*n→∞Jλ(un) = αλ(Ω) .*
*Consequently, un* *→ u*+
0 *strongly in H*01*(Ω) and Jλ*
¡
*u*+
0
¢
*= αλ(Ω) . From Lemma*
2.5*it follows that necessarily u*+_{0} *∈ M*+* _{λ}(Ω) and Jλ*¡

*u*+

_{0}¢

*= α*+

_{λ}*(Ω) = αλ(Ω) . Since*

*Jλ*¡

*u*+ 0 ¢

*= Jλ*¡¯

*¯u*+ 0 ¯ ¯¢

_{and}¯

*+ 0 ¯*

_{¯u}*¯ ∈ M*+

*λ(Ω) , by Lemma*2.4, we may assume that u+0 is

*a nonnegative (nontrivial) solution of equation (Eλ,f) . Moreover, by Lemmas*3.2,
*0 > Jλ*¡*u*+0
¢
*≥ −λ*
µ
*2 − q*
*2q∗ _{q}*
¶ ³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*

*.*

*We obtain Jλ*¡

*u*+ 0 ¢

*→ 0 as λ → 0.*

4. Proof of Theorem 1.1. First, we consider

*uε(x) =* *ε*

*(N −2)/2*

³

*ε*2* _{+ |x|}*2´

*(N −2)/2*

*, ε > 0 and x ∈ RN*

is an extremal function for the Sobolev inequality in R*N _{. Since f is a continuous}*
function on Ω and

*f*+_{(x) = max {f (x) , 0} 6≡ 0.}

Following the method of [5], let Σ = {x ∈ Ω | f (x) > 0} be a open set of positive measure. Without loss of generality, we may assume that the set Σ is a domain. We consider the test function

*wε,y(x) = ηy(x) uε,y(x) , x ∈ RN,*
*where y ∈ Σ, uε,y(x) = uε(x − y) and ηy∈ C∞*

0 (Σ) with

*ηy* *≥ 0 and ηy* *= 1 near y.*

*Let λ*0*> 0 as in Proposition* 1, then for λ ∈ (0, λ0) we have the following result.

Lemma 4.1. *Let u*+

0 *be the local minimum in Theorem3.4. Then for every l > 0*

*and a.e. y ∈ Σ, there exists ε*0*= ε*0*(l, y) > 0 such that*

*Jλ(u*+
0 *+ lwε,y) < αλ*(Ω) +
1
*NS*
*N*
2
*for all ε ∈ (0, ε*0*) .*

*Proof.* Since
*Jλ(u*+_{0} *+ lwε,y) =* 1
2
°
*°u*+
0 *+ lwε,y*
°
°2
*H*1*−*
*λ*
*q*
Z
Ω
*f (x)*¯*¯u*+_{0} *+ lwε,y*¯¯*qdx*
*−*1
2*∗*
Z
Ω
¯
*¯u*+
0 *+ lwε,y*
¯
¯2*∗*
*dx*
= 1
2
°
*°u*+
0
°
°2
*H*1+
*l*2
2 *kwε,yk*
2
*H*1+
*u*+_{0}*, lwε,y*
®
*H*1
*−λ*
*q*
Z
Ω
*f (x)*¯*¯u*+
0 *+ lwε,y*
¯
¯*q*
*dx −* 1
2*∗*
Z
Ω
¯
*¯u*+
0 *+ lwε,y*
¯
¯2*∗*
*dx.*(27)

A careful estimate obtained by Brezis–Nirenberg (see formulas (17) and (21) in [5])
shows that
Z
Ω
¯
*¯u*+
0 *+ lwε,y*
¯
¯2*∗*
*dx =*
Z
Ω
¯
*¯u*+
0
¯
¯2*∗*
*dx + l*2*∗*
Z
Ω
*|wε,y|*2
*∗*
*dx + 2∗l*
Z
Ω
¡
*u*+0
¢2*∗ _{−1}*

*wε,ydx*+2

*∗l*2

*∗−1*Z Ω

*(wε,y)*2

*∗−1u*+0

*dx + o*³

*εN −2*2 ´

*.*Also from [6] we have

*kwε,yk*2*H*1 *= B + O*
¡
*εN −2*¢ and
Z
Ω
*|wε,y|*2
*∗*
*dx = A + O*¡*εN*¢*,*
*where B = ku*1*k*2* _{H}*1

_{(R}

*N*

_{)}

*, A =*R R

*N*1 (

*1+|x|*2

_{)}

*Ndx and S = BA*

*2−N*

*N*

*. Substituting in (*27)

*and using the fact that u*+

0 *is a solution of equation (Eλ,f) , we obtain*

*Jλ(u*+_{0} *+ lwε,y*) = 1
2
°
*°u*+
0
°
°2
*H*1+
*l*2
2*B + l*
*u*+_{0}*, wε,y*
®
*H*1*−*
1
2*∗*
Z
Ω
¯
*¯u*+
0
¯
¯2*∗*
*dx*
*−l*
2*∗*
2*∗A − l*
Z
Ω
¡
*u*+_{0}¢2*∗−1wε,ydx − l*2
*∗ _{−1}*Z
Ω

*(wε,y*)2

*∗−1u*+

_{0}

*dx*

*−λ*

*q*Z Ω

*f (x)*¡

*u*+

_{0}

*+ lwε,y*¢

*qdx + o*³

*εN −2*2 ´

*= Jλ(u*+ 0) +

*l*2 2

*B −*

*l*2

*∗*2

*∗A − l*2

*∗*Z Ω

_{−1}*(wε,y)*2

*∗−1u*+ 0

*dx*

*−λ*

*q*Z Ω

*f (x)*¡

*u*+ 0

*+ lwε,y*¢

*q*

*dx +λ*

*q*Z Ω

*f (x)*¡

*u*+ 0 ¢

*q*

*dx*+

*λ*

*q*Z Ω

*f (x)*¡

*u*+ 0 ¢

*q−1*

*lwε,ydx + o*³

*εN −2*2 ´

*= Jλ(u*+ 0) +

*l*2 2

*B −*

*l*2

*∗*2

*∗A − l*2

*∗*Z Ω

_{−1}*(wε,y)*2

*∗−1u*+ 0

*dx*

*−λ*Z Ω

*f (x)*(Z

*lwε,y*0 h

*(u*+ 0

*+ s)q−1−*¡

*u*+ 0 ¢

*q−1*i

*ds*)

*dx*

*+o*³

*εN −2*2 ´

*.*From

*f > 0 in Σ and wε,y≡ 0 in Σc,*

we can conclude that
*Jλ(u*+0 *+ lwε,y) ≤ Jλ(u*+0) +

*l*2
2*B −*
*l*2*∗*
2*∗A − l*
2*∗ _{−1}*Z
Ω

*(wε,y)*2

*∗−1u*+0

*dx + o*³

*εN −2*2 ´

*.*Similar to the argument of Lemma 3.1 in Tarantello [17], we can conclude that for

*every l > 0 and a.e. y ∈ Σ, there exists ε*0

*= ε*0

*(l, y) > 0 such that*

*Jλ(u*+_{0} *+ lwε,y) < αλ*(Ω) + 1
*NS*

*N*

2 (28)

*for all ε ∈ (0, ε*0*) .*

The following proposition provides a precise description of the (PS)–sequence of
*Jλ.*

Proposition 2. *Each sequence {un} ⊂ M− _{λ}*

*(Ω) satisfying*

*(i) Jλ(un) = σ + o (1) with σ < αλ*(Ω) + 1

*NS*

*N*

2;

*(ii) J0*

*λ(un) = o (1) in H−1*(Ω)
*has a convergent subsequence.*

*Proof.* By Lemma 3.2 and the compact imbedding theorem, there exist a
*subse-quence {un} and u*0*∈ H*01*(Ω) is a solution of equation (Eλ,f*) such that

*un* u*0*weakly in H*01(Ω)

and

*un→ u*0 *strongly in Lq(Ω).* (29)

*First, we claim that u*0*6≡ 0. If not, by (*29) we have

Z
Ω
*f (x) |un|qdx → 0 as n → ∞.*
Thus, _{Z}
Ω
*|∇un|*2*dx =*
Z
Ω
*|un|*2
*∗*
*dx + o (1)* (30)
and
1
*N*
Z
Ω
*|un|*2
*∗*
*dx =* 1
2
Z
Ω
*|∇un|*2*dx −λ*
*q*
Z
Ω
*f (x) |un|qdx −* 1
2*∗*
Z
Ω
*|un|*2
*∗*
*dx + o (1)*
*= σ + o (1) ,*
*this contradicts σ < αλ*(Ω) + 1
*NS*
*N*

2*. Thus, u*_{0} *6≡ 0 and J _{λ}(u*

_{0}

*) ≥ αλ(Ω) . Write*

*un= u*0*+ vn* *with vn* ** 0 weakly in H*01*(Ω). By the Brezis-Lieb lemma [4], we have*

Z
Ω
*|un|*2
*∗*
*dx =*
Z
Ω
*|u*0*+ vn|*2
*∗*
*dx =*
Z
Ω
*|u*0*|*2
*∗*
*dx +*
Z
Ω
*|vn|*2
*∗*
*dx + o (1) .*
*Hence, for n large enough, we can conclude that*

*αλ*(Ω) + 1
*NS*
*N*
2 *> J _{λ}(u*

_{0}

*+ vn)*

*= Jλ(u*0) +1 2 Z Ω

*|∇vn|*2

*dx −*1 2

*∗*Z Ω

*|vn|*2

*∗*

*dx + o (1)*

*≥ αλ*(Ω) +1 2 Z Ω

*|∇vn|*2

*dx −*1 2

*∗*Z Ω

*|vn|*2

*∗*

*dx + o (1)*or 1 2 Z Ω

*|∇vn|*2

*dx −*1 2

*∗*Z Ω

*|vn|*2

*∗*

*dx <*1

*NS*

*N*2

*+ o (1) .*(31)

*Also from J0*

*λ(un) = o (1) in H−1(Ω), {un} is uniformly bounded and u*0 is a

*solution of equation (Eλ,f*) follows
*o (1) = h J0*
*λ(un) , uni*
=
Z
Ω
*|∇u*0*|*2*dx −*
Z
Ω
*|u*0*|qdx −*
Z
Ω
*|u*0*|*2
*∗*
*dx*
+
Z
Ω
*|∇vn|*2*dx −*
Z
Ω
*|vn|*2
*∗*
*dx + o (1)*
=
Z
Ω
*|∇vn|*2*dx −*
Z
Ω
*|vn|*2
*∗*
*dx + o (1) ,*
we obtain _{Z}
Ω
*|∇vn|*2*dx −*
Z
Ω
*|vn|*2
*∗*
*dx = o (1) .* (32)

We claim that (31) and (32) can hold simultaneously only if {vn*} admits a *
*subse-quence {vni} which converges strongly to zero. If not, the kvnkH*1 is bounded away

*from zero, that is kvnk _{H}*1

*≥ c for some c > 0. From (*32) then it follows

Z

Ω

*|vn|*2

*∗*

*dx ≥ SN*2 *+ o (1) .*

By (31) and (32) for n large enough
1
*NS*
*N*
2 *≤* 1
*N*
Z
Ω
*|vn|*2
*∗*
*dx + o (1)*
= 1
2
Z
Ω
*|∇vn|*2*dx −*
1
2*∗*
Z
Ω
*|vn|*2
*∗*
*dx + o (1)*
*<* 1
*NS*
*N*
2

*which is a contradiction. Consequently, un→ u*0 *strongly in H*01*(Ω) and Jλ(u*0) =

*σ.*

*Next, we establish the existence of a local minimum for Jλ* on M*− _{λ}*

*(Ω) .*

Theorem 4.2. *Let λ*0*> 0 as in Proposition1, then for λ ∈ (0, λ*0*), the functional*

*Jλ* *has a minimizer u−*0 *in M−λ* *(Ω) and it satisfies*
*(i) Jλ*¡*u−*
0
¢
*= α−*
*λ(Ω) < αλ*(Ω) + *N*1*S*
*N*
2;

*(ii) u−*_{0} *is a nonnegative (nontrivial) solution of equation (Eλ,f) .*
*Proof.* *First, we claim that α− _{λ}*

*(Ω) < αλ*(Ω) + 1

*NS*
*N*
2*. Let*
*U*1 =
½
*u ∈ H*1
0*(Ω) \ {0}*
¯
¯
¯
¯* _{kuk}*1

*H*1

*t−*µ

*u*

*kuk*1 ¶

_{H}*> 1*¾

*∪ {0} ,*

*U*2 = ½

*u ∈ H*01

*(Ω) \ {0}*¯ ¯ ¯ ¯

*1*

_{kuk}*H*1

*t−*µ

*u*

*kuk*1 ¶

_{H}*< 1*¾

*.*Then M

*−*

_{λ}*(Ω) disconnects H*1

0*(Ω) in two connected components U*1 *and U*2 and

*H*1

0*(Ω) \M−λ(Ω) = U*1*∪ U*2*. For each u ∈ M*+*λ(Ω) , we have 1 < t*max*(u) < t−(u) .*

*Since t− _{(u) =}* 1

*kuk*³

_{H1}t−*u*

*kuk*´

_{H1}*, then M*+

_{λ}*(Ω) ⊂ A*1

*. In particular, u*0

*∈ A*1

*. We*

*claim that there exists l*0*> 0 such that u*+0 *+ l*0*u*+*∈ A*2*. First, we find a constant*

*c > 0 such that 0 < t−*
µ
*u*+
0*+lwε,y*
*ku*+
0*+lwε,ykH1*
¶

*a sequence {ln} such that ln* *→ ∞ and t−*
µ
*u*+
0*+lnwε,y*
*ku*+
0*+lnwε,yk _{H1}*
¶

*→ ∞ as n → ∞. Let*

*vn*=

*u*+ 0

*+lnwε,y*

*ku*+0

*+lnwε,yk*

_{H1}. Since t*−*

_{(vn) vn}

_{∈ M}−*λ* *(Ω) ⊂ Mλ*(Ω) and by the Lebesgue
dominated convergence theorem,

Z
Ω
*v*2*∗*
*n* =
1
°
*°u*+
0 *+ lnwε,y*
°
°2*∗*
*H*1
Z
Ω
¡
*u*+
0 *+ lnwε,y*
¢2*∗*
*dx*
= ° 1
*°u*+
0 *+ lnwε,y*
°
°2*∗*
*H*1
Z
Ω
µ
*u*+_{0}
*ln* *+ wε,y*
¶2*∗*
*dx*
*→*
R
Ω*(wε,y)*
2*∗*
*dx*
*kwε,yk*2
*∗*
*H*1
*as n → ∞.*
We have
*Jλ*
¡
*t− _{(vn) vn}*¢

_{=}1 2 £

*t−*¤2

_{(vn)}*)]*

_{−}[t−(vn*q*

*q*

*λ*Z Ω

*f vq*

*ndx −*

*[t−*2

_{(vn)]}*∗*2

*∗*Z Ω

*v*2

*∗*

*n*

*dx*

*→ −∞ as n → ∞,*

*this contradicts that Jλ* is bounded below on Mλ(Ω). Let
*l*+_{0} =
¯
¯
*¯c*2* _{−}*°

*+ 0 ° °2*

_{°u}*H*1 ¯ ¯ ¯ 1 2

*kwε,yk*1

_{H}*+ 1,*then °

*°u*+ 0

*+ l*0

*wε,y*° °2

*H*1 = °

*°u*+

_{0}°°2

*1*

_{H}*+ (l*0) 2

*kwε,yk*2

*1*

_{H}*+ 2l*0

*u*+

_{0}

*, wε,y*®

*H*1

*>*°

*°u*+ 0 ° °2

*H*1+ ¯ ¯

*¯c*2

*−*°

*°u*+ 0 ° °2

*H*1 ¯ ¯

*¯ + 2l*0 µZ Ω

*u*+ 0

*wε,ydx + λ*Z Ω

*u*+ 0

*wε,ydx*¶

*> c*2

*>*"

*t−*Ã

*u*+0

*+ l*0

*wε,y*°

*°u*+ 0

*+ l*0

*wε,y*° °

*H*1 !#2

*,*

*that is u*+

0 *+ l*0*wε,y* *∈ U*2. Now, we define

*β = inf*
*γ∈Γs∈[0,1]*max *Jλ(γ (s)) ,*
where Γ = ©*γ ∈ C*¡*[0, 1] ; H*1
0(Ω)
¢
*| γ(0) = u*+_{0} *and γ(1) = u*+_{0} *+ l*0*wε,y*
ª
*. Define a*
*path γ*0*(s) = u*+0 *+ sl*0*wε,y* *for s ∈ [0, 1], then γ*0 *∈ Γ and there exists s*0 *∈ (0, 1)*

*such that γ*0*(s*0*) ∈ M−λ* *(Ω) , we have β ≥ α−λ(Ω) . Moreover, by Lemma*4.1
*α−*
*λ* *(Ω) ≤ β < αλ*(Ω) +
1
*NS*
*N*
2*.*

Analogously to the proof of Proposition1, one can show that the Ekeland variational
*principle gives a sequence {un} ⊂ M− _{λ}* (Ω) which satisfies

*Jλ(un) = α−λ* *(Ω) + o (1) and Jλ0* *(un) = o (1) in H−1(Ω) .*
*Since α− _{λ}*

*(Ω) < αλ*(Ω) + 1

*NS*

*N*

2, by Proposition 2, there exist a subsequence {un*}*

*and u−*

0 *∈ H*01(Ω) such that

*This implies u−*

0 *∈ M−λ*(Ω) and
*Jλ(un) → Jλ*

¡

*u−*_{0}¢*= α− _{λ}*

*(Ω) as n → ∞.*

*Since Jλ*¡*u−*_{0}¢ *= Jλ*¡¯*¯u−*_{0}¯¯¢ and ¯*¯u−*_{0}¯*¯ ∈ M− _{λ}*

*(Ω) , by Lemma*2.4, ¯

*¯u−*

_{0}¯¯ is also a

*solution of equation (Eλ,f) . Without loss of generality, we may assume that u−*

0 is

*nonnegative (nontrivial) solution of equation (Eλ,f*).

Now, we begin to show the proof of Theorem1.1: By Theorems 3.4and4.2the
*equation (Eλ,f) has two nonnegative (nontrivial) solutions u*+0 *and u−*0 such that

*u*+

0 *∈ M*+*λ(Ω) and u−*0 *∈ M−λ* (Ω). Since M+*λ* *(Ω) ∩ M−λ(Ω) = ∅. This implies that*
*u*+

0 *and u−*0 are distinct. Now, by Dr´abek–Kufner–Nicolosi [8, Lemma 2.1] we have

*u*+0*, u−*0 *∈ L∞(Ω) . Then we can apply the Harnack inequality due to Trudinger [19]*

*in order to get u*+_{0} *and u−*_{0} *are positive in Ω.*

5. Proof of Theorem 1.2. Finally in this section, we give a proof of Theorem 1.2. We need the following lemmas.

Lemma 5.1. *For each uλ* *∈ M−λ* *(Ω) , there is a unique su> 0 such that suλuλ* *∈*

M0*(Ω) ,*
*su≥*
*1 − λ kfk _{L}q∗*
µ
2

*∗*

_{− q}*S (2 − q)*¶

*2∗−q*

*2∗−2* 1

*2∗−2*

*and*

*su≤*

*1 + λ kfk*µ 2

_{L}q∗*∗*

_{− q}*S (2 − q)*¶

*2∗−q*

*2∗−2* 1

*2∗−2*

*.*

*Furthermore, suλ*

*→ 1 as λ → 0.*

*Proof.* *For uλ∈ M− _{λ}(Ω) , we have*
Z
Ω

*|∇uλ|*2

*dx − λ*Z Ω

*f (x) |uλ|qdx −*Z Ω

*|uλ|*2

*∗*

*dx = 0*and

*(2 − q)*Z Ω

*|∇uλ|*2

*dx < (2∗− q)*Z Ω

*|uλ|*2

*∗*

*dx.*

*Thus, there is a unique suλ*

*> 0 such that suλuλ∈ M*0(Ω) and so

*s*2
*uλ*
Z
Ω
*|∇uλ|*2*dx = s*2
*∗*
*uλ*
Z
Ω
*|uλ|*2
*∗*
*.*
Then by the H¨older inequality

*1 − λ kf k _{L}q∗kuλkq−2*

*∗*

*L2∗*

*≤ s*2

*∗*

_{−2}*uλ*= R Ω

*|∇uλ|*2

*dx*R Ω

*|uλ|*2

*∗*

*dx*= 1 +

*λ*R

_{Ω}

*f (x) |uλ|qdx*R Ω

*|uλ|*2

*∗*

*dx*

*≤ 1 + λ kf k*

_{L}q∗kuλkq−2*∗*

*L2∗*

*.*Since

_{Z}Ω

*|uλ|*2

*∗*

*>*

*2 − q*2

*∗*Z Ω

_{− q}*|∇uλ|*2

*dx ≥*

*S (2 − q)*2

*∗*

_{− q}*kuλk*2

*L2∗*or

*kuλkL2∗*

*>*µ

*S (2 − q)*2

*∗*¶ 1

_{− q}*2∗−2*

*.*

Therefore,
*su≥*
*1 − λ kfk _{L}q∗*
µ
2

*∗*

_{− q}*S (2 − q)*¶

*2∗−q*

*2∗−2* 1

*2∗−2*and

*su≤*

*1 + λ kfk*µ 2

_{L}q∗*∗*

_{− q}*S (2 − q)*¶

*2∗−q*

*2∗−2* 1

*2∗−2*

*.*This completes the proof.

*For c > 0, we define*
*J*0*c(u) =*
1
2
Z
Ω
*|∇u|*2*+ u*2*−* 1
2*∗*
Z
Ω
*c |u|*2*∗*;
M*c*0(Ω) =
©
*u ∈ H*01*(Ω) \ {0} |*
*(J*0*c*)*0(u) , u*
®
= 0ª;
M0(Ω) =
©
*u ∈ H*01*(Ω) \ {0} | hJ*0*0(u) , ui = 0*
ª
*.*

*Note that J*0 *= J*0*c* *for c = 1, and for each u ∈ M−λ(Ω) there is a unique su* *> 0*
*such that suu ∈ M*0*(Ω) . Furthermore, we have the following results.*

Lemma 5.2. *For each u ∈ M− _{λ}(Ω) , we have*

*(i) there is a unique sc _{(u) > 0 such that s}c_{(u) u ∈ M}c*

0*(Ω) and*
max
*t≥0J*
*c*
0*(tu) = J*0*c(sc(u) u) =*
1
*Nc*
*2−N*
2
Ã
*kuk*2* _{H}∗*1
R
Ω

*|u|*2

*∗*!

*N −2*2 ;

*(ii) for µ ∈ (0, 1) ,*

*Jλ(u) ≥ (1 − λµ)*

*N*2

*0*

_{J}*(suu) −λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*

*and*

*Jλ(u) ≤ (1 + λµ)*

*N*2

*0*

_{J}*(suu) +*

*λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*

*.*

*Proof.*

*(i) For each u ∈ M−*

_{λ}*(Ω) , let*

*f (s) = J*0*c(su) =*
1
2*s*
2* _{kuk}*2

*H*1

*−*1 2

*∗s*2

*∗*Z Ω

*c |u|*2

*∗,*

*then f (s) → −∞ as s → ∞, f0*2

_{(s) = s kuk}*H*1

*− s*2

*∗*R Ω

_{−1}*c |u|*2

*∗*

*and f00*

_{(s) =}*kuk*2

*1*

_{H}*− (2∗− 1) s*2

*∗*R Ω

_{−2}*c |u|*

*p*

*. Let*

*sc(u) =*Ã

*kuk*2

*1 R Ω*

_{H}*c |u|*2

*∗*! 1

*2∗−2*

*> 0.*

*Then f0*

_{(s}c_{(u)) = 0, s}c_{(u) u ∈ M}c0(Ω) and

*f00(sc(u)) = kuk*2* _{H}*1

*− (2∗− 1) kuk*

2

*H*1 *= (2 − 2∗) kuk*

2

*H*1 *< 0.*

*Thus, there is a unique sc _{(u) > 0 such that s}c_{(u) u ∈ M}c*

0(Ω) and
max
*s≥0J*
*c*
0*(su) = J*0*c(sc(u) u) =*
1
*Nc*
*2−N*
2
Ã
*kuk*2* _{H}∗*1
R
Ω

*|u|*2

*∗*!

*N −2*2

*.*

*(ii) For each u ∈ M−*

*λ(Ω) , let c = 1/ (1 − λµ), sc= sc(u) > 0 and su> 0 such that*
*sc _{u ∈ M}c*

0*(Ω) and suu ∈ M*0*(Ω) . For µ ∈ (0, 1) , by the H¨older inequality and the*

Young inequality,
Z
Ω
*f (x) |scu|qdx ≤ kf k _{L}q∗kscuk*

*q*

*L2∗*

*≤ kf kLq∗S*

*−q*2

*kscukq*

*H*1

*≤*

*2 − q*2 ³

*kf k*

_{L}q∗S*−q*2

*µ−q*2 ´ 2

*2−q*+

*q*2 ³

*µq*2

*kscukq*

*H*1 ´2

*q*=

*2 − q*2

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*+

*qµ*2

*ks*

*c*2

_{uk}*H*1

*.*

*Then by part (i) ,*
sup
*s≥0Jλ(su) ≥ Jλ(s*
*c _{u) ≥}*

*(1 − λµ)*2

*ks*

*c*2

_{uk}*H*1

*−*1 2

*∗*Z Ω

*|sc*2

_{u|}*∗*

*−λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*

*= (1 − λµ) J*

_{0}

*1/(1−λµ)(scu) −λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*

*= (1 − λµ)N*2 1

*N*Ã

*kuk*2

*1 R Ω*

_{H}∗*|u|*2

*∗*!

*N −2*2

*−λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*

*= (1 − λµ)N*2

*J*

_{0}

*(suu) −*

*λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*

*.*By Lemma2.5

*(i),*sup

*s≥0Jλ(su) = Jλ(u) .*Thus,

*Jλ(u) ≥ (1 − λµ)*

*2∗*

*2∗−2*0

_{J}*(suu) −*

*λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*

*.*Moreover,

*Jλ(su) ≤*

*(1 + λµ)*2

*ksuk*2

*H*1

*−*1 2

*∗*Z Ω

*|su|*2

*∗*+

*λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*

*= (1 + λµ) J*

_{0}

*1/(1+λµ)(su) +λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*and so

*Jλ(u) ≤ (1 + λµ)*

*N*2

*J*

_{0}

*(suu) +λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*

*.*This completes the proof.

Now, we begin to show the proof of Theorem1.2: Suppose that {λn*} is a sequence*
*of positive number such that λn* *→ 0 as n → ∞. Let u*(1)*n* *= u*+*0,nand u*

(2)

*n* *= u−0,n* be

positive solutions corresponding to Theorem1.1.
*(i) By Theorem*3.4*(iii) , we can conclude that*

°
°
*°u*(1)*n*
°
°
°
*H*1 *→ 0 as n → ∞.*

*(ii) By Lemmas* 5.1 and 5.2, for each n, there is a unique s* _{u}*(2)

*n*

*> 0 such that*

*s*(2)

_{u}*n*

*u*(2)

*n*

*∈ M*0

*(Ω) , s*(2)

_{u}*n*

*= 1 + o (1) and for µ ∈ (0, 1)*

*J*0 ³

*s*(2)

_{u}*n*

*u*(2)

*n*´

*≤*µ 1

*1 − λnµ*¶

*N*2 ·

*Jλn*³

*u*(2)

*n*´ +

*λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*¸

*.*(33) Since

*Jλn*³

*u*(2)

*n*´

*< Jλn(u*(1)

*n*) + 1

*NS*

*N*2

*.*(34) Then by (33) and (34)

*J*0 ³

*s*(2)

_{u}*n*

*u*(2)

*n*´

*<*µ 1

*1 − λnµ*¶

*N*2 ·

*Jλn(u*(1)

*n*) + 1

*NS*

*N*2 +

*λ (2 − q)*

*2q*

*µ*

*q*

*q−2*³

*kf k*

_{L}q∗S*−q*2 ´ 2

*2−q*¸

*.*

*Since Jλn(u*(1)

*n*

*) → 0 as n → ∞. Thus,*lim sup

*n→∞*

*J*0 ³

*s*(2)

_{u}*n*

*u*(2)

*n*´

*≤*1

*NS*

*N*2

*.*This implies lim

*n→∞J*0 ³

*s*(2)

_{u}*n*

*u*(2)

*n*´ = 1

*NS*

*N*2

*.*

*We can conclude that {s _{u}*(2)

*n* *u*
(2)

*n* *} is a minimizing sequence for J*0on M0*(Ω) . Since*

*s _{u}*(2)

*n*

*→ 1 as n → ∞. Thus,*Z Ω ¯ ¯

*¯∇u*(2)

*n*¯ ¯ ¯2

*dx =*Z Ω ¯ ¯

*¯u*(2)

*n*¯ ¯ ¯2

*∗*

*+ o (1) and J*0 ³

*u*(2)

*n*´ = 1

*NS*

*N*2

*+ o (1) .*

Similar to the method used in Wang–Wu [20, Lemma 7] (or Ekeland [9]), we have
*{u*(2)*n* *} is a (PS)*1

*NS*
*N*

2*–sequence for J*0 *in H*

1

0*(Ω) . Clearly, {u*(2)*n* *} is a bounded *
*se-quence. Then there exist a subsequence {u*(2)*n* *} and u*0*∈ H*01(Ω) such that

*u*(2)

*n* ** u*0 *weakly in H*01*(Ω) .*

*Since Ω is a bounded domain, we have u*0 *≡ 0. Moreover, by the concentration–*

compactness principle (see Lions [13] or Struwe [16, Theorem 3.1]), there exist two
*sequences {xn} ⊂ Ω, {Rn} ⊂ R*+ *and a positive solution v*0*∈ D1,2*

¡

R*N*¢_{of critical}
problem

*−∆u = |u|*2*∗−2u in RN _{,}*

*such that Rn*

*→ ∞ as n → ∞ and*

°
°
*°u*(2)*n* *(x) − R*
*N −2*
2
*n* *v*0*(Rn(x − xn))*
°
°
°
*D1,2*_{(R}*N*_{)}*→ 0 as n → ∞.*

This completes the proof of Theorem1.2.

Corollary 1. *If u− _{0,λ}∈ M−_{λ}(Ω) is a positive solution of corresponding to Theorem*

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Received September 2005; revised July 2007.