Global compactness for a class of quasi-linear elliptic problems

0.24 MB·

by  Carlo Mercuri

GLOBAL COMPACTNESS FOR A CLASS OF
QUASI-LINEAR ELLIPTIC PROBLEMS
CARLO MERCURI AND MARCO SQUASSINA
Abstract. We prove a global compactness result for Palais-Smale sequences associated with a
2 class of quasi-linear elliptic equations on exterior domains.
1
0
2

n
1. Introduction and main result
a
J Let Ω be a smooth domain of RN with a boundedcomplement and N > p > m > 1. The main
0
goal of this paper is to obtain a global compactness result for the Palais-Smale sequences of the
2
energy functional associated with the following quasi-linear elliptic equation

]
P (1.1) −div(L (Du))−div(M (u,Du))+M (u,Du)+V(x)|u|p−2u = g(u) in Ω,
ξ ξ s
A
1,p 1,m
where u ∈ W (Ω)∩D (Ω), meant as the completion of the space D(Ω) of smooth functions
. 0 0
h
with compact support, with respect to the norm kuk = kuk +kuk , having set
t W1,p(Ω)∩D1,m(Ω) p m
a kuk := kuk and kuk := kDuk . We assume that V is a continuous function on Ω,
m p W1,p(Ω) m Lm(Ω)
lim V(x) = V and inf V(x) = V > 0.
[ ∞ 0
|x|→∞ x∈Ω

2
As known, lack of compactness may occur due to the lack of compact embeddings for Sobolev
v
6 spaces on Ω and since the limiting equation on RN
4
(1.2) −div(L (Du))−div(M (u,Du))+M (u,Du)+V |u|p−2u= g(u) in RN,
0 ξ ξ s ∞
4 with u∈ W1,p(RN)∩D1,m(RN), is invariant by translations. A particular case of (1.1) is
.
7
1
0 (1.3) −∆ u−div(a(u)|Du|m−2Du)+ a′(u)|Du|m +V(x)|u|p−2u =|u|σ−2u in Ω,
1 p m
1
where ∆ u := div(|Du|p−2Du), for a suitable function a ∈ C1(R;R+), or the even simpler case
: p
v
where a is constant, namely
i
X
(1.4) −∆ u−∆ u+V(x)|u|p−2u = |u|σ−2u in Ω.
p m
r
a Since the pioneering work of Benci and Cerami [2] dealing with the case L(ξ) = |ξ|2/2 and
M(s,ξ) ≡ 0, many papers have been written on this subject, see for instance the bibliography
of [12]. Quite recently, in [12], the case L(ξ) = |ξ|p/p and M(s,ξ) ≡ 0 was investigated. The
main point in the present contribution is the fact that we allow, under suitable assumptions,
a quasi-linear term M(u,Du) depending on the unknown u itself. The typical tools exploited
in [2,12], in addition to the point-wise convergence of the gradients, are some decomposition
(splitting) results both for the energy functional and for the equation, along a given bounded
Palais-Smale sequence (u ). To this regard, the explicit dependence on u in the term M(u,Du)
n
requires a rather careful analysis. In particular, we can handle it for
ν|ξ|m ≤ M(s,ξ) ≤ C|ξ|m, p−1≤ m < p−1+p/N.
2000 Mathematics Subject Classification. 35D99, 35J62, 58E05, 35J70.
Key words and phrases. Quasi-linear equations, global compactness of Palais-Smale sequences.
Supported by Miur project: “Variational and Topological Methods in the Study of Nonlinear Phenomena”.
1
2 C.MERCURIANDM.SQUASSINA
The restriction on m, together with the sign condition (1.9) provides, thanks to the presence of
L, the needed a priori regularity on the weak limit of (u ), see Theorems 3.2 and 3.4.
n
Besides the aforementioned motivations, which are of mathematical interest, it is worth pointing
out that in recent years, some works have been devoted to quasi-linear operators with double
homogeneity, which arise from several problems of Mathematical Physics. For instance, the
reaction diffusion problem u = −div(D(u)Du)+ℓ(x,u), where D(u) = d |Du|p−2+d |Du|m−2,
t p m
d > 0 and d > 0, admitting a rather wide range of applications in biophysics [10], plasma
p m
physics [16] and in the study of chemical reactions [1]. In this framework, u typically describes
a concentration and div(D(u)Du) corresponds to the diffusion with a coefficient D(u), whereas
ℓ(x,u) plays the roˇle of reaction and relates to source and loss processes. We refer the interested
readerto[5]andtothereferencetherein. Furthermore,amodelforelementary particles proposed
by Derrick [9] yields to the study of standing wave solutions ψ(x,t) = u(x)eiωt of the following
nonlinear Schro¨dinger equation
iψ +∆ ψ−b(x)ψ+∆ ψ−V(x)|ψ|p−2ψ+|ψ|σ−2ψ = 0 in RN,
t 2 p
for which we refer the reader e.g. to [3].
In order to state the first main result, assume N > p > m ≥ 2 and
(1.5) p−1 ≤ m < p−1+p/N, p < σ < p∗,
and consider the C2 functions L : RN → R and M : R×RN → R such that both the functions
ξ 7→ L(ξ) and ξ 7→ M(s,ξ) are strictly convex and
(1.6) ν|ξ|p ≤ |L(ξ)| ≤ C|ξ|p, |L (ξ)| ≤ C|ξ|p−1, |L (ξ)| ≤ C|ξ|p−2,
ξ ξξ
for all ξ ∈ RN. Furthermore, we assume
(1.7) ν|ξ|m ≤ M(s,ξ)| ≤ C|ξ|m, |M (s,ξ)| ≤ C|ξ|m, |M (s,ξ)| ≤ C|ξ|m−1,
s ξ
(1.8) |M (s,ξ)| ≤ C|ξ|m, |M (s,ξ)| ≤ C|ξ|m−1, |M (s,ξ)| ≤ C|ξ|m−2,
ss sξ ξξ
for all (s,ξ) ∈ R×RN and that the sign condition (cf. [14])
(1.9) M (s,ξ)s ≥ 0,
s
holds for all (s,ξ) ∈ R×RN. Also, G : R → R is a C2 function with G′(s) := g(s) and
(1.10) |G′(s)| ≤ C|s|σ−1, |G′′(s)|≤ C|s|σ−2,
for all s ∈ R. We define
(1.11) j(s,ξ) := L(ξ)+M(s,ξ)−G(s),
1,p 1,m
and on W (Ω)∩D (Ω) with kuk = kuk +kuk the functional
0 0 W1,p(Ω)∩D1,m(Ω) p m
|u|p
φ(u) := j(u,Du)+ V(x) .
Z Z p
Ω Ω
Finally, on W1,p(RN)∩D1,m(RN) with kukW1,p(RN)∩D1,m(RN) = kukp +kukm we define
|u|p
φ (u) := j(u,Du)+ V .
∞ ∞
ZRN ZRN p
See Section 2 for some properties of the functionals φ and φ .
∞
The first main global compactness type result is the following
GLOBAL COMPACTNESS FOR QUASI-LINEAR PROBLEMS 3
Theorem 1.1. Assume that (1.5)-(1.11) hold and let (u ) ⊂ W1,p(Ω)∩D1,m(Ω) be a bounded
n 0 0
sequence such that
φ(u ) → c φ′(u )→ 0 in (W1,p(Ω)∩D1,m(Ω))∗
n n 0 0
Then, up to a subsequence, there exists a weak solution v ∈ W1,p(Ω)∩D1,m(Ω) of
0 0 0
−div(L (Du))−div(M (u,Du))+M (u,Du)+V(x)|u|p−2u= g(u) in Ω,
ξ ξ s
a finite sequence {v ,...,v } ⊂ W1,p(RN)∩D1,m(RN) of weak solutions of
1 k
−div(L (Du))−div(M (u,Du))+M (u,Du)+V |u|p−2u = g(u) in RN
ξ ξ s ∞
and k sequences (yi)⊂ RN satisfying
n
|yi|→ ∞, |yi −yj|→ ∞, i6= j, as n → ∞,
n n n
k
kun−v0− vi((·−yni)kW1,p(RN)∩D1,m(RN) → 0, as n → ∞,
Xi=1
k k
ku kp → kv kp, ku km → kv km, as n → ∞,
n p i p n m i m
Xi=0 Xi=0
as well as
k
φ(v )+ φ (v )= c.
0 ∞ i
Xi=1
Let us now come to a statement for the cases 1 < m ≤ 2 or 1 < p ≤ 2. Let us define
|L (ξ +h)−L (ξ)|
L(ξ,h) := ξ ξ , if 1 < p < 2,
|h|p−1
|G′(s+t)−G′(s)|
G(s,t) := , if 1 < σ < 2,
|t|σ−1
|M (s,ξ +h)−M (s,ξ)|
M(s,ξ,h) := ξ ξ , if 1 < m < 2.
|h|m−1
If either p < 2, σ < 2 or m < 2, we shall weaken the twice differentiability assumptions, by
requiring L ∈ C1(RN \{0}), G′ ∈ C1(R\{0}), M ∈ C1(R×(RN \{0})), M ∈ C0(R×RN)
ξ ξ sξ
and M ∈ C0(R×RN). Moreover we assume the same growth conditions for L,M,G and their
ss
derivatives, replacing only the growth assumptions for L ,M ,G′′ by the following hypotheses:
ξξ ξξ
(1.12) sup L(ξ,h) < ∞,
h6=0,ξ∈RN
(1.13) sup G(s,t) < ∞,
t6=0,s∈R
(1.14) sup M(s,ξ,h) < ∞.
h6=0,(s,ξ)∈R×RN
Conditions (1.12)-(1.13), in some more concrete situations, follow immediately by homogeneity
of L and G′ (see, for instance, [12, Lemma 3.1]). Similarly, (1.14) is satisfied for instance when
ξ
M is of the form M(s,ξ) = a(s)µ(ξ), being a : R → R+ a bounded function and µ : RN → R+ a
C1 strictly convex function such that µ is homogeneous of degree m−1.
ξ
Theorem 1.2. Under the additional assumptions (1.12)-(1.14) in the sub-quadratic cases, the
assertion of Theorem 1.1 holds true.
As a consequence of the above results we have the following compactness criterion.
4 C.MERCURIANDM.SQUASSINA
Corollary 1.3. Assume (2.1) below for some δ > 0 and µ > p. Under the hypotheses of Theorem
1.1 or 1.2, if c <c∗, then (u ) is relatively compact in W1,p(Ω)∩D1,m(Ω) where
n 0 0
p
c∗ := min δ,µ−pV min{ν,V∞} σ−p ,
∞
(cid:26)µ µp (cid:27)(cid:20) C S (cid:21)
g p,σ
and S and C are constants such that S kukσ ≥ kukσ and |g(s)| ≤ C |s|σ−1.
p,σ g p,σ p Lσ(RN) g
Remark 1.4. It would be interesting to get a global compactness result in the case L = 0 and
p = m, namely for the model case
1
(1.15) −div(a(u)|Du|m−2Du)+ a′(u)|Du|m +V(x)|u|m−2u= |u|σ−2u in Ω.
m
Notice that, even assuminga′ bounded,a′(u)|Du|m ismerely inL1(Ω)forW1,m(Ω)distributional
0
solutions. Ingeneral, inthis setting, thesplitting propertiesof theequation arehardtoformulate
in a reasonable fashion.
Remark 1.5. The restriction of between m and p in assumption (1.5) is no longer needed in the
case where M is independent of the first variable s, namely M ≡ 0.
s
Remark 1.6. We prove the above theorems under the a-priori boundedness assumption of (u ).
n
This occurs in a quite large class of problems, as Proposition 2.2 shows.
Remark 1.7. With no additional effort, we could cover the case where an additional term
W(x)|u|m−2u appears in (1.1) and the functional framework turns into W1,p(Ω)∩W1,m(Ω).
0 0
In the spirit of [11], we also get the following
Corollary 1.8. Let N > p ≥ m > 1 and assume that ξ 7→ L(ξ) is p-homogeneous, ξ 7→ M(ξ) is
m-homogeneous, L(ξ) ≥ p|ξ|p, M(ξ) ≥ m|ξ|m and set
L(Du) M(Du) V(x)
(1.16) S := inf + + |u|p,
Ω
kukLσ(Ω)=1ZΩ p m p
|Du|p |u|p
S := inf + ,
RN
kukLσ(RN)=1ZRN p p
with V(x)→ 1 as |x|→ ∞. Assume furthermore that
σ−p σ−p
(1.17) SΩ < σ SRN.
σ−m
(cid:16) (cid:17)
Then (1.16) admits a minimizer.
Remark 1.9. We point out that, some conditions guaranteeing the nonexistence of nontrivial
solutions inthestar-shapedcase Ω = RN can beprovided. For thesake of simplicity, assumethat
L is p-homogeneous and that ξ 7→ M(s,ξ) is m-homogeneous. Then, in view of [13, Theorem 3],
that holds for C1 solutions by virtue of the results of [8], we have that (1.1) admits no nontrivial
C1 solution well behaved at infinity, namely satisfying condition (19) of [13], provided that there
exists a number a ∈ R+ such that a.e. in RN and for all (s,ξ) ∈ R×RN
(N −p(a+1))L(ξ)+(N −m(a+1))M(s,ξ)+(asg(s)−NG(s))
(N −ap)V(x)+x·DV(x)
+ |s|p−aM (s,ξ)s ≥ 0,
s
p
holding, for instance, if there exists 0≤ a ≤ N−p such that
p
asg(s)−NG(s) ≥0, (N −ap)V(x)+x·DV(x) ≥ 0, M (s,ξ)s ≤0,
s
GLOBAL COMPACTNESS FOR QUASI-LINEAR PROBLEMS 5
fora.e. x ∈ RN andforall (s,ξ) ∈ R×RN. Also, in themoreparticular case whereg(s) = |s|σ−2s
and V(x) = V > 0, then the above conditions simply rephrase into
∞
σ ≥ p∗, M (s,ξ)s ≤ 0,
s
for every (s,ξ) ∈ R×RN. In fact, in (1.9), we consider the opposite assumption on M .
s
2. Some preliminary facts
It is a standard fact that, under condition (1.6) and (1.10), the functionals
u7→ L(Du), u7→ V(x)|u|p, u7→ G(u)
Z Z Z
Ω Ω Ω
are C1 on W1,p(Ω)∩D1,m(Ω). Analogously, although M depends explicitly on s, the functional
0 0
M :W1,p(Ω)∩D1,m(Ω)→ R, M(u)= M(u,Du),
0 0 Z
Ω
1,p 1,m
admits, thanks to condition (1.5), directional derivatives along any v ∈ W (Ω)∩D (Ω) and
0 0
M′(u)(v) = M (u,Du)·Dv+ M (u,Du)v,
ξ s
Z Z
Ω Ω
as it can be easily verified observing that p ≤ p ≤ p∗ and that, for u∈ W1,p(Ω)∩D1,m(Ω), by
p−m 0 0
Young’s inequality, for some constant C it holds
|M (u,Du)·Dv|≤ C|Du|m+C|Dv|m ∈L1(Ω),
ξ
p
|Ms(u,Du)v| ≤ C|Du|p+C|v|p−m ∈ L1(Ω).
Furthermore, if u → u in W1,p(Ω)∩D1,m(Ω) as k → ∞ then M′(u )→ M′(u) in the dual space
k 0 0 k
(W1,p(Ω)∩D1,m(Ω))∗, as k → ∞. Indeed, for kvk ≤ 1, we have
0 0 W1,p(Ω)∩D1,m(Ω)
0 0
|M′(u )(v)−M′(u)(v)|
k
≤ |M (u ,Du )−M (u,Du)||Dv|+ |M (u ,Du )−M (u,Du)||v|
ξ k k ξ s k k s
Z Z
Ω Ω
≤ kMξ(uk,Duk)−Mξ(u,Du)kLm′kDvkLm +kMs(uk,Duk)−Ms(u,Du)kLp/mkvkLp/(p−m)
≤ kMξ(uk,Duk)−Mξ(u,Du)kLm′ +kMs(uk,Duk)−Ms(u,Du)kLp/m.
Thisyieldsthedesiredconvergence, using(1.7)andtheDominatedConvergenceTheorem. Notice
thatthesameargumentcarriedoutbeforeapplieseithertointegralsdefinedonΩoronRN.Hence
the following proposition is proved.
Proposition 2.1. In the hypotheses of Theorems 1.1 and 1.2, the functionals φ and φ are C1.
∞
In addition to the assumptions on L,M and g,G set in the introduction, assume now that there
exist positive numbers δ > 0 and µ > p such that
(2.1) µM(s,ξ)−M (s,ξ)s−M (s,ξ)·ξ ≥ δ|ξ|m, µL(ξ)−L (ξ)·ξ ≥ δ|ξ|p, sg(s)−µG(s) ≥ 0,
s ξ ξ
for any s ∈ R and all ξ ∈ RN. This hypothesis is rather well established in the framework
of quasi-linear problems (cf. [14]) and it allows an arbitrary Palais-Smale sequence (u ) to be
n
1,p 1,m
bounded in W (Ω)∩D (Ω), as shown in the following
0 0
6 C.MERCURIANDM.SQUASSINA
Proposition 2.2. Let j be as in (1.11) and assume that (1.5) holds. Let (u ) ⊂ W1,p(Ω) ∩
n 0
D1,m(Ω) be a sequence such that
0
φ(u ) → c φ′(u )→ 0 in (W1,p(Ω)∩D1,m(Ω))∗
n n 0 0
Then, if condition (2.1) holds, (u ) is bounded in W1,p(Ω)∩D1,m(Ω).
n 0 0
Proof. Let(w ) ⊂ (W1,p(Ω)∩D1,m(Ω))∗ withw → 0asn → ∞besuchthatφ′(u )(v) =hw ,vi,
n 0 0 n n n
1,p 1,m
for every v ∈ W (Ω)∩D (Ω). Whence, by choosing v = u , it follows
0 0 n
j (u ,Du )·Du + j (u ,Du )u + V(x)|u |p = hw ,u i.
ξ n n n s n n n n n n
Z Z Z
Ω Ω Ω
Combining this equation with µφ(u )= µc+o(1) as n → ∞, namely
n
µ
µj(u ,Du )+ V(x)|u |p = µc+o(1),
n n n
Z p Z
Ω Ω
recalling the definition of j, and using condition (2.1), yields
µ−p
V(x)|u |p+δ |Du |p+δ |Du |m ≤ µc+kw kku k +o(1),
p ZΩ n ZΩ n ZΩ n n n W01,p(Ω)∩D01,m(Ω)
as n → ∞, implying, due to V ≥ V that
0
ku kp +ku km ≤ C +Cku k +Cku k +o(1),
n W1,p(Ω) n D1,m(Ω) n W1,p(Ω) n D1,m(Ω)
as n → ∞. The assertion then follows immediately. (cid:3)
From now on we shall always assume to handle bounded Palais-Smale sequences, keeping in mind
that condition (2.1) can guarantee the boundedness of such sequences.
Proposition 2.3. Let j be as in (1.11) and assume that 1 < m < p < N and p < σ < p∗. Let
(u ) ⊂ W1,p(Ω)∩D1,m(Ω) bounded be such that
n 0 0
φ(u ) → c φ′(u ) → 0 in (W1,p(Ω)∩D1,m(Ω))∗.
n n 0 0
Then, up to a subsequence, (u ) converges weakly to some u in W1,p(Ω)∩D1,m(Ω), u (x) → u(x)
n 0 0 n
and Du (x) → Du(x) for a.e. x ∈ Ω.
n
Proof. Itis sufficient to justify that Du (x) → Du(x) for a.e. x ∈ Ω. Given an arbitrary bounded
n
subdomainω ⊂ ω ⊂ Ω of Ω, from the fact that φ′(u ) → 0 in (W1,p(Ω)∩D1,m(Ω))∗, we can write
n 0 0
a(u ,Du )·Dv = hw ,vi+hf ,vi+ vdµ , for all v ∈ D(ω),
n n n n n
Z Z
ω ω
where (w ) ⊂ (W1,p(Ω)∩D1,m(Ω))∗ is vanishing, and hence in particular w ∈ W−1,p′(ω), with
n 0 0 n
w → 0 in W−1,p′(ω) as n → ∞ and we have set
n
a (x,s,ξ) := L (ξ)+M (s,ξ), for all (s,ξ) ∈ R×RN,
n ξ ξ
f := −V(x)|u |p−2u +g(u ) ∈ W−1,p′(ω), n ∈N,
n n n n
µ := −M (u ,Du ) ∈ L1(ω), n ∈ N.
n s n n
Due to the strict convexity assumptions on the maps ξ 7→ L(ξ) and ξ 7→ M(s,ξ) and the growth
conditions on L ,M ,M and g, all the assumptions of [6, Theorem 1] are fulfilled. Precisely,
ξ ξ s
|a (x,s,ξ)| ≤ |L (ξ)|+|M (s,ξ)| ≤ C|ξ|p−1+C|ξ|m−1 ≤ C +C|ξ|p−1,
n ξ ξ
GLOBAL COMPACTNESS FOR QUASI-LINEAR PROBLEMS 7
for a.e. x ∈ ω and all (s,ξ) ∈ R×RN, and
f → f, f := −V(x)|u|p−2u+g(u), strongly in W−1,p′(ω),
n
µ ⇀ µ, weakly* in M(ω), since supkM (u ,Du )k < +∞.
n s n n L1(ω)
n∈N
Then,itfollows thatDu (x) → Du(x)fora.e. x ∈ ω. Finally, asimpleCantordiagonal argument
n
allows to recover the convergence over the whole domain Ω. (cid:3)
Next we prove a regularity result for the solutions of equation (1.1).
Proposition 2.4. Let j be as in (1.11) and assume (1.5) and (1.9). Let u ∈W1,p(Ω)∩D1,m(Ω)
0 0
be a solution of (1.1). Then
u ∈ Lq(Ω), u∈ L∞(Ω) and lim u(x) = 0.
q\≥p |x|→∞
Proof. Let k,i ∈ N. Then, setting v (x):= (u (x))i with u (x) := min{u+(x),k}, it follows that
k,i k k
1,p 1,m
v ∈ W (Ω)∩D (Ω) can be used as a test function in (1.1), yielding
k,i 0 0
L (Du)·Dv + M (u,Du)·Dv
ξ k,i ξ k,i
Z Z
Ω Ω
+ M (u,Du)v + V(x)|u|p−2uv = g(u)v .
s k,i k,i k,i
Z Z Z
Ω Ω Ω
Taking into account that Dv is equal to iui−1Duχ , by convexity and positivity of the
k,i {0<u<k}
map ξ 7→ M(s,ξ) we deduce that M (u,Du)·Dv ≥ 0. Moreover, by the sign condition (1.9) it
ξ k,i
follows M (u,Du)v ≥ 0 a.e. in Ω. Then, we reach
s k,i
i(u )i−1L (Du )·Du + V(x)|u|p−2u(u (x))i ≤ g(u)(u (x))i,
k ξ k k k k
Z Z Z
Ω Ω Ω
yielding in turn, by (1.10), that for all k,i ≥ 1
(2.2) νi (u )i−1|Du |p ≤ C (u+(x))σ−1+i.
k k
Z Z
Ω Ω
If uˆ := min{u−(x),k}, a similar inequality
k
(2.3) νi (uˆ )i−1|Duˆ |p ≤ C (u−(x))σ−1+i,
k k
Z Z
Ω Ω
can be obtained by using vˆ := −(uˆ )i as a test function in (1.1), observing that by (1.9),
k,i k
M (u,Du)vˆ = −M (u,Du)χ (−u)i ≥ 0,
s k,i s {−k<u<0}
M (u,Du)·Dv = i(−u)i−1χ M (u,Du)·Du ≥0.
ξ k,i {−k<u<0} ξ
Once (2.2)-(2.3) are reached, the assertion follows exactly as in [15, Lemma 2, (a) and (b)]. (cid:3)
We now recall the following version of [7, Lemma 4.2] which turns out to be a rather useful
tool in order to establish convergences in our setting. Roughly speaking, one needs some kind of
sub-criticality in the growth conditions.
Lemma 2.5. Let Ω ⊂ RN and h : Ω×R×RN be a Carath´eodory function, p,m > 1, µ ≥ 1,
p ≤ σ ≤ p∗ and assume that, for every ε > 0 there exist a ∈ Lµ(Ω) such that
ε
(2.4) |h(x,s,ξ)| ≤ a (x)+ε|s|σ/µ +ε|ξ|p/µ +ε|ξ|m/µ,
ε
a.e. in Ω and for all (s,ξ) ∈ R×RN. Assume that u → u a.e. in Ω, Du → Du a.e. in Ω and
n n
(u ) is bounded in W1,p(Ω), (u ) is bounded in D1,m(Ω).
n 0 n 0
8 C.MERCURIANDM.SQUASSINA
Then h(x,u ,Du ) converges to h(x,u,Du) in Lµ(Ω).
n n
Proof. The proof follows as in [7, Lemma 4.2] and we shall sketch it here for self-containedness.
1,p 1,m
By Fatou’s Lemma, it immediately holds that u∈ W (Ω)∩D (Ω). Furthermore, there exists
0 0
a positive constant C such that
|h(x,s ,ξ )−h(x,s ,ξ )|µ ≤ C(a (x))µ+Cεµ|s |σ +Cεµ|s |σ
1 1 2 2 ε 1 2
+Cεµ|ξ |m+Cεµ|ξ |m+Cεµ|ξ |p+Cεµ|ξ |p,
1 2 1 2
a.e. in Ω and for all (s ,ξ ) ∈ R× RN and (s ,ξ ) ∈ R× RN. Then, taking into account the
1 1 2 2
boundednessof (Du ) in Lp(Ω)∩Lm(Ω) and of (u ) in Lσ(Ω) by interpolation being p ≤ σ ≤ p∗,
n n
the assertion follows by applying Fatou’s Lemma to the sequence of functions ψ : Ω → [0,+∞]
n
ψ (x) :=−|h(x,u ,Du )−h(x,u,Du)|µ +C(a (x))µ +Cεµ|u |σ +Cεµ|u|σ
n n n ε n
+Cεµ|Du |m+Cεµ|Du|m+Cεµ|Du |p+Cεµ|Du|p,
n n
and, finally, exploiting the arbitrariness of ε. (cid:3)
3. Proof of the result
3.1. Energy splitting. The next result allows to perform an energy splitting for the functional
1,p 1,m
J(u) = j(u,Du), u∈ W (Ω)∩D (Ω),
Z 0 0
Ω
1,p 1,m
along a bounded Palais-Smale sequence (u ) ⊂ W (Ω)∩D (Ω). The result is in the spirit of
n 0 0
the classical Brezis-Lieb Lemma [4].
Lemma 3.1. Let the integrand j be as in (1.11) and
p−1 ≤m < p−1+p/N, p ≤ σ ≤ p∗.
Let (u ) ⊂ W1,p(Ω)∩D1,m(Ω) with u ⇀ u, u → u a.e. in Ω and Du → Du a.e. in Ω. Then
n 0 0 n n n
(3.1) lim j(u −u,Du −Du)−j(u ,Du )+j(u,Du) = 0.
n n n n
n→∞Z
Ω
Proof. We shall apply Lemma 2.5 to the function
h(x,s,ξ) := j(s−u(x),ξ −Du(x))−j(s,ξ), for a.e. x ∈ Ω and all (s,ξ) ∈ R×RN.
Given x ∈ Ω, s ∈ R and ξ ∈ RN, consider the C1 map ϕ : [0,1] → R defined by setting
ϕ(t) := j(s−tu(x),ξ −tDu(x)), for all t ∈ [0,1].
Then, for some τ ∈[0,1] depending upon x ∈ Ω, s∈ R and ξ ∈ RN, it holds
h(x,s,ξ) = ϕ(1)−ϕ(0) = ϕ′(τ)
= −j (s−τu(x),ξ −τDu(x))u(x)−j (s−τu(x),ξ −τDu(x))·Du(x)
s ξ
= −L (ξ−τDu(x))·Du(x)
ξ
−M (s−τu(x),ξ −τDu(x))u(x)
s
−M (s−τu(x),ξ −τDu(x))·Du(x)+G′(s−τu(x))u(x).
ξ
GLOBAL COMPACTNESS FOR QUASI-LINEAR PROBLEMS 9
Hence, for a.e. x ∈ Ω and all (s,ξ) ∈ R×RN, it follows that
|h(x,s,ξ)| ≤ |L (ξ−τDu(x))||Du(x)|+|M (s−τu(x),ξ −τDu(x))||u(x)|
ξ s
+|M (s−τu(x),ξ −τDu(x))||Du(x)|+|G′(s−τu(x))||u(x)|
ξ
≤ C(|ξ|p−1+|Du(x)|p−1)|Du(x)|+C(|ξ|m+|Du(x)|m)|u(x)|
+C(|ξ|m−1+|Du(x)|m−1)|Du(x)|+C(|s|σ−1+|u(x)|σ−1)|u(x)|
≤ ε|ξ|p +C |Du(x)|p +ε|ξ|p +C |Du(x)|p +C |u(x)|p/(p−m)
ε ε ε
+ε|ξ|m+C |Du(x)|m +ε|s|σ +C |u(x)|σ
ε ε
= a (x)+ε|s|σ +ε|ξ|p +ε|ξ|m,
ε
where a :Ω → R is defined a.e. by
ε
a (x) := C |Du(x)|p +C |Du(x)|m +C |u(x)|p/(p−m) +C |u(x)|σ.
ε ε ε ε ε
Notice that, as p−1 ≤ m < p−1+p/N it holds p ≤ p/(p−m) ≤ p∗, yielding u ∈ Lp/(p−m)(Ω)
and in turn, a ∈ L1(Ω). The assertion follows directly by Lemma 2.5 with µ = 1. (cid:3)
ε
We have the following splitting result
Theorem 3.2. Let the integrand j be as in (1.11) and
p−1 ≤m ≤ p−1+p/N, p < σ < p∗.
Assume that (u ) ⊂ W1,p(Ω)∩D1,m(Ω) is a bounded Palais-Smale sequence for φ at the level
n 0 0
c∈ R weakly convergent to some u ∈W1,p(Ω)∩D1,m(Ω). Then
0 0
|u −u|p |u|p
n
lim j(u −u,Du −Du)+ V = c− j(u,Du)− V(x) ,
n n ∞
n→∞(cid:16)ZΩ ZΩ p (cid:17) ZΩ ZΩ p
namely
lim φ (u −u) = c−φ(u),
∞ n
n→∞
being u and u regarded as elements of W1,p(RN)∩D1,m(RN) after extension to zero out of Ω.
n
Proof. In light of Proposition 2.3, up to a subsequence, (u ) converges weakly to some function
n
1,p 1,m
u in W (Ω)∩D (Ω), u (x) → u(x) and Du (x) → Du(x) for a.e. x ∈ Ω. Also, recalling that
0 0 n n
by assumption V(x) → V as |x| → ∞, we have [4,17]
∞
(3.2) lim V(x)|u −u|p−V |u −u|p = 0,
n ∞ n
n→∞Z
Ω
(3.3) lim V(x)|u −u|p−V(x)|u |p+V(x)|u|p = 0.
n n
n→∞Z
Ω
Therefore, by virtue of Lemma 3.1, we conclude that
|u −u|p
n
lim φ (u −u)= lim j(u −u,Du −Du)+ V
∞ n n n ∞
n→∞ n→∞(cid:16)ZΩ ZΩ p (cid:17)
|u −u|p
n
= lim j(u −u,Du −Du)+ V(x)
n n
n→∞(cid:16)ZΩ ZΩ p (cid:17)
|u |p |u|p
n
= lim j(u ,Du )+ V(x) − j(u,Du)− V(x)
n n
n→∞(cid:16)ZΩ ZΩ p (cid:17) ZΩ ZΩ p
= lim φ(u )−φ(u) = c−φ(u),
n
n→∞
concluding the proof. (cid:3)
10 C.MERCURIANDM.SQUASSINA
Remark 3.3. In order to shed some light on the restriction (1.5) of m, it is readily seen that it
is a sufficient condition for the following local compactness property to hold. Assume that ω is a
smooth domain of Rn with finite measure. Then, if (u ) is a bounded sequence in W1,p(ω), there
h 0
exists a subsequence (u ) such that
hk
Υ(x,u ,Du ) converges strongly to some Υ in W−1,p′(ω),
hk hk 0
whereΥ(x,s,ξ) = g(s)−M (s,ξ)−V(x)|s|p−2s. Infact, takingintoaccount thegrowth condition
s
on g and M , this can be proved observing that, for every ε> 0, there exists C such that
s ε
N(p−1)+p
|Υ(x,s,ξ)| ≤ Cε+ε|s| N−p +ε|ξ|p−1+p/N,
for a.e. x ∈ ω and all (s,ξ) ∈ R×RN.
3.2. Equation splitting I (super-quadratic case). We shall assume that m,p ≥ 2 and that
conditions (1.7)-(1.8) hold. The following Theorem 3.4 and the forthcoming Theorem 3.5 (see
next subsection) are in the spirit of the Brezis-Lieb Lemma [4], in a dual framework. For the
particular case
|ξ|p
M(s,ξ) = 0 and L(ξ) = ,
p
we refer the reader to [12].
Theorem 3.4. Assume that (1.5)-(1.11) hold and that
p−1 ≤m < p−1+p/N, p < σ < p∗.
Assume that (u ) ⊂ W1,p(Ω)∩D1,m(Ω) is such that u ⇀ u, u → u a.e. in Ω, Du → Du a.e.
n 0 0 n n n
in Ω and there is (w ) in the dual space (W1,p(Ω)∩D1,m(Ω))∗ such that w → 0 as n → ∞ and,
n 0 0 n
for all v ∈ W1,p(Ω)∩D1,m(Ω),
0 0
(3.4) j (u ,Du )·Dv+ j (u ,Du )v+ V(x)|u |p−2u v = hw ,vi.
ξ n n s n n n n n
Z Z Z
Ω Ω Ω
Then φ′(u) = 0. Moreover, there exists a sequence (ξ ) that goes to zero in (W1,p(Ω)∩D1,m(Ω))∗,
n 0 0
such that
(3.5) hξ ,vi := j (u −u,Du −Du)v+ j (u −u,Du −Du)·Dv
n s n n ξ n n
Z Z
Ω Ω
− j (u ,Du )v− j (u ,Du )·Dv+ j (u,Du)v + j (u,Du)·Dv,
s n n ξ n n s ξ
Z Z Z Z
Ω Ω Ω Ω
for all v ∈ W1,p(Ω)∩D1,m(Ω).
0 0
Furthermore, there exists a sequence (ζ ) in (W1,p(Ω)∩D1,m(Ω))∗ such that
n 0 0
j (u −u,Du −Du)·Dv+ j (u −u,Du −Du)v+ V |u −u|p−2(u −u)v = hζ ,vi
ξ n n s n n ∞ n n n
Z Z Z
Ω Ω Ω
for all v ∈ W1,p(Ω)∩D1,m(Ω) and ζ → 0 as n → ∞, namely φ′ (u −u) → 0 as n → ∞.
0 0 n ∞ n
Proof. Fixed some v ∈ W1,p(Ω)∩D1,m(Ω), let us define for a.e. x ∈ Ω and all (s,ξ) ∈ R×RN,
0 0
f (x,s,ξ) := j (s−u(x),ξ −Du(x))v(x)
v s
+j (s−u(x),ξ −Du(x))·Dv(x)−j (s,ξ)v(x)−j (s,ξ)·Dv(x).
ξ s ξ
In order to prove 3.5 we are going to show that
(3.6) lim sup f (x,u ,Du )−f (x,u,Du) = 0.
v n n v
n→∞kvkW01,p(Ω)∩D01,m(Ω)≤1(cid:12)(cid:12)(cid:12)ZΩ (cid:12)(cid:12)(cid:12)