THE EXISTENCE OF POSITIVE SOLUTION FOR AN ELLIPTIC PROBLEM WITH CRITICAL GROWTH AND LOGARITHMIC PERTURBATION

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THE EXISTENCE OF POSITIVE SOLUTION FOR AN ELLIPTIC

PROBLEM WITH CRITICAL GROWTH AND LOGARITHMIC

PERTURBATION

Yinbin Deng 1, Qihan He 2, Yiqing Pan 3and Xuexiu Zhong 4

Abstract. We consider the existence and nonexistence of positive solution for the

following Br´ezis-Nirenberg problem with logarithmic perturbation:

(−∆u=|u|2∗−2u+λu +µu log u2x∈Ω,

u= 0 x∈∂Ω,

where Ω ⊂RNis a bounded smooth domain, λ, µ ∈R,N≥3 and 2∗:= 2N

N−2is the

critical Sobolev exponent for the embedding H1

0(Ω) →L2∗(Ω). The uncertainty

of the sign of slog s2in (0,+∞) has some interest in itself. We will show the

existence of positive ground state solution which is of mountain pass type provided

λ∈R, µ > 0 and N≥4. While the case of µ < 0 is thornier. However, for N= 3,4

λ∈(−∞, λ1(Ω)), we can also establish the existence of positive solution under

some further suitable assumptions. And a nonexistence result is also obtained for

µ < 0 and −(N−2)µ

2+(N−2)µ

2log(−(N−2)µ

2) + λ−λ1(Ω) ≥0 if N≥3. Comparing

with the results in Br´ezis, H. and Nirenberg, L. (Comm. Pure Appl. Math. 1983),

some new interesting phenomenon occurs when the parameter µon logarithmic

perturbation is not zero.

Keywords: Br´ezis-Nirenberg Problem, Critical exponents, Positive solution, Log-

arithmic perturbation

1. Introduction and main results

In this paper, we investigate the existence and nonexistence of positive solution for

the following Br´ezis-Nirenberg problem with a logarithmic term:

(−∆u=|u|2∗−2u+λu +µu log u2x∈Ω,

u= 0 x∈∂Ω,(1.1)

where Ω ⊂RNis a bounded smooth domain, λ, µ ∈R,N≥3, and 2∗=2N

N−2is the

critical Sobolev exponent for the embedding H1

0(Ω) →L2∗(Ω). Here H1

0(Ω) denotes

the closure of C∞

0(Ω) equipped with the norm kuk:= (RΩ|∇u|2dx)1

1School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China.

Email: ybdeng@ccnu.edu.cn.

2College of Mathematics and Information Science, Guangxi Center for Mathematical Research,

Guangxi University, Nanning, 530003, China, Email: heqihan277@gxu.edu.cn.

3College of Mathematics and Information Science, Guangxi Center for Mathematical Research,

Guangxi University, Nanning, 530003, China, Email: 13718049940@163.com.

4South China Research Center for Applied Mathematics and Interdisciplinary Studies, South

China Normal University, Guangzhou 510631, China, Email: zhongxuexiu1989@163.com.

arXiv:2210.01373v1 [math.AP] 4 Oct 2022

Our motivation to consider (1.1) is that it resembles some variational problems in

geometry and physics, which is lack of compactness. The most notorious example is

Yamabe’s problem: ﬁnding a function usatisfying

(−4N−1

N−2∆u=R0|u|2∗−2u−R(x)uon M,

u > 0 on M,

where R0is a constant, Mis an N-dimensional Riemannian manifold, ∆ denotes the

Laplacian and R(x) represents the scalar curvature. Some other examples we refer

to [2,8,10,13–15] and the references therein.

When λ=µ= 0, Eq.(1.1) is reduced to

(−∆u=|u|2∗−2u x ∈Ω,

u= 0 x∈∂Ω.(1.2)

Pohozaev [11] asserts that Eq.(1.2) has no nontrivial solutions when Ω is starshaped. But,

as Br´ezis and Nirenberg have shown in [3], a lower-order terms can reverse this cir-

cumstance. Indeed, they considered the following classical problem

(−∆u=|u|2∗−2u+λu x ∈Ω,

u= 0 x∈∂Ω,(1.3)

with λ∈R, N ≥3 and Ω ⊂RNis a bounded domain. They found out that the

existence of a solution depends heavily on the values of λand N. Precisely, they

showed that:

(i) when N≥4 and λ∈(0, λ1(Ω)), there exists a positive solution for Eq.(1.3);

(ii) when N= 3 and Ω is a ball, Eq.(1.3) has a positive solution if and only if

λ∈1

4λ1(Ω), λ1(Ω);

(iii) Eq. (1.3) has no solutions when λ < 0 and Ω is starshaped;

where λ1(Ω) denotes the ﬁrst eigenvalue of −∆ with zero Dirichlet boundary value.

Furthermore, Br´ezis and Nirenberg [3] also considered the following general case:

(−∆u=|u|2∗−2u+f(x, u)x∈Ω,

u= 0 x∈∂Ω,(1.4)

where f(x, u) satisﬁes some of the following assumptions :

(f1)f(x, u) = a(x)u+g(x, u), a(x)∈L∞(Ω);

(f2) lim

u→0+

g(x,u)

u= 0,uniformly in x∈Ω;

(f3) lim

u→+∞

g(x,u)

u2∗−1= 0,uniformly in x∈Ω;

(f4)∃α > 0 such that R(|∇v|2−a(x)v2)dx ≥αRv2dx for all v∈H1

0(Ω);

(f5)f(x, u)≥0 for a.e x∈ω0and for all u≥0, where ω0is some nonempty open

subset of Ω;

(f6)f(x, u)≥δ0>0 for a.e x∈ω0and for all u∈I, where ω0is given in (f5),

I⊂(0,+∞) is some nonempty open interval and δ0>0 is some constant;

(f7)f(x, u)≥δ1ufor a.e x∈ω1and for all u∈[0, A],or, f(x, u)≥δ1ufor a.e

x∈ω1and for all u∈[A, +∞],where ω1is some nonempty open subset of Ω

and δ1, A are two positive constants;

(f8) lim

u→+∞

f(x,u)

u3= +∞uniformly in x∈ω2, where ω2is some nonempty open

subset of Ω.

They showed that if the assumptions (f1)−(f4) hold and there exists some 0 ≤u0∈

0(Ω)\{0}such that sup

t≥0

I(tu0)<1

NSN

2, then Eq.(1.4) has a positive solution. More

precisely, they proved that:

(i) If N≥5, Eq.(1.4) has a positive solution provided (f1)−(f6);

(ii) If N= 4, Eq.(1.4) has a positive solution provided (f1)−(f5) and (f7);

(iii) If N= 3, Eq.(1.4) has a positive solution provided (f1)−(f5) and (f8).

Some similar results can be seen in [1,5,7]. Barrios et al. [1] proved the existence

of positive solution for a fractional critical problem with a lower-order term, and

Gao and Yang [5], Li and Ma [7] considered the existence of positive solution to a

Choquard equation with critical exponent and lower-order term in a bounded domain

Ω and in RN, respectively.

Remark 1.1. Compared with |u|2∗−2u,ulog u2is a lower-order term at inﬁnity.

However, we note that the situation we considered in present paper is not covered

above. Indeed, in the Eq. (1.1),f(x, u) = λu +µu log u2. So (f1)fails due to the fact

of lim

u→0+

ulog u2

u=−∞. That is λu =o(µu log u2)for uclose to 0. So it is natural to

believe that µu log u2has much more inﬂuence than λu on the existence of positive

solutions to Eq.(1.1). Hence, our main goal in present paper is to make clear this

guess.

To ﬁnd a positive solution to Eq.(1.1), we deﬁne a modiﬁed functional:

I(u) = 1

2ZΩ|∇u|2dx−1

2∗Z|u+|2∗

dx−λ

2Zu2

+dx−µ

2Zu2

+(log u2

+−1)dx, u ∈H1

0(Ω),

(1.5)

which can be rewritten by

I(u) = 1

2ZΩ|∇u|2dx−1

2∗Z|u+|2∗

dx−µ

2Zu2

+(log u2

++λ

µ−1)dx, u ∈H1

0(Ω),(1.6)

where u+= max{u, 0}, u−=−max{−u, 0}. It is easy to see that Iis well-deﬁned in

0(Ω) and any nonnegative critical point of Icorresponds to a solution of Eq.(1.1).

Before stating our results, we introduce some notations. Hereafter, we use Rto

denote RΩdx, unless speciﬁcally stated, and let Sand λ1(Ω) be the best Sobolev

constant of the embedding H1(RN)→L2∗(RN) and the ﬁrst eigenvalue of −∆ with

zero Dirichlet boundary value respectively, i.e,

S:= inf

u∈H1(RN)\{0}RRN|∇u|2dx

(RRN|u|2∗dx)2

2∗

and

λ1(Ω) := inf

u∈H1

0(Ω)\{0}RΩ|∇u|2dx

RΩ|u|2dx.

We also set

kvk2:= Z|∇v|2, v ∈H1

0(Ω),

N:= u∈H1

0(Ω) \ {0} | g(u) = 0,

and

cg:= inf

u∈N I(u), cM:= inf

γ∈Γmax

t∈[0,1] I(γ(t)),(1.7)

where

g(u) := Z|∇u|2−Z|u+|2∗−λZu2

+−µZu2

+log u2

and

Γ := {γ∈C([0,1], H1

0(Ω)) |γ(0) = 0, I(γ(1)) <0}.

Let

A0:= {(λ, µ)|λ∈R, µ > 0},

B0:= ((λ, µ)|λ∈[0, λ1(Ω)), µ < 0,1

Nλ1(Ω) −λ

λ1(Ω) N

2+µ

2|Ω|>0),

C0:= (λ, µ)|λ∈R, µ < 0,1

NSN

2+µ

2e−λ

µ|Ω|>0.

Here comes our main results.

Theorem 1.2. If (λ, µ)∈A0and N≥4, then problem (1.1)has a positive Mountain

pass solution, which is also a ground state solution.

Denote f(s) := |s|2∗−2s+λs +µs log s2which is of odd. It is easy to see that

N 6=∅and cM≥cgif problem (1.1) has a positive mountain pass solution. On the

other hand, when λ∈Rand µ > 0, f(s)

sis strictly increasing in (0,+∞) and strictly

decreasing in (−∞,0), which enable one to show that cM≤cg(See [18, Theorem

4.2]). Therefore, the ground state energy cgequals to the Mountain pass level energy

cM, which implies that the mountain pass solution must be a ground state solution.

So, in Theorem 1.2, we only need to show that problem (1.1) has a positive mountain

pass solution.

The case of µ < 0 is thorny. Indeed for (λ, µ)∈B0∪C0,I(u) still has the

mountain pass geometry (See Lemma 2.1). However, in such a case, it holds that

cg< cM. Since we can not check the (P S)cMcondition for I(u), we apply the

mountain pass theorem without (P S)cMcondition to gain a positive solution for

Eq.(1.1) when (λ, µ)∈B0∪C0. However, we don’t know whether this solution is of

mountain pass type or not.

Theorem 1.3. Problem (1.1)possesses a positive solution provided one of the fol-

lowing condition holds:

(i) N= 3,(λ, µ)∈B0∪C0;

(ii) N= 4,(λ, µ)∈B0∪C0with 32eλ

ρ2

max <1,where ρmax := sup{r > 0 : ∃x∈

Ωs.t. B(x, r)⊂Ω}.

For the nonexistence of positive solutions for problem (1.1), we have the following

partial result.

Theorem 1.4. Assume that N≥3.If µ < 0and −(N−2)µ

2+(N−2)µ

2log(−(N−2)µ

2) +

λ−λ1(Ω) ≥0, then problem (1.1)has no positive solutions.

The existence and nonexistence results given by Theorem 1.2 - Theorem 1.4 can be

described on the (λ, µ) plane by Figure 1. The pink regions stand for the existence

of positive solution, while the blue regions correspond the non-existence of positive

solution. Here τ1,η1,η2and η3are curves given by

τ1:−(N−2)µ

2+(N−2)µ

2log(−(N−2)µ

2) + λ−λ1(Ω) = 0,

η1:1

N(λ1(Ω) −λ

λ1(Ω) )N

2SN

2+µ

2|Ω|= 0,

η2:1

NSN

2+µ

2e−λ

µ|Ω|= 0,

η3: 32eλ

µ=ρ2

max, ρmax := sup{r∈(0,+∞) : ∃x∈Ωs.t. B(x, r)⊂Ω}.

(a) N= 3 (b) N= 4 (c) N≥5

Figure 1. existence and nonexistence

Remark 1.5. Comparing the results of [3] and FIGURE 1above, we ﬁnd that for

the case of N≥4: Eq.(1.1)possesses a positive solution only for λ∈(0, λ1(Ω))

if µ= 0. while it has a positive solution for all λ∈Rif µ > 0. So we see that

µu log u2(µ > 0) really plays a leading role (compared with λu) in the eﬀect on the

existence of positive solution to Eq.(1.1). A similar phenomenon occurs for N= 3:

Eq.(1.1)has a positive solution only for λ∈(λ∗, λ1(Ω)) ⊂(0, λ1(Ω)) if µ= 0, while

it has a positive solution for all λ∈(−∞, λ1(Ω)) if µ < 0.

Before closing the introduction, we give the outline of our paper. In Section 2,

we will check the mountain pass geometry structure for I(u), under diﬀerent speciﬁc

situations. We also give some other preliminaries. In Section 3, we are devoted to

estimate the mountain pass level cMfor diﬀerent parameters λ, µ and N. The proofs

of our main Theorems 1.2,1.3 and 1.4 are given in Section 4.

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THEEXISTENCEOFPOSITIVESOLUTIONFORANELLIPTICPROBLEMWITHCRITICALGROWTHANDLOGARITHMICPERTURBATIONYinbinDeng1,QihanHe2,YiqingPan3andXuexiuZhong4Abstract.WeconsidertheexistenceandnonexistenceofpositivesolutionforthefollowingBrezis-Nirenbergproblemwithlogarithmicperturbation:(u=juj22u+u+ulogu2x2;u=0x...

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THE EXISTENCE OF POSITIVE SOLUTION FOR AN ELLIPTIC PROBLEM WITH CRITICAL GROWTH AND LOGARITHMIC PERTURBATION

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