https://doi.org/10.33044/revuma.v60n2a03
PERIODIC SOLUTIONS OF EULER–LAGRANGE EQUATIONS IN AN ANISOTROPIC ORLICZ–SOBOLEV SPACE SETTING
FERNANDO D. MAZZONE AND SONIA ACINAS
Abstract. We consider the problem of finding periodic solutions of certain Euler–Lagrange equations which include, among others, equations involving thep-Laplace operator and, more generally, thepp, qq-Laplace operator. We employ the direct method of the calculus of variations in the framework of anisotropic Orlicz–Sobolev spaces. These spaces appear to be useful in for-mulating a unified theory of existence of solutions for such a problem.
1. Introduction
Let Φ : Rd
Ñ r0,`8qbe a differentiable, convex function such that Φp0q “ 0, Φpyq ą0 ify‰0, Φp´yq “Φpyq, and
lim
|y|Ñ8
Φpyq
|y| “ `8, (1)
where| ¨ | denotes the euclidean norm onRd. From now on, we say that Φ is an
N8 function if Φ satisfies the previous properties.
ForT ą0, we assume thatF:r0, Ts ˆRdÑRd(F “Fpt, xq) is a differentiable function with respect toxfor a.e.tP r0, Ts. Additionally, suppose thatF satisfies the following conditions:
(C) F and its gradient∇xF, with respect to xPRd, are Carath´eodory func-tions, i.e., they are measurable functions with respect to t P r0, Ts, for everyxPRd, and they are continuous functions with respect toxPRdfor a.e.tP r0, Ts.
(A) For a.e.tP r0, Ts, it holds that
|Fpt, xq| ` |∇xFpt, xq| ďapxqbptq,
whereaPC`Rd,r0,`8q˘and 0ďbPL1
pr0, Ts,Rq.
2010Mathematics Subject Classification. Primary 34C25; Secondary 34B15.
Key words and phrases. Periodic solutions; Orlicz spaces; Euler–Lagrange equations; Critical points.
The goal of this paper is to obtain existence of solutions for the following prob-lem:
# d
dt∇Φpu1ptqq “∇xFpt, uptqq, for a.e. tP p0, Tq,
up0q ´upTq “u1
p0q ´u1pTq “0. (PΦ)
Our approach involves the direct method of the calculus of variations in the framework of anisotropic Orlicz–Sobolev spaces. We suggest the article [27] for definitions and main results on anisotropic Orlicz spaces. These spaces allow us to unify and extend previous results on existence of solutions for systems like (PΦ).
We will find solutions of (PΦ) by finding extreme points of theaction integral Ipuq:“
żT
0
Φpu1ptqq `Fpt, uptqqdt. (IA)
In what follows, we shall denote byL“LΦ,F the function Φpyq `Fpt, xq, and we will call itLagrangian.
The classic book [21] deals mainly with problem (PΦ) with Φpxq “ Φ2pxq :“ |x|2{2, through various methods: direct, dual, saddle points, minimax, topological degree theory, etc. The results in [21] were extended and improved in several articles; see [29, 30, 31, 35, 38], to cite some examples. The case Φpxq “Φppyq:“ |y|p{p, for arbitrary 1
ăpă 8, were considered in [32, 33], among other papers. In this case, (PΦ) is reduced to thep-Laplacian system. If Φp1,p2:R
d
ˆRdÑ r0,`8q
is defined by
Φp1,p2py1, y2q:“
|y1|p1
p1
`|y2|
p2
p2
, (2)
then (PΦ) becomes app1, p2q-Laplacian system, see [18, 22, 23, 24, 25, 36, 37]. In
a previous paper (see [1]), we obtained similar results in an isotropic Orlicz frame-work. Hence (PΦ) contains several problems that have been considered by many
authors in the past. Our results still improve some results on pp1, p2q-Laplacian systems since we obtain existence of solutions for them under less restrictive con-ditions. For all this, we believe that anisotropic Orlicz–Sobolev spaces can provide a suitable framework to unify many known results. On the other hand, we point out that one of the most important aspects in our work is the possibility of dealing with functions Φ that grow faster than power functions.
Example 1.1. As an illustrative example, we obtain existence of solutions for
$ ’ ’ ’ ’ &
’ ’ ’ ’ %
d dt
”
u1ptqepu1 1ptqq
2`pu1 2ptqq
2ı
“Fx1pt, uptqq, for a.e. tP p0, Tq,
d dt
”
u2ptqepu1
1ptqq2`pu12ptqq2
ı
“Fx2pt, uptqq, for a.e. tP p0, Tq,
up0q ´upTq “u1p0q ´u1pTq “0,
(3)
whereFpt, xq “PptqQpxq, withP andQpolynomials (see Remark 4.10 below). As far as we know, the study of action integrals in an anisotropic Orlicz–Sobolev setting began in [8]. In that paper, the authors dealt with the differentiability of such action integrals assuming, for the sake of simplicity, that the convex function Φ and its complementary function Φ‹ satisfy the ∆
Φ and Φ‹are bounded from above and below by power functions (see Section 2 for
definitions).
Since we are interested in considering functions that grow faster than those that satisfy the ∆2-condition, in the present paper we develop another proof for
differentiability of action integrals. It is worth mentioning that in this situation the treatment requires more delicate techniques, due to the fact that the effective domain of the action integrals is not the whole Orlicz–Sobolev space.
It is appropriate to say that other problems similar to the one we are going to consider were treated in [2, 3, 4, 19, 20] using the Leray–Schauder degree theory. We point out that our approach is different because we use the direct method of the calculus of variations.
The paper is organized as follows. In Section 2, we summarize some known results about Orlicz and Orlicz–Sobolev spaces. In order to obtain existence of minimizers of action integrals it is necessary that the functionalI be coercive. In the past, several conditions on F have been useful to obtain coercivity of I for the functions Φpand Φp1,p2. In this paper we investigate the condition that in the literature was called sublinearity (see [30, 35, 38] for the Laplacian, [17, 32] for thep -Laplacian, and [18, 22, 23, 37] forpp1, p2q-Laplacian). In Section 3, we contextualize
the sublinearity within our framework (see (B) below) and we establish results of existence of minimizers of (IA) in Theorem 3.2. In Section 4, we establish conditions under which a minimum of (IA) is a solution of (PΦ). Our main result
is Theorem 4.9. This theorem unifies and extends several results obtained in the previously cited bibliography.
2. Anisotropic Orlicz and Orlicz–Sobolev spaces
In this section, we give a short introduction to Orlicz and Orlicz–Sobolev spaces of vector valued functions associated to anisotropicN8functions Φ :Rd
Ñ r0,`8q. References for these topics are [8, 9, 10, 14, 15, 27, 28, 34]. For the theory of convex functions in general we suggest [12]. Note that, unlike in [15], we do not require thatN8 functions be sublinear near 0, i.e., Φpxq{|x| Ñ0 when|x| Ñ0. However,
most of the results proved in [15] do not depend on this property.
If Φpyq is anN8 function which depends on |y| (Φpyq “ Φp|y|q), then we say
that Φ isradial.
We can use the following example to obtain new N8 functions from givenN8
ones.
Example 2.1. Let pd1, . . . , dkq P Zk`. Suppose that Φj : Rdj Ñ r0,`8q, j “
1, . . . , k, are N8 functions, and Oj P LpRd,Rdjq are bounded linear functions
satisfyingŞk
j“1kerOj “ t0u. Then
Φpyq:“
k
ÿ
j“1
ΦjpOjyq
Let us briefly show that Φ satisfies (1). Suppose that|yn| Ñ 8 and Φpynq{|yn| is bounded. If for some j “1, . . . , k there existą0 and a subsequencens such that |Ojyns| ě |yns|, then ΦjpOjynsq{|yns| Ñ 8, contrary to our assumption.
HenceOjyn{|yn| Ñ0 whennÑ 8. Passing to a subsequence, we can assume that there existsy PRd such that yn{|yn| Ñy. Theny PkerOj andy ‰0, which is a contradiction.
As a consequence, the function Φ :RdˆRdÑ r0,`8qdefined by Φpy1, y2q “e|y1´y2|´1` |y2|p,
with 1ăpă 8, is anN8 function.
Associated to Φ we have the complementary function Φ‹ which is defined at ζPRd as
Φ‹pζq “sup
yPRd
y¨ζ´Φpyq.
From the continuity of Φ and (1), we also have that Φ‹ :
Rd Ñ r0,8q. The complementary function Φ‹is anN
8 function (see [21, Ch. 2] and [27, Thm. 2.2]).
Now, Moreau’s theorem (see [12, Thm. 4.21]) implies that Φ‹‹“Φ.
Some useful properties which are satisfied byN8 functions are:
(P1) Φpλxq ďλΦpxq, for everyλP r0,1s,xPRd; (P2) if 0ă |λ1| ď |λ2|, then Φpλ1xq ďΦpλ2xq; (P3) x¨yďΦpxq `Φ‹pyq;
(P4) x¨∇Φpxq “Φpxq `Φ‹p∇Φpxqq.
We say that Φ :RdÑ r0,`8qsatisfies the ∆2-condition, and we denote ΦP∆2,
if there exists a constantCą0 such that
Φp2xq ďCΦpxq `1, xPRd.
Note that this definition is equivalent to the classic one, i.e., there existr0, C ą0 with Φp2xq ďCΦpxqfor|x| ąr0.
If there existsCą0 such that Φp2xq ďCΦpxqfor allxPRd, it is usually said that Φ satisfies the ∆2-condition globally (see [26]).
Throughout this article, we denote by C “ Cpλ1, . . . , λnq a positive constant that may depend onT, Φ (or otherN8 functions), and the parametersλ1, . . . , λn.
We assume that the value that C represents may change in different occurrences in the same chain of inequalities.
If Φ satisfies the ∆2-condition, then Φ satisfies the following properties:
(P5) There exists C ą0 such that for every x, y P Rd, Φpx`yq ď CpΦpxq `
Φpyqq `1.
(P6) For anyλą1 there existsCpλq ą0 such that Φpλxq ďCpλqΦpxq `1. (P7) There exist 1ăpă 8andCą0 such that Φpxq ďC|x|p
`1.
Let Φ1 and Φ2 beN8 functions. Following [34], we write Φ1JΦ2if there exist k, Cą0 such that
Φ1pxq ďC`Φ2pkxq, xPRd. (4)
We observe that Φ1JΦ2 implies that Φ‹2JΦ‹1. A similar assertion holds for the
relationÎ. If Φ‹P∆
2then Φ satisfies the∇2-condition, i.e., for every 0ără1 there exist l“lprq ą0 andC1“C1prq ą0 such that
Φpxq ďr
lΦplxq `C 1, x
PRd. (5)
It is easy to see that this definition is equivalent to the more usual, i.e.,r “1{2 and inequality (5) holding for|x| ąr0and a certain r0ą0.
We denote by M :“ M`r0, Ts,Rd˘, with d ě 1, the set of all measurable functions (i.e., functions which are limits of simple functions) defined on r0, Ts
with values onRd, and we writeu“ pu1, . . . , udqforuPM.
Given anN8 function Φ we define themodular function ρΦ:MÑR`Y t`8u
by
ρΦpuq:“ żT
0
Φpuqdt.
Now, we introduce the Orlicz classCΦ“CΦ`
r0, Ts,Rd˘by setting
CΦ:“ tuPM|ρΦpuq ă 8u.
TheOrlicz space LΦ“LΦ`r0, Ts,Rd˘is the linear hull ofCΦ; equivalently,
LΦ:“ tuPM| Dλą0 :ρΦpλuq ă 8u.
The Orlicz spaceLΦ equipped with theLuxemburg norm
}u}LΦ :“inf
! λ|ρΦ
´v λ ¯
dtď1
)
is a Banach space. The subspace EΦ
“EΦ`
r0, Ts,Rd˘is defined as the closure in LΦ of the sub-spaceL8`
r0, Ts,Rd˘
of allRd-valued essentially bounded functions. The equality
LΦ“EΦis true if and only if ΦP∆
2 (see [27, Cor. 5.1]).
A generalized version of H¨older’s inequality holds in Orlicz spaces (see [27, Thm. 7.2]). Namely, ifuPLΦ andvPLΦ‹ thenu¨vPL1 and
żT
0
v¨u dtď2}u}LΦ}v}LΦ‹.
Byu¨v we denote the usual dot product inRd betweenuandv. We consider the subset ΠpEΦ, rqofLΦgiven by
ΠpEΦ, rq:“ tuPLΦ|dpu, EΦq ăru “ tuPLΦ|dpu, L8 q ăru.
This set is related to the Orlicz classCΦby the inclusions
ΠpEΦ, rq ĂrCΦĂΠpEΦ, rq (6)
for any positive r. This relation is a trivial generalization of [27, Thm. 5.6]. If ΦP∆2, then the setsLΦ,EΦ, ΠpEΦ, rqand CΦ are equal.
As usual, if pX,} ¨ }Xq is a normed space and pY,} ¨ }Yqis a linear subspace of
states that LΦ
ãÑ “LΦ‹‰‹
, where a function v P LΦ is associated to ξ
v P
“ LΦ‹‰‹
given by
xξv, uy “
żT
0
v¨u dt.
It is easy to prove thatL8ãÑLΦãÑL1for anyN
8 function Φ.
SupposeuPLΦpr0, Ts,Rdqand considerK:“ρΦpuq `1ě1. Then, from (P1) we haveρΦpK´1uq ďK´1ρΦpuq ď1. Therefore, we conclude
}u}LΦ ďρΦpuq `1. (7) We highlight that LΦ`
r0, Ts,Rd˘
can be equipped with the weak‹ topology induced byEΦ‹`
r0, Ts,Rd˘becauseLΦ`r0, Ts,Rd˘““EΦ‹`
r0, Ts,Rd˘‰‹(see [15, Thm. 3.3]).
We define theOrlicz–Sobolev spaceW1LΦ`
r0, Ts,Rd˘by
W1LΦ`r0, Ts,Rd˘:“ u|uPAC`r0, Ts,Rd˘and u1PLΦ`r0, Ts,Rd˘(, where AC`r0, Ts,Rd˘
denotes the space of all Rd valued absolutely continuous functions defined on r0, Ts. The spaceW1LΦ`
r0, Ts,Rd˘ is a Banach space when equipped with the norm
}u}W1LΦ “ }u}LΦ` }u1}LΦ.
Let the functionAΦ:Rd Ñ r0,`8qbe the greatest convex radial minorant of Φ, i.e.,
AΦpxq “suptΨpxqu, (8)
where the supremum is taken over all the convex, non-negative, radial functions Ψ with Ψpxq ďΦpxq.
Proposition 2.1. AΦ is a radial andN8 function.
Proof. The convexity and radiality of AΦ is a consequence of the fact that the supremum preserves these properties. Then, it is only necessary to show that
AΦpxq ą 0 when x ‰ 0, and AΦpxq{|x| Ñ 8 when |x| Ñ 8. We write, for
r P R, r` “maxtr,0u. Since Φ is an N8 function, for every k ą 0 there exists
r0 ą 0 such that Φpxq ě kp|x| ´r0q`, for |x| ą r0. As kp|x| ´r0q` is a
non-negative, radial, convex function, it follows that AΦpxq ěkp|x| ´r0q`. Therefore lim inf|x|Ñ8AΦpxq{|x| ěkand consequently lim|x|Ñ8AΦpxq{|x| “ 8.
As Φ is an N8 continuous function, for every r ą 0 there exists kprq ą 0 such that Φpxq ě kprq|x| ě kprqp|x| ´rq`, when |x| ě r. This fact implies that
AΦpxq ą0 forx‰0.
By abuse of notation, we identify AΦwith a function defined onr0,`8q. This
function is invertible.
Corollary 2.2. LΦ
pr0, Ts,Rd
qãÑLAΦpr0, Ts, Rdq.
As is customary, we will use the decomposition u“ u`ur for a function u P L1
Lemma 2.3. Let Φ :Rd
Ñ r0,`8qbe an N8 function and let
uP W1LΦ`
r0, Ts,Rd˘
. Let AΦ : Rd Ñ r0,`8q be the isotropic function defined by (8). Then:
(1) Morrey’s inequality. For everys, tP r0, Ts withs‰t,
|uptq ´upsq| ď |s´t|A´1Φ ˆ
1
|s´t| ˙
}u1}
LΦ. (M.I.)
(2) Sobolev’s inequality.
}u}L8 ďA´1Φ
ˆ
1
T ˙
maxt1, Tu}u}W1LΦ. (S.I.)
(3) Poincar´e–Wirtinger’s inequality. We have urPL8`
r0, Ts,Rd˘
and
}u}r L8 ďT A ´1 Φ
ˆ
1
T ˙
}u1
}LΦ. (P-W.I.)
(4) If Φ is an N8 function, then the space W1LΦ`
r0, Ts,Rd˘
is compactly embedded in the space of continuous functionsCpr0, Ts,Rdq.
Proof. It is an immediate consequence of Corollary 2.2 and [1, Lemma 2.1, Cor.
2.2].
Lemma 2.3 gives us estimates for isotropic norms ofu. In these type of inequal-ities some information is lost. The following result gives us an estimate that takes into account the anisotropic nature of the spaceW1LΦ`
r0, Ts,Rd˘
. The proof is similar to that of [8, Thm. 4.5].
Lemma 2.4(Anisotropic Poincar´e–Wirtinger’s inequality). LetΦ :Rd
Ñ r0,`8q
be anN8 function and let uPW1LΦ `
r0, Ts,Rd˘. Then
Φpuptqq ď˜ 1
T żT
0
Φ`T u1prq˘
dr. (A.P-W.I.)
Proof. Applying Jensen’s inequality twice, we get
Φpuptqq “˜ Φ
˜
1
T żT
0
puptq ´upsqqds ¸
ď 1 T
żT
0
Φpuptq ´upsqqds
ď 1 T
żT
0
Φ
ˆżt
s
|t´s|u1prq dr |t´s|
˙ ds
ď 1 T
żT
0
1
|t´s| żt
s
Φ`|t´s|u1prq˘ dr ds.
From (P1) we have that Φprxq{r is increasing with respect to r ą 0 for a fixed
Remark 2.5. As a consequence of Lemma 2.3, we obtain that
}u}1W1LΦ “ |u| ` }u1}LΦ,
defines an equivalent norm to} ¨ }W1LΦ onW1LΦpr0, Ts,Rdq.
Another immediate consequence of Lemma 2.3 is the following result.
Corollary 2.6. Every bounded sequencetunuinW1LΦpr0, Ts,Rdqhas a uniformly convergent subsequence.
3. Existence of minimizers
It is well known that an important ingredient in the direct method of the calculus of variations is the coercivity of action integrals. In order to obtain coercivity for the action integralI, defined in (IA), it is necessary to impose more restrictions on the potentialF.
There are several restrictions that were explored in the past. The one we will study in this article is based on what is known in the literature as sublinearity (see [30, 35, 38] for the Laplacian, [17, 32] for the p-Laplacian, and [18, 22, 23, 37] for
pp1, p2q-Laplacian). In the present article we will use another denomination for this
property.
Definition 3.1. LetF :r0, Ts ˆRd ÑRbe a function satisfying (C) and (A). We say that F satisfies condition (B) if there exist anN8 function Φ0, with Φ0ÎΦ,
and a functiondPL1pr0, Ts,Rq, withdě1, such that Φ‹
ˆ ∇xF
dptq ˙
ďΦ0pxq `1. (B)
The condition (B) encompasses the sublinearity condition as it was introduced in the context of p-Laplacian or pp1, p2q-Laplacian systems. For example, in [18,
Thm. 1.1] Li, Ou and Tang considered a potential F : r0, Ts ˆRd ˆRd Ñ R satisfying (C) and (A) and the following condition (we recall thatp1“p{pp´1q):
(H) There existfi, gi, hiPL1pr0, Ts,R`q,αiP r0, pi{p1iq,i“1,2,β1P r0, p2{p11q,
andβ2P r0, p1{p1
2qsuch that |∇x1Fpt, x1, x2q| ďf1ptq|x1|
α1`g1ptq|x2|β1`h1ptq,
|∇x2Fpt, x1, x2q| ďf2ptq|x2|
α2`g2ptq|x1|β2`h2ptq.
We leave it to the reader to prove that (H) implies (B), with Φ “Φp1,p2, Φ0 “ Φp1,p2, where pi, i “ 1,2, are taken so that maxtα1p11, β2p12u ď p1 ă p1 and
maxtα2p1
2, β1p11u ď p2 ăp2, and d“Cp1` ř
itfi`gi`hiuq PL
1, with C ą0 chosen large enough.
Theorem 3.2. Let Φ be an N8-function whose complementary function Φ‹
sat-isfies the ∆2-condition. Let F be a potential that satisfies (C), (A), (B) and the
condition
lim
|x|Ñ8 şT
0 Fpt, xqdt
LetM be a weak‹ closed subspace ofLΦand letV
ĂCpr0, Ts,Rd
qbe closed in the
Cpr0, Ts,Rd
q-strong topology. ThenIattains a minimum onH “ tuPW1LΦ |uP V andu1PMu.
Proof. Step 1. The action integral is coercive. Letλbe any positive number with
λą2 maxtT,1u. Since Φ0ÎΦ, there exists Cpλq ą0 such that
Φ0pxq ďΦ´x 2λ
¯
`Cpλq, xPRd. (10)
By the decompositionu“u`u˜, the absolute continuity ofFpt, x`syqwith respect tosPR, Young’s inequality, (B), the convexity of Φ0, (P2), (10) and (A.P-W.I.)
we obtain:
J :“ ˇ ˇ ˇ ˇ ˇ żT 0
Fpt, uq ´Fpt, uqdt ˇ ˇ ˇ ˇ ˇ ď żT 0 ż1 0
|∇xFpt, u`suq˜ u|˜ ds dt
ďλ żT 0 dptq ż1 0 „ Φ‹`
d´1ptq∇xFpt, u`suq˜
˘ `Φ ˆ ˜ u λ ˙ ds dt ďλ żT 0 dptq ż1 0 „ 1
2Φ0p2uq ` 1
2Φ0p2˜uqds`Φ
ˆ ˜ u λ ˙ `1 ds dt ďλ żT 0 dptq ż1 0 „
Φ0p2uq `2Φ
ˆ ˜ u λ ˙ `Cpλq ds dt
ďC1Φ0p2uq `λC2 żT
0
Φ
ˆ T u1psq
λ ˙
ds`C1,
where C2 “ C2p}d}L1q and C1 “ C1p}d}L1, λq. Since Φ‹ P ∆2, we can choose λ
large enough so thatl “λT´1 satisfies (5) for r “ 1
2mintpC2Tq
´1,1u. Thus, we
have
J ďC1Φ0p2uq `1
2
żT
0
Φ`u1psq˘
ds`C1.
Then
Ipuq “ żT
0
Φpu1
q `Fpt, uqdt
“ żT
0
tΦpu1q ` rFpt, uq ´Fpt, uqs `Fpt, uqudt
ě 1
2
żT
0
Φpu1qdt´C1Φ0p2uq ` żT
0
Fpt, uqdt´C1
We take un PW1LΦ with}un}W1LΦ Ñ 8. From Remark 2.5, we can suppose that}u1
n}LΦ Ñ 8or|un| Ñ 8. In the first case, from (7) we have thatρΦpunq Ñ 8 and henceIpunq Ñ 8. In the second case,Ipunq Ñ 8as a consequence of (9).
Step 2. Suppose that un Ñ u uniformly and u1n
‹
á u1 in LΦ
pr0, Ts,Rd
q; then
Without loss of generality, passing to subsequences, we may assume that the lim inf is actually a lim. The embedding LΦ
pr0, Ts,Rd
q ãÑ L1
pr0, Ts,Rd
q implies that u1
n á u1 in L1pr0, Ts,Rdq. Now, applying [7, Thm. 3.6] we obtain Ipuq ď limnÑ8Ipunq.
Final step. The proof of the theorem is concluded with a usual argument. We take a minimizing sequenceunPH ofI. From the coercivity ofIwe have thatun is bounded on W1LΦ
pr0, Ts,Rdq. By Corollary 2.6 (passing to subsequences) we can suppose thatun converges uniformly to a functionuPV. On the other hand,
u1
n is bounded onLΦ“
“
EΦ‹‰‹. Thus, sinceEΦ‹ is separable (see [27, Thm. 6.3]),
from [5, Cor. 3.30] it follows that there exist a subsequence ofu1
n (we denote itu1n again) andv PM such that u1
n
‹
áv. From this fact and the uniform convergence ofun tou, we obtain that
żT
0 ϕ1
¨u dt“ lim nÑ8
żT
0 ϕ1
¨undt“ ´ lim nÑ8
żT
0 ϕ¨u1
n dt“ ´
żT
0
ϕ¨v dt,
for every functionϕPC8pr0, Ts,
Rdq ĂEΦ‹
withϕp0q “ϕpTq “0. Thus,uhas a derivative in the weak sense inLΦ. Taking into accountLΦ
ãÑL1 and [7, Thms.
2.3 and 2.17], we obtain uPW1LΦ andv
“u1 a.e.tP r0, Ts. Hence,uPH.
Finally, the semicontinuity ofI that was established in step 2 implies thatuis
a minimum ofI.
Remark 3.3. The results of this section can be extended without difficulty to any LagrangianL withLěLΦ,F (see [1]).
4. Regularity of minimizers and solutions of Euler–Lagrange equations
In this section, we will address the question of when minimizers of I are solu-tions of the associated Euler–Lagrange equasolu-tions. It is well known that, in virtue of the Lavrentiev phenomenon (see [7, Sec. 4.3]), this is a delicate matter. It is a consequence of Tonelli’s partial regularity theorem that under certain con-ditions on the Lagrangian function, if a minimizer is additionally Lipschitz then it is solution of Euler–Lagrange equations (see [7, Sec. 4.3]). The question then is to determine sufficient conditions on the Lagrangian L in order that minimiz-ers be a priori Lipschitz. In this direction, several conditions were discussed in the literature (see [6, 7, 11, 13]). We note that Lipschitz functions satisfy that
u1PL8, thereforedpu1, L8q “0. We will prove in Theorem 4.1 that the condition dpu1, L8pr0, Ts,
Rdqq ă1 will be sufficient for the minimizers of our functionalI to satisfy Euler–Lagrange equations.
We denote by Lippr0, Ts,Rdqthe set ofRd-valued Lipschitz continuous functions defined on r0, Ts. If X Ă LΦpr0, Ts,Rdq and u P LΦpr0, Ts,Rdq, we denote by
dpu, Xqthe distance fromutoX computed with respect to the Luxemburg norm. We recall thatuPLippr0, Ts,Rdqimplies thatdpu1, L8pr0, Ts,
Theorem 4.1. Assume thatF is as in Theorem 3.2 and Φ is strictly convex. If
uis a minimum of I on the set H “ tuPW1LΦ
pr0, Ts,Rd
q |up0q “ upTqu and
dpu1, L8pr0, Ts,
Rdqq ă1, thenuis solution of (PΦ).
Remark 4.2. We observe that H“ tuPW1LΦ
|uPV andu1PMu, where V :“ tuPCpr0, Ts,Rdq |up0q “upTqu, M :“LΦpr0, Ts,Rdq,
andV is Cpr0, Ts,Rd
q-closed. Therefore, from the results of the previous section the functional I given by (IA) has a minimum uon H and dpu1, L8q ď 1. The
last inequality follows fromρΦpu1q ă 8and (6). Then, the possible minima of I
that do not satisfy the hypotheses of Theorem 4.1 lie on the nowhere dense set
tu:dpu1, L8 q “1u.
The proof of the previous theorem depends on the Gateaux differentiability of the action integral on the spaceW1LΦpr0, Ts,Rdq. We will deal with a more general Lagrangian functionL:r0, Ts ˆRdˆRdÑR, which is assumed measurable int for eachpx, yq PRd
ˆRd and continuously differentiable atpx, yqfor almost every
tP r0, Ts. We consider
Ipuq “ILpuq “ żT
0
Lpt, uptq, u1ptqqdt, (IG)
the action integral associated to L. In order to obtain differentiability of I, it is necessary to impose some constraints on L. In the paper [8], Chmara and Maksymiuk obtained differentiability forI onW1LΦassuming a similar condition
to Definition 4.3 and additionally they supposed that Φ P ∆2 X∇2. For our
purpose, the condition Φ P ∆2 is a very serious limitation since it leaves out
of consideration functions that grow faster than power ones. According to our criterion, including Lagrangians with a faster growth than power functions is one of the greatest achievements of the present paper. For this reason, we present a proof of the results obtained in [8] without the assumption ΦP∆2. When ΦR∆2,
the differentiability ofI is somewhat more delicate since the effective domain ofI
is not the whole spaceW1LΦ.
Definition 4.3. We say that a LagrangianLsatisfies the condition (S) if
|L| ` |∇xL| `Φ‹
ˆ ∇yL
λ ˙
ďapxq ”
bptq `Φ
´y
Λ
¯ı
, (S)
for a.e.tP r0, Ts, whereaPC`Rd,
r0,`8q˘,bPL1
pr0, Ts,r0,`8qq, and Λ, λą0. Condition (S) includes structure conditions that have been previously considered in the literature in the case of p-Laplacian and pp1, p2q-Laplacian systems. For example, it is easy to see that, when Φpxq “ Φppxq “ |x|p{p, the condition (S) is equivalent to the structure conditions in [21, Thm. 1.4]. If Φ is a radial N8
function such that Φ‹ satisfies the ∆
2-condition, then (S) is related to conditions
(S) is a more compact expression than [33, Lemma 3.1, Eq. (3.1)] and moreover weaker, because (S) does not imply a control of|Dy1L|independent ofy2.
Remark4.4. We leave to the reader the proof of the fact that if a Lagrange function
Lsatisfies structure condition (S) and ΦJΦ0, thenLsatisfies (S) with Φ0instead
of Φ and possibly with other functionsb,aand constants Λ andλ.
Remark 4.5. The Lagrangian L “ LΦ,F “Φpyq `Fpt, xq satisfies condition (S), for every Λ ă1. In order to prove this, the only non-trivial fact that we should establish is that Φ‹p∇
yLq ďapxq tbptq `Φpy{Λqu. From (P4) and the fact that
pd{dtqΦptxq “∇Φptxq ¨xis a non-decreasing function of t, we obtain
Φ‹p∇Φpxqq ďx
¨∇Φpxq ď 1
Λ´1´1 żΛ´1
1 d
dtΦptxqdtď
1
Λ´1´1ΦpΛ ´1xq.
Therefore Φ‹p∇
yLq “Φ‹p∇Φpyqq ďΛp1´Λq´1Φpy{Λq, for every Λă1.
Given a functiona:Rd ÑR, we define the composition operator a:MÑM
byapuqpxq “ apupxqq. We will often use the following result, whose proof can be carried out as that of Corollary 2.3 in [1].
Lemma 4.6. If a P CpRd,R`q then a : W1LΦ
Ñ L8pr0, Tsq is bounded. More
concretely, there exists a non-decreasing functionA:r0,`8q Ñ r0,`8qsuch that
}apuq}L8pr0,TsqďAp}u}W1LΦq.
The following lemma will be applied several times. We adapted the proof of [1, Lemma 2.5] to the anisotropic case. For an alternative approach, we suggest [8].
Lemma 4.7. Let tununPN be a sequence of functions converging to uPΠpEΦ, λq
in the LΦ-norm. Then, there exist a subsequence unk and a real valued function
hPL1pr0, Ts,Rqsuch that unkÑua.e. andΦpunk{λq ďha.e.
Proof. Sincedpu, EΦ
q ăλandun converges to u, there exist a subsequence ofun (again denoted un), λ P p0, λq and u0 PEΦ such that dpun, u0q ă λ, n “1, . . .. As LΦ`r0, Ts,Rd˘ãÑL1`r0, Ts,Rd˘, the sequence un converges in measure to u. Therefore, we can extract a subsequence (denoted againun) such thatunÑua.e. and
λn:“ }un´un´1}LΦă
λ´λ
2n´1, forně2.
We can assume λn ą 0 for every n “ 1, . . .. We write λ1 :“ }u1´u0}LΦ and
λ0:“λ´ř8n“1λn, and we define h:r0, Ts ÑRby
hptq “ λ0 λΦ
ˆ u0 λ0
˙ `
8 ÿ
j“0 λj`1
λ Φ ˆ
uj`1´uj
λj`1 ˙
As Φp0q “0 and Φ is a convex function, we have for anyn“1, . . .
Φ
´un λ ¯ “Φ ˜ u0 λ ` n´1 ÿ j“0
uj`1´uj
λ ¸ ďλ0 λΦ ˆ u0 λ0 ˙ ` n´1 ÿ j“0 λj`1
λ Φ ˆ
uj`1´uj
λj`1 ˙
ďh.
Sinceu0PEΦĂCΦandEΦis a subspace, we get that Φpu0{λ0q PL1pr0, Ts,Rq. On the other hand,}uj`1´uj}LΦ“λj`1and therefore
żT
0
Φ
ˆ
uj`1´uj
λj`1 ˙
dtď1.
ThenhPL1pr0, Ts,Rq.
The proof of the next theorem follows the same lines as [1, Thm. 3.2], but with some modifications due to the lack of monotonicity of Φ with respect to the euclidean norm and the fact that the notion of absolutely continuous norm (used intensely in [1, Thm. 3.2]) does not work very well in the framework of anisotropic Orlicz spaces when ΦR∆2.
Theorem 4.8. Let L be a differentiable Carath´eodory function satisfying (S). Then the following statements hold:
(1) The action integral given by (IG) is finitely defined on the set EΦ Λ :“ W1LΦX tu|u1 PΠpEΦ,Λqu.
(2) The function I is Gateaux differentiable on EΦ
Λ and its derivative I1 is
demicontinuous fromEΦ Λ into
“
W1LΦ‰‹, i.e.,I1 is continuous whenEΦ Λ is
equipped with the strong topology and “W1LΦ‰‹ with the weak
‹ topology. Moreover,I1 is given by the expression
xI1puq, vy “ żT
0 “
∇xL
` t, u, u1˘
¨v`∇yL
` t, u, u1˘
¨v1‰
dt. (11)
(3) IfΦ‹P∆
2 thenI1 is continuous from EΛΦ into “
W1LΦ‰‹ when both spaces are equipped with the strong topology.
Proof. Let uPEΦ
Λ. From (6) we obtain that Φpu1ptq{Λq P L
1. Now, from (S) and
Lemma 4.6, we have
|Lpt, uptq, u1ptqq| ` |∇
xLpt, uptq, u1ptqq| `Φ‹
ˆ
∇yLpt, u, u1q
λ ˙
ďAp}u}W1LΦq
„
bptq `Φ
ˆ u1ptq
Λ
˙
PL1. (12) Thus, by integrating this inequality item (1) is proved.
We split up the proof of item (2) into four steps.
Step 1. The non-linear operator u ÞÑ ∇xLp¨, u, u1q is continuous from EΛΦ into
L1
LettununPNbe a sequence of functions in EΛΦand letuPE Φ
Λ such that un Ñu in W1LΦ. By (S.I.),u
n Ñuuniformly. As u1n Ñu1 PEΛΦ, by Lemma 4.7, there
exist a subsequence ofu1
n (again denotedu1n) and a functionhPL1pr0, Ts,Rqsuch thatu1
nÑu1 a.e. and Φpu1n{Λq ďha.e.
Sinceun,n“1,2, . . ., is a bounded sequence inW1LΦ, according to Lemma 4.6, there exists M ą0 such that }apunq}L8 ďM, n “1,2, . . .. From the previous facts and (12), we get
|∇xLp¨, un, u1nq| ďapunq
„ b`Φ
ˆ u1
n Λ
˙
ďMpb`hq PL1.
On the other hand, by the continuous differentiability ofL, we have
∇xLpt, unkptq, u
1
nkptqq Ñ∇xLpt, uptq, u
1ptqq for a.e. t
P r0, Ts.
Applying Lebesgue’s dominated convergence theorem we conclude the proof of step 1.
Step 2. The non-linear operator uÞÑ∇yLp¨, u, u1q is continuous fromEΛΦ with the
strong topology into“LΦ‰‹ with the weak
‹ topology. LetuPEΦ
Λ. From (12), it follows that ∇yLp¨, u, u1q PλCΦ
‹`
r0, Ts,Rd˘ĂLΦ‹`r0, Ts,Rd˘Ă“LΦ`r0, Ts,Rd˘‰‹. (13) Let un, u PEΛΦ such thatun Ñ uin the norm ofW1LΦ. We must prove that
∇yLp¨, un, u1nq w‹
á∇yLp¨, u, u1q. Assume, on the contrary, that there exist vPLΦ,
ą0, and a subsequence oftunu(denotedtunufor simplicity) such that
ˇ
ˇx∇yLp¨, un, u1nq, vy ´ x∇yLp¨, u, u1q, vy
ˇ
ˇě. (14)
We haveunÑuinLΦ andu1nÑu1 withu1PΠpEΦ,Λq. By Lemmas 2.6 and 4.7, there exist a subsequence oftunu(again denotedtunufor simplicity) and a function
hPL1
pr0, Ts,Rq such thatun Ñuuniformly,u1nÑu1 a.e. and Φpu1n{Λq ďha.e. As in the previous step, Lemma 4.6 implies thatapunptqqis uniformly bounded by a certain constant M ą0. Therefore, from inequality (12) with un instead of u, we have
Φ‹ ˆ
∇yLp¨, un, u1nq
λ
˙
ďMpb`hq “:h1PL1. (15) AsvPLΦthere existsλvą0 such that Φpv{λvq PL1. Now, by Young’s inequality and (15), we have
∇yLp¨, un, u1nq ¨vptq ďλλv
„
Φ‹ ˆ
∇yLp¨, un, u1nq
λ
˙ `Φ
ˆ v λv
˙
ďλλvMpb`hq `λλvΦ
ˆ v λv
˙ PL1.
Finally, from Lebesgue’s dominated convergence theorem, we deduce
żT
0
∇yLpt, un, u1nq ¨v dtÑ
żT
0
∇yLpt, u, u1q ¨v dt,
Step 3. We will prove (11). Note that (12), (13) and the imbeddingsW1LΦ ãÑ L8 andLΦ‹
ãÑ“LΦ‰‹ imply that the right-hand side of (11) defines an element of “
W1LΦ‰‹.
The proof follows similar lines as [21, Thm. 1.4]. ForuPEΛΦand 0‰vPW1LΦ,
we define the function
Hps, tq:“Lpt, uptq `svptq, u1ptq `sv1ptqq.
For|s| ďs0:“`Λ´dpu1, EΦ
q˘{}v}W1LΦ we have thatu1`sv1 PΠpEΦ,Λq. This fact implies, in virtue of Theorem 4.8 item 1, that Ipu`svqis well defined and finite for|s| ďs0.
We write s1 :“mints0,1´dpu1, EΦ
q{Λu. Let λv ą0 such that Φpv1{λvq PL1. Asu1PΠpEΦ,Λq, we have
d ˆ
u1
p1´s1qΛ, E
Φ ˙
“ 1
p1´s1qΛdpu
1, EΦ q ă1,
and consequentlyp1´s1q´1Λ´1u1 PCΦ. Hence, ifv1PLΦand|s| ďs1Λλ´1
v , from the convexity of Φ and (P2), we get
Φ
ˆ
u1`sv1
Λ
˙
ď p1´s1qΦ
ˆ u1
p1´s1qΛ
˙ `s1Φ
ˆ s s1Λv
1 ˙
ď p1´s1qΦ
ˆ u1
p1´s1qΛ
˙ `s1Φ
ˆ v1 λv
˙
“:hptq PL1.
(16)
We also have}u`sv}W1LΦ ď }u}W1LΦ`s0}v}W1LΦ; then, by Lemma 4.6, there exists M ą 0 independent of s, such that }apu`svq}L8 ď M. Now, applying Young’s inequality, (12), the fact thatvPL8, (16), and Φpv1{λ
vq PL1, we get |DsHps, tq| “
ˇ
ˇ∇xLpt, u`sv, u1`sv1q ¨v`∇yLpt, u`sv, u1`sv1q ¨v1
ˇ ˇ
ďM „
bptq `Φ
ˆ
u1`sv1
Λ
˙ |v|
`λλv
„
Φ‹ ˆ
∇yLpt, u`sv, u1`sv1q
λ ˙ `Φ ˆ v1 λv ˙ ďM „
bptq `Φ
ˆ
u1`sv1
Λ
˙
p|v| `λλvq `λλvΦ
ˆ v1 λv
˙
ďMpbptq `hptqq p|v| `λλvq `λλvΦ
ˆ v1 λv
˙ PL1.
Consequently,I has a directional derivative and
xI1puq, vy “ d
dsIpu`svq ˇ ˇ
s“0“ żT
0 “
∇xLpt, u, u1q ¨v`∇yLpt, u, u1q ¨v1
‰ dt.
Moreover, from the previous formula, (12), (13), and Lemma 2.3, we obtain
|xI1puq, vy| ď }∇
with an appropriate constant C. This completes the proof of the Gateaux dif-ferentiability of I. The previous steps imply the demicontinuity of the operator
I1:EΦ Λ Ñ
“
W1LΦd‰‹.
In order to prove item (3), it is necessary to see that the mapsuÞÑ∇xLpt, u, u1q anduÞÑ∇yLpt, u, u1qare norm continuous fromEΛΦintoL
1 andLΦ‹
, respectively. It remains to prove the continuity of the second map. To this purpose, we take
un, uPEΛΦ, n“1,2, . . ., with }un´u}W1LΦ Ñ0. As before, we can deduce the existence of a subsequence (denoted u1
n for simplicity) and h1 PL1 such that (15)
holds andunÑua.e. Since Φ‹P∆2, we have
Φ‹p∇yLp¨, un, u1nqq ďcpλqΦ‹
ˆ
∇yLp¨, un, u1nq
λ
˙
`1ďcpλqh1`1“:h2PL1.
Then, from (P5), we get
Φ‹`
∇yLp¨, un, u1nq ´∇yLp¨, u, u1q
˘
ďKph2`Φ‹p∇yLp¨, u, u1qqq `1. Now, by Lebesgue’s dominated convergence theorem, we obtain that∇yLp¨, un, u1nq isρΦ‹ modular convergent to∇yLp¨, u, u1qq, i.e., ρΦ‹pun´uq Ñ0. Since Φ‹P∆2,
modular convergence implies norm convergence (see [28]).
Proof of Theorem 4.1. Suppose thatdpu1, L8q ă1. Sincedpu1, EΦ
q “dpu1, L8q,
according to Remark 4.5 and Theorem 4.8, I is Gateaux differentiable at u. By Fermat’s rule (see [12, Prop. 4.12]), we havexI1puq, vy “0 for everyvPH.
There-fore
żT
0
∇Φpu1ptqq ¨v1ptqdt“ ´ żT
0
∇xFpt, uptqq ¨vptqdt. (17)
From Theorem 4.8, we have that∇xFpt, uptqq PL1pr0, Ts,Rdqand∇Φpu1ptqq P LΦ‹
pr0, Ts,Rq ãÑ L1
pr0, Ts,Rq. Identity (17) holds for every v P C8pr0, Ts,
Rdq
withvp0q “vpTq. Using [21, Fundamental Lemma, p. 6], we get that∇Φpu1ptqqis
absolutely continuous andpd{dtq p∇Φpu1ptqqq “∇
xFpt, uptqqa.e. onr0, Ts. More-over, ∇Φpu1
p0qq “∇Φpu1pT
qq. Since Φ isstrictly convex, ∇Φ :Rd ÑRd is a one-to-one map (see, e.g. [12, Ex. 4.17, p. 67]). Hence, we conclude thatu1
p0q “u1pT q.
Finally, Theorem 4.1 is proven.
The following is the main result in this article. In this theorem we give sufficient conditions for minimizers to be solutions of the Euler–Lagrange equations.
Theorem 4.9. Let Φ, F, and H be as in Theorem 4.1. Suppose some of the following condition holds: a) ΦP∆2, or b) Fpt, xqis differentiable with respect to pt, xq and
ˇ ˇ ˇ ˇ B BtFpt, xq
ˇ ˇ ˇ ˇ
ďapxqbptq, (18)
withaandbas in (A). Then ifuis a minimum ofIon the set H,uis solution of (PΦ).
Proof. The conditiondpu1, L8q ă1 is trivially satisfied when ΦP∆
2 because, in
this case,L8 is dense inLΦ
pr0, Ts,Rd
Suppose that b) holds anduis a minimum ofI. We note thatuis also minimum ofIon the set defined by a Dirichlet boundary condition
tvPW1LΦpr0, Ts,Rdq |vp0q “up0q, vpTq “upTqu.
Therefore, we can apply Proposition 3.1 in [13] (see also the remark which follows that Proposition) and we obtainu1 PL8.
Remark 4.10. Returning to the system (3) of Example 1.1, we note that theN8
function Φpy1, y2q “ exppy12`y22q ´1 has a complementary function which
sat-isfies the ∆2-condition (see [16, p. 28]). In addition, for every p ą 1 we have |py1, y2q|p Î Φpy1, y2q. Therefore Φ‹py1, y2q Î |py1, y2q|q for q “ p{pp´1q.
Consequently, if Fpt, x1, x2q “ PptqQpx1, x2q with P and Q polynomials, and dptq :“ Cmaxt1,|Pptq|u, then Φ‹pd´1ptq∇
xFq ď |px1, x2q|q `1, where p and C are chosen large enough. Hence Φ andF satisfy (B) with Φ0py1, y2q “ |py1, y2q|p. The conditions (C), (A) and (18) can be proved in a direct way. All these facts show that there exist solutions of the system in Example 1.1.
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F. D. MazzoneB
Dpto. de Matem´atica, Facultad de Ciencias Exactas, F´ısico-Qu´ımicas y Naturales, Universidad Nacional de R´ıo Cuarto, X5804BYA R´ıo Cuarto, C´ordoba, Argentina
fmazzone@exa.unrc.edu.ar
S. Acinas
Dpto. de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de La Pampa, L6300CLB Santa Rosa, La Pampa, Argentina
sonia.acinas@gmail.com
