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54
MSC 34K30, 35C15, 35J05, 35J25 DOI: 10.14529/mmpl50303
WEIGHTED TRUDINGER - MOSER INEQUALITIES AND APPLICATIONS
M. Calanchi, The University of Milan, Milan, Italy, [email protected], B. Ruf, The University of Milan, Milan, Italy, [email protected]
Trudinger - Moser inequalities provide continuous embeddings in the borderline cases of the standard Sobolev embeddings, in which the embeddings into Lebesgue Lp spaces break down. One is led to consider their natural generalization, which are embeddings into Orlicz spaces with corresponding maximal growth functions which are of exponential type. Some parameters come up in the description of these growth functions. The parameter ranges for which embeddings exist increase by the use of weights in the Sobolev norm, and one is led to consider weighted TM inequalities. Some interesting cases are presented for special weights in dimension two, with applications to mean field equations of Liouville type.
Keywords: Trudinger - Moser inequalities; Orlicz spaces; maximal growth functions; weighted TM inequalities.
To the memory of Alfredo Lorenzi, with affection, admiration and gratitude.
Qual e '1 geometra che tutto s'affige per misurar lo cerchio, e non ritrova, pensando, quel principio ond' elli indige, tal era io a quella vista nova: veder voleva come si convenne 1'imago al cerchio e come vi s'indova
Dante Alighieri, Par. XXXIII133-138
1. Introduction
1.1. Why Trudinger - Moser Inequalities?
For tt C RN a bounded domain, the classical Sobolev embeddings are
p N
Wl'p(tt) ^ Lq(tt) if p < N and 1 < q -= p*.
N — p
The Trudinger - Moser inequalities concern the borderline cases
p = N, for which form ally 1 < q < p* = which leads to the question whether
Wl'p(tt) C L^(tt).
The answer is no, as simple examples show. The natural question then is to find the maximal growth function 0 : R ^ R+ such that
if u e WFtt). then [0(u)ix is finite.
The answer was given by Yudovich [1], Peetre [2], Pohozaev [3] and Trudinger [4]: from their works one knows that (Q) embeds into the Orlicz space Lwith growth function
N
(so-called N -function) ((t) = e^ -1 — 1. This embedding is known by the name Trudinger inequality.
The proof is based on an expansion in power series of the exponential function, and
on a control of the Lp norm of each term of the series. This growth is optimal in the
sense that for any higher growth there exist functions u G Wq'n(Q) for which the integral
J ((u)dx becomes infinite. Trudinger's inequality was made more precise by J. Moser [5], n
who showed that the supremum on the unit ball in W^'N(Q)
N
sup
f |V«|Ndx<1 .
is bounded if and only if a < aN = Nu^^, where uN-1 is the N — 1-dimensional surface of the unit sphere. The integral is actually finite for any positive a, tat if a > aN it can
u
From now on we will consider the two dimensional case. Let us recall Moser's result (known as the Trudinger - Moser inequality) in this case
Theorem 1. (Moser 1970 [5]). Let N = 2; then
f { C|Q| if a < 4n sup eau <\ (1)
/ \vu\2dx<iJ I + ro if a >
n n
With a modification of the same arguments Moser also proved a corresponding inequality on the sphere S2 which is useful for application in conformal deformation theory on manifolds.
u 2 S2
that
J iVul2dn < 1, J ud/1 = 0, s2 s2
there exists a constant C such that
J e4nu2 d/ < C. (2)
s2
These inequalities are very useful to study 2-dimensional problems, such as the problem of prescribed Gauss curvature and mean field equation. In these contexts there is a more handy form.
1.2. Logarithmic Trudinger - Moser Inequalities
A variant of the TM inequality is the following logarithmic TM inequality.
Theorem 3. Let Q C R2 be a bounded domain. Then there exists a constant C > 0 such that
log/ eu dx < i^n / \Vu\2 dx + C , u e H1(Q), (3)
n n
while for the 2-dimensional unit sphere S2 we have
2 /\Vu\2 dx - 8nlog/ eu dx >-C, u e H 1(S2), /u = 0. (4)
S2 S2 S2
Proof. Using
a2
ab < — + e2b2 (5)
applied to b = \\yu\\2> a = ||Vu|2 and e2 = 4n, one has
ull 2
eu dx< i e "Vu"2dx< i e4^^2 + ^^"Vu"2dx <
n n n
< e |Vu|2 sup f e4nu2dx < Cet^|Vu|2 (6)
l|Vu||2<1 J
n
in virtue of the Trudinger - Moser inequality. Passing to the logarithm one obtains
log/eudx < ^||Vu||2 + logC (7)
n
so that (3) is proved.
□
The logarithmic TM inequality is crucial in the study of mean field equations of Liouville type (see [6]) of the form
eu
- Au = A -7— , in Q C R2
ieu w
u =0 on dQ.
Equation (8) has been derived by Caglioti, Lions, Marchioro and Pulvirenti [7,8] from the mean field limit of point vortices of the Euler flow, see also Chanillo - Kiessling [9] and Kiessling [10]. Equation (8) occurs also in the study of multiple condensate solutions for the Chern - Simons - Higgs theory, see Tarantello [11,12].
In particular, it has been shown (see e.g. [13,14]) that equation (8) has a solution which is a global minimizer of the associated energy functional if
A < 8n. (9)
Here we concentrate our attention on some extensions of the previous results.
1.3. Weighted Trudinger - Moser Inequalities
In some recent papers the influence of weights for such kind of inequalities has been studied.
In [15,16] the authors consider the effect of power weights on the maximal growth in the integral term. In [17-19] the interest is devoted to the impact of weights in the Sobolev norm.
More precisely, let w E L1(Q) be a non-negative function, and consider the weighted Sobolev space
Hl(tt,w) = cl{u E C^(tt) ; J\Vu\2w(x)dx < ro}. (10)
n
For a general embedding theory for weighted Sobolev spaces, see Kufner [20].
It turns out that for weighted Sobolev spaces of form (10) logarithmic weights have a particular significance. However, as we will see below (Remark 2), we will need to restrict attention to radial functions to obtain interesting results. We are therefore led to consider problems of the following type: let B C RN denote the unit ball in RN, and consider the weighted Sobolev space of radial functions
Ho,rad(B,w) = cl{u E CZad(B) ; IMlW := J \^u\2w(x)dx <
n
where
/1 \P ( e \в
w(x) = ( log -—г ) or w(x) = ( log -—г I , в > 0. (11)
V \x\/ \ \x\J
Theorem 4. (Calanchi - Ruf, [17]) Let в E [0,1) and, w0(x) = (log ^ or wi(x) (tog x)13■ Then
(a) elulYdx < for all u E H1rad(B,w) ^^ Y < Y ■= ^-ъ
J ' 1 — в
and
(b) sup I eaWdx <
IM|to<1,rad .
B
if and only if
a < a1 = 2 [2n(1 - /)]— = 2 [2^(2/7)]y/2 (critical growth).
Remark 1. This result extends the Trudinger - Moser inequality; indeed, for / = 0 we recover the classical TM inequality.
The proof is quite technical and we refer the reader to [17] for details. The idea is to transform the problem (which is one-dimensional by the radial symmetry) to an integral inequality on the half-line. In fact, by introducing the variable t by \x\ = e-t/2 and rescaling we obtain
^(t) = 2— [2n(1 - 3)]1/2 u(x) , a C
1
2 [2^(1 - в)] —
Then
oo . oo
I \Vu\2 \ log \xf dx = I ^^ dt and I eau dx = I ear'-t dt.
J J 1 - P m(B) J J
B 0 B 0
For the necessity of the conditions in (b) it is sufficient to test
e^1 -t dt
0
on the following Moser type functions
ti-ß
Vk (t) = ^ k X2ß
1-ß
k— t> k
0 < t < k
which satisfy \\nk\\w = 1- If Y = jrg> then
J eank-t dt > J e -t dt = eak-k ^^ if a > 1. 0 k
For the subcritical and critical cases, we need an estimate for functions ^ which satisfy
oo .
w = J dt < 1. In fact, by a radial lemma, see [17]), for such ^ it holds that 0
№(t)|< t1^.
With this estimate the subcritical case can be trivially proved. If y < j—g and a < 1, one easily obtains
oo oo
Ieár'"dt < Ie"" dt<
00
As concerns the critical exponent (that is, the most delicate case) y = j—g and <5=1, we follow an idea of Leckband [21], proving that the set of sublevels of the functions in the integral behaves well, and it is controlled by an integrable function on [0,
Going to the limiting case 3 = 1 in Theorem 4, one has that the exponent y of u in the integral goes to infinity, that is we are again in a borderline case. But again, the embedding does not go into L°, but we find a critical growth of double exponential type, as described in the following
Theorem 5. [17] Let wi(x) = | log (j^) | (¿.e. 3 = 1). Then,
I \u\2
(a) / eeU dx < , V u £ H^ rad(B,wl),
0,
B
while
(b) sup I eae2nu dx < ^^ a < 2.
IMUi <l,rad ,
2
B
Next, we consider the case 3 > 1. In this case the functions in H^ rad(B,w\) are bounded:
Theorem 6. [17] Let 3 > 1- Then we have the following embedding
Hi,rad(B,wi) ^ L~(B).
Also in these cases the idea is to reduce the problem to a one-dimensional integral inequality.
We conclude this first part with the following remark which explains why we have to restrict to radial functions.
Remark 2. If we do not restrict to radial function, that is, if we consider the whole weighted space H(B, w), and the problem to find a, 3,7 so that for u E Hl(B,w)^
sup I eaU dx < то , М» <1 ■
< B
then we note that for 7 = 2 and a > 4n the supremum is not finite. Indeed, translating Moser's functions to a region in which the presence of the weight is not substantial (see [17]) one concludes that the presence of the weight has no effect for increasing the range of the inequality.
1.4. Liouville Type Equations with Logarithmic Coefficients
In this article we will derive some new logarithmic Sobolev inequalities with weights, and will then apply them to prove existence results for associated mean field equations of Liouville type, with elliptic operators with logarithmic coefficients. In particular, we will prove
( e V
Theorem 7. Let w = w(x) = (log —J be a logarithmic weight, with в > 0. Then, for any X > 0, the equation
eu
—div(w(x)Vu) = X^--+ f(x) in B
Be
u =0 on dB
has a (weak) radial solution u E H(B,w) which is a global minimizer of the associated energy functional.
We recall that for the corresponding equation with в = 0 there exists a global
X
X < 8n. (13)
This is due to the fact that for в > 0 the growth eu is not the maximal possible. If we increase the growth to critical growth eu, then we find an analogous restriction to (13) to obtain a global minimizer for the associated equation:
Theorem 8. Let w(x) = (log X Y■ Then, with 9 = 2+y? equation —div(w(x) Vu) = A l l - . .„--+ f(x) in B
V V ) J j eya\e J \ ! ^
u =0 on dB
has a weak radial solution, which is a global minirnizer of the associated energy functional, A
2n 2+7
A < — (2 + y) ~. (15)
Note that for 9 = 1, i.e. y = 2 the condition (15) reduces to A < 8n, which is the condition (9) of the mean field Liouville equation.
Finally, we prove an existence theorem for the following elliptic mean field equation with a double exponential growth
Theorem 9. Let w = log Then the equation
eu ee
—div(logx v^ = Aloogjeeu J^ + f(x) in B m
jj
u =0 on dB
has a global radial minirnizer solution for every
A < n . (17)
2. Weighted Logarithmic TM Inequalities
In this section we prove some logarithmic TM inequalities with weights. We start with a simple inequality based on the previous weighted TM embeddings, and apply it to prove Theorem 7.
Proposition 1. Let w(x) = ^log -¡X^ with 3 > 0. Then for every A > 0 there exists a constant C = C(A, 3) which depends on A and 3 such that
2 Ml — A log / eu > 4\\u\\l — C (18)
j
where
Mi = / lvul2w(x) dx
j
Proof. Let 3 E (0,1). Reasoning as in Remark 1, using again inequality (5), we have
log / eu < log I / eii2(«w)2+e2|HlW dx I <
_j_Vj_)__
/|g Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 3, pp. 42-55
< e2\\u\\2w + log yj e (m» )2 dx
Now we choose e2 = There exists a constant C1 = C1(j, A) which depends only on e (hence on A) and 7 = i—g > 2, such that
J e(WW)2 dx < J eCl+a~<(WW)Y dx ; jj
here aY is the critical exponent given in Theorem 4. By (b) of the same theorem we have that
log ij e(WW)2 dx J < log J" eCl+a'f(WW)Y dx jj
< Ci(7, A) + log / eaY(mw>' dx < C.
в
With a similar argument (using Theorem 5 and Theorem 6) one can prove the result in the case 3 > 1-
□
3. Applications to Liouville Type Equations
Consider the following elliptic boundary value problem
eu
—div(w(x)Vu) = Ay—u + f in B
e
в
u =0 on dB
( e N в
where w = w(x) = (log -—-^j is a logarithmic weig ht with в > 0 , an d f E L2(B). We look for weak solutions of (LP), i.e. a function u E Hrad(B,w) satisfying
Jeuv dx
VuVv w(x)dx = ABу----+ fv dx, Vv E C0^rad(B) ,
j J eudx J
в в B
i.e. u is a weak solution if and only if it is a critical point of the functional J : Hirad(B,w) ^ R associated to equation (19) and given by
J(u) = 2 NI2 — A log J" eudx — J fu dx.
вв
Proposition 2. For every A > 0 problem (19) has a weak radial solution which corresponds to the minimum of the functional J(u).
Proof. We first prove that J is coercive for every A fixed. Indeed by Proposition 1, one has
J(u) > 4\\u\\l - C - D\\u\\w\\f ||2
for a suitable constant D. Then J is bounded from below. Let
m = inf J(u)
пен
and let {un}n be a minimizing sequence, i.e.
J(un) ^ m, as n ^
By the coercivity of J, un is bounded in Щrad(B,w). Therefore there exists u E Hq rad(B,w), such that (up to a subsequence)
un
u in Hqrad(B,w) , un ^ u in L1(B) , un ^ u a.e. , as n ^
Mn
Observe that the nonlinearity eu is subcritical and that there exists a constant Ci such that
| tef | < C^a ^. (20)
From Theorem 4 and this estimate we have that
eUn E L1 and J \eUn un\dx < C. j
We now apply the following result due to de Figueiredo - Miyagaki - Euf [22, lemma 2.1].
Lemma 1. Let (un)n be a sequence of functions in L1(Q) converging to u in L1(Q). Assume that F(un(x)) and F(u(x)) are also L1 functions. If
\F(un(x)) un(x)\ < C
then F(un(x)) converges to F(u(x)) in L1. We then obtain, for F(t) = el
eUn ^ en , as n ^ ,
в в
and by lower semi-continuity of the norm \\ • \\
m < J(u) < liminf J(un) = m .
n—
So that u is a global minimum (and then a critical point) of J. This completes also the proof of Theorem 7.
□
_^
4. A Critical Weighted TM Inequality
( 1
Let w(x) = ^log —J the logarithmic weight with 3 E (0,1). Consider the functional J\ : H rad(B,w) ^ R defined as follows
Jx(u) := 1 \\u\\w — A log ^ j eH* dx J — j fudx (21)
where \u\W = j |Vu(x)|2w(x) dx, 6 = 2+- (y is the critical growth given by (b), Theorem
b Y
4), and f E L2(B).
2^ 2+y
Proposition 3. J\ is coercive for all A < — (2 + 7) y .
Y 2
We need a preliminary inequality Lemma 2. Let 8, 8' two conjugate exponents. For every s, t > 0 one has
(as)6' ts
st < + -f^, Wa > 0. (22)
~ 8' 8as v '
Proof. This is a classical convexity inequality, which can be deduced by Jensen's inequality. Here we give an alternative proof. In fact, considering
sup{st--sp}
8>0 P
we get, by setting the derivative equal zero in the maximum point, t = sp-1, hence s(t) = and then
sup{st — 1 sP} = t1/(p-1)t — 1 tp/(p-1)
s>0 p p
= (1 — ^)t tp/(p-1) = 1 tq. p q
To obtain other constants in the inequality, it is sufficient to consider
1 (as)y ts (as)(— t) < + , Wa > 0.
v a ' " 8' 8as
□
Proof, (of Proposition 3) We first prove that
log / eJul'dx < lu^r + log I / ea( «Wdx
A(7)
BB
where AM = £ (2 + 7)^ and a7 = 2 [2n(2/7)]-^2 is the critical expo,rent giver in
Theorem 4 (notice that A(2) = 16n and a2 = 4n gives the classical TM inequality for the Liouville problem).
We use inequality (22), where 5,5' are conjugate exponents, such that
r 95 = 7 u . \ 5
< that is <
I 95' = 2 [5'
One has, by taking s = \\u\\dw and t = (pjp)e in (22),
I II II 2
u s„, u12
s (fit r+m
and the first inequality is proven. Let
Then
C = sup ea'1 ( hi™ )7 dx .
B
J\(u) := 1 \\u\\W — A log [I eu<> dx\ — j fudx
\B / B
2 — A(7)
and the proof is complete
" (2 — AT)) \\u\\w — AlogC — D\\f\\2\\ul
□
Let us now consider the following Liouville-type equation
W , VTT ) ^ \u\d-2ne\u\0 t . n
—div w(x)Vu = A-„ . --+ f in B
^ > * feI u I * J (23)
B
u =0 on dB
Again by the coercivity of the functional J\ one can prove the following existence result
, , 2n 2+7
Proposition 4. For every A < — (2 + 7) y the funetional J\ attains its minimum, and
Y 2
therefore problem (23) possesses a weak radial solution.
The proof is very similar to that of Theorem 7. This completes the proof of Theorem 8.
5. A Double Exponential Weighted Liouville Type Problem
Consider the functional I : H(B,w) — R defined as follows
Ia(u):=2 NI?og — A log log | I ^ dx | (24)
B
where ||u||2og = f \Vu(x)\2 ^log j^ dx We have the following log log-inequality:
Proposition 5. It holds
1 „2 , M , log Cm
loglog jj ee dxl < -\u\2og + log 8 + ^u E H1(B,w)
j J ^ e ^^ '
Therefore the functional I\ is coercive for every A < n and it has an absolute minimum. Proof.Taking a = \\u\\w, b = U" an(l £<2 = n in
a2
ab < f^ + e2b2 (25)
we have
so that
u< 4П Mû + 4Hû)
\2
e 8П Ml +2n( № ) [ _e Mil en( № )
log log \ J ee dx I < log log J ee dx = log log J ee ж 6 dx
в в в
Let
CMB = sup I e2e2nUl dx
11 ^^ ^ ^^ ^^ 1 ,rad J
Bi(0)
2
Now taking a = e^, b = en( hi™ ) and £2 = 2 in the inequality above, one gets loglog [j eeu dx j < loglog/ e 8e 21 'H'W +2e"( ^) dx =
\B / B
= log [log / e2"( ^ )2 dx + 8 e 2П 11 ( j < < log (log Cmb + 8 e 2^ I u 11 < \Hi + log ( | + .
Moreover
ад > (2 - A) N\w - A >o^8+lnCr)-
For A < n the functional (and every perturbation with a linear term) is coercive, therefore it is bounded from below. In order to prove the existence of a minimum it is sufficient to argue as in Theorem 2.
□
2
Finally we point out that any minimizer of I\ satisfies the following elliptic problem with a double exponential growth.
and hence the proof of Theorem 9 is complete. References
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Received January 24, 2015
УДК 517.954, 517.96 DOI: 10.14529/mmpl50303
ВЗВЕШАННЫЕ НЕРАВЕНСТВА ТРУДИНГЕРА - МОЗЕРА И ПРИЛОЖЕНИЯ
M. Каланки, Б. Руф
Неравенства Трудингера - Мозера обеспечивают непрерывные вложения в пограничных случаях стандартных вложенияй Соболева, в которых вложения в пространствах Лебега Lp отсутствуют. В этом случае приходится рассматривать их естественные обобщения, которые являются вложениями в пространства Орлича соответствующих функциям максимального роста, экспоненциального типа. При описании этих функций роста возникают некоторые параметры. Диапазоны параметров, для которых существуют вложения, можно увеличить с помощью введения весов в нормах Соболева, что и приводит к рассмотрению весовых неравенств ТМ. Представлены некоторые интересные случаи со специальными весами в двумерной области, с приложением к уравнениям среднего поля типа Лиувилля.
Ключевые слова: неравенства Трудингера - Мозера; пространства Орлича; максимальные функции роста; весовые неравенства ТМ.
Марта Каланки, кафедра математики, Миланский университет, Милан, Италия,
Бернард Руф, кафедра математики, Миланский университет, Милан, Италия,
Поступила в редакцию 24 января 2015 г.
