A uniqueness result for a Schrödinger–Poisson system with strong singularity

A uniqueness result for a Schrödinger–Poisson system with strong singularity

Shengbin Yu

and Jianqing Chen

1College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Qishan Campus, Fuzhou, Fujian 350117, P. R. China

2Department of Basic Teaching and Research, Yango University, Fuzhou, Fujian 350015, P. R. China

Received 20 April 2019, appeared 25 November 2019 Communicated by Dimitri Mugnai

Abstract. In this paper, we consider the following Schrödinger–Poisson system with strong singularity















−∆u+φu= f(x)u^−γ, x∈Ω,

−_∆φ=u², x∈_Ω,

u>0, x∈Ω,

u=φ=_0, x∈∂Ω,

where Ω ⊂_R³is a smooth bounded domain,γ > 1, f ∈ L¹(_Ω)is a positive function (i.e. f(x) > 0 a.e. in Ω). A necessary and sufficient condition on the existence and uniqueness of positive weak solution of the system is obtained. The results supplement the main conclusions in recent literature.

Keywords: Schrödinger–Poisson system, strong singularity, uniqueness, variational method, necessary and sufficient condition.

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

1 Introduction

In this paper, we consider the existence and uniqueness of positive solution for the following Schrödinger–Poisson system















−_∆u+φu= f(x)u^−γ, x∈_Ω,

−_∆φ= u², x∈_Ω,

u>0, x∈_Ω,

u=φ=0, x∈_∂Ω,

(SP)

BCorresponding author. Email: yushengbin.8@163.com

where Ω ⊂ _R³ is a smooth bounded domain, γ > 1, f ∈ L¹(_Ω) is a positive function (i.e.

f(x)>0 a.e. inΩ). System (SP) can be viewed as a special case of the following Schrödinger–

Poisson system with singularity















−_∆u+ηφu= f(x)u^−γ+g(x,u), x∈_Ω,

−_∆φ=u², x∈_Ω,

u>0, x∈_Ω,

u=φ=0, x∈_∂Ω,

(1.1)

which has been investigated recently. Wheng(x,u) =0, f(x) =µis a positive parameter and 0< γ< 1 (i.e. weak singularity), Zhang [28] obtained a sufficient condition on the existence, uniqueness and multiplicity of positive solutions for system (1.1) withη=±1. Whenη=−1, g(x,u) = λh(x)u+u³, f(x) = _|^µ

x|^β and 0 < γ < 1, Wang [25] considered the existence and multiplicity of positive solutions for system (1.1) under some suitable conditions by Nehari manifold. Combining with variational method and Nehari manifold method, Lei and Liao [7] generalized a part of the results in Zhang [28] to the critical problem and obtained two positive solutions of system (1.1) with η = 1, g(x,u) = u⁵, f(x) = ^µ

|x|^β and 0 < γ < 1.

Jiang and Zhou [5] established the existence and a priori estimate of positive solutions of non- autonomous Schrödinger–Poisson system with singular potential. In addition, Kirchhoff type of problems with singularity have been considered by many researchers, one could refer to [3, 8,9, 14–16,24, 26] and the references cited therein. In a more general sense, Lei, Suo and Chu [10] studied a class of Schrödinger–Newton systems with singular and critical growth terms in unbounded domains and established results on the existence and multiplicity of positive solutions. We [27] obtained the uniqueness and asymptotical behavior of solutions to a Choquard equation with singularity in unbounded domains. Mu and Lu [17], Li et al. [13] and Zhang [29] studied the existence, uniqueness and multiple results to singular Schrödinger–Kirchhoff–Poisson system.

However, investigations (see [3, 5, 7–10, 13–17, 24–29] and references therein) considered elliptic equations with singularity have mainly focused on weak singularity (i.e. 0 < γ < 1 ) and seldom with strong singularity (i.e. γ > 1 ) which have been studied extensively (see [1,2,4,6,11,12,18–23,30] and references therein). In 2013, Sun [20] considered the following nonlinear elliptic problem











−_∆u= f(x)u^−γ+k(x)u^q, x∈_Ω,

u>0, x∈_Ω,

u=0, x∈_∂Ω,

(1.2)

where Ω ⊂ _R^N_, N ≥ 1, is a bounded open set with smooth boundary ∂Ω, k ∈ L^∞(_Ω) _{is a} non-negative function,q∈(0, 1), γ>1 (i.e. strong singularity) and f ∈ L¹(_Ω)is positive (i.e.

f(x)>0 a.e. inΩ). By using variational method, Sun [20] has derived a compatible condition between coefficients and negative exponents, which is optimal forH¹₀(_Ω)-solutions of problem (1.2). The results obtained by Sun [20] supplement and improve the main conclusions in [9- 13]. WhenN≥3 andk(x)≡0, Sun [22] further obtained the existence of solutions of problem (1.2) and showed the reason on why 3 plays a crucial role in the study of elliptic equations with negative exponents. When k(x) ≡ 0 and−_∆u was replaced by−div(M(x)∇u) where M(x)is a bounded elliptic matrix, Tan and Sun [23] also proved the existence of a positive H₀¹(_Ω)-solutions of problem (1.2). Furthermore, Cong and Han [2], Li and Gao [11] both

considered the existence of positive solutions to elliptic boundary value problem with strong singularity and p-Laplace operator. As for Kirchhoff type equations with strong singularities, Li et al. [12], Tan and Sun [21] and Santos et al. [18] have obtained some perfect results.

However, to the best of our knowledge, Schrödinger–Poisson system with strong singularity has not been studied until now. Thus, the main purpose of this paper is to consider the existence and uniqueness of positive solution for system (SP) with strong singularity. Indeed, we obtain the following results.

Theorem 1.1. Assume that f ∈ L¹(_Ω)is a positive function (i.e. f(x) > 0 a.e. inΩ),γ > 1, then system(SP)admits a unique positive solution if and only if there exists a u₀ ∈ H₀¹(_Ω), such that

Z

Ω f(x)|u₀|^1−γdx< +_∞. (1.3) As a consequence of Theorem1.1, we also have the following.

Theorem 1.2. Suppose f₁, f₂ ∈ L¹(_Ω)are two positive functions (i.e. f_i(x)> 0, i= 1, 2a.e. inΩ) with R

Ω f_i(x)|u₀|^1−γdx < +_∞, i = 1, 2and u₁,u₂ are the corresponding solutions of system(SP) obtained in Theorem1.1, then f₁≥ f2implies u₁ ≥u2.

Theorem 1.3. Let Ω ⊂ _R³ be a smooth bounded domain containing 0. Suppose 0 < α < 3 and 1<γ<3,then















−_∆u+φu=|x|^−αu^−γ, x ∈_Ω,

−_∆φ=u², x ∈_Ω,

u>0, x ∈_Ω,

u=φ=0, x ∈_∂Ω,

admits a unique positive solution u∈ H₀¹(_Ω).

We then consider the property of theH₀¹(_Ω)-solution in Theorem1.3and get the following result.

Theorem 1.4. LetΩ ⊂ _R³ be a smooth bounded domain containing 0. Supposeα > 2and γ > 0,

then 













−_∆u+φu=|x|^−αu^−γ, x ∈_Ω,

−_∆φ=u², x ∈_Ω,

u>0, x ∈_Ω,

u=φ=0, x ∈_∂Ω,

admits no bounded positive solution.

Notations

• L^s(_Ω)is a Lebesgue space whose norm is denoted by|u|_s = (R

Ω|u|^sdx)^1s.

• H¹₀(_Ω)is the usual Sobolev space equipped with the normkuk² =R

Ω|∇u|²dx.

• u⁺=max{u, 0}andu⁻=min{u, 0}for any functionu.

• →denotes the strong convergence and*denotes the weak convergence.

• B_r(x₀)denotes the Euclidean ball of centerx₀and radiusr.

• C andC_i (i = 1, 2, . . .)denotes various positive constants, which may vary from line to line.

2 Proof of main results

Before proving our main results, we need the following lemma (see [28]).

Lemma 2.1. For each u∈ H₀¹(_Ω), there exists a uniqueφ_u∈ H¹₀(_Ω)solution of (−_∆φ=u², x ∈_Ω,

φ=0, x ∈_∂Ω.

Moreover, (i) kφuk² =R

Ωφuu²dx;

(ii) φ_u≥0. Moreover,φ_u>0when u6=0;

(iii) for each t6=0,φtu =t²φu; (iv) for any u∈ H¹₀(_Ω),

Z

Ωφuu²dx=

Z

Ω|∇φu|²dx≤ S⁻¹|u|⁴_12/5 ≤ S⁻¹|u|⁴₄|_Ω|^2/3 ≤ S⁻³kuk⁴|_Ω|, where S>0is the best Sobolev embedding constant.

(v) assume that u_n *u in H₀¹(_Ω), thenφ_un →φ_uin H₀¹(_Ω)andR

Ωφ_unu_nvdx →R

Ωφ_uuvdx for any v∈ H₀¹(_Ω);

(vi) we denoteΨ(u) =R

Ωφuu²dx, thenΨ :H₀¹(_Ω)→_Ris C¹and for any v∈ H¹₀(_Ω), h_Ψ⁰(u)_,vi=₄

Z

Ωφ_uuvdx;

(vii) for u,v∈ H₀¹(_Ω),R

Ω(φ_uu−φ_vv)(u−v)dx ≥ ¹

2kφ_u−φ_vk².

According to Lemma2.1, we substituteφ_uto the first equation of system (SP), then system (SP) transforms into the following equation











−_∆u+φuu= f(x)u^−γ, x ∈_Ω,

u>_0, x ∈_Ω,

u=0, x ∈_∂Ω.

(2.1)

The energy functional corresponding to equation (2.1) given by I(u) = ¹

2kuk²+¹ 4

Z

Ωφu|u|²dx− ¹ 1−γ

Z

Ω f(x)|u|^1−γdx, (2.2) and a functionuis called a solution of equation (2.1), i.e.(u,φu)is a solution of system (SP) if u∈ H₀¹(_Ω)such thatu>0 in Ωand for everyψ∈ H₀¹(_Ω),

Z

Ω∇u∇ψdx+

Z

Ωφuuψdx−

Z

Ω f(x)u^−γψdx=0. (2.3)

For the sake of simplicity, we just sayuinstead of(u,φ_u)is a solution of system (SP). In order to motivate our results, we consider the following two constrained sets:

N₁=

u∈ H₀¹(_Ω) : kuk²+

Z

Ωφ_u|u|²dx−

Z

Ω f(x)|u|^1−γdx≥0

, and

N₂=

u∈ H₀¹(_Ω) _: kuk²+

Z

Ωφ_u|u|²dx−

Z

Ω f(x)|u|^1−γdx=₀

. We now come to prove our main results.

Proof of Theorem1.1. (Necessity). Suppose u ∈ H¹₀(_Ω) is the solution of system (SP), then u>0 and satisfies (2.3). Choosingψ=uin (2.3) leads to

Z

Ω f(x)u^1−γdx =kuk²+

Z

Ωφ_uu²dx< +_∞, and the necessity is proved.

(Sufficiency) The proof will be complete in six steps.

Step 1. N_i 6=_∅, i =1, 2.

Fixu ∈ H¹₀(_Ω) withR

Ω f(x)|u|^1−γdx < +_{∞. For any} t > 0, according to Lemma2.1(iii), we have

I(tu) = ^t

2

2kuk²+ ^t

4

4 Z

Ωφ_u|u|²dx− ^t

1−γ

1−γ Z

Ω f(x)|u|^1−γdx.

Set g(t) =t^dI_dt^(tu), then

g(t) =t²kuk²+t⁴ Z

Ωφ_u|u|²dx−t^1−γ Z

Ω f(x)|u|^1−γdx.

Since γ > 1, one can easily obtain that g(_t) is increasing on (_0,+_∞) _{with limt→0}+g(_t) =

−_∞ and limt→+_∞g(t) = +∞. Thus, there exists a unique t(u) > 0 such that I(t(u)u) = min_t>0I(tu)andg(t(u)) =0, i.e.

t²(u)kuk²+t⁴(u)

Z

Ωφu|u|²dx−t^1−γ(u)

Z

Ω f(x)|u|^1−γdx=0,

that is t(u)u ∈ N₂. Specially, the assumption (1.3) implies that there exists a t(u₀) > 0 such that t(u₀)u₀ ∈ N₂ ⊂ N₁, and soN_i 6= _∅, i=1, 2.

Step 2. N₁ is an unbounded closed set in H₀¹(_Ω) and there exists a positive constant C1, such that kuk ≥C₁ for allu∈ N₁.

According to Step 1, tu ∈ N₁ for any t ≥ t(u₀), so N₁ is unbounded in H₀¹(_Ω). The closeness of N₁ follows easily from Lemma 2.1 (v) and Fatou’s lemma. We claim that there exists a positive constant C₁, such that kuk ≥ C₁ for all u ∈ N₁. Arguing by contradiction, there exists a sequence {u_n} ⊂ N_{1 satisfying}u_n →_{0 in} H₀¹(_Ω)_{. Since} γ> _{1 and}u_n ∈ N_{1, by} the reverse form of Hölder’s inequality and Lemma2.1(v), one can get

_Z

Ωf^γ1(x)dx γ_Z

Ω|u_n|dx 1−γ

≤

Z

Ωf(x)|u_n|^1−γdx≤ ku_nk²+

Z

Ωφ_un|u_n|²dx→0.

SinceR

Ωf^γ1(x)dx>0, we haveR

Ω|u_n|dx→∞, which is impossible. So there exists a positive constantC₁, such thatkuk ≥C₁for all u∈ N_1.

Step 3. Properties of the minimizing sequence{u_n}.

For anyu∈ N₁, according to Step 2, there exists a positive constantC₁such thatkuk ≥C₁, then by (2.2),γ>1 and Lemma2.1 (ii), one has

I(u) = ¹

2kuk²+ ¹ 4

Z

Ωφu|u|²dx− ¹ 1−γ

Z

Ω f(x)|u|^1−γdx≥ ¹ 2kuk²,

therefore,I(u)is coercive and bounded from below onN₁and so infN₁ I is well defined. Since N₁ is closed, applying the Ekeland variational principle to construct a minimizing sequence {un} ⊂ N₁satisfying:

(1) I(un)<infN₁ I+¹_n;

(2) I(z)≥ I(un)− ¹_nkun−zk,∀z∈ N₁.

The coerciveness of I on N₁ shows that kunk ≤ C2 uniformly for some suitable positive constant C₂. Hence, C₁ ≤ ku_nk ≤ C₂ and then there exists a subsequence of {u_n} (still denoted by{u_n}) and a function u∗ ∈ H₀¹(_Ω)such that

u_n *u∗ in H¹₀(_Ω),

un →u∗ in L^p(_Ω), p∈ [1, 6), un →u∗ a.e. in Ω.

Since I(|u|) = I(u), we could assume that u_n ≥ 0. By {u_n} ⊂ N₁, Lemma 2.1 (iv) and the boundness of {u_n}, we have R

Ωf(x)u¹_n^−γdx < +_∞ which implies that u_n(x) > 0 a.e. in Ω since f(x)> 0 a.e. in Ω, and γ > 1. Therefore, u∗(x) ≥0. Furthermore, by Fatou’s Lemma, we getR

Ω f(x)u¹∗^−γdx< +_∞which in turn impliesu∗(x)>0 a.e. inΩ.

Step 4.u_∗ ∈ N₂, infN₁I = I(u_∗),u_∗ >0 inΩand for any 0≤v∈ H₀¹(_Ω), Z

Ω∇u∗∇vdx+

Z

Ωφ_u∗u∗vdx−

Z

Ω f(x)u⁻∗^γvdx≥0.

To prove the above statements, we consider the following two cases regarding whether {u_n}belongs to N₁\ N₂ orN₂.

Case 1. Suppose that{un} ⊂ N₁\ N₂for allnlarge.

For any 0 ≤ v ∈ H₀¹(_Ω), since {u_n} ⊂ N₁\ N₂, f(x) > 0 a.e. in Ωand γ > 1, we can derive _Z

Ω f(x)(u_n+tv)^1−γdx ≤

Z

Ω f(x)u¹_n^−γdx<ku_nk²+

Z

Ωφ_unu²_ndx, ∀t≥0.

Therefore, we could chooset >0 small enough such that Z

Ω f(x)(un+tv)^1−γdx<kun+tvk²+

Z

Ωφun+tv(un+tv)²dx, that isu_n+tv∈ N₁. Applying condition (2) withz=u_n+tvleads to

ktvk

n ≥ I(u_n)−I(u_n+tv)

= ¹

2(ku_nk²− ku_n+tvk²) + ¹ 4

Z

Ω[φ_unu²_n−φ_un+tv(u_n+tv)²]dx + ¹

1−γ Z

Ω f(x)[(un+tv)^1−γ−u¹n^−γ]dx.

Dividing by t > 0 and passing to the liminf as t →0⁺, then we obtain from Fatou’s Lemma that

kvk n +

Z

Ω∇un· ∇vdx+

Z

Ωφ_ununvdx≥lim inf

t→0⁺

1 1−γ

Z

Ω f(x)(u_n+tv)^1−γ−u¹_n^−γ

t dx

≥

Z

Ωlim inf

t→0⁺

f(x) 1−γ

(u_n+tv)^1−γ−u¹_n^−γ

t dx

=

Z

Ω f(x)u⁻_n^γvdx, (sinceun>0 a. e. inΩ). Lettingn→∞, according to Lemma2.1(v) and Fatou’s Lemma again, one can get

Z

Ω∇u∗∇vdx+

Z

Ωφu∗u∗vdx ≥

Z

Ωf(x)u⁻∗^γvdx and Z

Ω f(x)u⁻∗^γvdx<+_∞. (2.4) Choose v = u∗ in (2.4), we get u∗ ∈ N₁, R

Ω f(x)u¹∗^−γdx < +_∞ and then Step 1 shows the existence of unique t(u∗) > 0 satisfying t(u∗)u∗ ∈ N₂ and I(t(u∗)u∗) = min_t>0I(tu∗). Hence, according to the weakly lower semi-continuity of the norm, Lemma2.1(v) and Fatou’s Lemma, one has

infN₁ I = lim

n→_∞I(un)

=lim inf

n→_∞

1

2kunk²+¹ 4

Z

Ωφunu²_ndx− ¹ 1−γ

Z

Ωf(x)u¹_n^−γdx

≥lim inf

n→_∞

1 2kunk²

+lim inf

n→_∞

1 4 Z

Ωφunu²_ndx

+lim inf

n→_∞

1 γ−1

Z

Ω f(x)u¹_n^−γdx

≥ ¹

2ku∗k²+ ¹ 4

Z

Ωφ_u∗u²_∗dx+ ¹ γ−1

Z

Ω f(x)u¹∗^−γdx

= I(u∗)≥ I(t(u∗)u∗)≥inf

N2

I ≥inf

N₁ I.

Thus, the above inequalities are actually equalities. By the uniqueness of t(u∗), we have t(u∗) =1, which implies that

u∗∈ N₂, inf

N₁ I = I(u∗). (2.5)

Moreover, we can also obtain that lim infn→_∞kunk² = ku∗k² and a subsequence of {un}(still denoted by{u_n}), such that lim_n→_∞ku_nk²= ku∗k². This together with the weak convergence of {u_n}_in H₀¹(_Ω)_impliesu_n→u∗strongly in H¹₀(_Ω)_.

Case 2. There exists a subsequence of{u_n}(still denoted by{u_n}) which belongs toN₂. For any 0≤ v ∈ H¹₀(_Ω), according to γ > 1, the boundness of{un}, Lemma2.1 (iv), we have

Z

Ω f(x)(u_n+tv)^1−γdx≤

Z

Ω f(x)u¹_n^−γdx= ku_nk²+

Z

Ωφ_unu²_ndx<+_∞, ∀t≥0, then Step 1 shows the existence of some functionsh_n,v(t):[0,+_∞)→(0,+_∞)corresponding to u_n+tvsuch that

h_n,v(₀) =_1, h_n,v(t)(u_n+tv)∈ N_2, ∀t≥_0.

The continuity of h_n,v(t) with respect to t follows from Lemma 2.1 (v) and the dominated convergence theorem sinceγ > 1 and R

Ω f(x)|un|^1−γdx < +∞. However, we have no idea whether or noth_n,v(t)is differentiable. For the sake of proof, we set

h⁰_n,v(₀) = _lim

t→0⁺

h_n,v(t)−₁

t ∈ [−_∞,+_∞]_.

If the above limit does not exist, we choose t_k → 0 (instead of t → 0) witht_k > 0 such that h⁰_n,v(0) = lim_k→∞ ^hn,v(_t^tk)−1

k ∈ [−_∞,+_∞]. According to u_n ∈ N₂, h_n,v(t)(u_n+tv) ∈ N₂ and Lemma2.1 (iii), we have

ku_nk²+

Z

Ωφ_unu²_ndx−

Z

Ω f(x)u¹_n^−γdx =0, h²_n,v(t)ku_n+tvk²+h⁴_n,v(t)

Z

Ωφ_un+tv(u_n+tv)²dx−h¹_n,v^−γ(t)

Z

Ω f(x)(u_n+tv)^1−γdx =_0.

Sinceγ>1, the above two equalities yield

0= [hn,v(t)−1]ⁿ[hn,v(t) +1]kun+tvk²−^h

1−γ n,v (t)−1 hn,v(t)−1

Z

Ω f(x)(un+tv)^1−γdx +h²_n,v(t) +1

[hn,v(t) +1]

Z

Ωφ_un+tv(un+tv)²dxo

+kun+tvk²− kunk² +

Z

Ω

φ_un+tv(u_n+tv)²−φ_unu²_n dx−

Z

Ω f(x)^h(u_n+tv)^1−γ−u¹_n^−γi dx

Dividing byt >0 and passing to the limit ast →0⁺, using Lemma2.1 (vi), the continuity of hn,v(t)andun∈ N₂, we obtain

0≥h⁰_n,v(₀)ⁿ₂ku_nk²+ (γ−₁)

Z

Ω f(x)u¹_n^−γdx+₄

Z

Ωφ_unu²_ndxo +2

Z

Ω∇un∇vdx+4 Z

Ωφununvdx

=h⁰_n,v(0)ⁿ(γ+1)ku_nk²+ (γ+3)

Z

Ωφ_unu²_ndxo +2

Z

Ω∇u_n∇vdx+4 Z

Ωφ_unu_nvdx We claim that there existsC₃>0, such thath⁰_n,v(0)≤C₃uniformly inn. Fixn, eitherh⁰_n,v(0)is nonnegative, orh⁰_n,v(₀)is negative. Ifh⁰_n,v(₀)≥0, then from the above inequality and Lemma 2.1(ii), one can get

0≥ (γ+₁)h⁰_n,v(₀)ku_nk²+₂

Z

Ω∇u_n∇vdx.

SinceC₁ ≤ ku_nk ≤C₂ by Step 3, we can conclude that

h⁰_n,v(0)≤C₃ uniformly in n (2.6) for some suitable constantC₃ >0 and

ku_nk

n −(γ+1)C²₁ γ−1 <₀

for n large enough. We also claim that there exists a constant C₄, such that h⁰_n,v(0) ≥ C₄ uniformly in alln large. If h⁰_n,v(0) < 0, then hn,v(t) < 1 for t > 0 small. Applying condition (2) withz=h_n,v(t)(u_n+tv)leads to

1

n[1−h_n,v(t)]ku_nk+ ^t

nh_n,v(t)kvk ≥ ¹

nku_n−h_n,v(t)(u_n+tv)k

≥ I(u_n)−I[h_n,v(t)(u_n+tv)].

(2.7)

Sinceu_n∈ N₂, Lemma2.1(iii) together with (2.7) leads to kvk

n h_n,v(t)≥ ^hn,v(t)−1 t

nku_nk

n −

1 2+ ¹

γ−₁

[h_n,v(t) +1]ku_n+tvk²

− 1

4+ ¹ γ−1

h²_n,v(t) +₁[h_n,v(t) +₁]

Z

Ωφ_un+tv(u_n+tv)²dxo

− 1

2+ ¹ γ−1

ku_n+tvk²− ku_nk² t

− 1

4+ ¹ γ−1

_Z

Ω

φun+tv(un+tv)²−φunu²_n

t dx.

Lettingt→0⁺, using Lemma2.1(vi), the continuity ofh_n,v(t)andC₁ ≤ ku_nk ≤C₂, we obtain kvk

n ≥h⁰_n,v(0)

ku_nk n −2

1 2 + ¹

γ−1

ku_nk²−4 1

4+ ¹ γ−1

_Z

Ωφ_unu²_ndx

−2 1

2 + ¹ γ−1

_Z

Ω∇u_n∇vdx−4 1

4 + ¹ γ−1

_Z

Ωφ_unu_nvdx

=h⁰_n,v(0)

kunk

n − ¹

γ−1

(γ+1)kunk²+ (γ+3)

Z

Ωφunu²_ndx

−

1+ ² γ−1

_Z

Ω∇u_n∇vdx−

1+ ⁴ γ−1

_Z

Ωφ_unu_nvdx

≥h⁰_n,v(0)

ku_nk

n − (γ+₁)_C1² γ−1

−

1+ ² γ−1

_Z

Ω∇u_n∇vdx

−

1+ ⁴ γ−1

_Z

Ωφ_unu_nvdx

since γ > 1 and h⁰_n,v(0)< 0. Then, from the construction of coefficient we see that h⁰_n,v(0) 6=

−_∞and cannot diverge to−_∞asn→∞, that is,

h⁰_n,v(0)6=−_∞and h⁰_n,v(0)≥C₄ uniformly in n large (2.8) for some suitable constantC₄. So, it follows from (2.6) and (2.8) that

h⁰_n,v(0)∈(−_∞,+_∞) and|h⁰_n,v(0)| ≤Cuniformly in n large,

where C = max{C₃,|C₄|} is independent of n. Furthermore, applying condition (2) with z=h_n,v(t)(u_n+tv)again leads to

|1−h_n,v(t)|

t

ku_nk n + kvk

n h_n,v(t)

≥ ¹

ntku_n−h_n,v(t)(u_n+tv)k ≥ ¹

t [I(u_n)−I(h_n,v(t)(u_n+tv))]

≥ ^hn,v(t)−1 t

n−^hn,v(t) +1

2 ku_n+tvk²+ ^h

1−γ n,v (t)−1 (₁−γ)[h_n,v(t)−₁]

Z

Ωf(x)(u_n+tv)^1−γdx

−¹ 4

h²_n,v(t) +1

[h_n,v(t) +1]

Z

Ωφ_un+_tv(u_n+tv)²dxo

−¹ 2

kun+tvk²− kunk² t

−¹ 4

Z

Ω

φ_un+tv(u_n+tv)²−φ_unu²_n

t dx+ ¹

1−γ Z

Ω f(x)(u_n+tv)^1−γ−u¹_n^−γ

t dx

Passing to the liminf ast→0⁺, then we get from Lemma2.1(vi), the continuity ofh_n,v(t)and Fatou’s Lemma that

|h⁰_n,v(0)| · kunk

n +kvk

n

≥ h⁰_n,v(0)ⁿ− kunk²+

Z

Ω f(x)u¹_n^−γdx−

Z

Ωφunu²_ndxo

−

Z

Ω∇u_n∇vdx−

Z

Ωφ_unu_nvdx+lim inf

t→0⁺

1 1−γ

Z

Ω f(x)(u_n+tv)^1−γ−u¹_n^−γ

t dx

≥ −

Z

Ω∇u_n∇vdx−

Z

Ωφ_unu_nvdx+

Z

Ω

f(x)

1−γlim inf

t→0⁺

(u_n+tv)^1−γ−u¹_n^−γ

t dx

= −

Z

Ω∇u_n∇vdx−

Z

Ωφ_unu_nvdx+

Z

Ω f(x)u⁻_n^γvdx,

sinceu_n ∈ N₂. Furthermore, by Lemma2.1(iv), fornlarge, we have Z

Ω f(x)u⁻_n^γvdx≤ |h⁰_n,v(0)| · ku_nk

n +kvk

n +

Z

Ω∇un∇vdx+

Z

Ωφununvdx

≤ ^C·C₂+kvk

n +

Z

Ω∇u_n∇vdx+

Z

Ωφ_unu_nvdx<+_∞,

thanks to C₁ ≤ ku_nk ≤ C₂ and |h⁰_n,v(0)| ≤ C uniformly in n large. Passing to the limit as n→_∞with using Lemma2.1 (v) and Fatou’s Lemma again leads to

Z

Ω f(x)u⁻_∗^γvdx ≤lim inf

n→_∞ Z

Ωf(x)u⁻_n^γvdx ≤

Z

Ω∇u∗∇vdx+

Z

Ωφ_u∗u∗vdx<+_∞, (2.9) for any 0≤v∈ H¹₀(_Ω). By the same argument as in Case 1, we can also obtain that

u∗ ∈ N₂, inf

N₁ I = I(u∗). (2.10)

in Case 2. Therefore, Combining (2.4), (2.5), (2.9) and (2.10), we could conclude that in either case, up to subsequence,u_n→u∗strongly in H¹₀(_Ω),u∗ ∈ N₂, inf_N1 I = I(u∗)and

Z

Ω∇u∗∇vdx+

Z

Ωφu∗u∗vdx−

Z

Ω f(x)u⁻∗^γvdx≥0, (2.11) for any 0≤ v∈ H₀¹(_Ω). Hence,−_∆u∗+φu∗u∗ ≥0 in the week sense. By Step 3, u∗(x)>0 a.e.

inΩand similar to the proof in [28], we getu∗ >0 inΩ.

Step 5.u∗ is a solution of system (SP).

For any ψ ∈ H₀¹(_Ω)\ {0}and ε > 0. Since 0 < u∗ ∈ N₂, applying inequality (2.11) with v= (u∗+εψ)⁺_{leads to}

0≤ ¹ ε

n^Z

Ω∇u∗∇(u∗+εψ)⁺dx+

Z

Ωφu∗u∗(u∗+εψ)⁺dx−

Z

Ω f(x)u⁻∗^γ(u∗+εψ)⁺dxo

= ¹ ε

Z

[u∗+εψ≥0]

n∇u∗∇(u∗+εψ) +φ_u∗u∗(u∗+εψ)− f(x)u⁻∗^γ(u∗+εψ)^odx

= ¹ ε

^Z

Ω−

Z

[u∗+εψ<0]

n∇u∗∇(u∗+εψ) +φu∗u∗(u∗+εψ)− f(x)u⁻∗^γ(u∗+εψ)^odx

≤ ¹ ε

nku∗k²+

Z

Ωφ_u∗u²_∗dx−

Z

Ω f(x)u¹_∗^−γdxo +ⁿ

Z

Ω∇u∗∇ψdx+

Z

Ωφu∗u∗ψdx−

Z

Ω f(x)u⁻∗^γψdxo

−¹ ε

Z

[u∗+εψ<0]

h∇u∗∇(u∗+εψ) +φ_u∗u∗(u∗+εψ)ⁱdx +¹

ε Z

[u∗+εψ<0] f(x)u⁻∗^γ(u∗+εψ)dx

≤ⁿ

Z

Ω∇u∗∇ψdx+

Z

Ωφ_u∗u∗ψdx−

Z

Ω f(x)u⁻_∗^γψdxo

−¹ ε

Z

[u∗+εψ<0]

h∇u∗∇u∗+φ_u∗u²_∗i dx−

Z

[u∗+εψ<0]

h∇u∗∇ψ+φ_u∗u∗ψ i

dx

≤ⁿ

Z

Ω∇u∗∇ψdx+

Z

Ωφ_u∗u∗ψdx−

Z

Ω f(x)u⁻∗^γψdxo

−

Z

[u∗+εψ<0]

h∇u∗∇ψ+φ_u∗u∗ψ i

dx.

Letting ε → 0⁺ to the above inequality and using the fact that meas[u∗+εψ < 0] → 0 as ε→0⁺, we have

Z

Ω∇u∗∇ψdx+

Z

Ωφ_u∗u∗ψdx−

Z

Ωf(x)u⁻_∗^γψdx≥_0, ∀ψ∈H₀¹(_Ω)_. This inequality also holds for −ψ, hence we obtain

Z

Ω∇u∗∇ψdx+

Z

Ωφ_u∗u∗ψdx−

Z

Ω f(x)u⁻_∗^γψdx=0, ∀ψ∈ H₀¹(_Ω). (2.12) Thus u∗ ∈ H₀¹(_Ω)is a solution of system (SP).

Step 6. u∗is a unique solution of system (SP).

Supposev∗ ∈ H₀¹(_Ω)is also a solution of system (SP), then for anyψ∈ H₀¹(_Ω), we have Z

Ω∇v∗∇ψdx+

Z

Ωφ_v∗v∗ψdx−

Z

Ω f(x)v⁻∗^γψdx =0, ∀ψ∈ H¹₀(_Ω). (2.13) Takingψ=u∗−v∗in both equations (2.12)–(2.13) and subtracting term by term, we obtain

0≥

Z

Ω f(x)(u⁻∗^γ−v⁻∗^γ)(u∗−v∗)dx

=ku∗−v∗k²+

Z

Ω(φ_u∗u∗−φ_v∗v∗)(u∗−v∗)dx

≥ ku∗−v∗k²+¹

2kφu∗−φv∗k² ≥ ku∗−v∗k² ≥0,

where we use Lemma2.1(vii). Soku∗−v∗k² =0, thenu∗ =v∗ andu∗ is the unique solution of system (SP).

Proof of Theorem1.2. Since u₁, u₂ ∈ H₀¹(_Ω)are two positive solutions of system (SP) corre- sponding to f₁and f2respectively, then for anyψ∈ H₀¹(_Ω), we have

Z

Ω∇u₁∇ψdx+

Z

Ωφu₁u₁ψdx−

Z

Ω f₁(x)u⁻₁^γψdx=0, Z

Ω∇u₂∇ψdx+

Z

Ωφ_u2u₂ψdx−

Z

Ω f₂(x)u⁻₂^γψdx=0.

SetΩ1 = {x|u₂(x) ≥ u₁(x), x ∈ _Ω}, then subtracting the above two equations and choosing ψ= (u₂−u₁)⁺∈ H₀¹(_Ω)yield

0≥

Z

Ω(f₂(x)u⁻₂^γ− f₁(x)u⁻₁^γ)(u₂−u₁)⁺dx

=k(u₂−u₁)⁺k²+

Z

Ω(φ_u2u₂−φ_u1u₁)(u₂−u₁)⁺dx

=k(u2−u₁)⁺k²+

Z

Ω1

(φu2u2−φu₁u₁)(u2−u₁)dx

≥ k(u2−u₁)⁺k²≥0,

where we use f₁ ≥ f₂,γ>1 and Lemma2.1(vii). So(u₂−u₁)⁺≡0 and hence u₁ ≥u₂. Proof of Theorem1.3. The proof is exactly the same as Sun and Tan [21]. We omit the details here.

Proof of Theorem1.4. We prove Theorem 1.4by contradiction that sup_Ωu< +∞. Motivated by Sun and Tan [21], Choose a sequence of test functions{ϕ_δ} ⊂C^∞₀ (_Ω)satisfying 0≤ ϕ_δ ≤1, ϕ_δ ≡0 inB_δ(0), ϕ_δ ≡1 inB_5δ/3(0)\B_4δ/3(0),ϕ_δ ≡0 inΩ\B_2δ(0)and|_∆ϕδ| ≤ ^C5

δ² inΩ. Thus,

we have _Z

Ω∇u∇ϕ_δdx+

Z

Ωφ_uuϕ_δdx−

Z

Ω|x|^−αu^−γϕ_δdx= 0. (2.14) According to the definition ofϕ_δ(x)andγ>0, we have

Z

Ω|x|^−αu^−γϕ_δdx=

Z

B2δ(0)\Bδ(0)

|x|^−αu^−γϕ_δdx

≥

sup

Ω u −γZ

B_2δ(0)\B_δ(0)

|x|^−αϕ_δdx

≥

sup

Ω u −γZ

B_5δ/3(0)\B_4δ/3(0)

|x|^−αdx

=

sup

Ω u −γ

4π 3−α

h 5 3

3−α

− 4

3 3−αi

δ^3−α. On the other hand, by Sobolev inequalities and Lemma2.1 (i), (iv), we have

Z

Ω∇u∇ϕ_δdx+

Z

Ωφ_uuϕ_δdx=−

Z

Ωu∆ϕ_δdx+

Z

Ωφ_uuϕ_δdx

≤

Z

Ωu|_∆ϕδ|dx+

Z

Ωφuuϕ_δdx

≤

sup

Ω u Z

Ω|_∆ϕδ|dx+

Z

Ωφ_uϕ_δdx

≤

sup

Ω u Z

B_2δ(0)\B_δ(0)

|_∆ϕ_δ|dx+

Z

B_2δ(0)\B_δ(0)φ_udx

≤

sup

Ω u 28πC₅δ

3 +C₆kuk²δ^5/2

.