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Journal of Mathematical Analysis and
Applications
journal homepage:www.elsevier.com/locate/jmaa
Factorization into
k-bubbles for Palais–Smale maps to
potential type energy functionals
Marcos Montenegro
a,∗, Gil F. Souza
baDepartamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil bDepartamento de Matemática, Universidade Federal de Ouro Preto, 35400-000, Campus Universitário, Ouro Preto, MG, Brazil
a r t i c l e i n f o
Article history:
Received 20 December 2012 Available online 30 March 2013 Submitted by Manuel del Pino
Keywords:
Critical Sobolev exponents Potential systems Bubbles Compactness
a b s t r a c t
We prove a decomposition into generalized bubbles for Palais–Smale sequences associated with potential energy functionals for vector-valued function spaces. The study is motivated by the compactness question for solutions of critical potential systems, for which the existence problem was recently addressed. We also present some examples of the existence of radial generalized bubbles.
©2013 Elsevier Inc. All rights reserved.
1. Introduction and main results
In 1984, Struwe established a compactness result for the well-known Brézis–Nirenberg problem [40]
−
1u= |
u|
n−42u+
λ
u inΩu
=
0 on∂
Ω,
(1)where
λ
is a real parameter. Throughout this paper,Ω⊂
Rndenotes a smooth bounded domain forn≥
2. Our starting point is the following well-known existence result due to Brézis and Nirenberg [11].Theorem A.Let
λ
1be the first eigenvalue of the Laplace operator under the Dirichlet boundary condition. If n≥
4and0< λ
< λ
1, then(1)admits at least one positive solution.This is a central result in the theory of elliptic equations as it addresses the existence of solutions for boundary problems involving critical Sobolev growth, which in turn leads to a loss of compactness from a variational viewpoint.
KnowingTheorem A, Struwe investigated as a particular case the behavior of bounded solutions inW01,2
(
Ω)
for(1). Before we state his main result, we first describe some notations.For each 1
<
p<
n, the Sobolev spaceW01,p(
Ω)
is defined as the completion ofC∞0
(
Ω)
under the norm∥
u∥
W1,p 0 (Ω):=
Ω
|∇
u|
pdx
1/p.
∗Corresponding author.
E-mail addresses:montene@mat.ufmg.br(M. Montenegro),gilsouza@iceb.ufop.br(G.F. Souza). 0022-247X/$ – see front matter©2013 Elsevier Inc. All rights reserved.
The analogous form for the whole space, denoted byD1,p
(
Rn)
, is the completion ofC∞0
(
Rn)
with respect to the norm∥
u∥
D1,p(Rn):=
Rn
|∇
u|
pdx
1/p.
Of course, we haveW01,p
(
Ω)
⊂
D1,p(
Rn)
.Given sequences
(
xα)
α∈
Ωand(
rα)
αof positive numbers with the propertyrα→ +∞
asα
→ +∞
, a 1-bubble is defined as a sequence(
Bα)
αof functionsBα
(
x)
=
(
rα)
n−2
2 u
(
rα(
x−
xα)),
obtained by renormalization of a nontrivial solutionu
∈
D1,2(
Rn)
of the equation−
1u= |
u|
n−42u inRn.
(2)We refer toxαandrαas the centers and weights of the 1-bubble
(
Bα)
α, respectively. We can write any positive solutionuof(2)as [13,37]
u
(
x)
=
an−22u0(
a(
x−
x0))
for alla
>
0, whereu0
(
x)
=
1
+
|
x|
2n
(
n−
2)
−n−22.
Struwe’s main result [40] concerns decomposition into 1-bubbles for Palais–Smale sequences associated with the energy functional of(1), namely,
Eλ
(
u)
=
1 2
Ω
(
|∇
u|
2−
λ
u2)
dx−
n−
2 2n
Ω
|
u|
n2−n2 dx.
Thus, we have the following theorem.
Theorem B. Let n
≥
3and let(
uα)
αbe a non-negative Palais–Smale sequence to Eλin W01,2(
Ω)
. Then there exists a solutionu0
∈
W1,20
(
Ω)
of (1)and1-bubbles(
Bjα)
α,
j=
1, . . . ,
l such that some subsequence(
uα)
αsatisfies
uα
−
u0−
l
j=1
Bjα
D1,2(Rn)
→
0 asα
→ +∞
.
Subsequent to the work by Brézis and Nirenberg [11], much effort has been devoted to other questions and extensions of(1)[41, Chapter 3]. The literature contains many discussions of this issue [3,12,15–18,23,28,29,39].
A particular extension that has been extensively investigated is
−
∆pu= |
u|
p∗−2
u
+
λ
|
u|
p−2u inΩ
u
=
0 on∂
Ω,
(3)where 1
<
p<
n,
∆pu=
div(
|∇
u|
p−2∇
u)
denotes thep-Laplace operator, andp∗=
nnp−pis the critical Sobolev exponentfor embedding ofW01,p
(
Ω)
intoLq(
Ω)
.In 1987, Azorero and Peral extendedTheorem A[4].
Theorem C. Let
λ
1,pbe the first eigenvalue of the p-Laplace operator under the Dirichlet boundary condition. If n≥
p2and0
< λ < λ
1,p, then(3)admits at least one positive solution.Several papers provide more details on the existence problem for(3)withp
̸=
2 and other interesting questions [2,4,21,27,31].
Inspired byTheorem C, Mercuri and Willem [35] extendedTheorem Bto problems of the type(3). To state this, we consider again sequences
(
xα)
α∈
Ωand(
rα)
αof positive numbers such thatrα→ +∞
asα
→ +∞
. A 1-bubble of orderpis simply a sequence
(
Bα)
αof functionsBα
(
x)
=
(
rα)
n−p p u
(
rα
(
x−
xα))
obtained by renormalization of a nontrivial solutionu
∈
D1,p(
Rn)
of the equation−
∆pu= |
u|
p∗−2
u inRn
.
(4)case of positive radial solutions. Precisely, any positive radial solutionuof(4)is of the form
u
(
x)
=
n
·
a
n
−
p p−
1
p−1
n−p p2
a
+ |
x|
p−p1
−n−ppfor all constantsa
>
0.The main result of Mercuri and Willem [35] concerns decomposition into 1-bubbles of orderpfor Palais–Smale sequences associated with the energy functional of(3):
Ep,λ
(
u)
=
1p
Ω
(
|∇
u|
p−
λ
|
u|
p)
dx−
1p∗
Ω
|
u|
p∗dx.
When the Palais–Smale sequence is non-negative, their main result yields the following theorem.
Theorem D. Let n
≥
2,
1<
p<
n, and let(
uα)
αbe a non-negative Palais–Smale sequence to Ep,λin W 1,p0
(
Ω)
. Then there existsa solution u0
∈
W1,p0
(
Ω)
of (3)and1-bubbles(
Bjα)
αof order p,
j=
1, . . . ,
l, such that some subsequence(
uα)
αsatisfies
uα
−
u0−
l
j=1
Bj
α
D1,p(Rn)
→
0 asα
→ +∞
.
Barbosa and Montenegro [5] established an extension ofTheorem Cdealing with potential (or gradient) elliptic systems, namely systems of the form
−
∆pu=
1
p∗
∇
F(
u)
+
1p
∇
G(
u)
inΩu
=
0 on∂
Ω,
(5)
where 1
<
p<
n,
u=
(
u1, . . . ,
uk),
∆pu=
(
∆pu1, . . . ,
∆puk)
, andF,
G:
Rk→
RareC1functions withF positive andhomogeneous of degreep∗andGhomogeneous of degreep. For physical reasons, the functionsF andGare known in the literature as potential functions.
After a succession of papers addressed systems of the type(5)[1,7,22,36], Barbosa and Montenegro proved the following existence result that simultaneously extendsTheorems AandC[5].
Theorem E.Let k
≥
1and let F,
G:
Rk→
Rbe C1functions with F positive and homogeneous of degree p∗and G homogeneousof degree p. If n
≥
p2,
MG
:=
maxt∈Skp−1G(
t) < λ
1,pand G(
t0) >
0for some maximum point t0of F onSk−1
p
:= {
t∈
Rk:
|
t|
p=
1}
, then(5)admits at least one nontrivial solution.Barbosa and Montenegro also presented some classes of potential systems that admit non-negative solutions [5, Section 5]. By a non-negative map, we mean one in which each coordinate is non-negative.
Whenk
=
1, note that(5)takes the form(3), since modulo constant factorsF(
t)
= |
t|
p∗andG(
t)
=
λ
|
t|
p. In particular,in this case, the conditionsMG
< λ
1andG(
t0) >
0 assumed inTheorem Ecorrespond toλ < λ
1andλ >
0, respectively. Whenk>
1, there are many homogeneous potential functions. The following are canonical examples.1. F
(
t)
= |
t|
pq∗,
F(
t)
= |
π
l(
t)
|
p∗
l−1
π
l(
t)
; and2. G
(
t)
= |
t|
pq,
G(
t)
= |
π
l(
t)
|
p
l−1
π
l(
t),
G(
t)
= |⟨
At,
t⟩|
(p−2)/2⟨
At,
t⟩
,where
|
t|
q:=
(
ni=1|
ti|
q)
1/q is the Euclideanq-norm forq≥
1, π
l is thelth elementary symmetric polynomial,l=
1
, . . . ,
k,
⟨·
,
·⟩
denotes the usual Euclidean inner product, andA=
(
aij)
is a realk×
kmatrix.Our main goal in this paper is to derive a compactness theorem for bounded non-negative solutions in the Sobolev
k-spaceW01,p
(
Ω,
Rk)
:=
W1,p0
(
Ω)
× · · · ×
W 1,p0
(
Ω)
with respect to the product norm of(5)for the full range 1<
p<
n. For this, we introduce the notion of generalized bubbles, the so-calledk-bubbles of orderp, and prove a factorization intok-bubbles of orderpfor Palais–Smale sequences associated with the energy functional of(5). Our theorem works well for bounded non-negative solutions of a family of potential systems whose corresponding potential functions converge in some sense toFandG.
Consider the Sobolevk-spaceD1,p
(
Rn,
Rk)
:=
D1,p(
Rn)
× · · · ×
D1,p(
Rn)
onRnendowed with the product norm. Obviously, we haveW01,p(
Ω,
Rk)
⊂
D1,p(
Rn,
Rk)
. We begin by taking sequences(
xα)
α∈
Ωand(
rα)
αof positive numbers satisfyingrα→ +∞
asα
→ +∞
. We define ak-bubble of orderpas a sequence(
Bα)
αof mapsBα
(
x)
=
rn−p p
α u
(
rα(
x−
xα))
(6)obtained by renormalization of a nontrivial solutionu
=
(
u1, . . . ,
uk)
∈
D1,p(
Rn,
Rk)
of the system−
∆pu=
1
p∗
∇
F(
u)
inR n.
(7)Our main result establishes a decomposition intok-bubbles of orderpfor non-negative Palais–Smale sequences associ-ated with the following energy functional of(5):
EF,G
(
u)
=
1
p
Ω
(
|∇
u|
p−
G(
u))
dx−
1p∗
Ω F
(
u)
dx,
where
Ω
|∇
u|
pdx:=
k
i=1
Ω
|∇
ui|
pdx.
Theorem 1.1. Let k
≥
1,
n≥
2,
1<
p<
n, andRk+
:= {
t∈
Rk:
ti≥
0}
. Let F,
G:
Rk→
Rbe C1functions with F positive,even, homogeneous of degree p∗and, for some i
,
DiF(
t) >
0for all t∈
R k+
\ {
0}
, and G homogeneous of degree p. Let(
uα)
αbe anon-negative Palais–Smale sequence toEF,Gin W01,p
(
Ω,
Rk
)
. Then there exists a solution u0∈
W1,p0
(
Ω,
Rk
)
of (5)and k-bubbles(
Bjα)
αof order p,
j=
1, . . . ,
l, such that some subsequence(
uα)
αsatisfies
uα
−
u0−
l
j=1 Bαj
D1,p(Rn,Rk)
→
0 asα
→ +∞
.
Theorem 1.1is a complete extension ofTheorems BandD. Following the ideas of Mercuri and Willem [35], it is possible to relax the assumption of non-negativity for
(
uα)
αby assuming only that the negative part of each component of(
uα)
α converges to zero inLp∗(
Ω)
.Note thatTheorems B,D, and1.1provide compactness results for bounded sequences of non-negative solutions of(1),
(3), and(5), respectively, since any such sequences are Palais–Smale sequences to each corresponding energy functional. A more general fact for the compactness of the solutions can be stated as a consequence ofTheorem 1.1.
Corollary 1.1. Let k
≥
1,
n≥
2, and1<
p<
n, and let(
Fα)
αand(
Gα)
αbe sequences of C1functions onRkconverging to Fand G in C1
loc
(
Rk)
, respectively. Assume that Fαand F are homogeneous of degree p∗, F is even, positive and, for some i, satisfiesDiF
(
t) >
0for all t∈
Rk+\ {
0}
, and Gαand G are homogeneous of degree p. Let(
uα)
α⊂
W01,p(
Ω,
Rk
)
be a bounded sequenceconstructed from non-negative solutions uαof the systems
−
∆pu=
1
p∗
∇
Fα(
u)
+
1p
∇
Gα(
u)
inΩ,
u
=
0 on∂
Ω.
(8)
Then there exists a solution u0
∈
W1,p0
(
Ω,
Rk)
of (5)and k-bubbles(
Bαj)
αof order p,
j=
1, . . . ,
l, such that some subsequence(
uα)
αsatisfies
uα
−
u0−
l
j=1 Bαj
D1,p(Rn,Rk)
→
0 asα
→ +∞
.
The proof of this corollary is quite simple. It suffices to note that the convergence of
(
Fα)
αand(
Gα)
αinCloc1(
Rk
)
implies that(
uα)
αis a Palais–Smale sequence toEF,GinW01,p(
Ω,
Rk)
.Compactness problems in PDEs still attract considerable interest, such as for singularly perturbed critical elliptic equations on bounded domains [14], critical anisotropic equations on bounded domains [32], critical elliptic equations on compact manifolds [25,38], critical potential systems on compact manifolds [24,26,33], and the Yamabe problem [8,9,34].
We conclude the paper with a classification result for certain solutions of(7), namely, those generated by solutions of(4). In other words, we provide an extension tok
>
1 of the result established by Ghoussoub and Yuan for(4)[30]. Druet et al. determined an explicit form of the positive solutions (i.e., each positive coordinate) forF(
t)
=
12∗
|
t|
2∗2 [26]. Fork
>
1 andp
=
2, Barbosa and Montenegro obtained a characterization of solutions of(7)that are extremal for a Sobolev inequality related to the potentialF[6].Theorem 1.2. Let k
≥
1,
n≥
2, and1<
p<
n, and let F:
Rk→
Rbe an even p∗-homogeneous positive C1function.Then(7)admits a nontrivial solution of the form tu, where t
=
(
t1, . . . ,
tk)
∈
Rkand u is a nontrivial solution of (4), if and onlyif the vectors tp
=
(
|
t1
|
p−2t1, . . . ,
|
tk|
p−2tk)
and∇
F(
t)
are parallel. In this case, for any vector t0parallel to t there exists a radialsolution u0of (7)satisfying u0
(
0)
=
t0. In particular, for F(
t)
=
p1∗|
t|
p∗
p and any vector t0
∈
Rk,(7)admits a unique radialsolution u0satisfying u0
(
0)
=
t0.Of course, there exist vectorst
∈
Rksuch thattpand∇
F(
t)
are parallel. To see this, it suffices to pick a maximum or mini-mum pointtof the functionFon thep-sphereSpk−1:= {
t∈
Rk: |
t|
pp=
ki=1
|
ti|
p=
1}
, as can easily be seen from Lagrangemultipliers.
2. Proof ofTheorem 1.1
In this section we prove the decomposition intok-bubbles for Palais–Smale sequences associated with the energy func-tionalEF,Gas described in the Introduction. We recall that a sequence
(
uα)
αinW1,p
0
(
Ω,
Rk)
is said to be Palais–Smale for EF,GifEF,G
(
uα)
is boundedand
DEF,G
(
uα)
→
0 inW01,p(
Ω,
Rk
)
∗.
The proof ofTheorem 1.1requires the following seven steps.
Step 1. Palais–Smale sequences forEF,Gare bounded in W
1,p
0
(
Ω,
Rk
)
.Step 1is used in the proof of the next step.
Step 2. Let
(
uα)
αbe a non-negative Palais–Smale sequence forEF,G. Then, up to a subsequence,(
uα)
αconverges weakly to u0inW01,p
(
Ω,
Rk)
. Moreover, u0is a non-negative weak solution of (5).Step 3. LetI
:
W01,p(
Ω,
Rk)
→
Rbe the energy functionalI
(
u)
=
1p
Ω
|∇
u|
pdx−
1p∗
Ω F
(
u)
dxassociated with the system
−
∆pu=
1
p∗
∇
F(
u)
inΩ,
u
=
0 on∂
Ω.
Let
(
uα)
αbe a Palais–Smale sequence forEF,Gconverging weakly to u0in W01,p(
Ω,
Rk)
. ThenEF,G
(
uα)
=
EF,G(
u0)
+
I(
uα−
u0)
+
o(
1)
and
(
uα−
u0)
αis a Palais–Smale sequence forI.In what follows, we letKF
(
n,
p)
be a sharp constant for the potential-type Sobolev inequality
Rn
F
(
u)
dx
p1∗≤
K
Rn
|∇
u|
pdx
1p.
(9)More precisely,
KF
(
n,
p)
=
sup
Rn
F
(
u)
dx
p1∗:
u∈
D1,p(
Rn,
Rk),
∥
u∥
D1,p(Rn,Rk)
=
1
.
Barbosa and Montenegro proved thatKF
(
n,
p)
=
M1 p∗
F K
(
n,
p)
[5], whereMFis the maximum ofFonSkp−1andK(
n,
p)
isthe sharp constant for the classical Sobolev inequality
Rn
|
u|
p∗dx
1p∗
≤
K
Rn
|∇
u|
pdx
1p.
Step 4. Let
(v
α)
αbe a Palais–Smale sequence forIconverging weakly to0in W01,p(
Ω,
Rk)
such thatI(v
α)
→
β
. Ifβ < β
∗:=
n−1KF(
n,
p)
−nthen
β
=
0and(v
α)
αconverges strongly to0in W 1,p0
(
Ω,
Rk)
.Step 5. Let u0
∈
D1,p(
Rn,
Rk)
be a nontrivial solution of the system(7). Then we haveJ(
u0)
≥
β
∗, whereJ:
D1,p(
Rn,
Rk)
→
Rdenotes the energy functional given byJ
(
u)
=
1p
Rn
|∇
u|
pdx−
1p∗
Rn
Step 6. Let H
= {
x∈
Rn:
xn>
0}
and let u∈
D1,p0
(
H,
Rk)
be a non-negative weak solution of the potential system−
∆pu=
1
p∗
∇
F(
u)
in H,
whereD01,p
(
H,
Rk)
denotes the completion of C0∞(
H,
Rk)
under the norm∥
u∥ :=
H
|∇
u|
pdx
1/p.
Then u
≡
0on H.Step 6is used in the proof of the next step.
Step 7. Let
(v
α)
αbe a non-negative Palais–Smale sequence forIconverging weakly to0in W01,p(
Ω,
Rk)
, but not strongly. Thenthere exists a sequence of points
(
xα)
αof Ωand a sequence of positive numbers(
rα)
αwith rα→ +∞
, a nontrivial solutionv
to(7)and a Palais–Smale sequence
(w
α)
forIin W 1,p0
(
Ω,
Rk
)
such that, modulo a subsequence(v
α
)
α, the following holds:w
α(
x)
=
v
α(
x)
− ˆ
Bα(
x)
+
o(
1),
whereB
ˆ
α(
x)
=
rn−p p
α
v(
rα(
x−
xα))
and o(
1)
→
0inD1,p(
Rn,
Rk)
. Moreover,I
(w
α)
=
I(v
α)
−
J(v)
+
o(
1)
and
rαdist
(
xα, ∂
Ω)
→ +∞
α
→ +∞
.
For the moment, we postpone the proofs ofSteps 1–7to present the following proof.
Proof of Theorem 1.1. ByStep 2,
(
uα)
αconverges weakly tou0inW 1,p0
(
Ω,
Rk
)
; if(
uα
)
αconverges strongly tou0, the proof is complete. Otherwise, by [35, Lemma 3.5], without loss of generality we can consider that(
uα−
u0)
αis non-negative, so we take the sequence(v
1α
)
αgiven byv
α1=
uα−
u0and evokeStep 7to find a sequence(
Bα1)
αofk-bubbles of orderpsuch that the sequence(v
2α
)
αdefined byv
2α=
v
1α−
B1αis Palais–Smale forI. If(v
α2)
αconverges strongly to 0 inW 1,p0
(
Ω,
Rk)
, the proof is complete. Otherwise, we proceed inductively by lettingv
α1=
uα−
u0 andv
αj=
uα−
u0−
j−1
i=1
Bαi
=
v
αj−1−
Bj−1 α
,
whereBαi
=
r n−pp
α ui
(
rα(
· −
xα))
andui∈
D1,p(
Rn,
Rk)
is a nontrivial solution of(7). BySteps 3and5, we obtainI
(v
αj)
=
EF,G(
uα)
−
EF,G(
u0)
−
j−1
i=1
J
(
ui)
≤
EF,G(
uα)
−
EF,G(
u0)
−
(
j−
1)β
∗.
We claim that this process stops afterlsteps. In fact, the preceding inequality andStep 4furnishI
(v
l+1α
)
≤
0 for some indexl≥
0. Thus,v
l+1α
=
uα−
u0−
li=1Bαi converges strongly to 0 inD1,p(
Rn,
Rk)
andEF,G
(
uα)
−
EF,G(
u0)
−
l
i=1
J
(
ui)
→
0.
Now we prove the seven steps.
Proof of Step 1. Let
(
uα)
αbe a Palais–Smale sequence forEF,G. Thanks to the homogeneity properties satisfied byFandG,we derive
DEF,G
(
uα)
·
uα=
Ω
|∇
uα|
p−
G(
uα)
−
F(
uα)
dx
=
o(
∥
uα∥
W1,p(Ω,Rk)),
(10)so that
EF,G
(
uα)
=
1n
Ω
F
(
uα)
dx+
1pDEF,G
(
uα)
·
uα=
1
n
Ω
F
(
uα)
dx+
o(
∥
uα∥
W1,p(Ω,Rk)).
SinceEF,G
(
uα)
≤
cfor some constantc>
0 independent ofα
, we obtain
Ω
F
(
uα)
dx≤
nc+
o
∥
uα∥
W1,p(Ω,Rk)Furthermore, sinceFis continuous, by Holder’s inequality, we easily deduce that
Ω
|
uα|
pdx≤
c+
o
∥
uα∥
p/p∗ W1,p(Ω,Rk)
,
wherec
>
0, like all the constants below, is independent ofα
. Writing
Ω
|∇
uα|
p−
G(
uα)
dx
=
pEF,G(
uα)
+
p p∗
Ω
F
(
uα)
dx,
we also obtain
Ω
|∇
uα|p−
G(
uα)
dx
≤
c+
o
∥
uα∥W1,p(Ω,Rk)
.
Noting by the continuity ofGthat
∥
uα∥
p
W1,p(Ω,Rk)
≤
Ω
|∇
uα|
p−
G(
uα)
dx
+
c∥
uα∥
p Lp(Ω,Rk)
,
it follows from the above equations that
∥
uα∥
p
W1,p(Ω,Rk)
≤
c+
o
∥
uα∥
W1,p(Ω,Rk)
+
o
∥
uα∥
p/p∗ W1,p(Ω,Rk)
.
However, this clearly implies that
(
uα)
αis bounded inW 1,p0
(
Ω,
Rk)
, which completes the proof ofStep 1.Proof of Step 2. ByStep 1and the Sobolev embedding theorems, modulo a subsequenceuα
⇀
u0inW1,p
0
(
Ω,
Rk
)
anduα
→
u0inLq(
Ω,
Rk)
for allq<
p∗, whereLq(
Ω,
Rk)
:=
Lq(
Ω)
× · · · ×
Lq(
Ω)
, is endowed with the product norm. Since(
uα)
αis a Palais–Smale sequence, we havek
i=1
Ω
|∇
uiα|
p−2⟨∇
uiα
,
∇
ϕ
i⟩
dx−
Ω
∇
G(
uα)
·
ϕ
dx−
Ω
∇
F(
uα)
·
ϕ
dx=
o(
1)
(11)for all
ϕ
=
(ϕ
1, . . . , ϕ
k)
∈
C0∞(
Ω,
Rk)
, whereuα=
(
u1α, . . . ,
ukα)
. The strong convergence of(
uα)
αinLq(
Ω,
Rk)
and the regularity and homogeneity conditions onFandGyield
Ω
∇
F(
uα)
·
ϕ
dx→
Ω
∇
F(
u0)
·
ϕ
dxand
Ω
∇
G(
uα)
·
ϕ
dx→
Ω
∇
G(
u0)
·
ϕ
dxas
α
→ +∞
. Conversely, the convergence of the first term of(11)is standard [38, Step 1.2 of Theorem 0.1]. Thus, we conclude from(11)thatu0is a weak solution of(5)and it is straightforward to show thatu0is non-negative.
Proof of Step 3. A standard fact is that
|∇
uiα|
p→ |∇
u0i|
pa.e. inΩfor alli[38, Step 1.2 of Theorem 0.1], so by the Brézis–Lieblemma [10] we have
Ω
|∇
uα|
pdx=
Ω
|∇
(
uα−
u0)
|
pdx+
Ω
|∇
u0|
pdx+
o(
1).
(12)According to the compactness,
Ω
G
(
uα)
dx=
Ω
G
(
uα−
u0)
dx+
Ω
G
(
u0)
dx+
o(
1)
(13)and by a version of the Brézis–Lieb lemma for maps [5],
Ω
F
(
uα)
dx=
Ω
F
(
uα−
u0)
dx+
Ω
F
(
u0)
dx+
o(
1).
(14)By setting
v
α=
uα−
u0and using(12)–(14), we can writeEF,G
(
uα)
=
1p
Ω
|∇
(v
α+
u0)
|
p−
G(v
α+
u0)
dx
−
1p∗
Ω
F
(v
α+
u0)
dx=
1p
Ω
|∇
v
α|
p+ |∇
u0|
p−
G(v
α)
−
G(
u0)
dx
−
1p∗
Ω
F
(v
α)
+
F(
u0)
dx
+
o(
1)
=
EF,G(
u0)
+
I(v
α)
+
1p
Ω
By the compactness and assumptions forG, the integral on the right-hand side goes to 0. In particular,
EF,G
(
uα)
=
EF,G(
u0)
+
I(v
α)
+
o(
1).
(15)To show that
(v
α)
αis a Palais–Smale sequence forI, we note first thatI
(v
α)
=
EF,G(
uα)
−
EF,G(
u0)
+
o(
1)
=
O(
1)
+
o(
1)
implies the boundedness of
(
I(v
α))
α. Arguing as inStep 2, we have
Ω
∇
F(
uα)
·
ϕ
dx=
Ω
∇
F(
u0)
·
ϕ
dx+
o(
1)
(16)and
Ω
∇
G(
uα)
·
ϕ
dx=
Ω
∇
G(
u0)
·
ϕ
dx+
o(
1)
(17)for any
ϕ
∈
C0∞(
Ω,
Rk)
.Combining Eqs.(12)–(14),(16)and(17), we compute
DEF,G
(v
α+
u0)
·
ϕ
−
DI(v
α)
·
ϕ
=
k
i=1
Ω
|∇
(v
αi+
u 0i)
|
p−2⟨∇
(v
iα+
u 0i),
∇
ϕ
i⟩
− ∇
G(v
iα+
u 0i)
·
ϕ
idx−
Ω
∇
F(v
α+
u0)
·
ϕ
dx−
k
i=1
Ω
|∇
v
αi|
p−2⟨∇
v
αi,
∇
Φi⟩
dx+
Ω
∇
F(v
α)
·
ϕ
dx=
k
i=1
Ω
⟨|∇
v
αi|
p−2∇
u0i+ |∇
u0i|
p−2∇
v
iα
,
∇
ϕ
i⟩
dx+
k
i=1
Ω
|∇
u0i|
p−2⟨∇
u0i,
∇
ϕ
i⟩
dx−
Ω
∇
G(
u0)
·
ϕ
dx−
Ω
∇
F(
u0)
·
ϕ
dx+
o
∥
ϕ
∥
W01,p(Ω,Rk)
.
Using the fact thatu0is a weak solution of(5), we can derive the desired result.
Proof of Step 4. ByStep 1, it follows that
(v
α)
αis bounded inW 1,p0
(
Ω,
Rk)
. Then we can writeDI
(v
α)
·
v
α=
Ω
|∇
v
α|
pdx−
Ω
F
(v
α)
dx=
o(
1)
and
I
(v
α)
=
1p
Ω
|∇
v
α|
pdx−
1p∗
Ω
F
(v
α)
dx=
β
+
o(
1).
From these relations, we obtain
Ω
F
(v
α)
dx=
nβ
+
o(
1)
and
Ω
|∇
v
α|
pdx=
nβ
+
o(
1).
In particular, we derive
β
≥
0. By its compactness, we can assume thatv
α→
0 inLp(
Ω,
Rk)
. TheF-Sobolev inequality [5]
Ω
F
(v
α)
dx
pp∗≤
KF(
n,
p)
p
Ω
|∇
v
α|
pdxleads to
(
nβ)
p p∗
≤
KWe assert that
β
=
0. Assume, by contradiction, thatβ >
0. Then(
nβ)
p
p∗−1
=
(
nβ)
−np≤
KF(
n,
p)
p,
so that
KF
(
n,
p)
p=
(
nβ
∗)
−p
n
< (
nβ)
− pn
≤
KF(
n,
p)
p.
Since
β
=
0, we have
Ω
|∇
v
α|
pdx=
o(
1).
In other words,
(v
α)
αconverges to 0 inW 1,p0
(
Ω,
Rk
)
, which completes the proof ofStep 4.
Proof of Step 5. Letu0
∈
D1,p
(
Rn,
Rk)
be a nontrivial solution of(7)Then, it follows directly from(7)that
Rn
|∇
u0|
pdx=
Rn
F
(
u0)
dx≤
KF(
n,
p)
p∗
Rn
|∇
u0|
pdx
p∗ p
.
However, this clearly implies
J
(
u0)
=
1p
Rn
|∇
u0|
pdx−
1p∗
Rn
F
(
u0)
dx≥
1nKF
(
n,
p)
−n
=
β
∗.
We proveStep 6using two lemmas. The first is the following weakened form of the divergence theorem presented by Mercuri and Willem [35].
Lemma 2.1. LetΩbe a smooth bounded domain inRnwith outer normal unit vector
ν(
·
)
and letv
∈
C(
Rn,
Rn)
be such that divv
∈
L1loc(
Rn)
. Then
Ω
div
v
dx=
∂Ω
v(σ )
·
ν(σ )
dσ .
Hereafter we denoteH
= {
x∈
Rn:
xn>
0}
. The next lemma is inspired by Mercuri and Willem [35] and its proofproceeds in the same spirit.
Lemma 2.2. Let k
≥
1,
n≥
2, and1<
p<
n, and let F:
Rk→
Rbe a function of the C1class that is positive, even, andhomogeneous of degree p∗. Let u
∈
D1,p0
(
H,
Rk)
be weak solution of the system−
∆pu=
1
p∗
∇
F(
u)
in H.
(18)Then Dnu
:=
∂∂xnu=
0everywhere on∂
H.Proof. Letu
∈
D01,p(
H,
Rk)
be a weak solution of(18). By the anti-reflection ofuinRn\
Hwith respect to∂
H, we can extenduto a mapv
∈
D1,p(
Rn,
Rk)
. SinceFis even, it follows thatv
is a weak solution of(7). By [5, Lemma 2.5], we havev
∈
Cloc1,α(
Rn,
Rk)
. Thus, since|∇
F(v)
| ∈
L∞loc
(
Rn)
, by [42, Proposition 1] we findv
∈
W 2,qloc
(
Rn,
Rk)
withq=
min{
p,
2}
. In particular,−
∆pv
=
1
p∗
∇
F(v)
almost everywhere inRn. Thus, we can easily derive
div
(
Dnv
i|∇
v
i|
p−2∇
v
i)
=
Dnv
i∆pv
i+ |∇
v
i|
p−2∇
v
i· ∇
(
Dnv
i)
∈
L1loc(
Rn
).
LetBρbe a ball of center 0 and radius
ρ
inRn. FromLemma 2.1, we have
H∩Bρ
Dnui div
(
|∇
ui|
p−2∇
ui)
dx=
∂(H∩Bρ)
Dnui
|∇
ui|
p−2∇
ui·
ν(σ )
dσ
−
H∩Bρ
|∇
ui|
p−2∇
ui· ∇
(
Dnui)
dx=
∂(H∩Bρ)
Dnui
|∇
ui|
p−2∇
ui·
ν(σ )
dσ
−
∂(H∩Bρ)
|∇
ui|
pp
ν
n(σ )
dσ
and 1
p∗
H∩Bρ
∇
F(
u)
·
Dnu dx=
1
p∗
∂(H∩Bρ)
F
(
u)ν
n(σ )
dσ
=
1
p∗
H∩∂Bρ
LetX
=
(
|∇
u1|
p−2∇
u1·
ν, . . . ,
|∇
uk|
p−2∇
uk·
ν)
. Thanks to(18), we obtain
1−
1p
∂H∩Bρ
|
Dnu|
pdσ
=
H∩∂Bρ
Dnu
·
X(σ )
dσ
−
H∩∂Bρ
|∇
u|
pp
ν
n(σ )
dσ
+
H∩∂Bρ
F
(
u)ν
n(σ )
dσ .
Note that the right-hand side is bounded by
M
(ρ)
=
1
+
1p
H∩∂Bρ
|∇
u|
pp d
σ
+
H∩∂Bρ
F
(
u)
dσ .
Since
∇
u∈
Lp(
H,
Rk)
andu∈
Lp∗(
H,
Rk)
, there exists a sequenceρ
α
→ ∞
such thatM(ρ
α)
→
0. The monotone convergence theorem then furnishes
∂H|
Dnu|
pdσ
=
0, which concludes the proof ofLemma 2.2.Proof of Step 6. Assume thatuis a nontrivial non-negative weak solution. SinceDiF
(
u) >
0 andui≥
0, we obtain∆pui≤
0and∆pui
̸≡
0. Sinceu∈
Cloc1,α(
H,
Rk
)
, by the strong maximum principle [43] we obtainDnui
>
0 on∂
H. Conversely, byLemma 2.2we haveDnui
=
0 on∂
H. This contradiction leads us to the conclusion ofStep 6.Proof of Step 7. We proveStep 7using three lemmas that are introduced during the proof. First, up to a subsequence, we can assume thatI
(v
α)
→
β
asα
→ +∞
. Moreover, by the density ofC0∞(
Ω,
Rk)
inW1,p
0
(
Ω,
Rk)
, we assume that each mapv
αis smooth. SinceDI(v
α)
→
0,1
n
Ω
|∇
v
α|
pdx=
I(v
α)
−
1p∗DI
(v
α)
·
v
α→
β,
and hence, byStep 4,
lim inf α→+∞
Ω
|∇
v
α|
pdx=
nβ
≥
KF(
n,
p)
−n.
(19)Fort
>
0, letµ
α(
t)
=
supx∈Ω
Bt(x)
|∇
v
α|
pdx
,
whereBt
(
x)
denotes the ball with radiustand centerxinRn. It follows from(19)thatµ
α(
t) >
0 and limt→+∞µ
α(
t)
≥
KF
(
n,
p)
−n. Let 0< δ <
KF(
n,
p)
−n. Sincev
α is smooth,µ
α(
·
)
is continuous. Thus, for anyλ
∈
(
0, δ)
, there existstα
∈
(
0,
+∞
)
such thatµ
α(
tα)
=
λ
. There also existsyα∈
Ωsuch that
Btα(yα)
|∇
v
α|
pdx=
λ.
In conclusion, we can choosexα
∈
Ωandrαsuch that the rescaling˜
v
α(
x)
=
r−n−pp α
v
α
x rα
+
xα
satisfies
˜
µ
α(
1)
=
supx∈Rn
x rα+xα∈Ω
B1(x)
|∇ ˜
v
α|
pdx=
B1(0)
|∇ ˜
v
α|
pdx=
12LKF
(
n,
p)
−n
,
(20)whereL
∈
Nis such thatB2(
0)
is covered byLballs of radius 1 centered onB2(
0)
. According to(19), there existsr0>
0 such thatrα≥
r0for allα
. Of course,∥ ˜
v
α∥
p
D1,p(Rn,Rk)
= ∥
v
α∥
p
W1,p(Ω,Rk)
→
nβ <
∞
,
so that
v
˜
α⇀
v
˜
0inD1,p(
Rn,
Rk)
up to a subsequence. Furthermore, by construction,v
˜
0≥
0. Our first lemma is as follows.Lemma 2.3. We have
v
˜
α→ ˜
v
0in W1,p(
Ω′,
Rk)
for anyΩ′⊂⊂
Rn.Proof. To prove this claim, it suffices to verify its validity forΩ′
=
B1(
x0)
for anyx0∈
Rn. By Fubini’s theorem, we have
21
∂Bρα(x0)
|∇ ˜
v
α|
pdσ
dr
≤
B2(x0)
By the mean value theorem, we obtain that there exists a radius
ρ
α∈ [
1,
2]
such that
∂Bρα(x0)
|∇ ˜
v
α|
pdσ
≤
2nβ
+
o(
1).
(21)Let p
ˆ
=
p−p1 and Wpˆ,p(∂
Ω,
Rk)
be the space product Wˆp,p(∂
Ω,
Rk)
=
Wpˆ,p(∂
Ω)
× · · · ×
Wˆp,p(∂
Ω)
endowedwith the product topology, whereWpˆ,p
(∂
Ω)
denotes the space of the trace function inW1,p(
Ω)
. By the compactnessof the embeddingW1,p
(∂
Bρα
(
x0),
Rk
) ↩
→
Wpˆ,p(∂
Bρα
(
x0),
Rk
)
[41, Appendix A], up to a subsequence we deduce thatv
˜
α converges strongly to
v
0inWˆp,p(∂
Bρα(
x0),
Rk)
. In addition, by the compactness of the trace operatorW1,p(
Bρα(
x0),
Rk) ↩
→
Lp
(∂
Bρα
(
x0),
Rk
)
, we havev
˜
0
=
v
0. We defineφ
α=
v
˜
α
− ˜
v
0 inBρα(
x0)
˜
w
α inB3(
x0)
\
Bρα(
x0)
0 otherwise
,
where
w
˜
αdenotes the solution of the Dirichlet problem
∆p
w
˜
α=
0 inB3(
x0)
\
Bρα(
x0)
˜
w
α= ˜
v
α− ˜
v
0 on∂
Bρα(
x0)
˜
w
α=
0 on∂
B3(
x0)
.
The existence of such
w
˜
αis guaranteed [38, Step 2.2 of Lemma 1.1]. The same step guarantees the existence of a constantc
>
0, independent ofρ
α,
w
˜
αandv
˜
α− ˜
v
0, such that∥ ˜
w
α∥
W1,p(B3(x0)\Bρα(x0),Rk)≤
C∥ ˜
v
α− ˜
v
0∥
Wˆp,p(∂Bρα(x0),Rk)
,
which gives us∥ ˜
w
α∥
W1,p(B3(x0)\Bρα(x0),Rk)→
0.
(22)Consider the rescaling
φ
ˆ
α(
x)
=
rn−p p
α
φ
α(
rα(
x−
xα))
. Since suppφ
α⊂
B3(
x0)
forα
large enough, we obtain suppφ
ˆ
α⊂
B3r−1 α
x0
rα
+
xα
⊂
Ω. Since(v
α)
αis a Palais–Smale sequence forI, we haveDJ
(
v
˜
α)
·
φ
α=
DI(v
α)
· ˆ
φ
α=
o(
1).
Thanks to the definition of
φ
α, Eqs.(12)and(14), the assumptions onF, the strong convergencev
˜
α→ ˜
v
0inLq(
Ω,
Rk)
withq<
p∗, and Eqs.(9)and(22), we deduce thato
(
1)
=
DJ(
v
˜
α)
·
φ
α=
k
i=1
Rn
|∇ ˜
v
iα|
p−2⟨∇ ˜
v
iα
,
∇
φ
i
α
⟩ −
1p∗
∂
iF(
v
˜
α)
·
φ
iα
dx
=
Bρα(x0)
|∇
(
v
˜
α− ˜
v
0)
|
p−
F(
v
˜
α− ˜
v
0)
dx
+
o(
1)
=
Rn
|∇
φ
α|
p−
F(φ
α)
dx
+
o(
1)
≥ ∥
φ
α∥
p
D1,p(Rn,Rk)
1
−
KF(
n,
p)
p∗
∥
φ
α∥
p∗−p
D1,p(Rn,Rk)
+
o(
1),
(23)whereo
(
1)
→
0 asα
→ +∞
. Conversely, by the definition ofφ
αand Eqs.(12),(21), and(22),
Rn
|∇
φ
α|
pdx=
Bρα(x0)
|∇
(
v
˜
α− ˜
v
0)
|
pdx+
B3(x0)\Bρα(x0)
|∇ ˜
w
α|
pdx+
o(
1)
=
Bρα(x0)
|∇
(
v
˜
α− ˜
v
0)
|
pdx+
o(
1)
≤
Bρα(x0)
(
|∇ ˜
v
α|p− |∇ ˜
v
0|
p)
dx+
o(
1)
≤
B2(x0)
|∇ ˜
v
α|
pdx+
o(
1)
≤
Lµ
˜
α(
1)
=
KF
