Multiplicity of positive solutions for a class of nonlocal problem involving critical exponent

Multiplicity of positive solutions for a class of nonlocal problem involving critical exponent

Xiaotao Qian

Department of Basic Teaching and Research, Yango University, Fuzhou 350015, P. R. China

Received 17 January 2021, appeared 30 July 2021 Communicated by Dimitri Mugnai

Abstract. In this paper, we study the following critical nonlocal problem







−

a−λb Z

Ω|∇u|²dx

∆u=λ|u|^p−2u+Q(x)|u|²u, x∈Ω,

u=_0, x∈∂Ω,

where a>0, b≥0, 2< p <4, λ>0 is a parameter, Ωis a smooth bounded domain in R⁴ and Q(x) ∈ C(_Ω) is a nonnegative function. By virtue of variational methods and delicate estimates, we prove that problem admits k positive solutions for λ > 0 sufficiently small, provided that the maximum of Q(x)is achieved atkinterior points inΩ.

Keywords: nonlocal problem, variational methods, critical nonlinearity, multiple posi- tive solutions.

2020 Mathematics Subject Classification: 35B33, 35J75.

1 Introduction

In this paper, we concern with the multiplicity of positive solutions to the nonlocal problem







−

a−λb Z

Ω|∇u|²dx

∆u=λ|u|^p−2u+Q(x)|u|²u, x ∈_Ω,

u=0, x ∈_∂Ω,

(1.1)

where a > 0, b≥ 0, 2 < p < 4, λ > 0 is a parameter, Ωis a smooth bounded domain in R⁴ (2^∗ =4 is the critical exponent in dimension four) andQ(x)∈C(_Ω)is a nonnegative function satisfying:

(Q₁) There existkdifferent pointsx¹,x², . . . ,x^k ∈ _Ωsuch thatQ(x^j)are strict local maximums and satisfy

Q(x^j) =Q_M = max{Q(x): x∈_Ω}>0, j=1, 2, . . . ,k;

BEmail: qianxiaotao1984@163.com

(Q₂) Q_M−Q(x) =O |x−x^j|²forxnear x^j, j=1, 2, . . . ,k.

In the past decade, the following Kirchhoff type problem involving critical growth on a bounded domainΩ⊂_R^N







−

a+b Z

Ω|∇u|²dx

∆u=g(x,u) +K(x)|u|^2∗−2u, x∈ _Ω,

u=0, x∈ _∂Ω,

(1.2)

has attracted considerable attention, where a,b > 0 are constants, 2^∗ = 2N/(N−2) with N ≥ _{3 and} K(x) is a nonnegative continuous function. Kirchhoff type problem is often viewed as nonlocal due to the presence of the term bR

Ω|∇u|²dx which implies that such problem is no longer pointwise identity. It is commonly known that Kirchhoff type problem has a mechanical and biological motivation, see [1,8]. Under different hypotheses on g(x,u) andK(x), there are many interesting results of positive solutions to (1.2) by using variational methods, see e.g. [6,7,15]. In particular, Fan [6] showed how the topology of the maximum set ofK(x)affects the number of positive solutions to (1.2) via Ljusternik–Schnirelmann category theory when N = 3 and f(x,u) = f(x)u^q with f(x) ∈ L^6−6q(_Ω) and 3 < q < 5. There are also several existence results for (1.2) in the whole spaceR^N, see [5,11,12] and the references therein.

In (1.2), if we replace a+bR

Ω|∇u|²dx by a−bR

Ω|∇u|²dx, it turns to be a new nonlo- cal one. This kind of nonlocal problem presents some interesting difficulties different from Kirchhoff type problem. Such nonlocal problem with subcritical growth







−

a−b Z

Ω|∇u|²dx

∆u= f_λ(x)|u|^p−2u, x∈_Ω,

u=0, x∈_∂Ω,

(1.3)

has been studied by some researchers, where f_λ(x)∈ L ²

∗

2∗ −p(_Ω)andΩ⊂_R^N is a bounded do- main. If f_λ(x)≡1 and 2< p<2^∗, Yin and Liu [23] obtained two nontrivial solutions to (1.3);

Qian [18] proved the existence and asymptotic behavior of ground state sign-changing solu- tions for (1.3); Wang et al. [22] proved that (1.3) has infinitely many sign-changing solutions.

For 1≤ p<2^∗, Duan et al. [4] established the existence of multiple positive solutions to (1.3).

In [10], the multiplicity result of positive solutions to (1.3) was obtained for 0< p<1. When f_λ(x)has indefinite sign, Lei et al. [9] and Qian and Chao [16] proved the existence of positive solution to (1.3) for 1 < p < 2 and 3 < p <6, respectively. For more results about (1.3) with general nonlinearities and its variants on unbounded domain, we refer the interested readers to [19,20,24]. To the best of our knowledge, there is little result for (1.3) when f(x,u)exhibits a critical exponent. Only Wang et al. [21] investigated the existence of two positive solutions for the following problem involving critical exponent







−

a−b Z

R⁴|∇u|²dx

∆u=λg(x) +|u|²u,x ∈_R⁴, u∈ D^1,2(_R⁴),

under the assumptions λ > 0 is sufficiently small and g(x) ∈ L^4/3(_R⁴) is a nonnegative function.

When a= 1, b= 0,R⁴ andQ(x)|u|²u are replaced byR^N andQ(x)|u|^2∗−2u, respectively, (1.1) is reduced to the following local one

(−_∆u=λ|u|^p−2u+Q(x)|u|^2∗−2u, x∈_Ω,

u=0, x∈_∂Ω, ^(1.4)

which does not depend on the nonlocal term R

Ω|∇u|²dx any more. The study by Cao and Noussair [3] is the first to investigate the effect of the shape of the graph of Q(x) on the number of positive solutions to (1.4) with p = 2. More precisely, they proved that for small enough λ > 0, (1.4) has k positive solutions if the maximum of Q(x) is achieved at exactly k different points of Ω, by applying Nehari manifold method. Liao et al. [13] extended the result of [3] in the sense that a more wider range of pis covered. In [17], Qian and Chen got a similar but more complicated result for (1.4) with an additional fast increasing weight.

Motivated by the idea of [3,6,21], it is natural and interesting to ask: can we apply the shape of the graph of Q(x) to prove the multiplicity of positive solutions for the critical nonlocal problem (1.1) as in Kirchhoff problem (1.2)? In the present paper, we will give a positive answer to this question.

Our main results can be stated as follows.

Theorem 1.1. Assume that a > 0, b ≥ 0,2 < p < 4 andΩis a smooth bounded domain inR⁴. If the conditions(Q₁)and(Q₂)hold, then there existsΛ0 >0, such that for eachλ∈ (0,Λ0),(1.1)has at least k positive solutions.

Since the result of Theorem1.1still holds forb=0, then we obtain the following corollary related to the multiplicity result of positive solutions for a semilinear problem with critical exponent.

Corollary 1.2. Assume that a > 0, 2 < p < 4 and Ω is a smooth bounded domain in R⁴. If the conditions (Q₁)and(Q₂)hold, then there existsΛ1 >0, such that for eachλ∈(_0,Λ1), the problem

(−a∆u=λ|u|^p−2u+Q(x)|u|²u, x∈_Ω,

u=0, x∈_∂Ω, ^(1.5)

has at least k positive solutions.

Associated with (1.1), we define the functional I_λ on H₀¹(_Ω)_by I_λ(u) = ^a

2kuk²− ^λb

4 kuk⁴− ^λ p

Z

Ω|u|^pdx−¹ 4

Z

ΩQ(x)|u|⁴dx, where kuk² = R

Ω|∇u|²dx. Then I_λ ∈ C¹ H₀¹(_Ω)_,R. Moreover, there exists a one to one correspondence between the critical points of I_λ on H₀¹(_Ω) and the weak solutions of (1.1).

Here, we say thatuis a weak solution of (1.1), ifu∈ H₀¹(_Ω)and for allv∈ H₀¹(_Ω), there holds (a−λbkuk²)

Z

Ω∇u∇vdx−λ Z

Ω|u|^p−2uvdx−

Z

ΩQ(x)|u|²uvdx=0.

The proof of Theorem 1.1 is based on variational methods. Since (1.1) has a negative nonlocal term, the approaches used in [6] to deal with Kirchhoff problem do not work here.

Indeed, we shall apply the ideas introduced by Cao and Noussair [3]. However, in the present paper, there are some new difficulties caused by the competing effect of the nonlocal term

with the nonlinear terms and the non-compactness due to the critical exponent. To overcome these difficulties, we need to add the factorλ of |u|^p−2u to the nonlocal term −bR

Ω|∇u|²dx in problem (1.1). This modification will play an important role in our arguments (see Lemma 2.2 below). Moreover, inspired by [21], we consider our problem in dimension 4 and make some delicate estimates in order to get the compactness condition. We also point out that it is not clear whether the multiplicity result in Theorem1.1still holds for critical problem (1.1) in other dimension, from which it follows that the critical exponent 2^∗is no longer equal to 4.

In Section 2, we present some lemmas which will be used to prove Theorem1.1. Section 3 is devoted to the proof of Theorem1.1.

2 Notations and preliminaries

Throughout the paper, for simplicity we write R

u instead of R

Ωu(x)dx. H₀¹(_Ω) and L^r(_Ω) are the usual Sobolev spaces equipped with the standard norms kuk and |u|_r, respectively.

D^1,2(_R⁴) = {u ∈ L⁴(_R⁴) : ∇u ∈ L²(_R⁴)}. Denote by Br(x) the ball centered at x with radiusr >0. LetB_r(x)and∂B_r(x)denote the closure and the boundary ofB_r(x), respectively.

We use → (*) to denote the strong (weak) convergence. O(ε^t) denotes |O(ε^t)|/ε^t ≤ C as ε →0, and o(ε^t)denotes |o(ε^t)|/ε^t →0 asε → 0. C andC_i denote various positive constants whose exact values are not essential. Let S be the best constant of the Sobolev embedding H₀¹(_Ω),→ L⁴(_Ω)_{, that is,}

S= _inf

u∈H₀¹(_Ω)\{0} Z

|∇u|² _Z

|u|⁴ _1/2. The Nehari manifold corresponding toI_λ is defined by

M_λ ={u∈ H₀¹(_Ω)\ {0} : hI_λ⁰(u),ui=0}.

By the condition (Q₁), we can take η > 0 sufficiently small such that B_2η(x^j) ⊂ _Ω are disjoint andQ(x) < Q(x^j) for x ∈ B_2η(x^j)\ {x^j}, j = 1, 2, . . . ,k. Following the argument of [3], we define a barycenter mapβ: H₀¹(_Ω)\ {0} →_R⁴ by setting

β(u) =

Z x|u|⁴

Z

|u|⁴ .

With the help of the map above, we will first separate the Nehari manifold M_λ, then study minimization problems of I_λ on its proper subset. We point out that, a key role of β is to insure that the minimizers of the considered minimization problems are distinct.

For j=1, 2, . . . ,k, we consider the following subsets ofM_λ,

M^j_λ ={u∈ M_λ : β(u)∈ Bη(x^j)} and O_λ^j ={u∈ M_λ: β(u)∈ ∂Bη(x^j)}. Correspondingly, study the following minimization problems

m_λ^j = inf

u∈M^j_λ

I_λ(u) and me^j_λ = inf

u∈O_λ^j

I_λ(u).

For allε>0 andx₀∈ _R⁴, we define

U_ε,x0 = (8)^1/2ε (ε²+|x−x₀|²)^,

which solves−_∆u=|u|²uinR⁴. Forj=1, 2, . . . ,kfixed, define a cut off functionϕ_j ∈C₀^∞(_R⁴) such that 0 ≤ ϕ_j ≤ _1, ϕ_j(x) = _{1 for} |x−x^j| < ρ and ϕ_j(x) = _{0 for} |x−x^j| ≥ _{2ρ with} 0<ρ <η/2. Let u_ε,j = ϕ_j(x−x^j)U_ε,xj(x). By [2], we have for 2< p<4,

ku_ε,jk² =S²+O(ε²),

|u_ε,j|⁴₄ =S²+O(ε⁴)_,

|u_ε,j|^p_p =O(ε^4−p). Lemma 2.1. For j=1, 2, . . . ,k andλ>0, we have

m_λ^j < ^a

2S²

4(λbS²+Q_M)^{. (2.1)}

Proof. It is easy to see that there exists a uniquetε >0 such thattεu_ε,j ∈ M_λ and I_λ(tεu_ε,j) = sup_t>0I_λ(tu_ε,j). By the symmetry of u_ε,j about x^j, we further obtain tεu_ε,j ∈ M^j_λ. Thus, to complete the proof of lemma, it suffices to prove that

sup

t>0

I_λ(tu_ε,j)< ^a

2S²

4(λbS²+QM)^{. (2.2)}

At this point, we can suppose that t_ε ≥ C₁ > 0 for any ε > 0 small. Otherwise, there is a sequence ε_n→0⁺ such thattεn →0. By the continuity of Iλ and the boundedness of{uε_n,j},

sup

t>0

I_λ(tu_εn,j) =I_λ(t_εnu_εn,j)→0< ^a

2S² 4(_λbS²+_QM)^,

that is, the proof is complete. Similarly, we also suppose that t_ε ≤ C₂ for some positive constantC₂and any ε>_{0 small.}

To proceed, set

h(t) = ^at

2

2 ku_ε,jk²− ^λbt

4

4 ku_ε,jk⁴− ^t

4

4 Z

Q_M|u_ε,j|⁴. We easily see thath(t)achieves its maximum at

t_max= ^aku_ε,jk² λbku_ε,jk⁴+Q_M|u_ε,j|⁴₄

!1/2

=

aS²+O(ε²) λbS⁴+Q_MS²+O(ε²)

1/2

=

aS² λbS⁴+Q_MS²

1/2

+O(ε²), with

h(t_max) = ^a

2S²

4(λbS²+Q_M)+O(ε²). (2.3)

Using condition(Q₂), we also have Z

(Q_M−Q(x))|u_ε,j|⁴ =O(ε²). (2.4) By (2.3) and (2.4),

sup

t>0

I_λ(tu_ε,j) = I_λ(t_εu_ε,j)

=h(t_ε) + ^t

4 ε

4 Z

(Q_M−Q(x))|u_ε,j|⁴−^λ pt_ε^p

Z

|u_ε,j|^p

≤h(tmax) + ^C

24

4 Z

(Q_M−Q(x))|u_ε,j|⁴− ^λ pC₁^p

Z

|u_ε,j|^p

= ^a

2S²

4(λbS²+Q_M)+O(ε²)−O(ε^4−p).

Since 2< p <4, (2.2) holds forε >0 small enough. This ends the proof.

Lemma 2.2. Assume that condition(Q₁)holds. Then there existsΛ0>0such that

me^j_λ > ^a

2S² 4Q_M for j=1, 2, . . . ,k,andλ∈(0,Λ0).

Proof. Let us argue by contradiction and suppose that there exist sequences λ_n → 0, and {u_n} ⊂ O^j_λ

n satisfying

I_λn(u_n)→c≤ ^a

2S² 4Q_M, and

a Z

|∇u_n|²−λ_nb _Z

|∇u_n|² 2

=λ_n Z

|u_n|^p+

Z

Q(x)|u_n|⁴. (2.5) By{un} ⊂ O_λ^j

n, one has for nlarge, c+1≥ Iλ_n(un)− ¹

phI_λ⁰_n(un),uni

=a 1

2− ¹ p

ku_nk²+λ_nb 1

p −¹ 4

ku_nk⁴+ 1

p −¹ 4

_Z

Q(x)|u_n|⁴

≥a 1

2− ¹ p

ku_nk²

which implies that {u_n}is bounded in H₀¹(_Ω). Using (2.5) and Sobolev embedding, we also have

akunk²= λnbkunk⁴+λn|un|^p_p+

Z

Q(x)|un|⁴≤ λnbkunk⁴+λnCkunk^p+Q_MS⁻²kunk⁴ from which we infer that

ku_nk ≥C₃ >_0.

Noting that λ_n→0, we then deduce from (2.5) that there is a constantC₄ >0 such that Z

Q(x)|u_n|⁴≥C₄>0,

for all n∈N. Thus, we are able to choosetn >0 such thatvn=tnun satisfies a

Z

|∇vn|² =

Z

Q_M|vn|⁴. (2.6)

This and Sobolev inequality give that _Q^aS2

M ≤ kvnk². Moreover,

tn =









 Z

Q(x)|un|⁴+λnb _Z

|∇un|² 2

+λn

Z

|un|^p

Z

Q_M|u_n|⁴











1/2

.

It follows that {t_n}is uniformly bounded. Then, we can assume lim_n→_∞t_n = t₀. ByQ(x)≤ QM, λn → 0 and the boundedness of{un}, we see that t0 ≤ 1. We show next that the case t₀ ≤1 leads to a contradiction. Since fort₀≤1, we have

a²S²

4Q_M ≤ lim

n→_∞

1 4a

Z

|∇vn|²= lim

n→_∞

1 4at²_n

Z

|∇un|²

= lim

n→_∞t²_n

"

1 2− ¹

4

a Z

|∇un|²−λnb Z

|∇un|² 2

−λn

Z

|un|^p

!

+λnb 1

2−¹ 4

Z

|∇un|² 2

+λn

1 2− ¹

p _Z

|un|^p

#

= _lim

n→_∞t²_nI_λn(u_n) =t²₀c≤c≤ ^a2S2 4QM, then it follows that

c= ^a

2S²

4Q_M and lim

n→_∞ Z

|∇vn|²= ^aS

2

Q_M. (2.7)

Letw_n=v_n/|v_n|_{4, then}|w_n|₄=1. Moreover, by (2.6) and (2.7),

nlim→_∞ Z

|∇w_n|²= lim

n→_∞

kv_nk²

|v_n|²₄ = lim

n→_∞

kv_nk²

(akv_nk²/Q_M)^1/2 =S,

namely, {w_n}is a minimizing sequence for S. According to [14], we can find a pointy₀ ∈ _Ω such that

|∇wn|² *dµ=Sδy₀ and |wn|₄*dν=δy₀ (2.8) with the above convergence holding weakly in the sense of measure, whereδ_y0 is a Dirac mass aty0. Then

β(un) =

Z

x|u_n|⁴

Z

|u_n|⁴

=

Z

x|v_n|⁴

Z

|v_n|⁴

=

Z

x|w_n|⁴

Z

|w_n|⁴

→y₀, asn→_∞.

This together with β(u_n) ∈ ∂B_η(x^j) imply that y₀ ∈ ∂B_η(x^j). Thus, from (2.6) and (2.8), we conclude that

nlim→_∞I_λn(u_n) = lim

n→_∞t²_n

"

1 2−¹

4

a Z

|∇u_n|²−λ_nb _Z

|∇u_n|² 2

−λ_n Z

|u_n|^p

!

+λ_nb 1

2−¹ 4

Z

|∇u_n|² 2

+λ_n 1

2− ¹ p

_Z

|u_n|^p

#

≤ lim

n→_∞

1 4 Z

Q(x)|u_n|⁴

= lim

n→_∞

1 4 Z

Q(x)|vn|⁴

= ^Q(y0) 4Q_M lim

n→_∞ Z

Q_M|vn|⁴

= ^Q(y₀) 4Q_M lim

n→∞a Z

|∇v_n|²

= ^Q(y0) 4Q_M

a²S² Q_M < ^a

2S² 4Q_M,

which contradicts with (2.7). This completes the proof.

Lemma 2.3. For any u ∈ M^j

λ, there exist ρ > 0 and a differential function g = g(w)defined for w∈ H₀¹(_Ω), w∈ B_ρ(₀)satisfying that

g(0) =1, g(w)(u−w)∈ M^j

λ

and

hg⁰(0),φi=

(2a−4λbkuk²)

Z

∇u∇φ−λp Z

|u|^p−2uφ−4 Z

Q(x)|u|²uφ akuk²−3λbkuk⁴−λ(p−1)

Z

|u|^p−3 Z

Q(x)|u|⁴ .

Proof. DefineF:R⁺×H¹₀(_Ω)→_Rby

F(t,w) =atku−wk²−λbt³ku−wk⁴−λt^p−1 Z

|u−w|^p−t³ Z

Q(x)|u−w|⁴. Byu∈ M^j_λ, we getF(1, 0) =0 and

F_t(1, 0) =akuk²−3λbkuk⁴−λ(p−1)

Z

|u|^p−3 Z

Q(x)|u|⁴

=a(2−p)kuk²−λb(4−p)kuk⁴−(4−p)

Z

Q(x)|u|⁴

<0.

Thus, we can use the implicit function theorem forFat the point(1, 0)and obtainρ >0 and a functionalg =g(w)>0 defined forw∈ H₀¹(_Ω),kwk<ρsatisfying that

g(₀) =_1, g(w)(u−w)∈ M_λ, ∀w∈ H₀¹(_Ω)_, kwk<_ρ.

By the continuity of the mapsgandβ, we can further takeρ>0 possibly smaller (ρ<ρ) such that

β(g(w)(u−w))∈ Bη(x^j), ∀w∈ H₀¹(_Ω), kwk< ρ, which means thatg(w)(u−w)∈ M_λ^j.

Moreover, we also have for allφ∈ H₀¹(_Ω), r >0, F(1, 0+rφ)−F(1, 0)

= aku−rφk²−λbku−rφk⁴−λ Z

|u−rφ|^p−

Z

Q(x)|u−rφ|⁴

−akuk²+λbkuk⁴+λ Z

|u|^p+

Z

Q(x)|u|⁴

= −a Z

2r∇u∇φ−r²|∇φ|² +λb

"

2 Z

|∇u|²

Z

2r∇u∇φ−r²|∇φ|²− Z

2r∇u∇φ−r²|∇φ|² 2#

−λ Z

|u−rφ|^p− |u|^p−

Z

Q(x)|u−rφ|⁴− |u|⁴. It follows that

hF_w,φi|_t=1,w=0 = lim

r→0

F(1, 0+rφ)−F(1, 0) r

= −(2a−4λbkuk²)

Z

∇u∇φ+pλ Z

|u|^p−2uφ+4 Z

Q(x)|u|²uφ.

Therefore,

hg⁰(0),φi= −hFw,φi F_t

t=1,w=0

=

(2a−4λbkuk²)

Z

∇u∇φ−λp Z

|u|^p−2uφ−4 Z

Q(x)|u|²uφ akuk²−3λbkuk⁴−λ(p−₁)

Z

|u|^p−₃

Z

Q(x)|u|⁴ .

The proof is completed.

Lemma 2.4. There existΛ0>0and a sequence{u_n} ⊂ M^j

λ such that un≥0, I_λ(un)→m^j_λ, I_λ⁰(un)→0, for j=1, 2, . . . ,k,andλ∈ (0,Λ0).

Proof. Note thatM^j_λ =M^j_λ∪ O_λ^j andO_λ^j is the boundary ofM^j_λ. In view of Lemmas2.1and 2.2, we know that there existsΛ0>0 such that

m^j_λ <m_eλ^j forλ∈(0,Λ0), j=1, 2, . . . ,k,. This implies that

m_λ^j =inf{I_λ(u):u∈ M^j_λ}.

Then, for each j = 1, 2, . . . ,k, we can apply Ekeland’s variational principle to construct a minimizing sequence {u_n} ⊂ M^j

λ satisfying the following properties :

(i) lim

n→_∞I_λ(u_n) =m^j_λ, (ii)I_λ(un)≤ I_λ(w) + ¹

nkw−unk, for each w∈ M^j_λ.

SinceI_λ(|u|) =I_λ(u), we may assumeu_n≥0. Using Lemma2.3withu= u_n, we getρ_n>_{0, a} differential functiong_n(w)defined forw∈ H₀¹(_Ω),w∈B_ρn(0)such thatg_n(w)(u_n−w)∈ M^j

λ. Let 0<δ <ρnand letw_δ = δuwithkuk=1. Fixnand setz_δ =gn(w_δ)(un−w_δ). Byz_δ ∈ M^j_λ and the property(ii), one has

I_λ(z_δ)−I_λ(un)≥ −¹

nkz_δ−unk. Then, by mean value theorem

hI_λ⁰(u_n),z_δ−u_ni+o(kz_δ−u_nk)≥ −¹

nkz_δ−u_nk. Thus,

hI_λ⁰(u_n),(u_n−w_δ) + g_n(w_δ)−1

(u_n−w_δ)−u_ni ≥ −¹

nkz_δ−u_nk+o(kz_δ−u_nk) which yields that

−δhI_λ⁰(un),ui+ gn(w_δ)−1

hI_λ⁰(un),un−w_δi ≥ −¹

nkz_δ−unk+o(kz_δ−unk). Combining this withhI_λ⁰(z_δ),g_n(w_δ)(u_n−w_δ)i=0, we obtain

hI_λ⁰(u_n),ui ≤ ¹ n

kz_δ−u_nk

δ + ^o(kz_δ−u_nk)

δ + ^gn(w_δ)−1

δ hI_λ⁰(u_n),u_n−w_δi. (2.9) By Lemma2.3and the boundedness of{un}, we easily see that

kz_δ−u^j_nk=k(g_n(w_δ)−1) (u^j_n−w_δ)−w_δk ≤ |g_n(w_δ)−1|C₅+δ and

lim

δ→0

|g_n(w_δ)−1| δ

=hg⁰_n(₀)_,ui ≤ kg⁰_n(₀)k ≤C₆. Therefore, for fixedn, we can conclude by passingδ→0 in (2.9) that

hI_λ⁰(u_n),ui ≤ ^C n,

which implies thatI_λ⁰(u_n)→0 asn→∞, and Lemma2.4is proved.

Lemma 2.5. For allλ>0, if{un} ⊂ M_λ is a sequence satisfying I_λ(un)→c< ^a

2S²

4(λbS²+Q_M) ^{and I}

0

λ(un)→0, as n→_∞, then{u_n}has a convergent subsequence.

Proof. As in the proof of Lemma2.2, it is easy to verify that{u_n}is bounded in H₀¹(_Ω). Hence, we may assume that for some u∗ ∈ H₀¹(_Ω),

un*u∗ in H₀¹(_Ω),

u_n→u∗ in L^r(_Ω)_{, 1}≤r<_4, u_n→u∗ a.e. inΩ.

Denotev_n=u_n−u∗ and we claim thatkv_nk →0. If not, there is a subsequence (still denoted by{vn}) such thatkvnk →Lwith L>0. By hI_λ⁰(un),u∗i=o(1)and the weak convergence of u_n, we see that

0= aku∗k²−λb(L²+ku∗k²)ku∗k²−λ Z

|u∗|^p−

Z

Q(x)|u∗|⁴_{. (2.10)} Moreover, by hI_λ⁰(un),uni=0, we can apply the Brézis–Lieb Lemma to get

0= a(kv_nk²+ku∗k²)−λb(kv_nk⁴+₂kv_nk²ku∗k²+ku∗k⁴)

−λ Z

|u∗|^p−

Z

Q(x)|vn|⁴−

Z

Q(x)|u∗|⁴+o(1).

(2.11)

Combining (2.10) and (2.11), we have

o(₁) =akv_nk²−λbkv_nk⁴−λbkv_nk²ku∗k²−

Z

Q(x)|v_n|⁴ _(2.12) and consequently,

akv_nk²−λbkv_nk⁴−λbkv_nk²ku∗k² =

Z

Q(x)|v_n|⁴+o(1)≤Q_MS⁻²kv_nk⁴+o(1). Passing the limit as n→∞, we obtain that

L² ≥ ^S

2(a−λbku∗k²)

λbS²+Q_M ≥0. (2.13)

By (2.10) and (2.13), we have I_λ(u∗) = ^a

2ku∗k²−^λb

4 ku∗k⁴− ^λ p Z

|u∗|^p− ¹ 4

Z

Q(x)|u∗|⁴

= ^λb

4 ku∗k⁴+^λb

2 L²ku∗k²+λ 1

2− ¹ p

_Z

|u∗|^p+¹ 4

Z

Q(x)|u∗|⁴

≥ ^λb

4 ku∗k⁴+^λb 2

S²(a−λbku∗k²) λbS²+Q_M ku∗k²

= ^λb(λbS²+Q_M)ku∗k⁴

4(λbS²+Q_M) + ^λabS

2ku∗k²

2(λbS²+Q_M)− ^λ

2b²S²ku∗k⁴ 2(λbS²+Q_M)

≥ ^λabS

2ku∗k²

2(λbS²+Q_M)− ^λ

2b²S²ku∗k⁴ 4(λbS²+Q_M)^.

(2.14)

Furthermore, using (2.12)–(2.14), we deduce that c+o(1) = I_λ(u_n)

= ^a

2ku_nk²−^λb

4 ku_nk⁴− ^λ p

Z

|u_n|^p−¹ 4

Z

Q(x)|u_n|⁴

= ^a

2ku∗k²− ^λb

4 ku∗k⁴− ^λ p

Z

|u∗|^p−¹ 4

Z

Q(x)|u∗|⁴ + ^a

2kvnk²− ^λb

4 kvnk⁴− ^λb

2 kvnk²ku∗k²− ¹ 4

Z

Q(x)|vn|⁴+o(1)

= I(u∗) + ^a

4kvnk²− ^λb

4 kvnk²ku∗k²+o(1)

= I(u∗) + ^a−λbku∗k²

4 L²+o(1)

≥ I(u∗) + ^a

2S²

4(λbS²+Q_M)− ^λabS

2ku∗k²

2(λbS²+Q_M)+ ^λ

2b²S²ku∗k⁴

4(λbS²+Q_M)+o(1)

≥ ^a

2S²

4(λbS²+Q_M)+_o(₁) a contradiction to the assumptionc< ^a2S2

4(λbS²+Q_M). Therefore, the claim holds, namely,un→u∗

inH₀¹(_Ω). This completes the proof of Lemma2.5.

3 Proof of Theorem 1.1

Proof of Theorem1.1. By Lemma 2.4, we know that there exists Λ0 such that for each λ ∈ (0,Λ0) and j = 1, 2, . . . ,k, there is a minimizing sequence {u_n^j} ⊂ M_λ^j satisfying u^j_n ≥ 0, I_λ(u^j_n)→ m^j_λ and I_λ⁰(u^j_n) → 0. From Lemmas2.1and2.5, it follows thatu_n^j → u^j andu^j ≥ 0 is a weak solution of (1.1). Furthermore, standard elliptic regularity argument and strong maximum principle imply thatu^j is a positive solution. Finally,u^j,j=1, 2, . . . ,k, are different positive solutions sinceβ(u^j)∈ B_η(x^j)andB_η(x^j)are disjoint. The proof is completed.

Acknowledgements

The author would like to thank the referee for careful reading of this paper and for the helpful comments. This research was supported by the National Natural Science Foundation of China (11871152).

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