On eigenvalue problems involving the critical Hardy potential and Sobolev type inequalities with logarithmic weights in two dimensions

2025-05-02 0 0 258.13KB 34 页 10玖币

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arXiv:2210.10282v1 [math.AP] 19 Oct 2022

On eigenvalue problems involving the critical Hardy

potential and Sobolev type inequalities with logarithmic

weights in two dimensions

Megumi Sanoa, Futoshi Takahashib,1,

aLaboratory of Mathematics, School of Engineering, Hiroshima University, Higashi-Hiroshima,

739-8527, Japan

bDepartment of Mathematics, Graduate School of Science, Osaka Metropolitan University,

Sumiyoshi-ku, Osaka, 558-8585, Japan

Abstract

We consider the two-dimensional eigenvalue problem for the Laplacian with the

Neumann boundary condition involving the critical Hardy potential. We prove the

existence of the second eigenfunction and study its asymptotic behavior around

the origin. A key tool is the Sobolev type inequality with a logarithmic weight,

which is shown in this paper as an application of the weighted nonlinear potential

theory.

Keywords: Second eigenvalue problem, Critical Hardy inequality, logarithmic

weight

2010 MSC: 35A23, 35J20, 35A08

1. Introduction

Let Ω⊂R2be a smooth bounded domain with 0 ∈Ω. For simplicity, we

assume supx∈Ω|x|=1 without loss of generality. Let a≥1. In this paper, we

consider the following linear eigenvalue problem

(N)









−∆u=λu

|x|2log a

|x|2in Ω,

∂u

∂ν =0 on ∂Ω,

Email addresses: smegumi@hiroshima-u.ac.jp (Megumi Sano), futoshi@omu.ac.jp

(Futoshi Takahashi)

1Corresponding author.

Preprint submitted to Elsevier October 20, 2022

here νdenotes the unit outer normal vector to ∂Ω. The problem stems from

the critical Hardy inequality on a bounded domain Ω⊂R2for functions in the

Sobolev space H1

0(Ω): for any u∈H1

0(Ω), it holds that

4ZΩ

|x|2log a

|x|2dx ≤ZΩ|∇u|2dx.(1)

Moreover, the constant 1

4on the left-hand side is best possible and is not attained.

We recall that the Sobolev space H1(Ω) is a set of functions u∈L2(Ω) such that

its distributional gradient ∇uis also in L2(Ω). H1(Ω) is a Hilbert space with an

inner product (u,v)H1(Ω)=RΩ(∇u· ∇v+uv)dx, and H1

0(Ω) is a closure of C∞

0(Ω)

with respect to the norm kukH1(Ω)=(u,u)1/2

H1(Ω). For the inequality (1), we refer the

readers to [11], [15], [17] [18] and the references there in.

In a higher dimensional case, we know the subcritical Hardy inequality for

functions in H1

0(Ω):

HNZΩ

|x|2dx ≤ZΩ|∇u|2dx

holds for any u∈H1

0(Ω), here Ωis a bounded domain in RN,N≥3, with 0 ∈Ω.

The constant HN=N−2

22is optimal and is never attained by a non zero func-

tion in H1

0(Ω). In [6], Chabrowski, Peral and Ruf consider the linear eigenvalue

problem











−∆u=λu

|x|2in Ω⊂RN,N≥3,

∂u

∂ν =0 on ∂Ω.

Clearly, the ﬁrst eigenvalue is λ=0 and constant functions are the ﬁrst eigenfunc-

tions. To seek the nontrivial solution in H1(Ω), the authors in [6] introduce the

minimization problem

λH=inf 









RΩ|∇u|2dx

RΩ|u|2

|x|2dx 

u∈H1(Ω)\ {0},ZΩ

|x|2dx =0









and prove that if λH<HN, then λHis attained and the minimizer corresponds the

second eigenfunction of the above eigenvalue problem. Also the authors obtain

several examples of domains such that the condition λH<HNholds true. Es-

pecially, they establish the existence of the second eigenfunction on balls in RN,

N≥7. Moreover, they study the asymptotic behavior of the second eigenfunc-

tions around the origin in the case 0 ∈Ω. To obtain the asymptotic estimate of

the second eigenfunction near the origin, they use De Giorgi-Nash-Moser type

procedure and the Caﬀarelli-Kohn-Nirenberg inequality [4].

The aim of this paper is to extend the results in [6] to the two-dimensional

problem (N). Let a≥1 and 0 ∈Ω⊂R2. We consider the minimization problem

λa=inf 









RΩ|∇u|2dx

RΩ|u|2

|x|2log a

|x|2dx



u∈H1(Ω)\ {0},ZΩ

|x|2log a

|x|2dx =0









.(2)

We show λa>0 for a>1, see (12). We seek for suﬃcient conditions to assure

the existence of minimizers, which yiedls the second eigenfunction of the problem

(N). Our suﬃcient condition claims that if a>1 and λa<1

4, then λais attained by

a non trivial function in H1(Ω). We also study the asymptotic behavior near the

origin of the second eigenfunctions. We remark that, unlike [6], we can treat the

case 0 ∈∂Ωtoo. Furthermore, since our Hardy potential involves the logarithmic

weights, it is diﬃcult to control the weights by Caﬀarelli-Kohn-Nirenberg type

inequality, which was useful for treating power type weights. Therefore, we need

to establish the Sobolev type inequality with logarithmic weights. Combining

this inequality with the De Giorgi-Nash-Moser procedure, we obtain the expected

asymptotic behavior of the second eigenfunctions. To obtain the Sobolev type in-

equality with logarithmic weights, we exploit weighted nonlinear potential theory

by [1]. We believe that this part is also interesting in itself.

In the following, Ls(Ω) will denote the standard Lebesgue spaces. Also for

a given nonnegative weight function ωand 1 ≤s<∞, the weighted Lebesgue

space Ls(Ω, ω(x)dx) is the set of functions usuch that RΩ|u|sw(x)dx <∞.Brwill

denote a ball in R2of radius rwith center the origin. “→” and “⇀” will denote the

strong and weak convergence in Banach spaces, respectively. (Possibly diﬀerent)

general positive constants are denoted by C.

2. The critical Hardy type inequalities for H1(Ω).

Let Ω⊂R2be a smooth bounded domain with 0 ∈Ωand supx∈Ω|x|=1. In §6,

we prove the following Hardy-Sobolev type inequality with logarithmic weights.

Theorem 1. Let a >1, p ≥2, B <1, A ≥1+p

2(1 −B). Then there exists a

positive constant Cp,A,Bsuch that the inequality

Cp,A,B



ZΩ

|u|p

|x|2(log a

|x|)Adx





≤ZΩ log a

|x|!B

|∇u|2dx (3)

holds for any u ∈C∞

c(Ω).

Note that if we set p=2, A=2, B=0, then (3) is nothing but the critical

Hardy inequality (1). Also we remark that if A=1+p

2(1 −B) and Ω = B1, then

the inequality (3) has the scale invariance under the following scaling C∞

c(B1)∋

u7→ uλfor any λ≤1, where

uλ(x)=









λ−1−B

2u|x|

aλ−1xfor x∈Baλ−1

λ,

0 for x∈B1\Baλ−1

λ.

When A<1+p

2(1−B), (3) does not have this scale invariance and by letting λ→0,

we can easily show that the inequality does not hold when A<1+p

2(1 −B).

The proof of Theorem 1 is postponed to §6.

To prove the existence of the second eigenfunctions for the problem (N), we

need the critical Hardy inequality for functions in H1(Ω), and also for functions in

H1(Ω) with average zero. Also to treat the case 0 ∈∂Ω, we need the next lemma.

Lemma 1. Let x =(x1,x2)∈R2, h ∈C1(R),0<r≤1and h(0) =0, and

h′(0) =0. Set Bh

r=Br∩ {x∈R2|x2>h(x1)}. Let a,p,A,B as in Theorem 1.

Then, for any ε > 0, there exists δ > 0such that if |h′(x1)| ≤ δfor any x1∈(−r,r),

then the inequality

p−1Cp,A,B



ZBh

|u|p

|x|2(log a

|x|)Adx





≤(1 +ε)ZBh

r log a

|x|!B

|∇u|2dx

holds for any u ∈H1(Br)with supp u⊂Br, where Cp,A,Bis given in Theorem 1.

Especially, we have

ZBh

|x|2log a

|x|2dx ≤(4 +ε)ZBh

r|∇u|2dx

for any u ∈H1(Br)with supp u⊂Br.

Proof. We follow the argument of the proof of [8] Lemma 2.1.

(I) Assume that h(x1)≡0. Since the value of u(x) are irrelevant for x2<0, we

may suppose that u(x) is even in x2. By the inequality (3) in Theorem 1, we have

Cp,A,B



ZB0

|u|p

|x|2log a

|x|Adx





=Cp,A,B





2ZBr

|u|p

|x|2log a

|x|Adx





≤2−2

pZBr log a

|x|!B

|∇u|2dx

=21−2

pZB0

r log a

|x|!B

|∇u|2dx.

(II) We consider the case where h(x1).0. Then we set y1=x1,y2=x2−h(x1)

and ˜u(y1,y2)=u(x1,x2). From (I), we have

p−1Cp,A,B



ZB0

|˜u|p

|y|2log a

|y|Ady





≤ZB0

r log a

|y|!B

|∇˜u|2dy.(4)

Direct calculation implies that

|∇˜u(y1,y2)|2=

∂u

∂x1

+∂u

∂x1

h′(x1)

+

∂u

∂x2

=|∇u(x1,x2)|2+2∂u

∂x1

∂u

∂x2

h′(x1)+

∂u

∂x2

|h′(x1)|2

≤(1 +δ)2|∇u(x1,x2)|2.(5)

Also, for any B∈[0,1)

log a

|y|!B

=



log a

p|x|2+|h(x1)|2−2x2h(x1)





≤



log a

p|x|2−2δ|x2||h(x1)|





≤ log a

|x|!B





1+





log 1

1−2δ

2 log a





B



≤1+CδB log a

|x|!B

(6)

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arXiv:2210.10282v1[math.AP]19Oct2022OneigenvalueproblemsinvolvingthecriticalHardypotentialandSobolevtypeinequalitieswithlogarithmicweightsintwodimensionsMegumiSanoa,FutoshiTakahashib,1,aLaboratoryofMathematics,SchoolofEngineering,HiroshimaUniversity,Higashi-Hiroshima,739-8527,JapanbDepartmentofMathe...

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