Low dimension anomalies and solvability in higher dimensions for some perturbed Pohozaev equation

Low dimension anomalies and solvability in higher dimensions for some perturbed Pohozaev equation G. Mancini∗ Dipartimento di Matematica, Universit´a di Roma Tre Abstract In this talk, we will present a new solvability condition in higher dimension for a Brezis Nirenberg type equation. This result, which we believe to be sharp in some sense, has been obtained in collaboration with Adimurthi

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Introduction

Very surprising ”low dimension phenomena” have been observed by Brezis and Nirenberg in their pioneering work [BN] concerning perturbations of the ”Pohozaev equation”

(P )

N +2

 

−∆u = u N −2 + p(u) u>0 in Ω u=0 in ∂Ω



in Ω

Here, Ω ⊂ 3, (P ) has a solution ∀Ω (ii)if N = 3, (P ) is not solvable in general (in fact (P ) has a solution in a ball Br iff r is large enough) As for (i) it follows from the very general sufficient condition they found (assuming p ≥ 0, but this is non really relevant)

(∗) lim  →0

4−N 2

Z

|x|≤ √1

P ((

N −2 1 ) 2 ) = +∞  + |x|2

Rs N where P (s) = 0 p(t)dt. Of course, p(s) = s N −2 does not satisfy (∗) if N = 3. This example shows two facts : 0∗

Supported by M.U.R.S.T.

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- anomalies may occurr in dimension 3 - condition (∗) is sharp in dimension 3. In view of this example, two related questions naturally arise: - is condition (∗) sharp also in higher dimensions ? - do anomalies occur also in higher dimensions ? We also refer to [PS], [BG] for related problems and interpretation of ”low dimension phenomena”. In a recent paper [AY], Adimurthi and Yadava gave an adfirmative answer to the question above in dimensions 4, 5, 6: they exibited a class of compactely supported perturbations PN such that (k) if N = 3, 4 and p ∈ PN , there are anomalies (i.e. there are solutions on Br iff r is large) R (kk) if N = 5, 6, and p ∈ PN and, in addition, P ( |x|N1 −2 ) = 0, then there are anomalies. (kkk) if N ≥ 7, and p ∈ PN , then (P ) is solvable on any ball. Of course, in cases (k) − (kk) Rcondition (∗) is violated: in case (k), because obviously P ( |x|N1 −2 ) < ∞, while, in case (kk), because R R N −2 1 2 ) = 0() (see Lemma 4.1 below). P ( |x|N1 −2 ) = 0 implies |x|≤ √1 P (( +|x| 2) 

In the attempt to extend the existence result in (kkk) to more general perturbations and domains, Adimurthi and myself [AM] realized that condition (∗) is not sharp in dimension N ≥ 7. To be more precise, let us first give a closer look to condition (∗) , limiting ourselves, for sake of simplicity, to perturbations p in the class P := C0∞ ((0, ∞)). If we denote, for a given p ∈ P

Ik = Ik (p) :=

Z


[

1 dk P ( N −2 )](|x|2 )dx k dt t 2

it is easy to see that (j) If Ik = 0 ∀k , orI0 = I1 ... = Ik−1 and Ik < 0 for some k, (∗) fails in any dimension N (jj) If I0 = I1 = ... = Ik−1 = 0, and Ik > 0 for some k = 0, 1... then (∗) holds iff N > 2k + 4.In fact, 4−N R N −2 1 2 )= if I0 = I1 = ... = Ik−1 = 0 then lim→0  2 |x|≤ √1 P (( +|x| 2)  k N −2 4+2k−N R d 1 2 ) lim→0  2 P (( +|x| 2) 0, and this explains the ”surprising” existence result (kkk)). 2

In particular: R I0 = 0 and 0 > I1 = − N 2−2 p( |x|N1 −2 ) |x|1N ⇒ (∗) fails ∀N . On the other hand, we will prove that, if

(∗∗) I0 = 0 and

Z

1 N

|∇H|2 −

Z

p(

1 1 ) >0 |x|N −2 |x|N

then(P ) has a solution on any Ω if N ≥ 7. Here, H is the finite energy solution of − ∆u = p(

1 ) in
(1.1)

So, we see that, in case I0 = 0, a much weaker condition than (∗) insures, in dimension N ≥ 7, the solvability of (P ) on any domain, i.e. (∗) is not sharp in dimension N ≥ 7. Of course, all this does not answer the question whether the anomaly described above occurs in dimension N ≥ 7. Even if we do believe that such anomalies occur in any dimension ( i.e. ”anomaly” is not a low dimension phenomena!), we content ourselves with the weaker conjecture N

S 2 if (P ) has solutions with energy smaller than N (S := best Sobolev con2N 1 N N −2 stant in the embedding H → L (< )) on any small ball Br , then

Z

1 |∇H| − N 2

Z

p(

1 1 ) ≥0 |x|N −2 |x|N

In other words, we suspect that a weak inequality in (∗∗) is necessary for ”low energy” solutions to exist on any small ball.

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The main result and the variational approch

For sake of simplicity we will assume p ∈ C0∞ ((0, ∞)). Let H be the unique solution in D1,2 of − ∆H = p(

1 |x|N −2

) in
R Theorem 2.1 If P ( |x|N1 −2 ) > 0 and N ≥ 5 , (P ) has a solution. R R R If P ( |x|N1 −2 ) = 0 and |∇H|2 > N1 p( |x|N1 −2 ) |x|1N , then - if N ≥ 7, (P ) has a solution ∀Ω - if N = 3, 4, 5, 6, (P ) has a solution if Ω contains a large ball. 3

By standard arguments, a non zero critical point of 1 ˜ E(u) := 2

Z

N −2 |∇u| − 2N 2

Z

+

(u )

2N N −2

−

Z

P (u+ ) u ∈ H01 (Ω)

˜ enjoies the mountain pass is a solution of (P ). The energy functional E geometry, and, again by standard arguments, a mountain pass level is critical N

S 2 (S := best Sobolev constant). provided it is strictly smaller than N In fact it is enough to prove that, for some V ∈ H01 (Br ) , Br ⊂ Ω, it results N

S 2 max E(tV ) < (2.1) t≥0 N R R R 2N −2 where E(u) = 21 R|∇u|2 − N2N |u| N −2 − P (u) , u ∈ H01 (Br ) and P is s the even extension of 0 p(t)dt, s ≥ 0. This is because, after extending V equal to zero outside Br , it results ˜ E(t|V |) ≤ E(tV ).

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Choise of the test function and basic estimates

To get (2.1), we will choose V = V , for  suitably small, where V ∈ H01 (Br ) satisfies N +2

− ∆V = UN −2 + p(U ) on Br ⊂ Ω

(3.1)

N −2

Here U (x) = − 2 U ( x ) p N −2 cN 2 , cN = N (N − 2) U (x) = ( 1+|x| 2) N +2

Recall that −∆U = UN −2 in

2

|∇U | =

Z

2N

UN −2 = S

N 2


ˆ N −2 and In what follows, we will always assume  < 4δ r2 , where δ = c−1 N δ ˆ ˆ δ < 1 is such that p(s) ≡ 0 for s ≤ δ. We will denote w := V − U , so that −∆w = p(U ) in Br and w ≡ −b N −2 N 2 . on ∂Br where b = ( 2c+r 2) N −2

We will also write sometimes c˜N instead of cN 2 4

Lemma 3.1 Let V be given by (3.1).Then

E(V ) ≤

N 2

S N

N −2 Z N +2 cN 2  U N −2 ]( )N −2 − r 2 N Z < Z 1 + o(1) − |∇w |2 − P (U ) 2 Br Br

+ +[o(1) +

(3.2)

where o(1) are small in  uniformly with respect to r ∈ (1, ∞) Proof. Multiplying (3.1) by V and integrating by parts, and using the inequality 2N

2N

2N

|V | N −2 − UN −2 ≥

N −2 2N w , N −2 U

we get, by straightforward computations:

N +2 N R S 2 E(V ) ≤ N − 12 [ |x|≤r UN −2 w − p(U )U + p(U )w ]− R R − Br P (U ) − Br ×[0,1] (1 − t)p0 (U + tw )w2 N +2

Now, from −∆U = UN −2 , −∆w = p(U ), and Green formulas, we get Z

N +2 N −2

U Br Z

|x|≤r

Z

R

−

1 2

Z Br

UN −2 ≤ 

S N Z |∇w |2 −

E(V ) ≤

(3.3)

Br

N +2

and hence, since

N +2

p(U )(U − w UN −2 ) , Br Z p(U )w = |∇w |2 − w p(U ) w =

N 2

+

|x|≤r

N −2 2

R

N +2


c˜N  N −2 ( ) 2 r Z P (U ) +

U N −2 : Z

N +2

U N −2 + b


Z

p(U ) −

Br

(1 − t)p0 (U + tB )w2

(3.4)

Br ×[0,1]

To complete the proof of the Lemma, we just use (3.4) and the estimates gathered in the following Lemma 3.2 R N R (i) p(U ) =  2 [
Proof 1 To see (i), we just make the change of variable y = − 2 x : R R N −2 N −2 N cN N 2 ) =  2 2 )dy p(( 2c+|x| p(( +|y| 2) 2) Br
because

N −2 2

cN p(( +|y| 2)

≤ r2 , |y| ≥

r √ 

cN ⇒ ( +|y| 2)

N −2 2

≤ ( crN2 )

N −2 2

≤ (δcN )

N −2 2

= δˆ ⇒

)=0

R N −2 N N 2 ) =  2 (o(1) + Hence p(( 2c+|x| p( |y|c˜NN−2 )) where o(1) is uniform 2)
Z (

2N

|w | N −2 )

N −2 N

Br

≤ [(

1 S

Z

1

|∇w |2 ) 2 + b (volBr )

N −2 2N

]2 ≤

Br

1 ≤ 2( S

Z

N −2 −2  N −2 |∇w |2 + ωNN cN ( ) ) N r

(3.5)

R where ωN := B1 dx. To see (iii), we can use the representation formula (recall that p(U ) ≡ 0 outside B√  ⊂ Br ) : δ

c−2 w (x) + b = N ωN

Z

1 1 √  p(U (y))( |x − y|N −2 − |x − y|N −2 )dy |y|≤ δ

(3.6)

where x is any R ppoint on ∂Br . p Now , |x| ≥ 2 δ ⇒ |x − y| ≥ δ ⇒ |y|≤√  |x − y|2−N ≤ δ R p N −2 N ( δ ) 2 ( δ ) 2 ωN = ωN δ , while |x| ≤ 2 δ implies |y|≤√  |x − y|2−N δ

≤

R

|y|≤3

√  |z|2−N = 9 ωN  . 2 δ δ

N −2

So, for any x we obtain from (3.6) |w (x)| ≤ ( crN2 ) 2 + dN  (dN just 1 depending on N and p) that is k w k∞ = 0( 2 ) uniformly for r ≥ 1. Finally, by H˘ older inequality and (ii), we have (t − 1)|p0 (U + tw )w2 | R 2N N −2 R 1 R N 2 ≤ ( |w | N −2 ) N 0 ( Br |p0 (U + tw )| 2 ) N R R R N 2 1 ≤ 0(1)[ |∇w |2 + ( r )N −2 ] 0 ( Br |p0 (U + tw )| 2 ) N dt |

R

Br ×[0,1]

R1 R N 2 By dominated convergence, 0 ( Br |p0 (U + tw )| 2 ) N →→0 0 for any given r, because k w kL∞ (Br ) →→0 0 for every r by elliptic theory. Furthermore, by p (iii), if r ≥ 1 , p0 (U + tw )(x)) ≡ 0 for |x| ≥ δ and  < N , for some N 6

R N 2 depending only on N and not on r, and hence we get ( Br |p0 (U + tw )| 2 ) N ≤ 4 δ

4

2

k p0 k∞ ωNN . This complets the proof of (iiii).

Higher order estimates and the proof of the Theorem

We first estimate the last two terms in (3.2). Lemma 4.1 N +2 R R N +2 2 2 (c 2 |∇w | =  |∇H|2 + o(1))  N B
(i) (ii)

Furthermore, all estimates are uniform in r ∈ (1, ∞). Proof √ (i) Let w ˜ (x) := −1 w ( x) in B √r , so that − ∆w ˜ = p((

N −2 cN ) 2 ) in B √r . 2  + |x|

Now, if H ∈ D1,2 (
N −2 cN ) 2 ) in
clearly w ˜ − H ≡ const on B √r , and hence R N +2 R N +2 R |∇w |2 =  2 B r |∇w ˜ |2 =  2 B Br √



r √ 

|∇H |2 = 

N +2 2

R ( |∇H0 |2 +

o(1)), where

− ∆H0 = p(

R

c˜N ) in
and o(1) is uniform with respect to r in [1, ∞). Then (i) follows from N +2 R |∇H0 |2 = cN 2 |∇H|2 . 1 (ii) After the change of variable y = − 2 x we have 7

Z

N

P (U ) =  2 +1 (

Br

1 

Z

P ((

B √r

N −2 cN ) 2 ) 2  + |x|



where the second integral is in fact taken over B √1 , because  ≤ δr2 . First δ

lim

→0

Z

P ((

B √r

N −2 cN ) 2 )= 2  + |x|

Z

P(

c˜N ). |x|N −2

(4.1)



If this integral vanishes, lim→0

1 

R

= lim→0

cN P (( +|x| 2)

|y|≤ √1

R

δ

|y|≤ √1

= − N 2−2 c˜N

R

δ


N −2 2

)=

cN c˜N 2−N 2 p(( +|y|2 )

p( |y|c˜NN−2 ) |y|1N

N −2 2

N

1 2 = )( +|y| 2) R = − N 2−2 c˜N
where the limit is obviously uniform in r ≥ 1. Finally, we prove (iii). We have R N −2 N −2 N R cN cN 2 ) = 2 p(( lim→0 − 2 Br U p(U ) = lim→0 |y|≤ √r ( +|y| 2) +|y|2 )  R c˜N c˜N =
N +2

U p(U ) =  2 [o(1) − c˜N + |x|N −2 p0 ( |x|c˜NN−2 )]dx] Br

N −2 ˜N 2 c

R

1 |x|N

[p( |x|c˜NN−2 )+

Proof of the theorem We have to show how to deriveR(2.1) from Lemma 3.1 and 4.1. We will limit ourselves to the ”vanishing case” P ( |x|N1 −2 ) = 0. Inserting the estimates (i) - (ii) of Lemma 4.1 in (3.4), we get N

S2  + (aN + o(1))( )N −2 − N r N +2 Z Z cN 2 N +2 1 1 1 2 2 [o(1) + |∇H| − −  p( )] 2 N |x|N |x|N −2 E(V ) ≤

8

(4.2)

where o(1) are uniform in r ≥ R non depend on r. R 1 and aN does Since, by assumption, I1 := |∇H|2 − N1 |x|1N p( |x|N1 −2 ) > 0, we see from (4.2) that for  smaller than some (r)

E(V ) ≤

S N

N 2

− N −2 (

I1 6−N cN  2 − N −2 ) 2 r

and in fact (r) can be taken independent on r if r ≥ 1. Now, (2.1), and hence the result, follows from the

Claim Let E(t V ) = maxt>0 E(tV ) ,  < (r). Then (t − 1)2 = o(1)(

N +2 2

 + ( )N −2 ) r

where o(1) is uniform with respect to r ∈ [1, ∞).

Assuming the claim, we have 1

d2 E((tt + (1 − t))V ) = dt2 0 N +2  = 0(1)(1 − t )2 = o(1)( 2 + ( )N −2 ) r

0 ≤ E(t V ) − E(V ) = −

Z

t

uniformly in r ≥ 1, and hence, from (4.3): N

S2 I1 6−N 1 + cN max E(tV ) ≤ − N −2 (  2 − N −2 ) t>0 N 4 r for  ≤ (r) ∀r, or  ≤ 0 := inf r≥1 (r) > 0, if r ≥ 1. So N

S2 if N ≥ 7 then, ∀r > 0 : max E(tV ) < for  < ˜(r) t 2 N S2 if 3 ≤ N ≤ 6 then max E(tV ) < for  ≤ 0 and r ≥ r() t>0 N Proof of the Claim 9

(4.3)

Let us first observe that Z

|∇V |2 = S

Z

|V | N −2 = S

N 2

+ 0(1)(

N +2 2

1  + ( )N −2 ) 2 r

(4.4)

1  + ( )N −2 ) 2 r R R R R In fact, |∇V |2 = |∇U |2 + |∇w |2 + 2 ∇V ∇w = 2N

N 2

+ 0(1)(

N +2 2

(4.5)

R N +2 0(1)( r )N −2 + 0( 2 ) + 2 B√ p(U )V  δ R 2N N −2 N +2 R and | p(U )V | ≤ LN  4 ( |V | N −2 ) 2N , for some pure constant LN . Also,

R

2N

|V | N −2 =

N +2 + N2N −2 N −2

R

R

2N

UN −2 +

2N N −2

R

4

|U + tw | N −2 (U + tw )w 4

Br ×[0,1]

(1 − t)|U + tw | N −2 w2 = S

N 2

+ 0(1)(

N +2 2

1

+ ( r )N −2 ) 2

by H˘ older inequality, Lemma 3.2-ii and Lemma 4.1-i. Also, all estimates in (4.4) - (4.5) are uniform in r ≥ 1. Now, from (4.4)-(4.5) and

0=

Z Z Z 4 2N d 1 p(t V )V ] E(tV )|t=t = t [ |∇V |2 − tN −2 |V | N −2 − t dt

we get 4

|1 − tN −2 | = 0(1)( R because | p(t V )V | ≤ LN |V |

N +2 2

N

2N N −2

1  + ( )N −2 ) 2 r

( 2 )

N +2 2N

This proves the claim, because |t − 1| ≤

4

N −2 N −2 2 |t

− 1|.

References . [AM] Adimurthi and Mancini, A sharp condition in higher dimensions for a Brezis Nirenberg type equation, in preparation

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[AY] Adimurthi and Yadava, Critical Sobolev exponent problem in a ball with nonlinear perturbation changing sign, Advances in Differential equation 2,(1997), 161-182. [BG] Bernis F. and H. -Ch. Grunau, Critical exponents and multiple critical dimensions for polyharmonic operators, J.Differ Equations 117, (1995), 469486. [BN] Brezis H. and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical exponents, Comm.Pure. Appl. math.36 (1983), 437-477. [PS] Pucci P. and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures at Appl. 69 (1990), 55-83.

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