Al Detalla Theory

INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES

IMPROVED RELLICH INEQUALITIES

MISSING TERMS IN CLASSICAL

INEQUALITIES

ALNAR L. DETALLA

Department of MathematicsCollege of Arts and SciencesCentral Mindanao University

8710 Musuan, Bukidnon

May 21, 2010

A. L. Detalla Missing Terms in Classical Inequalities



INRODUCTION

In 1920, G. H. Hardy proved the following:




INRODUCTION

In 1920, G. H. Hardy proved the following:




INRODUCTION

In 1920, G. H. Hardy proved the following:Let 1 p− 1




INRODUCTION


F (t) :=

∫ t


∫∞t


f : is a non-negative measurable function on (0,∞). Then




INRODUCTION


F (t) :=

∫ t


∫∞t



∫ ∞

0

F p(t)tǫ−pdt ≤ C

∫ ∞

0

f p(t)tǫdt (1)




INRODUCTION


F (t) :=

∫ t


∫∞t



∫ ∞

0


∫ ∞

0

f p(t)tǫdt (1)

where C > 0 independent of f .




∫ ∞

0


∫ ∞

0

f p(t)tǫdt (1)




∫ ∞

0


∫ ∞

0

f p(t)tǫdt (1)




∫ ∞

0


∫ ∞

0

f p(t)tǫdt (1)

C =(

p

|ǫ−p+1|

)p

by: E. Landau (1930)




∫ ∞

0


∫ ∞

0

f p(t)tǫdt (1)

C =(

p

|ǫ−p+1|

)p


NOTE: (1) can be written as




∫ ∞

0


∫ ∞

0

f p(t)tǫdt (1)

C =(

p

|ǫ−p+1|

)p



∫ ∞

0

|u(t)|ptǫ−pdt ≤(

p

|ǫ− p+ 1|

)p ∫ ∞

0

|u′(t)|ptǫdt (2)




∫ ∞

0


∫ ∞

0

f p(t)tǫdt (1)

C =(

p

|ǫ−p+1|

)p



∫ ∞

0


p

|ǫ− p+ 1|

)p ∫ ∞

0

|u′(t)|ptǫdt (2)

where u′(t) := dudt, u ∈ Cc((0,∞)), p > 1, ǫ 6= p− 1.




∫ ∞

0


∫ ∞

0

f p(t)tǫdt (1)

C =(

p

|ǫ−p+1|

)p



∫ ∞

0


p

|ǫ− p+ 1|

)p ∫ ∞

0

|u′(t)|ptǫdt (2)

where u′(t) := dudt, u ∈ Cc((0,∞)), p > 1, ǫ 6= p− 1.

We call (2) the classical Hardy inequality




for ǫ = 0, (2) becomes









∫ ∞

0

|u′(t)|pdt ≥(

p− 1

p

)p ∫ ∞

0

|u(t)|p|t|p dt (3)





∫ ∞

0

|u′(t)|pdt ≥(

p− 1

p

)p ∫ ∞

0

|u(t)|p|t|p dt (3)

from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.





∫ ∞

0

|u′(t)|pdt ≥(

p− 1

p

)p ∫ ∞

0

|u(t)|p|t|p dt (3)

from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.For n > 2,





∫ ∞

0

|u′(t)|pdt ≥(

p− 1

p

)p ∫ ∞

0

|u(t)|p|t|p dt (3)


∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx, ∀u ∈ W 1,20 (Ω). (4)





∫ ∞

0

|u′(t)|pdt ≥(

p− 1

p

)p ∫ ∞

0

|u(t)|p|t|p dt (3)


∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx, ∀u ∈ W 1,20 (Ω). (4)

Lp-version of (4) where 1 ≤ p < n is





∫ ∞

0

|u′(t)|pdt ≥(

p− 1

p

)p ∫ ∞

0

|u(t)|p|t|p dt (3)


∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx, ∀u ∈ W 1,20 (Ω). (4)

Lp-version of (4) where 1 ≤ p < n is

∫

Ω

|∇u(x)|pdx ≥(

n− p

p

)p ∫

Ω

|u(x)|p|x|p dx, ∀u ∈ W 1,p

0 (Ω).

(5)A. L. Detalla Missing Terms in Classical Inequalities



A higher order generalization of (4) was proven by Rellich




A higher order generalization of (4) was proven by Rellich




A higher order generalization of (4) was proven by RellichFor n > 4,





∫

Ω

|∆u(x)|2dx ≥ n2(n− 4)2

16

∫

Ω

|u(x)|2|x|4 dx,

∀u ∈ W 2,20 (Ω).

(6)





∫

Ω

|∆u(x)|2dx ≥ n2(n− 4)2

16

∫

Ω

|u(x)|2|x|4 dx,

∀u ∈ W 2,20 (Ω).

(6)

Lp-version of (6) is





∫

Ω

|∆u(x)|2dx ≥ n2(n− 4)2

16

∫

Ω

|u(x)|2|x|4 dx,

∀u ∈ W 2,20 (Ω).

(6)

Lp-version of (6) is

∫

Ω

|∆u(x)|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx,

∀u ∈ W 2,p0 (Ω).

(7)




W l,p(Ω):function space on Ω whose generalized derivatives∂γu of order ≤ l satisfies









||u||W l,p(Ω) =∑

|γ|≤l

(∫

Ω

|∂γu(x)|pdx)

1p

< ∞





||u||W l,p(Ω) =∑

|γ|≤l

(∫

Ω

|∂γu(x)|pdx)

1p

< ∞

W l,p0 (Ω) denotes the completion of C∞

0 (Ω) in W l,p(Ω).




Hardy-Sobolev Inequalities









∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)





∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)

∫

Ω

|∇u(x)|pdx ≥(

n− p

p

)p ∫

Ω

|u(x)|p|x|p dx, ∀u ∈ W 1,p

0 (Ω) (5)





∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)

∫

Ω

|∇u(x)|pdx ≥(

n− p

p

)p ∫

Ω

|u(x)|p|x|p dx, ∀u ∈ W 1,p

0 (Ω) (5)

Hardy-Sobolev-Rellich Inequalities





∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)

∫

Ω

|∇u(x)|pdx ≥(

n− p

p

)p ∫

Ω

|u(x)|p|x|p dx, ∀u ∈ W 1,p

0 (Ω) (5)






∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)

∫

Ω

|∇u(x)|pdx ≥(

n− p

p

)p ∫

Ω

|u(x)|p|x|p dx, ∀u ∈ W 1,p

0 (Ω) (5)


∫

Ω

|∆u(x)|2dx ≥ n2(n− 4)2

16

∫

Ω

|u(x)|2|x|4 dx, ∀u ∈ W 2,2

0 (Ω) (6)





∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)

∫

Ω

|∇u(x)|pdx ≥(

n− p

p

)p ∫

Ω

|u(x)|p|x|p dx, ∀u ∈ W 1,p

0 (Ω) (5)


∫

Ω

|∆u(x)|2dx ≥ n2(n− 4)2

16

∫

Ω

|u(x)|2|x|4 dx, ∀u ∈ W 2,2

0 (Ω) (6)

∫

Ω

|∆u(x)|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx (7)




IMPROVEMENT OF INEQUALITY (4)

NOTATIONS





NOTATIONS





NOTATIONS

For t > 0 and ρ ≥ 2,





NOTATIONS


A1(t) = log Rt, A2(t) = logA1(t), . . . Aρ(t) = logAρ−1(t)





NOTATIONS



and





NOTATIONS



and

e1 = e, e2 = ee1 , . . . eρ = eeρ−1




THEOREM 1 ( A. Detalla, T. Horiuchi and H. Ando(2005))









Assume γ ≥ 2 and n ≥ 2. If R ≥ supΩ |x|ek then thereexists sharp remainder terms such that

∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx+1

4

∫

Ω

u(x)2

|x|2[

A1(|x|)−γ +

(

A1(|x|)A2(|x|))−γ

+ · · ·+(

A1(|x|)A2(|x|) . . . Ak(|x|))−γ]

dx.

(8)

for any u ∈ W 1,20 (Ω).




∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx+1

4

∫

Ω

u(x)2

|x|2[

A1(|x|)−γ +

(

A1(|x|)A2(|x|))−γ

+ · · ·+(

A1(|x|)A2(|x|) . . . Ak(|x|))−γ]

dx (8)




∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx+1

4

∫

Ω

u(x)2

|x|2[

A1(|x|)−γ +

(

A1(|x|)A2(|x|))−γ

+ · · ·+(

A1(|x|)A2(|x|) . . . Ak(|x|))−γ]

dx (8)




∫

Ω

|∇u(x)|2dx ≥(

n− 2

2

)2 ∫

Ω

u(x)2

|x|2 dx+1

4

∫

Ω

u(x)2

|x|2[

A1(|x|)−γ +

(

A1(|x|)A2(|x|))−γ

+ · · ·+(

A1(|x|)A2(|x|) . . . Ak(|x|))−γ]

dx (8)

REMARK

In inequality (8), 14is best constant for all k-missing terms

and γ ≥ 2 is sharp









THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))

Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.







Noncritical Case(1 0 depending on

n, p, and R such that for any u ∈ W 1,p0 (Ω) and γ ≥ 2








Let R ≥ supΩ

(

|x|e 2p

)

. Then there exist K > 0 depending on

n, p, and R such that for any u ∈ W 1,p0 (Ω) and γ ≥ 2

∫

Ω

|∇u(x)|pdx ≥(

n− p

p

)p ∫

Ω

|u(x)|p|x|p dx+

K

∫

Ω

|u(x)|p|x|p

(

logR

|x|

)−γ

dx(9)

















Critical Case (p = n)















Let R ≥ supΩ

(

|x|e 2n

)

. Then for any u ∈ W 1,n0 (Ω)








Let R ≥ supΩ

(

|x|e 2n

)

. Then for any u ∈ W 1,n0 (Ω)

∫

Ω

|∇u(x)|ndx ≥(

n− 1

n

)n ∫

Ω

|u(x)|n|x|n

(

logR

|x|

)−n

dx

(10)




Corollary

Let 1 < p < n, and let

Fp =

f : Ω → R+|f ∈ L∞

loc(Ω \ 0) with

lim sup|x|→0

|x|pf(x)(

log1

|x|

)2

< ∞




Corollary


Fp =

f : Ω → R+|f ∈ L∞

loc(Ω \ 0) with

lim sup|x|→0

|x|pf(x)(

log1

|x|

)2

< ∞




Corollary


Fp =

f : Ω → R+|f ∈ L∞

loc(Ω \ 0) with

lim sup|x|→0

|x|pf(x)(

log1

|x|

)2

< ∞

If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 1,p0 (Ω)

∫

Ω

|∇u|pdx ≥(

n− p

p

)p ∫

Ω

|u(x)|p|x|2p dx+λ(f)

∫

Ω

|u(x)|pf(x)dx




Corollary


Fp =

f : Ω → R+|f ∈ L∞

loc(Ω \ 0) with

lim sup|x|→0

|x|pf(x)(

log1

|x|

)2

< ∞


∫

Ω

|∇u|pdx ≥(

n− p

p

)p ∫

Ω


∫

Ω

|u(x)|pf(x)dx

If f /∈ Fp and if |x|pf(x)(

log 1|x|

)2

tends to ∞ as |x| → 0,

then no inequality of the above type can hold.







APPLICATION

The results will be used to analyze the behaviour of thefirst eigenvalue of the weighted eigenvalue problem for theoperator




APPLICATION





APPLICATION


L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)




APPLICATION


L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)

as µ →(

n−p

p

)p

.




L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)




L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)

L∇µis related to the variational problem




L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)





L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)


infu∈K

∫

Ω

(

|∇u|p − µ|u|p|x|p

)

dx




L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)


infu∈K

∫

Ω

(


)

dx

where K is given by




L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)


infu∈K

∫

Ω

(


)

dx

where K is given by

K =

u ∈ W 1,p0 (Ω) :

∫

Ω

|u(x)|pf(x)dx = 1

,

f : is a weight function.




L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)




L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)

Consider the weighted eigenvalue problem

L∇µu = λ|u|p−2uf in Ω

u = 0 on ∂Ω




L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)



u = 0 on ∂Ω




L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)



u = 0 on ∂Ω

µ →(

n−p

p

)p




L∇µu = −

(

div(

|∇u|p−2∇u)

+µ

|x|p |u|p−2u

)



u = 0 on ∂Ω

µ →(

n−p

p

)p

The expression L∇µu = λ|u|p−2uf is related to the

improved Hardy-Sobolev inequality.









∫

Ω

|∆u(x)|2dx ≥ n2(n− 4)2

16

∫

Ω

|u(x)|2|x|4 dx, ∀u ∈ W 2,2

0 (Ω) (6)





∫

Ω

|∆u(x)|2dx ≥ n2(n− 4)2

16

∫

Ω

|u(x)|2|x|4 dx, ∀u ∈ W 2,2

0 (Ω) (6)





∫

Ω

|∆u(x)|2dx ≥ n2(n− 4)2

16

∫

Ω

|u(x)|2|x|4 dx, ∀u ∈ W 2,2

0 (Ω) (6)

ON GOING RESEARCH




∫

Ω

|∆u(x)|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx (7)




∫

Ω

|∆u(x)|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx (7)

Here the best constant(

n−2pp

)p (np−n

p

)p

is given by the

infimum of




∫

Ω

|∆u(x)|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx (7)


n−2pp

)p (np−n

p

)p

is given by the

infimum of I(u) =∫Ω |∆u|pdx

∫Ω

|u(x)|p

|x|2pdx.




∫

Ω

|∆u(x)|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx (7)


n−2pp

)p (np−n

p

)p

is given by the


∫Ω

|u(x)|p

|x|2pdx.

No extremal function in W 2,p0 (Ω) which attains the infimum

of I(u).




∫

Ω

|∆u(x)|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx (7)


n−2pp

)p (np−n

p

)p

is given by the


∫Ω

|u(x)|p

|x|2pdx.


of I(u).




∫

Ω

|∆u(x)|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx (7)


n−2pp

)p (np−n

p

)p

is given by the


∫Ω

|u(x)|p

|x|2pdx.


of I(u).

The candidates of the extremals are singular at the originwhich are not in W 2,p

0 (Ω).




∫

Ω

|∆u(x)|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx (7)


n−2pp

)p (np−n

p

)p

is given by the


∫Ω

|u(x)|p

|x|2pdx.


of I(u).


0 (Ω).




∫

Ω

|∆u(x)|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx (7)


n−2pp

)p (np−n

p

)p

is given by the


∫Ω

|u(x)|p

|x|2pdx.


of I(u).


0 (Ω).

Therefore it is natural to consider that there exists amissing terms in the right hand side of (7).




OBJECTIVE




OBJECTIVE

To achieve an optimal improvement of inequality (7) byadding sharp terms in the right hand side involving a

singular weight of type(

log 1|x|

)−2




OBJECTIVE



log 1|x|

)−2




OBJECTIVE



log 1|x|

)−2

Optimal in the sense that the improved inequality holds for

this weight function(

log 1|x|

)−2

but fails for any weight

more singular than this.




MAIN RESULTS




MAIN RESULTS


Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.




MAIN RESULTS



Noncritical Case(1 0 such that if R > supΩ |x| andfor any u ∈ W 2,p

0 (Ω),




MAIN RESULTS





0 (Ω),

∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx(11)




MAIN RESULTS




MAIN RESULTS






MAIN RESULTS



Critical Case(p = n2)




MAIN RESULTS







MAIN RESULTS




There exists K∗ = K∗(n) > 0 and C∗ = C∗(n) > 0 such

that if R > K∗ supΩ |x| and for any u ∈ W2,n

20 (Ω),




MAIN RESULTS




There exists K∗ = K∗(n) > 0 and C∗ = C∗(n) > 0 such

that if R > K∗ supΩ |x| and for any u ∈ W2,n

20 (Ω),

∫

Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx(12)




∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)




∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)

REMARK 1

In inequality (11) the exponent 2 of the weight function isoptimal.




∫

Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)




∫

Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)

REMARK 2

In inequality (12) the exponent n2of the weight function is

optimal and the constant(

n−2√n

)n

is sharp.




∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)

∫

Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)




∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)

∫

Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)

REMARK 3

C and C∗ depends on R in a weak sense since(

log R|x|

)−2

and(

log R|x|

)−n2−1

tends to zero as R → ∞.






0 (Ω),

∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)






0 (Ω),

∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)






0 (Ω),

∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)

REMARK 4

At first we prove inequality (11) assuming that

R > e1p supΩ |x| because of technical reason. Namely, the

weight function g(r) = r−2p(

log Rr

)−2should be monotone

decreasing. Then we can extend (11) for any R > supΩ |x|




Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and

C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫

Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)






Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)






Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)

REMARK 4

In the proof of the critical case (12), we used decreasing

rearrangement argument, hence g∗(r) = r−n(

log Rr

)−n2−1

should be monotone decreasing and R ≥ re12+ 1

n .






Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)

REMARK 4



log Rr

)−n2−1


n .






Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)

REMARK 4



log Rr

)−n2−1


n .Moreoverwe need the condition to absorb the error terms in the righthand side of (12) with C∗ > 0, hence K∗ ≥ e

12+ 1

n




Corollary

Let 1 < p < n2, and let

Fp =

f : Ω → R+|f ∈ L∞

loc(Ω \ 0) with

lim sup|x|→0

|x|2pf(x)(

log1

|x|

)2

< ∞




Corollary


Fp =

f : Ω → R+|f ∈ L∞

loc(Ω \ 0) with

lim sup|x|→0

|x|2pf(x)(

log1

|x|

)2

< ∞




Corollary


Fp =

f : Ω → R+|f ∈ L∞

loc(Ω \ 0) with

lim sup|x|→0

|x|2pf(x)(

log1

|x|

)2

< ∞


∫

Ω

|∆u|pdx ≥ Λn,p

∫

Ω

|u(x)|p|x|2p dx+ λ(f)

∫

Ω

|u(x)|pf(x)dx(13)




Corollary


Fp =

f : Ω → R+|f ∈ L∞

loc(Ω \ 0) with

lim sup|x|→0

|x|2pf(x)(

log1

|x|

)2

< ∞


∫

Ω

|∆u|pdx ≥ Λn,p

∫

Ω

|u(x)|p|x|2p dx+ λ(f)

∫

Ω

|u(x)|pf(x)dx(13)

If f /∈ Fp and if |x|2pf(x)(

log 1|x|

)2


then no inequality of type (13) can hold.A. L. Detalla Missing Terms in Classical Inequalities






APPLICATION

We use the results to determine exactly when the firsteigenvalue of the weighted eigenvalue problem for theoperator




APPLICATION





APPLICATION


L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u




APPLICATION


L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u

will tend to 0 as µ →(

n−2pp

)p (np−n

p

)p

.




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u

L∆µis related to the variational problem




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u





L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u


infu∈K

∫

Ω

(

|∆u|p − µ|u|p|x|2p

)

dx




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u


infu∈K

∫

Ω

(

|∆u|p − µ|u|p|x|2p

)

dx

where K is given by




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u


infu∈K

∫

Ω

(

|∆u|p − µ|u|p|x|2p

)

dx

where K is given by

K =

u ∈ W 2,p(Ω) ∩W 1,p0 (Ω) :

∫

Ω

|u(x)|pf(x)dx = 1

,

f : is a weight function which will be specified later.




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u


L∆µu = λ|u|p−2uf in Ω

u = ∆u = 0 on ∂Ω




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u



u = ∆u = 0 on ∂Ω




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u



u = ∆u = 0 on ∂Ω

µ →(

n−2pp

)p (np−n

p

)p




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u



u = ∆u = 0 on ∂Ω

µ →(

n−2pp

)p (np−n

p

)p




L∆µu = ∆

(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u



u = ∆u = 0 on ∂Ω

µ →(

n−2pp

)p (np−n

p

)p

The expression L∆µu = λ|u|p−2uf is related to the

improved Rellich inequality.




REARRANGEMENT

For a domain Ω we define a ball Ω∗ such that |Ω∗| = |Ω|with center at the origin.




REARRANGEMENT





REARRANGEMENT


u∗(x) : spherically symmetric decreasing rearrangement offunction u(x)




REARRANGEMENT


u∗(x) : spherically symmetric decreasing rearrangement offunction u(x)

u∗(x) = inf

t ≥ 0 : µ(t) < B|x|n

in Ω∗

µ(t) = |x ∈ Ω : |u(x)| > t|

B: Volume of unit ball




LEMMA (Talente)

Let f ∈ C∞0 (Ω). If u is a solution of

−∆u = f in Ω

u = 0 on ∂Ω

and v is a solution of

−∆v = f ∗ in Ω∗

v = 0 on ∂Ω∗

⇒v≥ u∗.




From this lemma we can show that∫

Ω

|∆u|pdx =

∫

Ω∗

|∆v|pdx

∫

Ω

|u|p|x|2pdx ≤

∫

Ω∗

|v|p|x|2pdx





Ω

|∆u|pdx =

∫

Ω∗

|∆v|pdx

∫

Ω

|u|p|x|2pdx ≤

∫

Ω∗

|v|p|x|2pdx





Ω

|∆u|pdx =

∫

Ω∗

|∆v|pdx

∫

Ω

|u|p|x|2pdx ≤

∫

Ω∗

|v|p|x|2pdx

Hence we have∫

Ω|∆u|pdx

∫

Ω|u|p|x|2pdx

≥∫

Ω∗ |∆v|pdx∫

Ω∗

|v|p|x|2pdx





Ω

|∆u|pdx =

∫

Ω∗

|∆v|pdx

∫

Ω

|u|p|x|2pdx ≤

∫

Ω∗

|v|p|x|2pdx

Hence we have∫

Ω|∆u|pdx

∫

Ω|u|p|x|2pdx

≥∫

Ω∗ |∆v|pdx∫

Ω∗

|v|p|x|2pdx

From this we can assume that u is radial and Ω is a ball.




ASSUMPTIONS

Ω : unit ball B1

u : radially nonincreasing,

−∆u > 0

u > 0




OUTLINE OF THE PROOF

1. Noncritical Case(1 < p < n2)











1a) We prove inequality (11)

∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)







∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)

1b) We show the sharpness of(

n−2pp

)p (np−n

p

)p







∫

Ω

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

Ω

|u(x)|p|x|2p dx

+ C

∫

Ω

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)


n−2pp

)p (np−n

p

)p

1c) We show the optimality of the exponent 2.





2. Critical Case(p = n2)












∫

Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)







∫

Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)


n−2√n

)n







∫

Ω

|∆u|n2 dx ≥(

n− 2√n

)n ∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

Ω

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)


n−2√n

)n

2c) We show the optimality of the exponent n2.




LEMMA 2

For any R > 1, q ≤ 0, ν ∈ (0, 1) satisfying 2ν − 1 + q = 0

∫ 1

0

|h′(r)|2(

logR

r

)q

rdr ≥ ν2

∫ 1

0

|h(r)|2(

logR

r

)q−2dr

r(14)

holds for any h ∈ C([0, 1]) ∩ C1(0, 1), with h(0) = h(1) = 0.




1a) PROOF OF INEQUALITY (11)

For u ∈ C20(B1), u > 0, radially nonincreasing , we define

v(r) = u(r)rnp−2 r = |x|.

then apply this to






v(r) = u(r)rnp−2 r = |x|.

then apply this to






v(r) = u(r)rnp−2 r = |x|.

then apply this to

∫

B1

|∆u|pdx−(

n− 2p

p

)p(np− n

p

)p ∫

B1

|u(x)|p|x|2p dx

we get





∫

B1

|∆u|pdx−(

n− 2p

p

)p(np− n

p

)p ∫

B1

|u(x)|p|x|2p dx

≥ 4ωn

α

(

n− 2p

p

)p(np− n

p

)p(p− 1

p

)∫ 1

0

|(

vn2 (r)

)′ |2rdr

By Lemma 2(

v = 12, q = 0

)





∫

B1

|∆u|pdx−(

n− 2p

p

)p(np− n

p

)p ∫

B1

|u(x)|p|x|2p dx

≥ 4ωn

α

(

n− 2p

p

)p(np− n

p

)p(p− 1

p

)∫ 1

0

|(

vn2 (r)

)′ |2rdr

By Lemma 2(

v = 12, q = 0

)





∫

B1

|∆u|pdx−(

n− 2p

p

)p(np− n

p

)p ∫

B1

|u(x)|p|x|2p dx

≥ 4ωn

α

(

n− 2p

p

)p(np− n

p

)p(p− 1

p

)∫ 1

0

|(

vn2 (r)

)′ |2rdr

By Lemma 2(

v = 12, q = 0

)

LEMMA 2

∫ 1

0

|h′(r)|2(

logR

r

)q

rdr ≥ ν2

∫ 1

0

|h(r)|2(

logR

r

)q−2dr

r





we get

∫

B1

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

B1

|u(x)|p|x|2p dx

+ C

∫

B1

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)





we get

∫

B1

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

B1

|u(x)|p|x|2p dx

+ C

∫

B1

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)





we get

∫

B1

|∆u|pdx ≥(

n− 2p

p

)p(np− n

p

)p ∫

B1

|u(x)|p|x|2p dx

+ C

∫

B1

|u(x)|p|x|2p

(

logR

|x|

)−2

dx (11)

where C =(

n−2pp

)p (np−n

p

)p (p−1p

)




For ǫ > 0 sufficiently small, let us define

uǫ =

0, 0 < r < ǫ2

log r

ǫ2

rn−pp log 1

ǫ

, ǫ2 < r < ǫ

log 1r

rn−pp log 1

ǫ

, ǫ < r < 1




Let wǫ =∫ 1

ruǫ(ρ)dρ. Direct calculation gives




Let wǫ =∫ 1





Let wǫ =∫ 1


wǫ =

(

p

n−2p

)21−2ǫ

2−np +ǫ

2(2−np )

log 1ǫ

, 0 < r < ǫ2

p

n−2p

r2−n

p log r

ǫ2

log 1ǫ

+(

p

n−2p

)21−2ǫ

2−np +r

2−np

log 1ǫ

, ǫ2 < r < ǫ

p

n−2p

r2−n

p log 1r

log 1ǫ

+(

p

n−2p

)21−r

2−np

log 1ǫ

, ǫ < r < 1




∆wǫ =

0, 0 < r < ǫ2

1p

r−np

(

−p+ n(1− p) log rǫ2

)

(

log 1ǫ

)−1, ǫ2 < r < ǫ

1p

r−np

(

p+ n(1− p) log 1r

)

(

log 1ǫ

)−1, ǫ < r < 1




Direct calculation gives









∫

B1

|∆wǫ|pdx =2

p+ 1

(

n(p− 1)

p

)p

ωn log1

ǫ+O

(

(

log1

ǫ

)−1)

∫

B1

|wǫ|p|x|2p dx ≥ 2

p+ 1

(

p

n− 2p

)p

ωn log1

ǫ+O

(

(

log1

ǫ

)−1)




∫

B1

|∆wǫ|pdx−(

n− 2p

p

)p(np− n

p

)p ∫

B

|wǫ|p|x|2p dx

≤ O

(

(

log1

ǫ

)−1)




∫

B1

|∆wǫ|pdx−(

n− 2p

p

)p(np− n

p

)p ∫

B

|wǫ|p|x|2p dx

≤ O

(

(

log1

ǫ

)−1)




∫

B1

|∆wǫ|pdx−(

n− 2p

p

)p(np− n

p

)p ∫

B

|wǫ|p|x|2p dx

≤ O

(

(

log1

ǫ

)−1)

and




∫

B1

|∆wǫ|pdx−(

n− 2p

p

)p(np− n

p

)p ∫

B

|wǫ|p|x|2p dx

≤ O

(

(

log1

ǫ

)−1)

and∫

B1

|wǫ|p|x|2p

(

logR

|x|

)−γ

≥ O

(

(

log1

ǫ

)1−γ)




1b. SHARPNESS OF(

n−2pp

)p (np−n

p

)p

limǫ→0

I(wǫ) = limǫ→0

∫

B1|∆wǫ|pdx

∫

B1

|wǫ|p|x|2p dx

≤(

n− 2p

p

)p(np− n

p

)p




1b. SHARPNESS OF(

n−2pp

)p (np−n

p

)p

limǫ→0

I(wǫ) = limǫ→0

∫

B1|∆wǫ|pdx

∫

B1

|wǫ|p|x|2p dx

≤(

n− 2p

p

)p(np− n

p

)p




1b. SHARPNESS OF(

n−2pp

)p (np−n

p

)p

limǫ→0

I(wǫ) = limǫ→0

∫

B1|∆wǫ|pdx

∫

B1

|wǫ|p|x|2p dx

≤(

n− 2p

p

)p(np− n

p

)p

But by Rellich inequality

limǫ→0

I(wǫ) ≥(

n− 2p

p

)p(np− n

p

)p




1b. SHARPNESS OF(

n−2pp

)p (np−n

p

)p

limǫ→0

I(wǫ) = limǫ→0

∫

B1|∆wǫ|pdx

∫

B1

|wǫ|p|x|2p dx

≤(

n− 2p

p

)p(np− n

p

)p

But by Rellich inequality

limǫ→0

I(wǫ) ≥(

n− 2p

p

)p(np− n

p

)p

hence

limǫ→0

I(wǫ) =

(

n− 2p

p

)p(np− n

p

)p




1c. OPTIMALITY OF THE EXPONENT 2

Assume 0 < γ < 2.





Assume 0 < γ < 2.





Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf

u∈W 2,p0 (B1)\0

Iγ(u) = 0






u∈W 2,p0 (B1)\0

Iγ(u) = 0 where

Iγ(u) =

∫

B1|∆u|pdx−

(

n−2pp

)p (np−n

p

)p∫

B1

|u|p|x|2pdx

∫

B1

|u|p|x|2p

(

log R|x|

)−γ

dx






u∈W 2,p0 (B1)\0

Iγ(u) = 0 where

Iγ(u) =

∫

B1|∆u|pdx−

(

n−2pp

)p (np−n

p

)p∫

B1

|u|p|x|2pdx

∫

B1

|u|p|x|2p

(

log R|x|

)−γ

dx

Hence using a family wǫ ∈ W 2,p0 (B1)






u∈W 2,p0 (B1)\0

Iγ(u) = 0 where

Iγ(u) =

∫

B1|∆u|pdx−

(

n−2pp

)p (np−n

p

)p∫

B1

|u|p|x|2pdx

∫

B1

|u|p|x|2p

(

log R|x|

)−γ

dx

Hence using a family wǫ ∈ W 2,p0 (B1) we have

limǫ→0 Iγ(wǫ) = 0.






u∈W 2,p0 (B1)\0

Iγ(u) = 0 where

Iγ(u) =

∫

B1|∆u|pdx−

(

n−2pp

)p (np−n

p

)p∫

B1

|u|p|x|2pdx

∫

B1

|u|p|x|2p

(

log R|x|

)−γ

dx

Hence using a family wǫ ∈ W 2,p0 (B1) we have

limǫ→0 Iγ(wǫ) = 0. Thus optimality follow. i.e. γ ≥ 2.




LEMMA 3

Assume f ∈ C2(B1) and u ∈ C20(B1) are radial satisfying

f(r) > 0,∆f(r) ≤ 0, u(r) > 0, and −∆u > 0 where r = |x|.Set u(r) = f(r)v(r), then for any u ∈ C2

0(B1)




LEMMA 3

∫

B1

|∆u|n2 dx ≥

n(n− 2)

4ωn

∫ 1

0

(v′(r))2v

n−42 (r)rn−1|∆f(r)|n−2

2 f(r)dr

+ ωn

∫ 1

0

vn2 (r)

rn−1|∆f(r)|n2 +

∂r

[

rn−1

(

|∆f(r)|n−22 f ′(r)− ∂r|∆f(r)|n−2

2 f(r)

)]

dr




2a. PROOF OF INEQUALITY (12)

From Lemma 3, we denote

S1 =n(n− 2)

4ωn

∫ 1

0

(v′(r))2v

n−42 (r)rn−1|∆f(r)|n−2

2 f(r)dr

and

S2 =ωn

∫ 1

0

vn2 (r)

rn−1|∆f(r)|n2 +

∂r

[

rn−1

(

|∆f(r)|n−22 f ′(r)− ∂r|∆f(r)|n−2

2 f(r)

)]

dr





Then∫

B1|∆u|n2 dx ≥ S1 + S2.





Then∫

B1|∆u|n2 dx ≥ S1 + S2.





Then∫

B1|∆u|n2 dx ≥ S1 + S2. For f(r) =

(

log Rr

)a,

0 < a < 1





Then∫

B1|∆u|n2 dx ≥ S1 + S2. For f(r) =

(

log Rr

)a,

0 < a < 1

S1 ≥a

n−22 (1− a)2(n− 2)

n2

4ωn

∫ 1

0

un2

(

logR

r

)−n2−1

dr

r

S2 ≥a

n−22 (1− a)(n− 2)

n2+1

2ωn

∫ 1

0

un2

(

logR

r

)−n2

(

1 +O

(

logR

r

)−2)

dr

r





Then∫

B1|∆u|n2 dx ≥ S1 + S2. For f(r) =

(

log Rr

)a,

0 < a < 1

S1 ≥a

n−22 (1− a)2(n− 2)

n2

4ωn

∫ 1

0

un2

(

logR

r

)−n2−1

dr

r

S2 ≥a

n−22 (1− a)(n− 2)

n2+1

2ωn

∫ 1

0

un2

(

logR

r

)−n2

(

1 +O

(

logR

r

)−2)

dr

r

set Q(a) = an−22 (1− a),





Then∫

B1|∆u|n2 dx ≥ S1 + S2. For f(r) =

(

log Rr

)a,

0 < a < 1

S1 ≥a

n−22 (1− a)2(n− 2)

n2

4ωn

∫ 1

0

un2

(

logR

r

)−n2−1

dr

r

S2 ≥a

n−22 (1− a)(n− 2)

n2+1

2ωn

∫ 1

0

un2

(

logR

r

)−n2

(

1 +O

(

logR

r

)−2)

dr

r

set Q(a) = an−22 (1− a), Q takes its maximum at a = n−2

2





∫

B1

|∆u|n2 dx ≥(

n− 2√n

)n ∫

B1

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

B1

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)





∫

B1

|∆u|n2 dx ≥(

n− 2√n

)n ∫

B1

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

B1

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)





∫

B1

|∆u|n2 dx ≥(

n− 2√n

)n ∫

B1

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

B1

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)

where C∗ =(

n−2√n

)n (

(n− 2)−1 − k(logR)−1)





∫

B1

|∆u|n2 dx ≥(

n− 2√n

)n ∫

B1

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

B1

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)

where C∗ =(

n−2√n

)n (

(n− 2)−1 − k(logR)−1)

> 0





∫

B1

|∆u|n2 dx ≥(

n− 2√n

)n ∫

B1

|u(x)|n2|x|n

(

logR

|x|

)−n2

dx

+ C∗∫

B1

|u(x)|n2|x|n

(

logR

|x|

)−n2−1

dx (12)

where C∗ =(

n−2√n

)n (

(n− 2)−1 − k(logR)−1)

> 0 if

R > e(n−2)k, k = k(n)




2b. SHARPNESS OF(

n−2√n

)n

To show sharpness, we use the test function

zǫ =

(

log1

r + ǫ

)n−2n

−(

log1

1 + ǫ

)n−2n




2b. SHARPNESS OF(

n−2√n

)n


zǫ =

(

log1

r + ǫ

)n−2n

−(

log1

1 + ǫ

)n−2n




2b. SHARPNESS OF(

n−2√n

)n


zǫ =

(

log1

r + ǫ

)n−2n

−(

log1

1 + ǫ

)n−2n

then we can show

limǫ→0

∫

B1|∆zǫ|

n2 dx

∫

B1

|zǫ|n2

|x|n

(

log R|x|

)−n2dx

=

(

n− 2√n

)n




2c. OPTIMALITY OF THE EXPONENT n2

We use the same test function uǫ with p = n2, and

wǫ =∫ 1

ruǫ(ρ)dρ. Then for 0 < γ < n

2

limǫ→0

∫

B1|∆wǫ|

n2 dx

∫

B1

|wǫ|n2

|x|n

(

log R|x|

)γ

dx= 0

Thus optimality follow. i.e. γ ≥ n2




APPLICATION

Consider the weighted eigenvalue problem with a singularweight

∆(

|∆u|p−2∆u)

− µ

|x|2p |u|p−2u = λ|u|p−2uf in Ω

u = ∆u = 0 on ∂Ω (15)

Here f ∈ Fp

Fp =

f : Ω → R+| lim

|x|→0|x|2pf(x) = 0, f ∈ L∞

loc

(

Ω \ 0)

,

1 < p < n2, 0 ≤ µ <

(

n−2pp

)p (np−n

p

)p

and λ ∈ R.




We look for a weak solution

u ∈ W = W 2,p(Ω) ∩W 1,p0 (Ω)

of problem ♯15 and study the asymptotic behaviour of thefirst eigenvalues for different singular weights as µ increases

to(

n−2pp

)p (np−n

p

)p

.




We look for a weak solution

u ∈ W = W 2,p(Ω) ∩W 1,p0 (Ω)

of problem ♯15 and study the asymptotic behaviour of thefirst eigenvalues for different singular weights as µ increases

to(

n−2pp

)p (np−n

p

)p

.

Definition

u ∈ W is said to be a weak solution of (15) iff for anyφ ∈ C2(Ω) with φ = 0 on ∂Ω

∫

Ω

(

|∆u|p−2∆u∆φ− µ

|x|2p |u|p−2uφ

)

dx = λ

∫

Ω

|u|p−2ufφdx.




LEMMA

For u ∈ W ∃ v ∈ W such that v > 0 and satisfies

∫

Ω|∆u|pdx− λ

∫

Ω|u|p|x|2pdx

∫

Ω|u|pfdx ≥

∫

Ω|∆v|pdx− λ

∫

Ω|v|p|x|2pdx

∫

Ω|v|pfdx .




LEMMA

For u ∈ W ∃ v ∈ W such that v > 0 and satisfies

∫

Ω|∆u|pdx− λ

∫

Ω|u|p|x|2pdx

∫

Ω|u|pfdx ≥

∫

Ω|∆v|pdx− λ

∫

Ω|v|p|x|2pdx

∫

Ω|v|pfdx .

REMARK

Since λ is first eigenvalue and u is the correspondingeigenfunction, by using the above lemma, we can assumeu > 0 in Ω. Then by the elliptic regularity theory, u issmooth near the boundary. From the definition of weaksolution one can derive the boundary condition of (15) byusing integration by parts.




THEOREM

For all 1 0.




THEOREM

For all 1 0.




THEOREM

For all 1 0.

Moreover, as µ →(

n−2pp

)p (np−n

p

)p

,




THEOREM

For all 1 0.


n−2pp

)p (np−n

p

)p

,

If lim sup|x|→0

|x|2pf(x) = 0, ⇒ λ1µ(f) → λ(f) ≥ 0




THEOREM

For all 1 0.


n−2pp

)p (np−n

p

)p

,

If lim sup|x|→0

|x|2pf(x) = 0, ⇒ λ1µ(f) → λ(f) ≥ 0

If lim sup|x|→0

|x|2pf(x)(

log1

|x|

)2

< ∞, ⇒ λ(f) > 0




THEOREM

For all 1 0.


n−2pp

)p (np−n

p

)p

,

If lim sup|x|→0

|x|2pf(x) = 0, ⇒ λ1µ(f) → λ(f) ≥ 0

If lim sup|x|→0

|x|2pf(x)(

log1

|x|

)2

< ∞, ⇒ λ(f) > 0

If lim sup|x|→0

|x|2pf(x)(

log1

|x|

)2

= ∞, ⇒ λ(f) = 0




REMARK

In the proof of the theorem u will be characterize as asolution of variational problem defined by

Jµ(u) =

∫

Ω

(

|∆u|p − µ|u|p|x|2p

)

dx

and the problem (♯15) stated earlier becomes Euler-Lagrange equation of this variational problem.




SKETCH OF THE PROOF OF THE THEOREM









We define Jµ(u) =∫

Ω

(

|∆u|p − µ |u|p|x|2p

)

dx






Ω

(

|∆u|p − µ |u|p|x|2p

)

dx






Ω

(

|∆u|p − µ |u|p|x|2p

)

dx

We minimize Jµ over M = u ∈ W |∫

Ω|u(x)|pf(x)dx = 1

and let λ1µ > 0 be the infimum.






Ω

(

|∆u|p − µ |u|p|x|2p

)

dx


Ω|u(x)|pf(x)dx = 1







Ω

(

|∆u|p − µ |u|p|x|2p

)

dx


Ω|u(x)|pf(x)dx = 1


We choose minimizing sequence (um)m∈N ⊂ M such thatJµ(um) → λ1

µ.













For a subsequence umkof um, umk

u weakly in W whereu ∈ W ∩M and

Jµ(umk) → λ1

µ = λ

J ′µ(umk

) → 0







Jµ(umk) → λ1

µ = λ

J ′µ(umk

) → 0







Jµ(umk) → λ1

µ = λ

J ′µ(umk

) → 0

By Fatou’s lemma, we get

umk→ u strongly in W

umk→ u strongly in Lp (Ω, |x|−2p)













Hence

Jµ(umk) → Jµ(u) = λ1

µ = λ

J ′µ(umk

) = 0





Hence


µ = λ

J ′µ(umk

) = 0





Hence


µ = λ

J ′µ(umk

) = 0

u satisfies Euler- Lagrange equation in a distribution sense.





Hence


µ = λ

J ′µ(umk

) = 0






Hence


µ = λ

J ′µ(umk

) = 0


Since u ∈ W , it is a weak solution of problem (♯15).





Hence


µ = λ

J ′µ(umk

) = 0







Hence


µ = λ

J ′µ(umk

) = 0



The remaining part of the proof follows from the corollaryof the main theorem.




Corollary


Fp =

f : Ω → R+|f ∈ L∞

loc(Ω \ 0) with

lim sup|x|→0

|x|2pf(x)(

log1

|x|

)2

< ∞


∫

Ω

|∆u|pdx ≥ Λn,p

∫

Ω


∫

Ω

|u(x)|pf(x)dx (13)

If f /∈ Fp and if |x|2pf(x)(

log 1|x|

)2


then no inequality of type (13) can hold.A. L. Detalla Missing Terms in Classical Inequalities



As µ →(

n−2pp

)p (np−n

p

)p

, λ1µ → λ(f) where




As µ →(

n−2pp

)p (np−n

p

)p





As µ →(

n−2pp

)p (np−n

p

)p


λ(f) = infu∈W (Ω\0)

∫

Ω

(

|∆u|p −(

n−2pp

)p (np−n

p

)p |u|p|x|2p

)

dx∫

Ω|u|pfdx ≥ 0




As µ →(

n−2pp

)p (np−n

p

)p



∫

Ω

(

|∆u|p −(

n−2pp

)p (np−n

p

)p |u|p|x|2p

)

dx∫

Ω|u|pfdx ≥ 0

if f ∈ Fp ⇒ λ(f) > 0




As µ →(

n−2pp

)p (np−n

p

)p



∫

Ω

(

|∆u|p −(

n−2pp

)p (np−n

p

)p |u|p|x|2p

)

dx∫

Ω|u|pfdx ≥ 0

if f ∈ Fp ⇒ λ(f) > 0if f 6= Fp ⇒ λ(f) = 0




THANK YOU VERY MUCH!

Al Detalla Theory

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