On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent

On existence and asymptotic behavior of solutions of elliptic equations with nearly critical exponent

and singular coefficients

Shiyu Li

, Gongming Wei

and Xueliang Duan

1College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

2School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

Received 5 August 2020, appeared 8 September 2021 Communicated by Dimitri Mugnai

Abstract. In this paper we study the existence and asymptotic behavior of solutions of

−_∆u=µ u

|x|²+|x|^αu^{p(α)−1−ε}, u>0 inBR(0)

with Dirichlet boundary condition. Here,−2<α<0, p(α) = ^2(N+α)_N−2 , 0<ε<p(α)−1 and p(α)−1−εis a nearly critical exponent. We combine variational arguments with the moving plane method to prove the existence of a positive radial solution. Moreover, the asymptotic behaviour of the solutions, asε→0, is studied by using ODE techniques.

Keywords: asymptotic behavior, critical Sobolev exponent, Hardy exponents, singular coefficient.

2020 Mathematics Subject Classification: 35A15, 35A24, 35B40.

1 Introduction

In this paper, we consider the following elliptic problem:











−_∆u=µ u

|x|² +|x|^αu^{p(α)−1−ε}, x∈ _Ω,

u>0, x∈ _Ω,

u=0, x∈ _∂Ω,

(1.1)

where Ω is a ball BR(0) in R^N(N ≥ 3), −2 < α < 0, p(α) = ²⁽_N^N₋⁺₂^α), 0 < ε < p(α)−1, 0≤µ<µ= ^N−₂²².

The equation in problem (1.1) is the Euler–Lagrange equation of the energy functional E:H₀¹(_Ω)→_Rdefined by

E(u) = ¹ 2

_Z

Ω|∇u|²−µ u²

|x|²

− ¹

p(α)−ε Z

Ω|x|^αu^p(α)−ε, ∀u∈ H¹₀(_Ω).

BCorresponding author. Email: gmweixy@163.com

It is known that critical points of functionalE(u)correspond to solutions of (1.1).

We denote

kuk, _Z

Ω|∇u|²−µ u²

|x|^{2 1}₂

, ∀u∈ H₀¹(_Ω).

Let us recall the Sobolev–Hardy inequality (see Lemma2.1in this paper), which using the fact 0≤µ<µ¯ implies thatkukis equivalent to the norm ofH₀¹(_Ω).

In the caseµ=0 andα=0, a prototype of problem (1.1) is







−_∆u=u^{2∗−1−ε}, x ∈_Ω, u >0, x ∈_Ω,

u =0, x ∈_∂Ω.

(1.2)

Whenε=0, it is well known that the solution of problem (1.2) is bounded in the neighbor- hood of the origin. Gidas, Ni and Nirenberg [17] proved that all the solutions with reasonable behavior at infinity, namely

u=O(|x|^2−N), (1.3)

are radially symmetric about some point. So, the form of the solutions may be assumed as u(x) = [N(N−2)λ²]^N−42

(λ²+|x−x₀|²)^N2−1 for someλ>0 andx₀ ∈_R^N.

Later in [7, Corollay 8.2] and [9, Theorem 2.1], the growth assumption (1.3) was removed, which implies that, for positiveC² solutions of problem (1.2), we have the same result.

When ε > 0, Atkinson and Peletier [2] used ODE arguments to obtain exact asymptotic estimates of the radially symmetric solution of problem (1.2) asε→0. The following are their principal results

lim

ε→0εu²(0,ε) = ⁴

N−2{N(N−2)}^{N−22 Γ}(N) Γ(^N₂)²

1 R^N−2 and forx6=0

lim

ε→0ε⁻

1

2u(x,ε) = ¹

2N^N4−2(N−2)^N4R^R−22 Γ(^N₂) [_Γ(N)]¹²

1

|x|^N−2 − ¹ R^N−2

. In the caseµ=0 andα>0, problem (1.1) is known as the Hénon equation







−_∆u=|x|^αu^p−1, x∈_Ω,

u>0, x∈_Ω,

u=0, x∈_∂Ω,

(1.4)

where p ∈ (2, 2^∗). Equation (1.4) was proposed by Hénon when he studied rotating stellar structures and readers can refer to Ni [24], Smets [26] and Cao–Peng [11]. Among these works, for equations with critical, supercritical and slightly subcritical growth, the existence and multiplicity of non-radial solutions, the symmetry and asymptotic behavior of ground states were studied by variational method (forp→ _N^2N_{−2 or}α→_∞).

In the case 0≤µ<µ= ^N₂⁻²² andα=0, problem (1.1) can be written as











−_∆u=µ u

|x|² +u^{2∗−1−ε}, x∈_Ω,

u>0, x∈_Ω,

u=0, x∈_∂Ω.

(1.5)

By using Moser iteration and a generalized comparison principle, Cao and Peng [10] proved u(x)∈ H₀¹(_Ω)satisfying

(u(x)|x|^ν ≥C₁, ∀x∈ _Ω⁰ ⊂⊂_Ω, u(x)|x|^ν ≤C2, ∀x∈ _Ω,

where C₁ and C₂ are two positive constants, ν = ^pµ−^pµ−_{µ. When Ω} = B_R, u(x) _is radially symmetric. Hence, they converted (1.5) to ODE and obtained the following:

lim

ε→0 lim

|x|→0εu²_ε|x|^2ν=4(2p

µ−µ)^N−1N^N−22(N−2)^{−N+22 Γ}(N) Γ(^N₂)²

1 R²

√

µ−µ

and for x6=0 lim

ε→0ε⁻

1

2uε(x) = ¹ 2√

2(2p

µ−µ)^N−23N^N−42(N−2)^−N4−6R

√

µ−µ Γ(^N₂) [_Γ(N)]¹²

× ¹

|x|

√

µ+√

µ−µ

− ¹

|x|

√

µ−√

µ−µ|R|²

√

µ−µ

! .

Motivated by the previous works and remark 4.2 in [10], we first prove the existence and radial symmetry of positive solution of (1.1). Then we focus on the asymptotic behavior of the solutions of problem (1.1) asε→0.

To state our main results, for convenience, we set p = p(α)−1−ε, ν = ^pµ−^pµ−µ, Ω= B_R = {x ∈ _R^N : |x| < R}, R > 0. We denote byu_ε(x)the solution of (1.1) andΓ(x)is the Gamma function.

Theorem 1.1. Suppose that−2< α< 0, 0≤ µ<µ, 0¯ < ε< p(α)−1. Then problem(1.1) has a radially symmetric solution in H₀¹(_Ω).

For the proof of this Theorem1.1, we first obtain a solution by the Mountain Pass Lemma.

Then, by moving plane method for elliptic equations with variable coefficients in [14], we can prove that the posotive solution is radially symmetric. For problem (1.2), the solution satisfies Gidas–Ni–Nirenberg Theorem in [17] and hence all solutions of (1.2) are radial symmetric.

However, here we cannot use Gidas–Ni–Nirenberg theorem directly since problem (1.1) in- cludes the hardy termµ_|x^u_|2 and singular coefficient |x|^α. Luckily, through a transformation of the original solution u_ε(x), the new equation satisfied by the new solution v(x)satisfies the conditions of a Corollary in [14] and we obtain the result. To be more precise, set

v(x) =|x|⁻

√

µ+√

µ−µu_ε(x),

using Moser iteration and a generalized comparison principle introduced by Merle and Peletier [22], we prove that v ∈ L^∞(_Ω) and is bounded from below and above. Thus we obtain that the precise singularity of u_ε(x)at the origin is like |x|⁻

√

µ+√

µ−µ

. Then applying

Lemma2.5 in Section 2 to this new equation, we deduce thatv(x)is radially symmetric and satisfies the following ODE:











v⁰⁰+ ^N−1−2ν

r v⁰+ ¹

r^{(p(α)−2−ε)ν−α}v^{p(α)−1−ε} =0, for 0<r <R,

v(r)>0, for 0<r <R,

v(R) =0.

(1.6)

Because (1.6) is still singular at the origin, we can use the well-known shooting argument introduced by Atkinson and Peletier [2] to convert (1.6) to the following ODE:







y⁰⁰(t) =−t^{−k(α,ε)yp(α)−1−ε}, y(t)>0, forT< t<_∞, y(T) =0,

(1.7)

wherek(_α,ε) = ^2m_m−⁺₁^α −^(p(α_m⁾⁻₋²₁^−ε)ν,m=₁+₂^pµ−µ= _N−2ν−1,T = (^m_R⁻¹)^m−1_,p(α)−1= 2k(α,ε)−3−_m^2νε₋₁.

Till now, study on behaviors and precise properties of the original solution u_ε(x)can be reduced to deal with (1.7). Based on this, we have

Theorem 1.2. Let u_ε(x)∈ H₀¹(_Ω)be a solution of problem(1.1). Then lim

ε→0 lim

|x|→0εu²_ε|x|^2ν =2(α+2)(2p

µ−µ)^{2Nα++α2−2}(N+α)^Nα+−22(N−2)^{−2α+α+N2+2 Γ}(^2(N+2)

α+2 ) Γ(^N_α+⁺₂^α)²

1 R²

√

µ−µ

. Theorem 1.3. Let uε(x)∈ H₀¹(_Ω)be a solution of problem(1.1). Then, for every x 6=0,

lim

ε→0ε⁻

1

2u_ε(x) = ¹

2(α+2)⁻¹²(2p

µ−µ)^{2N2α−+α4−6}(N+α)^2αN−+24(N−2)^{2α2α−N++46}R

√

µ−µ Γ(^N+α

α+2) hΓ

2(N+α) α+2

i¹₂

× ¹

|x|

√

µ+√

µ−µ

− ¹

|x|

√

µ−√

µ−µ|R|²

√

µ−µ

! . Notations:

• C,C_i,i=0, 1, 2, . . . denote positive constants, which may vary from line to line;

• k · k and k · k_L^q denote the usual norms of the spaces H₀¹(_Ω) and L^q(_Ω), respectively, Ω∈_R^N_;

• Some of the notations that will appear in the following paragraphs:

m=1+2p

µ−µ= N−2ν−1, T =^m−1 R

m−1

, k=k(α,ε) = ^2m+α

m−1 − (p(α)−2−ε)ν

m−1 , k₁(α,ε) = (k−1)^k−12, k₂(α,ε) = ^k−1

k−2, Tα,ε = ^γ

p(α)−2−ε k−2

k₁(α,ε) = ^γ

2−_(m^m₋⁻₁¹₎₍⁻_k^2ν₋₂₎ε

k₁(α,ε) ^, τ(α,ε) = ^t

T_α,ε k−2

, ϕ(α,ε) = ^m−1+2ν

(m−1)(k−2)^ε,

C_α,β,ε = ^β

(1+β^k−2)^{k−12, dα,β,ε}

= (1−C_α,β,ε)(1+2ν/(m−1)) C_α,β,ε^{2+(1+2ν/(m−1))ε} .

2 Preliminary results and existence of solution

In this section,we shall provide some preliminaries which will be used in the sequel and prove the existence of solution to problem (1.1).

Lemma 2.1(see [16, Lemma 3.1 and 3.2]). Suppose−2< α<0,2≤ q≤ p(α), and0≤ µ< µ.

Then

(i) (Hardy inequality)

Z

Ω

u²

|x|² ≤ ¹ µ

Z

Ω|∇u|², ∀u∈ H₀¹(_Ω); (ii) (Sobolev–Hardy inequality)

there exists a constant C>0such that Z

Ω|x|^α|u|^{q 1}_q

≤Ckuk, ∀u∈ H₀¹(_Ω)_; (iii) the map u 7→ |x|^αqu from H₀¹(_Ω)into L^q(_Ω)is compact for q< p(α).

Lemma 2.2(see [5, Theorem 2.2]). Let J be a C¹function on a Banach space X. Suppose there exists a neighborhood U of0in X and a constantρsuch that J(u)≥ρfor every u in the boundary of U,

J(₀)< ρ and J(_v)<ρ for some v∈/U.

Set

c= inf

g∈_Γmax

ω∈g J(ω)≥ρ, whereΓ={g ∈C([0, 1],X):g(0) =0,g(1) =v,J(g(1))< ρ}.

Conclusion: there is a sequence{un}in X such that J(un)→c and J⁰(un)→0in X^∗.

Lemma 2.3(The Caffarelli–Kohn–Nirenberg inequalities, see [8] and [12]). For all u∈ C₀^∞(_R^N), Z

R^N|x|^−bq|u^q| ^p_q

≤C_a,b Z

R^N|x|^−ap|Du|^pdx, where (i) for n > p,

−_∞< a< ⁿ⁻_p^p, 0≤ b−a≤1, and q= ^np

n−p+p(b−a)

and (ii) for n≤ p,

−_∞< a< ⁿ⁻_p^p_, ^p−_pⁿ ≤b−a ≤_1, and q= ^np

n−p+p(b−a).

Lemma 2.4(see [25, page 4]). Suppose V is a reflexive Banach space with normk · k, and let M⊂V be a weakly closed subset of V. Suppose E : M → _R∪ {+_∞}is coercive and (sequentially) weakly lower semi-continuous on M with respect to V, that is, suppose the following conditions are fulfilled:

(1) (coercive) E(u)→_∞askuk →_{∞, u}∈ M.

(2) (W.S.L.S.C) For any u ∈ M, any sequence {u_m} in M such that u_m + u weakly in V there holds:

E(u)≤lim inf

m→_∞ E(u_m).

Then E is bounded from below on M and attains its infimum in M.

Lemma 2.5(see [14, Corollary 1.6]). Let u be a bounded C²(B_R\{0})∩C¹(B_R\{0})solution of











∂_i(|x|^b∂_iu) +K|x|^au^q=0, x ∈B_R\{0}, u>0, x ∈BR\{0},

u=0, x ∈∂B_R\{0},

where K is a positive constant.Then u is radially symmetric in B_Rprovided q≥1, b(¹₂b+N−2)≤0 and¹₂b≥ ^a_q.

Proof. Whenb<0, we have|x|^bis singular at origin.

It’s clear that S(_x)

|x|^b − ^S(_x^λ)

|x^λ|^b = ¹ 2b

1

2b+N−2

(|x|^b−2− |x^λ|^b−2)≥0 and

K |x|^b

|x^λ|^{b 1}₂

|x^λ|^au^q−K|x|^a



 |x|^b

|x^λ|^{b 1}₂

u





q

= K|x|^12b|x^λ|^a−12b



1− |x^λ|

|x|

¹₂bq−a

u^q≥_0, whereS(x) = ¹₂(_∆|x|^b− ¹

2|x|^b|∇|x|^b|²). From [14], we have

h_λ(x) =

|x^λ|^b

|x|^{b 1}₂

u(x^λ)−u(x), wherex^λ = (2λ−x₁,x₂, . . . ,x_N).

By Lemma 4.2 in [14], we can obtainuhas a positive lower bound near the origin. Hence, we can get the estimate ofh_λ(x)near the origin. Furthermore, ifx^λ =0, we haveh_λ(x) =_∞.

Now, we consider the case of u ∈ C²(B₁\{0})∩C¹(B₁\{0}) in Proposition 1.3 of [14].

Analogically, we can also obtain u(x₁,x₂, . . . ,x_N) ≤ u(−x₁,x₂, . . . ,x_N) _for x₁ ∈ (−1, 0) _and x₁ ∈(0, 1). Hence, uis symmetric inx₁. By Lemma 1.1 in [14], the above analysis and scaling transformation,u(x)is radially symmetric inB_R.

Next, we shall prove the existence of solution to the problem (1.1). To start with, we prove the existence of nonnegative solution to the following Dirichlet problem:

(−_∆u=µ_|x^u_|2 +|x|^α|u|^{p(α)−2−ε}u, x ∈_Ω,

u=0, x ∈∂Ω, (2.1)

whereΩ is a ball inR^N(N ≥ 3) centered at the origin,−2 < α< 0, p(α) = ²⁽_N^N₋⁺₂^α), 0< ε <

p(α)−1,0≤ µ<µ= ^N−₂²².

The energy functional corresponding to problem (2.1) is J(u) = ¹

2kuk²− ¹ p(α)−ε

Z

Ω|x|^α|u|^p(α)−ε, u∈ H₀¹(_Ω). Lemma 2.6. The function J satisfies(PS)_ccondition for every c∈_R.

Proof. Takec ∈ _R and assume that {u_n} is a Palais–Smale sequence at level c, namely such that

J(un)→c and J⁰(un)→0 (in H⁻¹(_Ω)). This implies that there is a constant M>0 such that

|J(u_n)| ≤ M. (2.2)

FromJ⁰(u_n)→0, we obtain

o(1)kunk=hJ⁰(un),uni=kunk²−

Z

Ω|x|^α|un|^p(α)−ε. (2.3) Calculating (2.2) − _p(¹

α)−ε(2.3), we have M+o(1)kunk ≥ ¹

2kunk²− ¹ p(α)−ε

Z

Ω|x|^α|un|^p(α)−ε

− ¹

p(α)−εku_nk²+ ¹ p(α)−ε

Z

Ω|x|^α|u_n|^p(α)−ε

= 1

2− ¹

p(α)−ε

ku_nk²_,

which implies the boundedness of {un}. By usual arguments, we can assume that up to a subsequence, there existsu∈ H₀¹(_Ω)such that

• u_n*uin H¹₀(_Ω);

• |x|^p(αα)−εun → |x|^p(αα)−εuin L^p(α)−ε(_Ω);

• ·u_n→ufor almost everyx ∈_Ω.

We now show that the convergence ofu_nto uis strong.

First of all, from the above convergence properties, we obtain

u_n|x|^p(αα)−ε −u|x|^p(αα)−ε

L^p(α)−ε(_Ω)→0, n→_∞.

As J⁰(un) → 0 and un * u, we also have hJ⁰(un),un− ui → 0 and obviously hJ⁰(u),u_n−ui →0. Then, asn→∞, on the one hand,

hJ⁰(un)−J⁰(u),un−ui ≤ |hJ⁰(un),un−ui|+|hJ⁰(u),un−ui|=o(1). On the other hand,

hJ⁰(_un)−J⁰(_u)_,un−ui

=

Z

Ω|∇u_n− ∇u|²−

Z

Ωµ|u_n−u|²

|x|² −

Z

Ω|x|^α(|u_n|^{p(α)−2−ε}u_n− |u|^{p(α)−2−ε}u)(u_n−u)

=ku_n−uk²−

Z

Ω|x|^α(|u_n|^{p(α)−2−ε}u_n− |u|^{p(α)−2−ε}u)(u_n−u). We claimR

Ω|x|^α(|u_n|^{p(α)−2−ε}u_n− |u|^{p(α)−2−ε}u)(u_n−u)→_0.

Indeed, by Hölder’s inequality, Z

Ω|x|^α|un|^{p(α)−2−ε}un(un−u)

≤

Z

Ω|x|^α|u_n|^{p(α)−1−ε}|u_n−u|

=

Z

Ω|x|^α·p

(_α)−1−ε

p(_α)−ε |u_n|^{p(α)−1−ε}|x|^{α·p(α1)−ε}|u_n−u|

≤



 Z

Ω

|x|^α·p

(_α)−1−ε

p(_α)−ε |u_n|^{p(α)−1−ε} _p(^p(α)−ε

α)−1−ε





p(α)−1−ε p(_α)−ε _Z

Ω

|x|^{α·p(α1)−ε}|u_n−u|^p(α)−ε _p(α¹_)−ε

= _Z

Ω|x|^α|u_n|^p(α)−ε

^p(_p^α₍^)−1−ε

α)−ε

|x|^p(αα)−ε|u_n−u|

L^p(α)−ε(_Ω)

≤Cku_nk^{p(α)−1−ε}u_n|x|^p(αα)−ε −u|x|^p(αα)−ε

L^p(α)−ε(_Ω)

=_o(₁)_.

(2.4)

By (2.4), similar calculation also gives Z

Ω|x|^α|u|^{p(α)−2−ε}u(u_n−u) =o(1). (2.5) From the above analysis, we obtain

o(1) =hJ⁰(un)−J⁰(u),un−ui=kun−uk²+o(1),

which impliesu_n→uinH₀¹(_Ω)and proves thatJsatisfies(PS)_ccondition for everyc∈_R.

Lemma 2.7. The function J admits a(PS)_csequence in the cone of nonnegative function at the level c= inf

g∈_Γmax

t∈[0,1]J(g(t)), whereΓ={g∈C([0, 1],H₀¹(_Ω)): g(0) =0,J(g(1))<0}.

Proof. We next prove thatJsatisfies all the hypotheses of the mountain pass lemma. Obviously, J(0) =0.

From the Sobolev–Hardy inequality, we obtain J(u) = ¹

2kuk²− ¹ p(α)−ε

Z

Ω|x|^α|u|^{p(α)−1−ε}u

≥ ¹

2kuk²−C₁kuk^p(α)−ε.

For anyα, we can chooseεsmall enough such thatp(α)−ε>2. From the above analysis,there exist ρ,e > 0 such that J(u) ≥ ρ,∀u ∈ {u ∈ H¹₀(_Ω) : kuk = e}. Furthermore, for any u∈ H₀¹(_Ω),

J(tu) = ^t

2

2kuk²− ^t

p(α)−ε

p(α)−ε Z

Ω|x|^α|u|^{p(α)−1−ε}u.

We obtain J(tu) → −_∞ as t → ∞. Hence, we can choose t₀ > 0 such that J(t₀u) < 0.

Therefore, by Lemma 2.2, we infer that J admits a (PS)_c sequence at level c, such sequence may be chosen in the set of nonnegative functions becauseJ(|u|)≤ J(u)_{for all}u∈ H₀¹(_Ω)_.

By Lemma 2.6, 2.7 and mountain pass lemma, we get a nonnegative solution u ∈ H₀¹(_Ω) for (1.1), this solution is positive by the maximum principle.

3 Estimate of the singularity

First, we fix p = p(α)−1−ε > 0 in problem (1.1) and study the singularity and radial symmetry of the solution u_ε(x) ∈ H₀¹(_Ω). By standard elliptic regularity theory, u_ε(x) ∈ C²(_Ω\{0})^TC¹(_Ω\{0}). Hence the singular point ofu_ε(x)should be the origin.

Suppose thatuε(x)∈ H₀¹(_Ω)satisfies problem (1.1).

Letv(x) =|x|^νuε(x), then

−_∆u= (−ν²−2ν+Nν)|x|^−ν−2v(x) +2ν|x|^−ν−2x∇v(x)− |x|^−ν_∆v(x), µ u

|x|² +|x|^αu^{p(α)−1−ε} =µ|x|^−ν−2v(x) +|x|^{α−(p(α)−1−ε)ν}v(x)^{p(α)−1−ε}. From equation in (1.1),

(−ν²−2ν+Nν)|x|^−ν−2v(x) +2ν|x|^−ν−2x∇v(x)− |x|^−ν_∆v(x)

=µ|x|^−ν−2v(x) +|x|^{α−(p(α)−1−ε)ν}v(x)^{p(α)−1−ε}. Multiply both sides of the above equation by|x|^−ν, then we get

[−ν²+ (N−2)ν]|x|^−2ν−2v(x)−div(|x|^−2ν∇v(x)) =µ|x|^−2ν−2v(x) +|x|^{α−(p(α)−ε)ν}v(x)^{p(α)−1−ε}. For ν=^pµ−^pµ−µ, we obtain











−div(|x|^−2ν∇v) =|x|^{−(p(α)−ε)ν+α}v^{p(α)−1−ε}, x∈_Ω,

v >0, x∈_Ω,

v =0, x∈_∂Ω.

(3.1)

By the regularity theory of elliptic equations,v_ε(x)∈C²(_Ω\{0})^TC¹(_Ω\{0}). Moreover, we have

Lemma 3.1.

(i) v(x)∈ H₀¹(_Ω,|x|^−2ν). (ii) v(x)is bounded in Ω.

Proof. (i) For anyu(x)∈ H₀¹(_Ω)satisfying problem (1.1), by Hardy inequality, we have Z

Ω|x|^−2ν|∇v|²=

Z

Ω|x|^−2ν||x|^ν∇u+ν|x|^ν−2ux|²

≤2 _Z

Ω|∇u|²+ν² Z

Ω

u²

|x|²

≤ C.

Hence, we claimv(x) =|x|^νu(x)∈ H¹₀(_Ω,|x|^−2ν).

(ii) From Caffarelli–Kohn–Nirenberg inequality mentioned in Lemma2.3, we have

_Z

Ω|x|^m1|∇u|^{2 1}₂

≥C_{m1,n1 Z}

Ω|x|ⁿ¹|u|^p(m1,n1) _p(m¹

1,n1)

, ∀u∈ H₀¹(_Ω,|x|^m1), (3.2)

where

m₁= −2ν, n₁= −(p(α)−ε)ν+α, p(m₁,n₁) = p(α) + _p^εν µ−µ. Note that _Z

Ω|x|^m1∇v· ∇ϕ=

Z

Ω|x|ⁿ¹v^pϕ, ∀ϕ∈ H₀¹(_Ω,|x|^m1).

For s,l > 1, define v_l(x) = min{v(x),l}. Taking ϕ = v·v²_l^(s−1) ∈ H₀¹(_Ω,|x|^m1)in the above equation, we have

Z

Ω|x|^m1|∇v|²v²_l^(s−1)+2(s−1)

Z

Ω|x|^m1|∇v_l|²v²_l^(s−1) =

Z

Ω|x|ⁿ¹v^p+1v²_l^(s−1). Hence,

_Z

Ω|x|ⁿ¹(v·v^s_l⁻¹)^p(m1,n1) _p(m²

1,n1)

≤C_m⁻₁²_,n1 Z

Ω|x|^m1|∇(v·v^s_l⁻¹)|²

≤2C_m⁻₁²_,n1

(s−1)²

Z

Ω|x|^m1|∇v_l|²v²_l^(s−1)+

Z

Ω|x|^m1|∇v|²v²_l^(s−1)

≤2C_m⁻₁²_,n1s Z

Ω|x|ⁿ¹v^p+2s−1.

(3.3)

From (3.3) and Levi’s theorem, we see thatv∈L^p+2s−1(_Ω,|x|ⁿ¹)impliesv∈L^sp(m1,n1)(_Ω,|x|ⁿ¹). For j=0, 1, 2, . . . , by induction we define

(p−1+2s₀= p(m₁,n₁),

p−1+2s_j+1= p(m₁,n₁)s_j, (3.4)







M0= (C·C_m⁻²_1,n1)^p

(m1,n1)

2 ,

M_j+1= (2C_m⁻₁²_,n1s_jM_j)^p

(m1,n1)

2 ,

(3.5) whereCis a fixed number such thatR

Ω|x|^m1|∇v|² ≤C.

From (3.4), we see that

s_j = (2⁻¹p(m₁,n₁))^j+1(p(m₁,n₁)−p−1) +p−1 p(m₁,n₁)−₂ ^. From (3.5), similar to the computation in [21], we can see that

∃d>0 anddis independent of j, such that M_j ≤e^dsj−1.

Since 2< p+1< p(m₁,n₁), it follows thats_j >1 for all j≥0, s_j →+_∞as j→+_∞.

By (3.3), (3.4) and (3.5), Z

Ω|x|ⁿ¹v^p+2s1−1≤(2C_m⁻₁²_,n1s0)^p

(m1,n1) 2

Z

Ω|x|ⁿ¹v^{p+2s0−1 p}

(m1,n1) 2

≤(2C_m⁻₁²_,n1s₀)^p(m1,2n1)

C^p(m1,2n1)C⁻_m1^p_,n^(m₁^1,n1)

^p(m1,₂ⁿ¹⁾

≤(2C_m⁻₁²_,n1s₀M₀)^p(m1,2n1)

≤ M₁.

Similarly,

Z

Ω|x|ⁿ¹v^p+2sj−1 ≤ M_j.

Hence, byp+2s_j+1−1= p(m₁,n₁)s_j, denotingC(_Ω,n₁) =max_x∈_Ω|x|⁻ⁿ¹, we obtain

|v|

L^p(m1,n1)sj(_Ω)≤ Z

Ω|v|^p(m1,n1)sj|x|ⁿ¹· |x|⁻ⁿ¹ _p(m¹

1,n1)_sj

≤ C(_Ω,n₁)

1 p(m1,n1)_sj

|v|

1 p(m1,n1)_sj

L^p(m1,n1)sj(_Ω,|x|ⁿ¹)

≤ C(_Ω,n₁)

1 p(m1,n1)_sj

M

1 p(m1,n1)_sj j+1

≤ C(_Ω,n1)

1 p(m1,n1)_sj

e

d p(m1,n1).

Taking limit on each side of the above inequality and usings_j →+_{∞, as}j→+_{∞, we have}

|v|_L∞(_Ω)≤e^p(m1,dn1), which implies the conclusion.

From Lemma 3.1, we can see that v(x) = |x|^νu(x)is bounded form above in Ω. For the lower bound ofv(x) =|x|^νu(x), we have

Lemma 3.2. Suppose that u(x) ∈ H₀¹(_Ω) satisfies problem (1.1) and 0 ≤ µ < µ, then for any Bρ⊂⊂_Ωthere exists a C(ρ)>0, such that

u(x)≥C(ρ)|x|^−ν, ∀x∈ B_ρ ⊂⊂_Ω. Proof. Let f(x) =min{|x|^αu^{p(α)−1−ε}(x),l}withl>0, then f ∈ L^∞(_Ω).

Letu₁ ≥0 andu₁∈ H₀¹(_Ω)be the solution of the following linear problem







−_∆u₁ =µ u₁

|x|² + f, x ∈_Ω,

u₁=0, x ∈_∂Ω.

(3.6)

SetU=u−u₁, thenU∈ H₀¹(_Ω)andUsatisfies the following problem







−_∆U= µ U

|x|² +g, x∈_Ω,

U=0, x∈_∂Ω,

(3.7)

where g≥0 and 0≤ µ<µ= (^N−₂²)².

From Lemma 2.4, there exist solutions for problem (3.6) and (3.7). From the Hardy in- equality and the comparison principle proved in [15], we obtain that u is a super-solution of problem (3.6) and 0 ≤ u₁ ≤ u. Actually we can prove this as follows. Multiplying U⁻:=max{0,−U(x)}on both side of equation in (3.7) and integrating by parts, we have

−

Z

Ω|∇U⁻|² =−

Z

Ωµ(U⁻)²

|x|² +

Z

ΩgU⁻. It follows thatU⁻ =_{0 inΩand hence} U≥_0.