Advanced Nonlinear Studies 5 (2005), 337–350
Non-radial Maximizers For Functionals With
Exponential Non-linearity in R2
Marta Calanchi, Elide Terraneo
Dipartimento di Matematica ”F. Enriques”
Università di Milano, via Saldini, 50, 20133 Milano, Italy
e-mail: [email protected], [email protected]
Received 1 October 2004
Communicated by Jean Mawhin
Abstract
We consider the functional F : H01 (B(0, 1)) → R
Z
γ
F (u) =
|x|α (ep|u| − 1 − p|u|γ ) dx
B(0,1)
where α > 0, p > 0, 1 < γ ≤ 2, and B(0, 1) is the unit ball in R2 . We prove that for
any p > 0, 1 < γ < 2 and 0 < p < 4π, γ = 2 no maximizer of F (u) on the unit ball
in H01 is radially symmetric provided that α is large enough. This extends a result of
Smets, Su and Willem concerning the existence of non-radial ground state solutions for
the Rayleigh quotient related to the Hénon equation with Dirichlet boundary conditions.
2000 Mathematics Subject Classification. 35J60.
Key words. non-radial solutions, exponential growth.
1
Introduction
The present work is mainly motivated by a paper due to Smets, Su, Willem (see [14]) in
which the authors consider the problem of the symmetry of positive solutions for the Hénon
equation with Dirichlet boundary conditions
337
338
M. Calanchi, E. Terraneo

α p−1
in B(0, 1),
 −∆u = |x| u
u > 0 in B(0, 1),

u|∂B(0,1) = 0,
(1.1)
where B(0, 1) denotes the unit ball in RN , N ≥ 2, centered at the origin, α > 0, p > 2.
The existence of solutions in the Sobolev space H01 for N ≥ 3 was first considered by Ni
in 1982 [11]. He realizes that the presence of the weight |x|α widens the range of p for
which a solution exists. Ni proves via an application of the Mountain Pass Theorem in
the space of radial functions in H01 that problem (1.1) possesses a radial solution for any
2N
∗
p ∈ (2, 2∗ + N2α
−2 ), where 2 = N −2 is the critical exponent of the Sobolev embedding if
N ≥ 3. Moreover thanks to the Pohozaev identity, one can show that there are no solution
if p ≥ 2∗ + N2α
−2 . In contrast with the case in which the weight is a non-increasing radial
function (see [8]), (1.1) has not only radial solutions. Indeed in [14] Smets, Su, Willem
prove that for any p ∈ (2, 2∗ ), there exists at least a non-radial solution of (1.1) provided
that α is large enough. This result also holds in the case of dimension N = 2 where
2∗ = +∞. They study the ground state solutions of (1.1), i.e. functions which minimize
the Rayleigh quotient
R
|∇u|2 dx
B(0,1)
p
Rα (u) = u ∈ H01 ,
u 6= 0.
(1.2)
p2 ,
R
α |u|p dx
|x|
B(0,1)
Under suitable rescalings, the minimizers of (1.2) are solutions of (1.1). Then they establish that for α sufficiently large the infimum of (1.2) is attained by a non-radial function.
Theorem [Smets, Su, Willem [14]]. Assume N ≥ 2. For any p ∈ (2, 2∗ ) there
exists α∗ such that any minimizer of (1.2) is non-radial provided α > α∗ .
They also give some information about the behaviour of α∗ as p goes to 2 or to 2∗ . In
particular they show that α∗ goes to 0 when p tends to 2∗ and α∗ goes to +∞ if p tends to
2.
In [13] Serra investigates the existence of non-radial solutions to (1.1) in the critical
case p = 2∗ when the spatial dimension N is greater than or equal to 4. He proves the
following theorem:
Theorem [Serra [13]]. Let N ≥ 4. Then for every α > 0 large enough, the problem

α 2∗ −1
in B(0, 1),
 −∆u = |x| u
u > 0 in B(0, 1),

u|∂B(0,1) = 0,
admits at least one non-radial solution.
(1.3)
339
Non-radial maximizers
Of course this solution is not a ground state of the Rayleigh quotient. The idea of the
author is to find it as a critical point of the Rayleigh quotient (1.2) restricted to a wellchosen subspace of H01 invariant under the action of some subgroup of O(N ).
In this paper we are interested in the critical case when the spatial dimension N is equal
to 2. In this case any polynomial growth is allowed and the critical growth is given by the
Trudinger-Moser inequality (see [16], [10]), namely
Z
2
sup
eβu dx ≤ C(β) ≤ C(4π) < +∞ for β ≤ 4π,
(1.4)
kukH 1 ≤1
0
B(0,1)
Z
2
eβu dx = +∞ for β > 4π.
sup
kukH 1 ≤1
0
(1.5)
B(0,1)
The second relation (1.5) can be easily established by testing the functional
Z
2
eβu dx
B(0,1)
on a family of functions called Moser’s sequence defined as follows:
p


 log k
|x| <
1
k
1
wk (x) = √
log( 1 ) 1
2π 

 √ |x|
≤ |x| < 1.
log k k
We recall that while the best Sobolev constant in the critical case
Z
∗
sup
|u|2 dx = SN ,
kukH 1 ≤1
0
(1.6)
(1.7)
B(0,1)
is not assumed, the value C(4π) is attained (for this result we refer the reader to the works
of Carleson and Chang [4], De Figueiredo, do Ó and Ruf [5], and Flucher [7]).
In view of the Trudinger-Moser inequality, following [5], we say that a nonlinearity g(s)
has
• subcritical growth, if lim|s|→∞
• critical growth, if lim|s|→∞
g(s)
e4πs2
g(s)
e4πs2
= 0;
= 1.
Here we investigate an analogue of the Theorem of Smets, Su and Willem for nonlinearities with exponential growth in dimension N = 2. We concentrate our attention on the
following problem. Let F : H01 (B(0, 1)) → R be the functional
Z
γ
F (u) =
|x|α ep|u| − 1 − p|u|γ dx
(1.8)
B(0,1)
where α > 0, p > 0 and 1 < γ ≤ 2. We are interested in understanding for which values
of γ, p and α the supremum of F (u) on the set S1 = {u ∈ H01 : kukH01 ≤ 1} is finite
340
M. Calanchi, E. Terraneo
and attained. Then, as in the polynomial case, the natural question is to know whether it is
achieved by a radial or by a non-radial function. We denote by
Tα,p,γ = sup F (u)
u∈ S1
and by
R
Tα,p,γ
=
sup
F (u),
u∈ S1 , u rad
namely the supremum considered only over the sub-space of radial functions. Thanks to the
Trudinger-Moser inequality, Tα,p,γ is finite in the subcritical cases 1 < γ < 2, p > 0 and
γ = 2, 0 < p < 4π, and in the critical case γ = 2 and p = 4π. For these cases we prove
R
that the supremum Tα,p,γ
on radial functions differs from that on non-radial functions,
provided that α is large enough. Indeed we establish the theorem:
Theorem 1.1 For p > 0, 1 < γ < 2 and for 0 < p ≤ 4π, γ = 2, there exists α∗ > 0 such
R
that Tα,p,γ > Tα,p,γ
for any α > α∗ .
By an argument similar to that used in Lemma 2.1 in [6] one can prove that in the subcritical
R
are
cases p > 0, 1 < γ < 2 and 0 < p < 4π, γ = 2, both suprema Tα,p,γ and Tα,p,γ
achieved. Then in these cases an analogue of the result of Smets, Su and Willem holds:
Corollary 1.1 For p > 0, 1 < γ < 2 and 0 < p < 4π, γ = 2 there exists α∗ > 0 such
that no maximizer of F (u) on S1 is radial provided that α > α∗ .
Then we analyse the supercritical case, namely γ = 2 and p > 4π, and we show that
the supremum over the whole space H01 is not finite. In order to prove this it is enough to
evaluate the functional F on a suitable family of Moser’s type functions supported in balls
of radius tending to zero and centered at points approaching the boundary. In this way the
action of the weight |x|α becomes negligible and we are in the same situation described
by the Trudinger-Moser inequality (1.5). Moreover, in analogy with the result due to Ni in
[11], we can prove that for 4π < p ≤ 4π + 2πα the supremum over the subspace of radial
functions in H01 is finite. In fact we establish the following result:
Proposition 1.1 i) Tα,p,2 = +∞
if p > 4π;
R
R
ii) Tα,p,2
< +∞ if 0 < p ≤ 4π + 2πα and Tα,p,2
= +∞
if p > 4π + 2πα.
Finally we propose for γ = 2 a sort of generalization of the result described in Theorem
1.1. We consider the functional:
Z
2
Gλ (u) =
|x|α (epu − 1 − λpu2 )dx
(1.9)
B(0,1)
341
Non-radial maximizers
where λ ∈ [0, 1], α > 0 and 0 < p ≤ 4π and denote
Tα,λ =
sup
kukH1 ≤1
0
Gλ (u) =
sup
Gλ (u)
(1.10)
kukH1 =1
0
(where the last equality is due to the monotonicity properties of the functional Gλ ).
We prove the following:
Theorem 1.2 Let 0 < p ≤ 4π. Then there exists 0 ≤ λ∗ < 1 such that for any
R
λ ∈ [λ∗ , 1], Tα,λ > Tα,λ
, for large values of α.
The presence of the term λpu2 in the functional (1.9) seems to us to be in a certain way
necessary: in fact we do not know if λ∗ can be equal to zero.
R
For the subcritical case we also prove that both suprema Tα,λ and Tα,λ
are achieved
and so we obtain:
Corollary 1.2 For any 0 < p < 4π, there exists 0 ≤ λ∗ < 1 such that for any
λ ∈ [λ∗ , 1] no maximizer of Gλ (u) on S1 is radial for large value of α.
Finally we point out that any maximizer of (1.10) satisfies for some constant c the
following elliptic problem

2

 −∆u = c|x|α u(epu − λ) in B(0, 1),
u > 0 in B(0, 1),

 u|
∂B(0,1) = 0.
(1.11)
Of course the constant c cannot be computed explicitly because of the non-homogeneity
of the exponential function. So there is no direct correspondence between equation and
ground state solutions (as it happens for the Hénon equation).
The question of existence of non-symmetric solutions for symmetric problems has been
widely investigated (we refer the reader to the survey of Brezis ([2]) for a complete bibliography). Here we only mention the paper of Brezis and Nirenberg [3], in which the authors
prove that if the domain is an annulus there exists a non-radial solution for (1.1) in the
autonomous case (i.e α = 0). The results of existence for the Hénon equation in [14] and
[13] and in Theorem 1.1 above suggest that the coefficient |x|α has a similar effect to the
presence of a ”hole” in B(0, 1), when α is large enough. We also want to remark that in
spite of these ”simmetry breaking phenomena” the maximizers may still present a weaker
symmetry, as suggested in the paper of Smets and Willem [15].
342
2
M. Calanchi, E. Terraneo
Proof of Theorem 1.1
In the present section we consider the subcritical cases 1 < γ < 2, p > 0 and γ = 2,
0 < p < 4π, and the critical case γ = 2 and p = 4π and we prove that the supremum on
R
non-radial functions Tα,p,γ is greater than the supremum on radial functions Tα,p,γ
at least
R
for α large enough. Our first result is an asymptotic estimate of Tα,p,γ for α → +∞.
Lemma 2.1 (Asymptotic estimate). For p > 0, 1 < γ < 2 and 0 < p ≤ 4π,
γ
2
R
, for α → +∞ where S(2γ) is the Sobolev constant
γ = 2, then Tα,p,γ
∼ 2 S(2γ)p
1+γ
α
R
2γ
S(2γ) = supkukH 1 ≤1 B(0,1) |u| dx.
0
The proof of Lemma 2.1 relies on the following lemma.
Lemma 2.2 There exists a positive constant C such that for any u ∈ H10 and any
q ≥ 2 the following inequality holds:
i q1
h q
+1
kukLq ≤ C Γ
kukH10 ,
2
where Γ(λ) =
R +∞
0
(2.12)
e−x xλ−1 dx is the Gamma function.
Proof of Lemma 2.2. Since H10 ,→ Leu2 −1 , where Leu2 −1 is the Orlicz space defined
2
via the convex function ϕ(t) = et − 1, t ∈ [0, +∞), the inequality (2.12) is a direct
consequence of
i q1
h q
+1
kukL u2 ,
(2.13)
kukLq ≤ Γ
e
−1
2
established in [12]. For the convenience of the reader we recall in the appendix the definition of the Orlicz space Leu2 −1 and the proof of the inequality (2.13).
Proof of Lemma 2.1. For a radially symmetric function consider the transformation
√
2
, introduced in the paper of Smets, Su and Willem [14].
u(|x|β ) = β w(|x|) , β = α+2
By an easy computation we have
Z
Z
γ γ
γ
γ
|x|α ep|u| − 1 − p|u|γ dx = β
epβ 2 |w| − 1 − pβ 2 |w|γ dx;
B(0,1)
B(0,1)
and
Z
|∇w(x)|2 dx ≤ 1.
B(0,1)
So we obtain
R
Tα,p,γ
=
Z
sup
kwkH1 ≤1, w rad
0
β
B(0,1)
γ
γ
γ
epβ 2 |w| − 1 − pβ 2 |w|γ dx.
(2.14)
343
Non-radial maximizers
The control from below of the integral in (2.14) comes from the following inequality:
R
Tα,p,γ
≥
β 1+γ p2
2
kwkH 1 ≤1
Z
|w|2γ dx =
sup
0
B(0,1)
β 1+γ p2
S(2γ).
2
R
In order to obtain an estimate from above of Tα,p,γ
we expand the exponential function in
series to get:
Z
β
+∞ Z
γ
X
γ
γ
epβ 2 |w| − 1 − pβ 2 |w|γ dx = β
B(0,1)
=
k=2
β 1+γ p2
2
Z
|w|2γ dx + β
B(0,1)
+∞ Z
X
k=3
X
β 1+γ p2
S(2γ) + β
2
k=3
B(0,1)
B(0,1)
k
k
γ
B(0,1)
+∞ Z
≤
γ
(pβ 2 |w|γ )
dx
k!
(pβ 2 |w|γ )
dx
k!
γ
k
(pβ 2 |w|γ )
dx.
k!
Now we estimate the series: thanks to Lemma 2.2 we have
γk
kukγk
γk ≤ C Γ(
γk
+ 1)kukγk
H10
2
so in order to control the integral in (2.14) from above it is enough to show that the series
γ k
+∞
X
γk
(pβ 2 )
Γ
+ 1 C γk
k!
2
k=3
converges, since kukH10 ≤ 1. We recall that the Gamma function has a minimum localized
between 1 and 2 and it is increasing afterwards. So the last series is controlled by
γ
+∞
X
pβ 2
k=3
k
C γk Γ(k + 1)
3γ
= O(β 2 )
k!
(2.15)
for β small enough. So we obtain the estimate from above
R
Tα,p,γ
≤
3γ
β 1+γ p2
S(2γ) + O(β 1+ 2 ).
2
Now we are able to prove Theorem 1.1.
Proof of Theorem 1.1. Let u be a positive smooth function with support in B(0, 1)
and such that kukH10 = 1. Following Smets, Su and Willem let us consider uα (x) =
344
M. Calanchi, E. Terraneo
u(α(x − xα )), where xα = (1 − α1 , 0). Since kuα kH10 = 1 and it has support in the ball
B(xα , α1 ), by the change of variables y = α(x − xα ) we obtain:
Z
γ
Tα,p,γ ≥
|x|α ep|uα | − 1 − p|uα |γ dx
B(0,1)
Z
γ
|x|α ep|uα | − 1 − p|uα |γ dx
=
1
)
B(xα , α
≥
2
α
1−
α
1
α2
Z
γ
ep|u| − 1 − p|u|γ
dy
B(0,1)
2γ S(2γ)p2
,
α1+γ
R
and so Tα,p,γ ≥ αC2 , for α → +∞. Since, by Lemma 2.1, Tα,p,γ
∼
R
we are able to conclude that Tα,p,γ > Tα,p,γ for α large enough.
as α → +∞
Proof of Corollary 1.1. In view of Theorem 1.1 it is enough to show that both suprema
R
Tα,p,γ and Tα,p,γ
are achieved. For simplicity we only consider the case of the supremum
Tα,p,γ for γ = 2, 0 < p < 4π. Let {un } be a maximizing sequence such that kun kH01 ≤ 1.
Then, up to a subsequence, there exists u ∈ H01 such that un * u weakly in H01 and
strongly in Lq , ∀q ≥ 1 and un (x) → u(x) a.e. in B(0, 1). By using Lemma 2.1 in [6] in
order to prove that
Z
Z
2
2
α
pun
2
|x| e
− 1 − pun dx →
|x|α epu − 1 − pu2 dx as n → +∞
B(0,1)
B(0,1)
it is enough to show that
Z
2
|x|α epun |un |dx ≤ C.
(2.16)
B(0,1)
Since p < 4π, there exists q > 1 such that pq < 4π. By the Hölder inequality, if q 0 is the
conjugate exponent of q, one has
Z
α pu2n
|x| e
B(0,1)
Z
|un |dx ≤
! 10
q
q0
|un |
B(0,1)
! q1
Z
e
pqu2n
≤C
B(0,1)
which is a direct a consequence of the Trudinger-Moser inequality.
Remark. We do not know if Corollary 1.1 holds in the case p = 4π and γ = 2. In fact
thanks to equality (2.14) and to the Theorem of de Figueiredo, do Ó and Ruf below, it is
R
easy to show that the supremum Tα,4π,2
is achieved. On the other hand it is not clear if the
supremum Tα,4π,2 is attained or not.
Theorem [de Figueiredo, do Ó, Ruf [5]] Suppose that g satisfies:
i) g ∈ C 1 (R);
ii) g is increasing on R+ and g(s) = g(|s|);
345
Non-radial maximizers
2
iii) 0 ≤ g(s) ≤ e4πs − 1, for any s ≥ 0;
iv) g(s) has subcritical growth.
Then
Z
sup
kukH 1 ≤1
0
g(u)dx
B(0,1)
is attained.
3
A remark on the supercritical case
In the present section we prove that in the supercritical case γ = 2, p > 4π the supremum
Tα,p,2 is not finite. In order to prove this we evaluate the functional F (u) on some particular
functions obtained by a suitable translation and dilation of Moser’s functions in a region of
B(0, 1) far from the origin where the presence of |x|α can be neglected.
Proof of Proposition 1.1 (Case i). Consider the following family of functions
p

log k





1
)
1  log( α|x−x
α|
wk,α (x) = √
√
2π 
log k





0
|x − xα | <
1
kα
1
1
≤ |x − xα | <
kα
α
1
|x − xα | ≥
α
(3.17)
where xα = (1 − α1 , 0), α > 2, k > 2. We observe that kwk,α kH01 = 1. Moreover,
Z
2
2
|x|α epwk,α − 1 − pwk,α
dx
Tα,p,2 ≥
B(0,1)
Z
2
2
|x|α epwk,α − 1 − pwk,α
dx
≥
1
B(xα , kα
)
α
≥
1−
=
1−
2
α
Z
2
α
α p
e 2π log k − 1 −
1
B(xα , kα
)
p
k 2π − 1 −
p
log k dx
2π
π
p
log k
2π
α2 k 2
and the last term tends to +∞ when k → +∞ since p > 4π.
If we restrict our attention to the supremum on radial functions the situation is different.
In this case the factor |x|α , α > 0 improves the convergence properties of the integral and
we have the following result:
Proof of Proposition 1.1 (Case ii). For a radially symmetric function, using the same
346
M. Calanchi, E. Terraneo
change of variables as in Proposition 2: u(|x|β ) =
R
Tα,p,2
=
Z
sup
β
||w||H 1 ≤1, w rad
√
βw(|x|) , β =
2
α+2 ,
we get
2
epβw − 1 − pβw2 dx.
(3.18)
B(0,1)
0
Thanks to the Trudinger-Moser inequality quoted above this quantity is finite if and only if
p verifies 0 < p ≤ 4π
β = 2πα + 4π.
R
Remark. We point out that the supremum on radial functions Tα,p,2
is attained for any
0 < p < 4π + 2πα, α > 0. Indeed, this result is a direct consequence of equality (3.18)
and of the Theorem of de Figueiredo, do Ó, Ruf [5] quoted at the end of the second section.
In the limit case p = 4π + 2πα, α > 0 it is not known if the supremum is achieved (see
Theorem 5 in [5]).
4
A generalization
In the present section we consider
Z
Tα,λ =
2
|x|α epu − 1 − pλu2 dx
sup
kukH1 ≤1
0
(4.19)
B(0,1)
where α > 0, 0 < p ≤ 4π and 0 ≤ λ ≤ 1 and we prove Theorem 1.2, which generalizes
Theorem 1.1 for γ = 2.
Proof of Theorem 1.2. The case λ = 1 was treated in Theorem 1.1; so here we only
analyse the case λ < 1. At first, in a similar way as in Lemma 2.1, we establish the
following asymptotic estimate for the supremum on radial functions in the unit ball of H01 :
R
Tα,λ
∼
4p(1 − λ)
α 2 λ1
as α → +∞,
where λ1 is the first eigenvalue of the Laplacian in H01 (B(0, 1)). Indeed, thanks to the
√
2
, we obtain
transformation u(|x|β ) = β w(|x|), β = α+2
R
Ta,λ
Z
=
sup
kwkH 1 ≤1,w rad
0
β
2
epβw − 1 − pβλw2 dx.
B(0,1)
Now the estimate from below comes from the inequality:
R
Ta,λ
≥
sup
kwkH 1 ≤1
0
β 2 p(1 − λ)
Z
B(0,1)
w2 dx =
pβ 2 (1 − λ)
.
λ1
(4.20)
347
Non-radial maximizers
On the other hand we control the supremum from above by using a Taylor expansion of the
exponential term, namely
Z
2
β
epβw − 1 − pβλw2 dx =
B(0,1)
Z
2
2
= β p(1 − λ)
w dx + β
B(0,1)
+∞ k k Z
X
p β
k=2
k!
(4.21)
w
2k
dx.
B(0,1)
Then, thanks to Lemma 2.2, there exists a constant C > 0 independent of w such that
2k
2k
kwk2k
2k ≤ C Γ(k + 1)kwkH 1 ,
0
so, since kwkH01 ≤ 1, we get
+∞
X (pβC 2 )k
β 2 p(1 − λ)
+β
Γ(k + 1)
λ1
k!
R
Tα,λ
≤
k=2
(4.22)
p2 C 4
β 2 p(1 − λ)
+ β3
=
λ1
1 − pβC 2
2
for β = α+2
small enough.
For the supremum on the whole unit ball in H01 , since λ ≤ 1, one has
Z
Tα,λ =
2
|x|α epu − 1 − pλu2 dx
sup
kukH1 ≤1
0
B(0,1)
Z
≥
sup
kukH1 ≤1
0
2
|x|α epu − 1 − pu2 dx
B(0,1)
and as in the proof of Theorem 1.1 we can show that there exists C > 0 independent of
λ such that Tα,λ ≥ αC2 , for α large enough. The conclusion of the theorem follows by
∗
)
< C.
choosing λ∗ such that 4p(1−λ
λ1
Proof of Corollary 1.2. The proof follows the same lines of Corollary 1.1.
As mentioned in the introduction we are not able to establish Theorem 1.2 if λ = 0. Indeed
in this case we obtain that
R
Tα,0
∼
4p
, as α → +∞;
α 2 λ1
on the other hand for the supremum on all functions Tα,0 we give an estimate from below,
namely Tα,0 ≥ αC2 for α big enough with a constant C < λ4p1 . So the same argument as in
Theorem 1.2 cannot apply.
348
5
M. Calanchi, E. Terraneo
Appendix
In this last section we recall the definition of the Orlicz space Leu2 −1 and we give the proof
of equality (2.13) established in [12]. We refer to [1] and [9] for a wide presentation of
the Orlicz spaces.
Let ϕ : [0, +∞[→ [0, +∞[ be a convex, strictly increasing function such that lim+ ϕ(s) =
s→0
ϕ(0) = 0 and lim ϕ(s) = +∞.
s→+∞
Definition 5.1 The Orlicz space Lϕ is defined by
Lϕ = {u measurable on B(0, 1) such that ∃ K > 0
)
u(x) dx < +∞ .
with
ϕ K B(0,1)
Z
It can be proven that Lϕ is a vector space (see [9]). For every u ∈ Lϕ , let us define the
functional
)
(
Z
u(x) dx ≤ 1
kukϕ = inf K > 0 such that
ϕ K B(0,1)
One can prove that k · kϕ is a norm on Lϕ and (Lϕ , k · kϕ ) is a Banach space (see [9]).
It is easy to verify that the function ϕ(u) = |u|p , 1 < p < +∞ defines the Lp space.
2
In the following we will consider the Orlicz space defined by ϕ(u) = eu − 1. For this
particular space we have:
Lemma 5.1 For every p ≥ 2 we have Leu2 −1 ,→ Lp and
h p
i p1
+1
kukL 2 .
kukLp ≤ Γ
eu −1
2
where Γ(λ) =
R +∞
0
(5.23)
e−x xλ−1 dx.
Proof. Let K = kukL
2
eu −1
> 0, otherwise inequality (5.23) is obvious. We will use the
property that
ex − 1 ≥
xρ
,
Γ(ρ + 1)
(5.24)
for every ρ ≥ 1 and x ≥ 0. This is easily proven when ρ is an integer. For x ∈ (0, 1]
and for every ρ > 1 the inequality follows by using the fact that Γ(ρ + 1) ≥ 1 if ρ ≥ 1,
and ex − 1 ≥ xρ . For x ∈ [1, +∞) and 1 < ρ < 2 the inequality (5.24) is implied by
the fact that ex − 1 − x2 ≥ 0 and the property Γ(ρ + 1) ≥ 1. The general inequality for
x ∈ [1, +∞) and 2 < ρ follows from the case 1 < ρ < 2 by an easy computation, since
349
Non-radial maximizers
Γ(ρ + 1) = ρΓ(ρ). By the theorem of monotone convergence for K = kukL
have
Z
1≥
2
eu −1
> 0 we
|u| 2
e( K ) − 1 dx
B(0,1)
and by (5.24)
Z
1≥
B(0,1)
for every ρ ≥ 1. Thus we obtain 1 ≥
|u|
K
2ρ
Γ(ρ + 1)
kuk2ρ
2ρ
2ρ
K Γ(ρ+1)
dx
and we conclude that K ≥
kuk2ρ
1
(Γ(ρ+1)) 2ρ
.
Acknowledgments. The authors would like to thank Bernhard Ruf and Enrico Serra for
bringing this problem to their attention and for stimulating discussions.
References
[1] R.A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic
Press, New York-London, 1975.
[2] H. Brezis, Symmetry in Nonlinear PDE’s, Differential Equations: La Pietra 1996
(Florence), Volume 65 of Proc. Sympos. Pure Math., pages 1–12, Amer. Math. Soc.,
Providence, RI, 1999.
[3] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations
involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (4) (1983);
437–477.
[4] L. Carleson and S.A. Chang, On the existence of an extremal function for an
inequality of J. Moser, Bull. Sci. Math. (2) 110 (2) (1986); 113–127.
[5] D.G. de Figueiredo, J.M. do Ó, and B. Ruf, On an inequality by N. Trudinger and
J. Moser and related elliptic equations, Comm. Pure Appl. Math. 55 (2) (2002);
135–152.
[6] D.G. de Figueiredo, O.H. Miyagaki, and B. Ruf, Elliptic equations in R2 with
nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations
3 (2) (1995); 139–153.
[7] M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv. 67 (3) (1992); 471–497.
[8] B. Gidas, W.M. Ni, and L. Nirenberg, Symmetry and related properties via the
maximum principle, Comm. Math. Phys. 68 (3) (1979); 209–243.
350
M. Calanchi, E. Terraneo
[9] L. Maligranda, Orlicz Spaces and Interpolation. IMECC, 1989.
[10] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J.
20 (1970/71); 1077–1092.
[11] W.M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications,
Indiana Univ. Math. J. 31 (6) (1982); 801–807.
[12] B. Ruf and E. Terraneo, The Cauchy Problem For a Semilinear Heat Equation With
Singular Initial Data, Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), Volume 50, Progr. Nonlinear Differential Equations Appl., pages 295–
309. Birkhäuser, Basel, 2002.
[13] E. Serra, Non radial positive solutions for the Hénon equation with the critical
growth, To appear in Calc. Var. PDEs.
[14] D. Smets, J. Su, and M. Willem, Non-radial ground states for the Hénon equation,
Commun. Contemp. Math. 4 (3) (2002); 467–480.
[15] D. Smets, and M. Willem, Partial symmetry and asymptotic behavior for some
elliptic variational problems, Calc. Var. PDEs 18 (2003); 57–75.
[16] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J.
Math. Mech., 17 (1967); 473–483.