New multiple positive solutions for elliptic equations with singularity and critical growth

New multiple positive solutions for elliptic equations with singularity and critical growth

Hong-Min Suo

, Chun-Yu Lei

and Jia-Feng Liao

1School of Sciences, GuiZhou Minzu University, Guiyang, Guizhou, 550025, P. R. China

2College of Mathematics Education, China West Normal University, Nanchong, Sichuan, 637002, P. R. China

Received 2 January 2019, appeared 15 March 2019 Communicated by Dimitri Mugnai

Abstract. In this note, the existence of multiple positive solutions is established for a semilinear elliptic equation −_∆u = _u^λ_γ +u^2∗−1, x ∈ _Ω, u = 0,x ∈ _{∂Ω, where Ω} is a smooth bounded domain in R^N (N ≥ 3), 2^∗ = _N−2^2N , γ ∈ (0, 1) and λ > 0 is a real parameter. We show by the variational methods and perturbation functional that the problem has at least two positive solutionsw₀(x)_andw₁(x)_withw₀(x)<w₁(x)_inΩ.

Keywords: semilinear elliptic equations, critical growth, singularity, positive solution.

2010 Mathematics Subject Classification: 35A15, 35B09, 35B33, 35J75.

1 Introduction

The singular bounded value problem of the type

(−_∆u=λf(x)u^−γ+µg(x)u^p−1, inΩ,

u=0, on∂Ω, (1.1)

where Ω is a bounded smooth domain in R^N (N ≥ 3), γ ∈ (_{0, 1}) _and f,g satisfying some certain conditions, was extensively investigated. Such problem describes naturally several physical phenomena, therefore, only the positive solutions are relevant in most cases.

Singular elliptic problems have been intensively studied in the last decades. For example, in the case when µ=0, the existence or uniqueness of positive solutions to problem (1.1) has been studied extensively (see [6,7,12,13,18,24] and the references therein).

For the case of µ > 0. When 1 < p < 2^∗, Sun, Wu and Long [21] established two positive solutions to problem (1.1) by using the Nehari manifold provided λ > 0 is enough small. For singular elliptic problems with subcritical growth, please see [2–5,8,9,19] and the references therein. For the case of critical growth, there are many interesting results, see [10,11,15,20,22,23]. In particular, Yang [23] considered the problem

(−_∆u=λu^−γ+u^2∗−1, in Ω,

u=0, on ∂Ω. (1.2)

BCorresponding author. Email: liaojiafeng@163.com

The author firstly proved that problem (1.2) has a positive local minimizer solution u_λ for λ > 0 enough small. After that, with the helps of the sub-supersolutions and variational arguments, and a second positive solutionv_λ was obtained withu_λ <v_λ inΩ. In additional, in problem (1.2), ifuis replaced byλ

1+1_γv, problem (1.2) reduces to (−_∆v =v^−γ+µv^2∗−1, in Ω,

v=0, on ∂Ω,

here µ = λ

2∗ −2

1+γ. By using the Nehari manifold, Sun and Wu [20] proved that there was an exactµ∗ such that the problem has two positive solutions for allµ ∈ (0,µ∗) and no solution for µ > µ∗. In the case when 0 < γ ≤ 1, by using the variational method, Hirano, Saccon and Shioji showed the existence of two positive solutions for problem (1.2) with λ> 0 small enough, see [10].

Thus, observing the all above studies, it is natural to ask whether problem (1.2) has multi- ple positive solutions by other methods? We shall give a positive answer to this question, the main technical approaches are based on the variational and perturbation functional. Now, the main result can be stated as follows.

Theorem 1.1. Assume that γ ∈ (_{0, 1}). Then there exists λ∗ > 0, such that for any λ ∈ (_0,λ∗)_, problem(1.2)has at least two positive solutions w₀(x)and w₁(x)with w₀(x)< w₁(x)inΩ.

Remark 1.2. Compared with [23], with the help of a perturbation functional, we give a simple and direct method to obtain the size relation of the two positive solutions.

Throughout this paper, we make use of the following notations:

• the space H₀¹(_Ω)is equipped with the normkuk² = R

Ω|∇u|²dx, which is equivalent to the usual norm. The norm inL^p(_Ω)is denoted by|u|^p_p= R

Ω|u|^pdx;

• C,C₁,C₂, . . . denote various positive constants, which may vary from line to line;

• we denote by Br (respectively, ∂Br) the closed ball (respectively, the sphere) of center zero and radiusr, i.e.,B_r={u∈ H₀¹(_Ω):kuk ≤r},∂B_r ={u∈ H₀¹(_Ω):kuk=r};

• u=u⁺+u⁻,u^±=±max{±u, 0};

• letSbe the best Sobolev constant, i.e., S:= inf

u∈H¹₀(_Ω)\{0}

R

Ω|∇u|²dx (R

Ω|u|^2∗)^22∗.

2 Existence of the first positive solution of problem (1.2)

We define the energy functional of problem (1.2) by I(u) = ¹

2kuk²− ^λ 1−γ

Z

Ω(u⁺)^1−γdx− ¹ 2^∗

Z

Ω(u⁺)^2∗dx, ∀u∈ H₀¹(_Ω).

In general, a functionuis called a positive solution of problem (1.2) ifu ∈ H₀¹(_Ω)and for all v∈ H₀¹(_Ω)it holds

Z

Ω(∇u,∇v)dx−λ Z

Ωu^−γvdx−

Z

Ωu^2∗−1vdx=0.

From [23] and [10], we obtain the following result.

Theorem 2.1. For 0 < γ < 1, there existsΛ0 > 0 such that problem (1.2) has a positive solution w0 ∈ L^∞(_Ω)∩C^∞(_Ω)with I(w0)<0whenλ∈(0,Λ0).

3 Existence of a second positive solution of problem (1.2)

Up to now, we get that problem (1.2) has a positive solutionw₀. Next we will prove that there is another positive solution for problem (1.2) by a translation argument. For α> 0, we define aC¹functional J_α : H₀¹(_Ω)→_Rby

Jα(v) = ¹

2kvk²− ¹ 2^∗

Z

Ω[(v⁺+w₀)^2∗−w²₀^∗−2^∗w²₀^∗−1v⁺]dx

− ^λ 1−γ

Z

Ω

(v⁺+w₀+α)^1−γ−(w₀+α)^1−γ−(1−γ)^v

+

w^γ₀

dx,

forv∈ H₀¹(_Ω). Now, we show that the functional J_α satisfies the mountain-pass lemma.

Lemma 3.1. There exist r,ρ>₀such that Jαsatisfies the following conditions for anyλ>_0, (i) Jα(u)> ρfor any u ∈∂Br;

(ii) there existsζ ∈ H₀¹(_Ω)withkζk>r such that J_α(ζ)<0.

Proof. (i) For u ∈ H₀¹(_Ω) _with u⁺ 6= 0, by the mean value theorem and the Lebesgue domi- nated convergence theorem, one has

tlim→0⁺

Jα(tu)

t = − ¹ 2^∗ lim

t→0⁺

Z

Ω

(tu⁺+w₀)^2∗−w²₀^∗−2^∗w²₀^∗−1tu⁺

t dx

−λ lim

t→0⁺

Z

Ω

(tu⁺+w₀+α)^1−γ−(w₀+α)^1−γ−(1−γ)w⁻₀^γtu⁺

(1−γ)t dx

= − _lim

t→0⁺

Z

Ω[(ηtu⁺+w₀)^2∗−1−w²₀^∗−1]u⁺dx

−λ lim

t→0⁺

Z

Ω[(ξtu⁺+w₀+α)^−γ−w⁻₀^γ]u⁺dx

=λ Z

Ω

u⁺

w^γ₀ − ^u

+

(w₀+α)^γ

dx

>0,

which implies that there existρ,r >0 such thatJ_α|_kuk=r ≥ρ>0 for eachλ>0.

(ii) Fora,b≥0, there holds

(a+b)^2∗ ≥a^2∗+b^2∗−2^∗a^2∗−1b.

Therefore, foru∈ H₀¹(_Ω),u⁺6=0 andt >0, one has Jα(tu)≤ ^t

2

2kuk²− ^t

2^∗

2^∗ Z

Ω(u⁺)^2∗dx+tλ Z

Ω

u⁺ w₀^γdx

→ −_∞

as t → +∞. Therefore we can easily find ζ ∈ H₀¹(_Ω)with kζk >r, such that J_α(ζ)< 0. The proof is complete.

Lemma 3.2. The functional Jα satisfies the (PS)_c condition with c < _N¹S^N2 −Dλ, where D = D(|_Ω|,N,S,γ,|w₀|_∞).

Proof. Let{v_n} ⊂ H₀¹(_Ω)be a(PS)_csequence for Jα, namely,

J_α(v_n)→c, J_α⁰(v_n)→_{0, as}n→_{∞. (3.1)} Set

H₁(vn) =

Z

Ω[(v⁺_n +w0)^2∗−1(vn+w0)−w²₀^∗−1(vn+w0)]dx

−

Z

Ω[(v⁺_n +w₀)^2∗−w²₀^∗−2^∗w²₀^∗−1v⁺_n]dx, and

H₂(v_n) =λ Z

Ω

vn+w₀

(v⁺_n +w₀+α)^γ − ^vn+w₀ w^γ₀

dx

− ²

∗λ 1−γ

Z

Ω

(v⁺_n +w₀+α)^1−γ−(w₀+α)^1−γ−(1−γ)^v

+n

w^γ₀

dx.

Then

H₁(vn) =

Z

vn≥0

[(v⁺_n +w0)^2∗−w²₀^∗−1v⁺_n −w²₀^∗]dx

−

Z

vn≥0

[(v⁺_n +w₀)^2∗−w²₀^∗−2^∗w²₀^∗−1v⁺_n]dx

= (2^∗−1)

Z

Ωw²₀^∗−1v⁺_ndx

≥0, and

H₂(v_n)≥ −λ Z

Ω

|v_n|+w₀

(v⁺_n +w₀+α)^γdx−λ Z

Ω

|v_n|+w₀

w₀^γ dx− ²

∗λ 1−γ

Z

Ω(v⁺_n)^1−γdx

≥ −λ Z

Ω

|vn|+w₀

w^γ₀ dx−λ Z

Ω

|vn|+w₀

w^γ₀ dx− ²

∗λ 1−γ

Z

Ω|vn|^1−γdx

= −2λ Z

Ω

|v_n|

w^γ₀ dx−2λ Z

Ωw¹₀^−γdx− ²

∗λ 1−γ

Z

Ω|v_n|^1−γdx.

It follows from (3.1) that

2^∗c+o(kv_nk)≥2^∗Jα(v_n)− hJ_α⁰(v_n),v_n+w₀i

= ²

N−2kvnk²−

Z

Ω(∇w0,∇vn)_dx+_H1(_vn) +_H2(_vn)

= ^N

N−2kvnk²−

Z

Ω(λw₀^−γ+w²₀^∗−1)vndx+H₁(vn) +H₂(vn)

≥ ²

N−2kvnk²−

Z

Ωw²₀^∗−1|vn|dx−3λ Z

Ω

|vn| w^γ₀ dx

−2λ Z

Ωw¹₀^−γdx− ²

∗λ 1−γ

Z

Ω|v_n|^1−γdx

≥ ²

N−2kvnk²−C₁kvnk − ²

∗λ 1−γ|_Ω|²

∗_γ−1

2∗ S^−1−2γkvnk^1−γ−C₂kw₀k^1−γ,

which implies that {v_n} is bounded in H₀¹(_Ω). Moreover, by using the concentration com- pactness principle (see [16,17]), there exist a subsequence, say {vn} and v∗ ∈ H₀¹(_Ω) such that

Z

Ω|∇v_n|²dx*dµ≥

Z

Ω|∇v∗|²dx+

∑

j∈K

µ_jδ_xj, Z

Ω(v⁺_n)^2∗dx *dη=

Z

Ω(v⁺_∗)^2∗dx+

∑

j∈K

η_jδ_xj,

where K is an at most countable index set, δx_j is the Dirac mass at x_j, and x_j ∈ _Ω is in the support of µ,η. Moreover, there holds

µ_j ≥Sη_j^22∗ for allj∈K. (3.2)

For ε>0, letψ_ε,j(x)be a smooth cut-off function centered atx_j such that 0≤ψ_ε,j(x)≤1, ψ_ε,j(x) =1 inB x_j,ε/2

, ψ_ε,j(x) =0 inΩ\B(x_j,ε), |∇ψ_ε,j(x)| ≤ ² ε. Sinceψ_ε,jvnis bounded in H₀¹(_Ω), according to (3.1), there holds

0= _lim

ε→0 lim

n→_∞hJ_α⁰(v_n)_,v_nψ_ε,ji

= lim

ε→0 lim

n→_∞

_Z

Ω∇vn∇(vnψ_ε,j)dx

−

Z

Ω[(v⁺_n +w₀)^2∗−1vnψ_ε,j−w²₀^∗−1vnψ_ε,j]dx

−λlim

ε→0 lim

n→_∞ Z

Ω

v_nψ_ε,j

(v⁺_n +w₀+α)^γ −^vnψε,j w^γ₀

dx

= lim

ε→0 lim

n→_∞

_Z

Ω∇v_n∇(v_nψ_ε,j)dx−

Z

Ω(v⁺_n)^2∗ψ_ε,jdx

−

Z

Ω[(v⁺_n +w₀)^2∗−1v_nψ_ε,j−(v⁺_n)^2∗−1v_nψ_ε,j−w²₀^∗−1v_nψ_ε,j]dx

−λlim

ε→0 lim

n→_∞ Z

Ω

vnψ_ε,j

(v⁺_n +w₀+α)^γ −^vnψε,j w^γ₀

dx.

(3.3)

Note that{vn}is bounded inH₀¹(_Ω). Then lim

ε→0 lim

n→_∞

Z

Ω[(v⁺_n +w₀)^2∗−1v_nψ_ε,j−(v⁺_n)^2∗−1v_nψ_ε,j]dx

≤lim

ε→0 lim

n→_∞ Z

Ω[(v⁺_n +w₀)^2∗−1+ (v⁺_n)^2∗−1]|v_n|ψ_ε,jdx

≤lim

ε→0 lim

n→_∞C Z

B(x_j,ε)

|v_n|dx

=0.

Similarly, one has

lim

ε→0 lim

n→_∞ Z

Ω

v_nψ_ε,j

(v⁺_n +w₀+α)^γ − ^vnψε,j w^γ₀

dx=0,

and

lim

ε→0 lim

n→_∞ Z

Ωw²₀^∗−1vnψ_ε,jdx =0, lim

ε→0 lim

n→_∞ Z

Ωvn∇vn∇ψ_ε,jdx=0.

From the above information, by (3.3), we have η_j =µ_j, this combining (3.2) we deduce that

η_j ≥S^N2 or η_j =0.

Next we show that η_j ≥S^N2 is impossible. By contradiction, there exists some j₀ ∈ J such thatη_j0 ≥S^N2. By (3.1), there holds

c= lim

n→_∞

Jα(vn)− ¹

2hJ_α⁰(vn),vn+w0i

= −¹ 2

Z

Ω(∇vn,∇w0)dx+ ¹ N

Z

Ω(|v⁺_n +w0|^2∗−w²₀^∗)dx+¹ 2

Z

Ωw²₀^∗−1v⁺_ndx

− ^λ 1−γ

Z

Ω

(v⁺_n +w₀+α)^1−γ−(w₀+α)^1−γ−(1−γ)^v

+n

w₀^γ

dx

+^λ 2

Z

Ω

v_n+w₀

(v⁺_n +w₀+α)^γ − ^vn+w₀ w^γ₀

dx+o(1/n)

≥ ¹ N

Z

Ω(_v⁺_n)^2∗_dx+ ¹ 2

Z

Ωw²₀^∗−1v⁺_ndx−¹ 2

Z

Ωw²₀^∗−1vndx+λ Z

Ω

v_n w^γ₀dx

− ^λ 1−γ

Z

Ω

(v⁺_n +w₀+α)^1−γ−(w₀+α)^1−γ−(₁−γ)^v

+n

w₀^γ

dx

+^λ 2

Z

Ω

vn+w₀

(v⁺_n +w₀+α)^γ − ^vn+w0

w^γ₀

dx+o(1/n)

≥ ¹ N

Z

Ω(v⁺_n)^2∗dx+H₃(v_n) +o(1/n), where

H₃(vn), ^λ 2

Z

Ω

vn+w₀

(v⁺_n +w₀+α)^γ −^vn+w₀ w^γ₀

dx− ^λ 2 Z

Ω

vn

w₀^γdx+λ Z

Ω

v⁺_n w^γ₀dx

− ^λ 1−γ

Z

Ω

h

(v⁺_n +w₀+α)^1−γ−(w₀+α)^1−γidx

= ^λ 2

Z

Ω

v⁺_n +v⁻_n +w₀

(v⁺_n +w₀+α)^γ −w¹₀^−γ

dx−λ Z

Ω

v_n

w^γ₀dx+λ Z

Ω

v⁺_n w^γ₀dx

− ^λ 1−γ

Z

Ω

h

(v⁺_n +w0+α)^1−γ−(w0+α)^1−γidx

= ^λ 2

Z

Ω

v⁺_n +v⁻_n +w₀

(v⁺_n +w₀+α)^γ + ^v

−n

(v⁺_n +w₀+α)^γ −w¹₀^−γ

dx−λ Z

Ω

v⁻_n w^γ₀dx

− ^λ 1−γ

Z

Ω

h

(v⁺_n +w₀+α)^1−γ−(w₀+α)^1−γidx

≥ ^λ 2

Z

Ω

v⁺_n +v⁻_n +w₀

(_v⁺_n +_w0+α)^γ −w¹₀^−γ

dx+^λ 2

Z

Ω

v⁻_n

w^γ₀dx−λ Z

Ω

v⁻_n w₀^γdx

− ^λ 1−γ

Z

Ω

h

(v⁺_n +w₀+α)^1−γ−(w₀+α)^1−γidx

≥ − 1

1−γ

−¹ 2

λ

Z

Ω(v⁺_n +w₀+α)^1−γdx +

1 1−γ−¹

2

λ Z

Ω(w₀+α)^1−γdx− ^λ 2

Z

Ω

α

(v⁺_n +w₀+α)^γdx

≥ − 1

1−γ−¹ 2

λ

Z

Ω(v⁺_n +w₀+α)^1−γdx− ^λ 1−γ

Z

Ω(w₀+α)^1−γdx

− ^λ 2

Z

Ω(v⁺_n +w₀+α)^1−γdx

= − ^λ 1−γ

Z

Ω(v⁺_n)^1−γdx− ^2λ 1−γ

Z

Ω(w0+α)^1−γdx

≥ − ^λ 1−γ

Z

Ω(v⁺_n)^1−γdx− ^2λ

1−γ(|_Ω|+|_Ω||w₀|¹_∞^−γ). Then, by the Sobolev inequality and Young inequality, we have

c≥ ¹ N

Z

Ω(v⁺_n)^2∗dx+H₃(vn) +o(1/n)

≥ ¹ N

Z

Ω(v⁺_∗)^2∗dx+

∑

j∈J

η_j

!

+H₃(v_n) +o(1/n)

≥ ¹ Nη_j0+

Z

Ω(v⁺_∗)^2∗dx− ^λ 1−γ|_Ω|²

∗+γ−1 2∗

_Z

Ω(v⁺_∗)^2∗dx ¹₂⁻_∗^γ

−Aλ+o(1/n)

≥ ¹

NS^N2 −A₁λ

2∗

2∗+_γ−1 −Aλ+o(1/n)

≥ ¹

NS^N2 −A₁λ−Aλ+o(1/n)

= ¹

NS^N2 −Dλ+o(1/n), where A = ₁₋²

γ(|_Ω|+|_Ω||w₀|¹_∞^−γ)_, A₁ = A₁(N,γ,S,|_Ω|)_, D = A₁+A. Therefore, we get

1

NS^N2 −Dλ≤c< _N¹S^N2 −Dλ, which contradicts to the assumption. It implies thatKis empty, soR

Ω(_v⁺_n)^2∗_dx→R

Ω(_v⁺_∗)^2∗_dxasn→∞. Recalling that for p≥2, there holds

|x|^p−1x− |y|^p−1y

≤C_p|x−y|(|x|+|y|)^p−1, C_p >0.

As a result, there holds 0≤

Z

Ω[(v⁺_n +w₀)^2∗−(v⁺_∗ +w₀)^2∗]dx

≤Cp

Z

Ω|v⁺_n −v⁺_∗|(v⁺_n +v⁺_∗ +2w₀)^2∗−1dx

≤C_p|v⁺_n −v⁺_∗|₂^∗|v⁺_n +v⁺_∗ +2w₀|²₂^∗∗⁻¹

≤C|v⁺_n −v⁺_∗|₂^∗

→0, which implies that R

Ω(v⁺_n +w₀)^2∗dx → R

Ω(v⁺_∗ +w₀)^2∗dx asn → ∞. Note that J_α⁰(v_n) → _{0, it} follows

Z

Ω(∇v∗,∇φ)dx=

Z

Ω[(v⁺_∗ +w₀)^2∗−1φ−w²₀^∗−1φ]dx +λ

Z

Ω

φ

(v⁺∗ +w₀+α)^γ − ^φ w₀^γ

dx

(3.4)

for eachφ∈ H₀¹(_Ω). Taking the test functionφ= v∗+w₀in (3.4), we have Z

Ω(∇v∗,∇(v∗+w₀))dx=

Z

Ω[(v⁺_∗ +w₀)^2∗−w²₀^∗−1(v⁺_∗ +w₀)]dx +λ

Z

Ω

v∗+w₀

(v⁺∗ +w0+α)^γ − ^v∗+w₀ w^γ₀

dx.

(3.5)

According tohJ_α⁰(v_n),v_n+w₀i →0, one has Z

Ω(∇v_n,∇(v_n+w₀))dx =

Z

Ω[(v⁺_n +w₀)^2∗−1(v_n+w₀)−w²₀^∗−1(v_n+w₀)]dx +λ

Z

Ω

vn+w₀

(v⁺_n +w₀+α)^γ −^vn+w₀ w^γ₀

dx+o(1/n). Consequently,

Z

Ω(∇v∗,∇w0)dx+kvnk²=

Z

Ω[(v⁺_∗ +w0)^2∗−w₀^2∗−1(v⁺_∗ +w0)]dx +λ

Z

Ω

v∗+w₀

(v⁺_∗ +w₀+α)^γ − ^v∗+w₀ w^γ₀

dx+o(1/n).

(3.6)

It follows from (3.5) and (3.6) that

kvnk² → kv∗k², asn→_∞.

Hence, we havevn→v∗ in H₀¹(_Ω). The proof is complete.

It is known that the function

U_ε(x) = [N(N−2)ε²]^N−42

(ε²+|x|²)^{N−22 , x}∈_R^N, ε>0 satisfies

−_∆Uε =U_ε^2∗−1 inR^N.

We choose a function ϕ ∈ C₀^∞(_Ω) such that 0 ≤ ϕ(x) ≤ 1 in Ω, ϕ(x) = 1 near x = 0 and it is radially symmetric. Letu_ε(x) = ϕ(x)U_ε(x). Moreover, from [10], there exist two constants m,M>0 such thatm≤ w0(x)≤ Mfor each x∈ suppϕ, and

(ku_εk²≤ S^N2 +O(ε^N−2) +O(ε^N), R

Ω|uε|^2∗dx=S^N2 −O(ε^N). Moreover, one has





 R

Ωw₀u²_ε^∗−1dx=Cε^N−22 +O(ε

N+2 2 ), R

Ω uε

w^γ₀dx =O(ε^N

−2

2 ). (3.7)

Lemma 3.3. For every0<α<1andλ>0small, there holds sup

t≥0

Jα(tuε)< ¹

NS^N2 −Dλ, where D is defined by Lemma3.2.

Proof. Fort ≥0, there holds Jα(tuε) = ^t

2

2kuεk²− ¹ 2^∗

Z

Ω[(tuε+w₀)^2∗−w₀^2∗−2^∗w²₀^∗−1tuε]dx

− ^λ 1−γ

Z

Ω

(_tuε+_w0)^1−γ−w¹₀^−γ−(₁−γ)^tuε w^γ₀

dx

≤ ^t

2

2kuεk²− ^t

2^∗

2^∗ Z

Ωu²_ε^∗dx−t^2∗−1 Z

Ωw₀u²_ε^∗−1dx+tλ Z

Ω

uε

w^γ₀dx, where the following inequality is used

(a+b)^2∗ ≥a^2∗+b^2∗−2^∗a^2∗−1b−2^∗ab^2∗−1, ∀a,b>0.

Set

g(t) = ^t

2

2kuεk²− ^t

2^∗

2^∗ Z

Ωu²_ε^∗dx−t^2∗−1 Z

Ωw₀u²_ε^∗−1dx+tλ Z

Ω

uε

w^γ₀dx.

As limt→+_∞g(tu_ε) = −∞, similar to the paper [14], we can prove that there existt_ε > _{0 and} positive constantst₀,t₁ which are independent ofε,λ, such that sup_t≥0g(t) = g(tε)and

0<t0 ≤tε ≤t₁<_∞. (3.8)

Therefore, from (3.7) and (3.8), there holds sup

t≥0

g(t)≤ sup

t≥0

t²

2kuεk²− ^t

2^∗

2^∗ Z

Ωu²_ε^∗dx

−t²₀^∗−1 Z

Ωw₀u²_ε^∗−1dx+t₁λ Z

Ω

uε

w^γ₀dx

≤ ¹

NS^N2 +C₁ε^N−2−C₂ε

N−2

2 +C₃λε

N−2 2 ,

where C_i >0 (independent of ε,λ),i= 1, 2, 3. Therefore, letε =λ

1

N−2,λ< _Λ1 = _C ^C2

1+C3+D

2

, then

C₁ε^N−2−C₂ε

N−2

2 +C₃λε

N−2

2 = C₁λ+C₃λ

3

2 −C₂λ

1 2

≤ λ

C₁+C3−C2λ

−1 2

< −Dλ.

From the above information, it holds that sup

t≥0

g(t)≤ ¹

NS^N2 +C₁ε^N−2−C₂ε

N−2

2 +C₃λε

N−2 2

< ¹

NS^N2 −Dλ, which implies that

sup

t≥0

J_α(tu_ε)< ¹

NS^N2 −Dλ for any 0< λ< _Λ1. The proof is complete.

Lemma 3.4. For given0<α<1andλ>0is sufficiently small, there exists v_α ∈ H₀¹(_Ω)such that J_α⁰(v_α) =₀and J_α(v_α)>0.

Proof. Letλ^∗ =min

Λ0,Λ1,^S

N2

ND, 1 and 0< λ<λ^∗. By Lemma3.1, the functionalJα satisfies the geometry of the mountain-pass lemma. Applying the mountain-pass lemma [1], there exists a sequence{v_n} ⊂H₀¹(_Ω), such that

Jα(vn)→c>ρ and J_α⁰(vn)→0, (3.9) where

c= inf

γ∈_Γmax

t∈[0,1]J_α(γ(t)), Γ=ⁿγ∈ C([0, 1],H¹₀(_Ω)):γ(0) =0,γ(1) =ζ o

.

By Lemma3.2 and Lemma3.3, {v_n} ⊂ H₀¹(_Ω) has a convergent subsequence, say {v_n}, we may assume thatv_n →v_α in H¹₀(_Ω)asn→_∞. Hence, from (3.9), it holds

Jα(_vα) = _lim

n→_∞Jα(_vn) =_c>_0.

Furthermore, we have J_α⁰(v_α) =0. The proof is complete.

Now, we give the proof of Theorem1.1.

Proof of Theorem1.1. Let 0< λ< λ^∗, where λ^∗ is defined by Lemma3.4. Since v_α is a critical forJα, for eachφ∈ H₀¹(_Ω)_{, one has}

Z

Ω(∇v_α, ∇φ)dx=

Z

Ω[(v⁺_α +w₀)^2∗−1φ−w²₀^∗−1φ]dx +λ

Z

Ω

φ

(v⁺_α +w₀+α)^γ − ^φ w^γ₀

dx,

(3.10)

which implies thatv_α satisfies the following equation







−_∆vα = (v⁺_α +w₀)^2∗−1−w²₀^∗−1+ ^λ

(v⁺_α +w₀+α)^γ − ^λ

w^γ₀, in Ω,

vα =0, on ∂Ω.

Moreover, we can easily prove that{v_α}is bounded inH₀¹(_Ω), thus there exist a subsequence, still denoted by{vα}, andv₀ ∈ H₀¹(_Ω)such that











v_α *v₀, weakly in H₀¹(_Ω),

vα →v₀, strongly in L^p(_Ω) (1≤ p <2^∗), v_α(x)→v₀(x), a.e. inΩ,

asα→0. Note thatw₀ fulfilling







−_∆w₀ =w²₀^∗−1+ ^λ

w₀^γ, in Ω,

w₀ =0, on ∂Ω.

Since 0<w₀ ≤ M, from the above information, we have

−_∆(vα+w₀) = (v⁺_α +w₀)^2∗−1+ ^λ (v⁺_α +w₀+α)^γ

≥(v⁺_α)^2∗−1+ ^λ (v⁺_α +M+α)^γ

≥min

1, λ

(M+2)^γ

.

(3.11)