Equations of p-Laplacian Type in Unbounded
Domains
P. De Nápoli and M.C. Mariani
Abstract
This work is devoted to study the existence of solutions to equations
of the p-Laplacian type in unbounded domains. We prove the existence
of at least one solution, and under further assumptions, the existence
of innitely many solutions. We apply the mountain pass theorem in
weighted Sobolev spaces.
1
Introduction
In the literature ([10], [13], [6]) the mountain pass theorem ([2]) has been applied
to nd solutions to quasilinear elliptic equations of the form
−∆p u = f (x, u)
Elliptic problems involving the p-Laplacian have been studied by several authors, see [7], [8], [10], [11]. In [7], we have considered the problem
−div(a(x, ∇u)) = f (x, u)
u = 0
in
on
Ω
∂Ω
(1)
involving more general elliptic operators in divergence form, in a bounded domain
Ω ⊂ RN .
In order to apply to these equations the mountain pass theorem,
it is necessary to check that the associated functional veries the Palais-Smale
condition, and we had to introduce a notion of uniformly convex functional.
In this work, we consider an analogous problem, but in an unbounded domain. When studying problems in unbounded domains, it is frequently useful
to work in weighted Sobolev spaces.
the space
W 1,p (Ω)
This is due to the fact that in general
is not compactly embedded into
Lq (Ω)
for any
q
if
Ω
is not
bounded.
Under suitable assumptions, we prove the existence of at least one solution
to equation (1) with Dirichlet condition (theorem 5.1), in unbounded domains.
Finally, we prove the existence of innitely many solutions to problem (1) if the
nonlinearity
f
is odd (theorem 6.1).
Our results can be applied to equations involving dierent kinds of operators
of the p-Laplacian type, like the generalized mean curvature operator
div((1 + |∇u|2 )(p−2)/2 ∇u)
1
when
p≥2
(For
p=1
this operator corresponds to the prescribed mean curva-
ture equation for a surface in nonparametric form, see [3],[4]).
2
Uniformly Convex Functionals
In this section, we review some denitions and results from ([7]), that we will
use in order to prove that the functional associated to problem (1) veries the
Palais-Smale condition.
X
will be a Banach space.
Denition 2.1 We shall say that the convex functional A : X → R is uniformly
convex on the (convex) set C ⊂ X , if for any ε > 0 there exists δ(ε) > 0 such
that
A
x+y
2
≤
1
1
A(x) + A(y) − δ(ε)
2
2
for x, y ∈ C and kx − yk > ε. If A is uniformly convex on every ball of X , we
will say that A is locally uniformly convex.
Denition 2.2 An operator a : X → X ∗ veries the (S+ ) condition if for any
sequence (xn )n∈N such that xn * x and
lim sup ha(xn ), xn − xi ≤ 0
n→∞
we have that xn → x strongly.
Proposition 2.1 Suppose that A : X → R is a C 1 locally uniformly convex
functional that is locally bounded. Then a = DA : X → X ∗ veries the (S+ )
condition.
3
Let
Weighted Sobolev Spaces
Ω ⊂ RN
w ∈ L1loc (Ω) such that w ≥ 0
p
Lw (Ω) is the set of measurable functions u : Ω → R
be a domain. A weight is a function
a.e. The space
p
L (Ω; w) =
such that the norm:
1/p
|u| w(x) dx
Z
kukp,Ω,w =
p
Ω
w with the measure:
Z
w(E) =
w dx E ⊂ Ω
is nite. If we identify the weight
E
p
p
then Lw is the space L associated with this measure. The weighted Sobolev
1,p
space W
(Ω; w0 , w1 ) is then dened as the space of functions in Lpw0 (Ω) with
2
distributional derivatives
∂u
∂xi (1
≤ i ≤ N)
Lpw1 (Ω).
in
The corresponding norm
is dened by:
Z
kuk1,p,w0 ,w1 =
The space
W01,p (Ω; v0 , v1 )
Z
p
|∇u| w1 (x) dx +
Ω
1/p
|u| w0 (x)dx
p
(2)
Ω
is dened as the clousure in
W 1 (Ω; v0 , v1 )
of
C01 (Ω).
We note that in the literature there are some other denitions of weighted
Sobolev spaces that are not always equivalent. For example if we dene
∞
C (Ω) with respect to the
H p (Ω; w0 , w1 ) = W 1,p (Ω; w0 , w1 ), when w0 and w1
class Ap (see [9]).
as the completation of
3.1
H p (Ω; w0 , w1 )
norm (2), we have that
are in the Muckenhoupt
Sobolev embedding in Weighted Sobolev spaces: KufnerOpic theorem
In this section we recall a embedding theorem for weighted Sobolev spaces. We
Ωn = {x ∈ Ω : |x| < n} = Ω ∩ B(0, n) and Ωn = {x ∈ Ω : |x| > n}.
ñ
N
assume that there exist ñ such that Ω = {x ∈ R
: |x| > ñ}.
ñ
Let r(x) be a positive function dened in Ω such that:
note
c−1
r
with
cr
r(x) ≤ |x|
3
≤ r(y)/r(x) ≤ cr
for almost all
for almost all
We
x ∈ Ωñ
x ∈ Ω , y ∈ B(x, r(x))
ñ
a constant.
Theorem 3.1 ([12],Theorem 2.1) Let 1 ≤ p ≤ q < ∞, Nq − Np + 1 ≥ 0 and
assume that there are constants 0 ≤ k0 ≤ K0 , 0 ≤ c0 ≤ C0 , 0 ≤ c1 ≤ C1 ,
and positive measurable functions b0 , b1 dened on Ωñ , such that the weights
v0 , v1 , w satisfy:
k0 v0 (x) ≤ v1 (x)r−p (x) ≤ K0 (x)v0 (x)
c0 b0 (x) ≤
w(y)
≤
C0 b0 (x)
c1 b1 (x) ≤
v1 (y)
≤
C1 b1 (x)
a.e.
x ∈ Ωñ
ñ
a.e. x ∈ Ω , y ∈ B(x, r(x))
a.e. x ∈ Ωñ , y ∈ B(x.r(x))
Assume that the embedding W 1,p (Ωn ; v0 , v1 ) ,→ Lq (Ωn ; w) is compact for every
n ∈ N. Then the embedding
W 1,p (Ω; v0 , v1 ) ,→ Lq (Ω; w)
is compact if and only if
lim Bn = 0,
n→∞
with Bn = sup
x∈Ωn
b0 (x)1/q ( Nq − Np +1)
r
b1 (x)1/p
Furthermore this embedding is continuous if the embedding W 1,p (Ωn ; v0 , v1 ) ,→
Lq (Ωn ; w) is continuous for every n ∈ N and limn→∞ Bn < ∞.
3
3.1.1 Applications of the Kufner-Opic theorem
1. Let
Ω ⊂ RN
be a domain as above,
α, β ∈ R,
and let us dene
v0 (x) = (1 + |x|)β−p , v1 (x) = (1 + |x|)β , w(x) = (1 + |x|)α
Choosing
b0 = w, b1 = v1
and
r(x) = (1+|x|)/6, the conditions of theorem
3.1 are satised if the following inequalities hold:
N
α β
N
N
N
−
+ 1 > 0 and − +
−
+1<0
q
p
q
p
q
p
Hence the embedding
2. Let us dene
b1 = w1
and
W 1,p (Ω; v0 , v1 ) ,→ Lq (Ω; w)
β|x|
v0 (x) = v1 (x) = e
r(x) = 1, then the
and
is compact.
α|x|
w(x) = e
. If we choose
b0 = w ,
conditions of theorem 3.1 are satised if
N
N
α β
−
+ 1 > 0 and − < 0
q
p
p
q
3.2
Hardy inequalities
Another useful tools for studying p-Laplacian type equations in unbounded
domains are the Hardy inequalities.
Theorem 3.2 ([12],Theorem 2.3) Let Ω ⊂ RN be a domain, with the property that x ∈ Ω and t ≥ 1 implies tx ∈ Ω. Assume that v0 and v1 are radial
weights, i.e.: v0 (|x|), v1 (|x|), and that they are bounded in each compact subinterval of (0, ∞). Assume furthermore that there exist constants k,B ,t0 > 0
verifying that:
v0 (t) ≥ kv1 (t)t−p for almost all t > t0
Z
1/p Z
x
B(x) =
v0 (t)t
N −1
∞
dt
−1
−1/(p−1) − N
p−1
v1 (t)
0
t
1/p0
dt
≤B
(3)
x
Then there exists a constant C > 0 such that:
Z
|u|p v0 (x) dx ≤ C
Ω
Example:
Z
|∇u|p v1 (x) dx ∀ u ∈ W01,p (Ω; v0 , v1 )
Ω
If we take
v0 (x) = (1 + |x|)β−p , v1 (x) = (1 + |x|)β
with
0 ≤ β ≤ p,
N > p − β,
and
then
v0 (t) ≥
1
v1 (t)t−p
2p
From
Z
B(x) ≤
x
1/p Z
tβ−p+N −1 dt
0
∞
t
−β−(N −1)
p−1
0
condition (3) is satised. Then inequality (4) holds.
4
1/p0
dt
≤C
(4)
4
The Action of the Nemitsky Operator in Weighted
Spaces
Proposition 4.1 Let f : Ω×R → R be a Caratheodory function. Let 1 ≤ p, q <
and 0 < α ≤ pq , and let w1 and w2 be two weights. Assume that f veries
the growth condition:
∞
|f (x, s)| ≤ f0 (x) + f1 (x)|s|α
with f0 ∈ Lq (Ω; w2 ) and
• f1 ∈ Lpq/(p−αq) (Ω; w̃)
with w̃ = w2p/(p−αq) w1−αq/(p−αq) if 0 < α <
• f1 (x)q w2 (x) ≤ Cw1 (x)
a.e. for some constant C if α =
p
q
p
q
Then the Nemitsky operator Nf : u 7→ f (x, u) is continuous from Lp (Ω; w1 ) to
Lq (Ω; w2 ).
For the proof, we need the following lemma:
Lemma 4.1 ([5], theorem IV.9) Let (Ω, M, µ) be a measure space and 1 ≤
p < ∞. If un → u in Lp (Ω, µ), then there exist a subsequence (un ) and a
function g ∈ Lp (Ω, µ) such that
k
un (x) → u(x) for a.e. x ∈ Ω
and |un (x)|, |u(x)| ≤ g(x).
Proof:
We rst show that
Nf
maps
Lp (Ω; w1 )
into
Lq (Ω; w2 ).
If
r > 1,
from
Hölder inequality we have that:
Z
Z
q
|f (x, u)| w2 (x)dx ≤
Ω
≤C
2q [f0 (x)q + f1 (x)q |u|αq ]w2 (x)dx
Ω
q
kf kLq (Ω;w2 ) +C
Z
qr 0
f1 (x)
0
−1/(r−1)
w2r (x)w1
1/r0 Z
1/r
dx
|u|αqr w1 (x)dx
Ω
Ω
p
p
1
0
We choose αqr = p, or: r =
αq . Then r = p−αq , r−1
p
0 < α < q . This proves that Nf (u) ∈ Lq (w2 )
α=
If
Z
Ω
p
q , then
r0 = ∞
=
αq
p−αq , and
r>1
if
and we have:
q
|f (x, u)|q w2 (x)dx ≤ C kf kLq (Ω;w2 ) + C f1 (x)q w2 (x)w1−1 L∞ (Ω) kukLp (Ω;w1 )
Then
Nf (u) ∈ Lq (w2 ).
Lp (Ω; w1 ) to Lq (Ω; w2 ). Let
(un ) be a sequence such that un → u in L (Ω; w1 ). We want to show that
Nf (un ) converges to Nf (u). It is enough to show that any subsequence (unk )
Now we will prove that
Nf
is continuous from
p
5
of
(un )
u. From lemma 4.1 there exists a
(unkj ) of (unk ) such that unkj → u almost everywhere with respect
p
measure w1 , and there exists g ∈ L (Ω; w1 ) such that |un (x)| ≤ g(x).
has a subsequence that converges to
subsequence
to the
Then:
|f (x, u) − f (x, unkj )|q ≤ 2q [|f (x, u)|q + |f (x, unkj )|q ]
≤ 2q+1 [f0 (x)q + f1 (x)q |g(x)|qα ] = h(x)
Using Hölder inequality as before, we have that:
h ∈ L1 (Ω; w2 ).
Therefore we
can use the Lebesgue bounded convergence theorem to see that
Z
Ω
since
|f (x, u) − f (x, unkj )|q w2 (x)dx → 0
f (x, unkj ) → f (x, u)
In the special case
almost everywhere (with respect to
q = p0 , w1 = w
and
w2 = w−1/(p−1) ,
w2 ).
we have that:
w̃ = w−p/[(p−αq)(p−1)] w−αq/(p−αq)
and
αq
p0 + αq
p0 (1 + α)
1+α
p
+
=
= 0
=
(p − αq)(p − 1) p − αq
p − αq
p (p − 1 − α)
p−1−α
p
p
pq
p−1
=
=
1
p − αq
p−1−α
1 − α p−1
Corollary 4.1 Let f : Ω×R → R be a Caratheodory function. Let 1 ≤ p, q < ∞
, 0 < α ≤ p − 1, and let w be a weight. Assume that f veries the growth
condition:
|f (x, s)| ≤ f0 (x) + f1 (x)|s|α
with f0 ∈ Lp (Ω; w−1/(p−1) ) and
0
• f1 ∈ Lp/(p−1−α) (Ω; w̃)
• f1 (x) ≤ Cw(x)
with w̃ = v−(1+α)/(p−1−α) if 0 < α <
p
q
a.e. for some constant C if α = p − 1
Then the Nemitsky operator Nf : u 7→ f (x, u) is continuous from Lp (Ω; w) to
Lq (Ω; w).
Remark:
In the previous result, the case
in [12].
6
α = p−1
generalizes lemma 3.1
5
p-Laplacian Type Equations in Unbounded Domains
In this section, we will prove that, under suitable assumptions, problem (1) has
at least one solution.
Theorem 5.1 Assume that the following conditions hold:
1. 2 ≤ p < N , Ω ⊂ RN is a domain (possibly unbounded), and the weights
v0 ,v1 and w are such that the Sobolev embedding:
W01,p (Ω; v0 , v1 ) ⊂ Lq (Ω; w)
is continuous if p ≤ q ≤ p∗ and compact if p < q < p∗ (with p∗ =
and that the Hardy inequality:
Z
Z
p
|u| v0 (x) dx ≤ C
Ω
(5)
pN
N −p
),
|∇u|p v1 (x) dx ∀ u ∈ W01,p (Ω; v0 , v1 )
Ω
holds. In particular, this means that
Z
kuk =
1/p
|∇u| v1 (x) dx
p
Ω
is a norm in W01 (Ω; v0 , v1 ) that is equivalent to the norm given by (2).
2. A : Ω × RN → R is a continuous function in Ω × RN , with continuous
derivative with respect to ξ , a = DA = A0 , that veries:
(a) A(x, 0) = 0 ∀ x ∈ Ω.
(b) a satises the growth condition:
|a(x, ξ)| ≤ a0 (x) + a1 (x)|ξ|p−1 ∀ x ∈ Ω, ξ ∈ RN
(6)
with a0 , a1 positive functions that satisfy:
0
−1/(p−1)
a0 ∈ Lp (Ω, v1
), a0 (x) ≤ c1 v1 (x), a1 (x) ≤ c2 v1 (x)
(7)
for some positive constants c1 , c2 .
(c) A is p-uniformly convex with respect to the weight v1 (x), in the following sense: There exists a constant c3 > 0 such that:
1
1
ξ+η
≤ A (x, ξ)+ A(x, η)−c3 v1 (x)|ξ−η|p ∀ x ∈ Ω, ξ, η ∈ RN
A x,
2
2
2
(8)
(d) A is p-subhomogeneous:
a(x, ξ) · ξ ≤ pA(x, ξ) ∀ x ∈ Ω, ξ ∈ RN
7
(9)
(e) A satises the ellipticity condition:
A(x, ξ) ≥ Λ|ξ|p v1 (x) ∀ x ∈ Ω, ξ ∈ RN
(10)
with Λ a positive constant.
3. f : Ω × R → R is a Caratheodory function that veries:
(a) The subcritical growth condition:
|f (x, s)| ≤ f0 (x) + f1 (x)|s|q−1 ∀ x ∈ Ω, s ∈ R
with p < q < p∗ =
Np
N −p
(11)
, and f0 ,f1 are positive functions that satisfy:
0
f0 (x) ∈ Lq (Ω, w−1/(q−1) ), f0 (x) ≤ Cw(x), f1 (x) ≤ Cw(x)
(12)
with C a positive constant.
R
(b) The Ambrosetti-Rabinowitz condition: F (x, s) = 0s f (x, t) dt is θsuperhomogeneous:
0 ≤ θF (x, s) ≤ f (x, s)s ∀s ∈ R, x ∈ Ω
(c)
(13)
with p < θ ≤ q.
f (x, 0) = lim
t→0
f (x, t)
=0
|t|p−1 v0 (x)
(14)
uniformly for x ∈ Ω.
(d) There exist an open set O ⊂ Ω and s0 positive such that
F (x, s) > 0 ∀x ∈ O, |s| ≥ s0 .
(15)
Then the Dirichlet problem:
−div(a(x, ∇u))
u
= f (x, u)
= 0
in Ω
on ∂Ω
has at least one non trivial weak solution in W01,p (Ω; v0 , v1 ).
5.1
Proof of theorem 5.1
In order to prove theorem 5.1, we apply the mountain pass theorem to the
functional
Z
Z
A(x, ∇u) dx −
J(u) =
Ω
working in the weighted Sobolev space
F (x, u) dx
Ω
W01,p (Ω; v0 , v1 ).
As a consequence of corollary 4.1 and from the Sobolev embeddings with
weights that hold by hypothesis, it is easy to see that
(as in lemma 3.2 in [12]), and we have that:
8
J ∈ C 1 (W01,p (Ω; v0 , v1 ))
Z
0
hJ (u), vi =
Z
a(x, ∇u) · ∇v dx −
Ω
f (x, u)v dx
Ω
We will check the conditions of the mountain pass theorem in the following two
lemmas:
Lemma 5.1 J veries the Palais-Smale condition.
Proof: If (un ) ⊂ W01,p (Ω; v0 , v1 ) is a Palais-Smale
J(un ) → c
and
J 0 (un ) → 0.
First we show that
(un )
sequence, it veries that:
is bounded:
Z 1
1 0
J(un ) − hJ (un ), un i =
A(x, ∇un ) − a(x, ∇un )∇un dx+
θ
θ
Ω
Z
1
f (x, un )un − F (x, un ) dx
Ω θ
From condition (9), we obtain that:
Z
p
1 0
A(x, ∇un ) dx+
J(un ) − hJ (un ), un i ≥ 1 −
θ
θ Ω
Z 1
f (x, un )un − F (x, un ) dx
Ω θ
From Ambrosetti-Rabinowitz condition (13), we deduce that:
p
1−
θ
Z
A(x, ∇un ) ≤ J(un ) −
Ω
1 0
hJ (un ), un i
θ
From the ellipticity condition (10), we conclude that:
Z
|∇un |p v1 (x) dx ≤ J(un ) −
C
Ω
with
C = 1−
p
θ
Λ > 0,
1 0
hJ (un ), un i
θ
i.e.:
p
C kun k ≤ J(un ) +
1 0
kJ (un )k kun k
θ
kJ 0 (un )k the norm in the dual space of W01,p (Ω; v0 , v1 ).
(un ) is not bounded, for a subsequence we may assume that:
with
If
kun k → ∞
Then dividing by
that
(un )
kun k and letting n → ∞ we have a contradiction.
W01,p (Ω; v0 , v1 ) we may extract a weakly convergent sub* u. In order to show that (unk ) converges strongly to u, we
From the reexivity of
sequence:
This proves
is bounded.
u nk
note that from condition (8),
Z
A(x, ∇u) dx
J0 (u) =
Ω
9
W01,p (Ω; v0 , v1 ). From
0
proposition 2.1 it follows that its derivative J0 veries condition (S+ ), then it
is enough to see that:
is locally uniformly convex in the weighted Sobolev space
Z
a(x, ∇un ) · (∇un − ∇u) dx ≤ 0
lim sup
n→∞
Ω
We have that:
Z
Z
0
a(x.∇un ) · (∇un − ∇u) dx = hJ (un ), un − ui +
Ω
f (x, un ) · (un − u) dx
(16)
Ω
W01,p (Ω; v0 , v1 ) ,→ Lq (Ω; w) is compact, we get that: un → u strongly in L (Ω; w), and by the growth condition
q0
−1/(q−1)
on f , f (x, un ) is bounded in L (Ω; w
). Hence,
Z
f (x, un )(un − u) dx → 0
Since by hypothesis, the Sobolev embedding
q
Ω
J 0 (un )
1,p
W0 (Ω; v0 , v1 ), we deduce that:
On the other hand, since
→0
by hypothesis and
un − u
hJ 0 (un ), un − ui → 0
From (16), we conclude that:
Z
a(x.∇un ) · (∇un − ∇u) dx → 0
Ω
and then by condition
Lemma 5.2
J
(S+ ), un → u
strongly in
W01,p (Ω; v0 , v1 ).
has the mountain pass lemma geometry, i.e.:
1. There exists r > 0 such that:
inf J(u) = b > 0 ∀ u ∈ W01,p (Ω; v0 , v1 )
kuk=r
2. There exists u0 ∈ W01,p (Ω; v0 , v1 ) such that
J(tu0 ) → −∞ as t → ∞
Proof:
From (14), given any
ε>0
we have that:
|f (x, t)| < ε|t|p−1 v0 (x) if |t| < δ
therefore
|F (x, t)| ≤ ε
|t|p
v0 (x) if |t| < δ
p
10
is bounded in
As a consequence:
Z
Z
|F (x, u)| dx ≤
{x∈Ω:|u|<δ}
ε
Ω
ε
|u|p
v0 (x) dx ≤ kukp
p
p
On the other hand, from condition (11), we obtain that:
|F (x, t)| ≤ f0 (x)|t| + f1 (x)
|t|q
q
hence, using the Sobolev embedding (5):
Z
Z
|F (x, u)| dx ≤
{x∈Ω:|u|≥δ}
{x∈Ω:|u|≥δ}
Z
|u|q
f0 (x)|u| + f1 (x)
q
Z
q
Ω
dx ≤
|u|q w(x) dx ≤ C2 kukq
(f0 (x) + f1 (x))|u| dx ≤ C1
Cδ
Ω
We conclude that:
Z
Z
A(x, u) dx −
J(u) =
F (x, u) dx ≥
Ω
Ω
Z
ε
Λv1 (x)|∇u|p dx − kukp − C2 kukq = φ(r)
p
Ω
with
r = kuk.
Since
q > p,
we deduce that
φ(r) > 0
if
r
and
ε
are positive and
small.
u0 ∈ W 1,p (Ω; v0 , v1 )
(15)). Since A is pt > 1 we have, as in
In order to prove that the second condition holds, we choose
such that
u0 (x) ≥ s0 ∀x ∈ O (where s0 and O satisfy
F (x, s) is θ-superhomogeneous, for
subhomogeneous and
[7], that:
Z
Z
A(x, t∇u0 ) dx −
J(tu0 ) =
F (x, tu0 ) dx
Ω
≤t
p
Ω
Z
A(x, ∇u0 ) dx − t
Ω
Therefore, as
5.2
θ > p,
we conclude that
θ
Z
F (x, u0 ) dx
O
J(tu0 ) → −∞
as
t → +∞.
An Example
We consider the problem

= f (x, u)
 −∆p u
u
= 0

lim|x|→∞ u(x) = 0
with
2 ≤ p < N , Ω = {x ∈ RN : |x| > R}
assumptions of theorem 5.1.
11
for some
in
in
Ω
∂Ω
R > 0,
(17)
and
f
verifying the
We choose the weights
v0 (x) = (1 + |x|)−p , v1 (x) = 1, w(x) = (1 + |x|)−α (α > 0)
In order to have a compact embedding
W 1,p (Ω; (1 + |x|)−p , 1) ⊂ Lq (Ω, (1 + |x|)−α )
we assume that
−
α N
N
+
−
+1<0
q
q
p
and in order to have the Hardy inequality, we have to choose
p < N.
Under
these conditions, theorem 5.1 can be applied to this example, and we conclude
that problem (17) has at least one weak nontrivial solution.
6
Multiple Solutions
In this section, we will prove the existence of innitely many solutions to problem
(1) if the nonlinearity
f
is odd.
Theorem 6.1 Assume that the conditions of theorem 5.1 are satised, and
furthermore that the functions f and a are odd with respect to their second
arguments and that the weight w ∈ L1 (Ω). Then problem (1) has innitely
many solutions in W01,p (Ω; v0 , v1 ).
We will need the following Z2 -symmetric variant (for even functionals) of the
mountain pass theorem (see [14], teorema 9.12).
Theorem 6.2 Let X be an innite dimensional real Banach space and let I ∈
C 1 (X, R) be even, satisfying the (P S) condition and I(0) = 0. Assume that the
following conditions hold:
• (I1 )
There exist constants ρ, α > 0 such that I(x) ≥ α if kxk = ρ.
• (I2 ) For each nite dimensional subspace X1 ⊂ X , the set {x ∈ X : I(x) ≥
0} is bounded.
Then I possesses an unbounded sequence of critical values.
In order to prove our result, we will need the following lemma:
Lemma 6.1 The conditions of theorem 6.1 imply that if X1 ⊂ W01,p (Ω) is a
nite dimensional subspace, the set S = {u ∈ X1 : J(u) ≥ 0} is bounded in
W01,p (Ω; v0 , v1 ).
Proof:
From condition (6), we have that:
|A(x, ξ)| ≤ a0 (x)|ξ| +
12
|ξ|p
a1 (x)
p
Then, we deduce that:
Z
Z
1
A(x, ∇u) dx ≤
a0 (x)|∇u| dx +
p
Ω
Ω
Z
a1 (x)|∇u|p dx
Ω
By Hölder inequality, we have that:
Z
Z
0 −1/(p−1)
a0 (x)p v1
a0 (x)|∇u| dx ≤
Ω
1/p0 Z
dx
Ω
1/p
|∇u| v1 (x) dx
p
Ω
Therefore,
Z
A(x, ∇u) dx ≤ C1 kuk + C2 kukp
Ω
−1/(p−1)
since a0 ∈ L (Ω; v1
) and a1 (x) ≤
we have that, for any ε > 0, there exists
p0
|F (x, s)| ≤ ε
On the other hand, since
F
is
Cv1 (x), by assumption (7).
δ > 0 such that
|s|p
v0 (x) if |s| < δ
p
θ-superhomogeneous,
F (x, s) ≥ γ(x)|s|θ
with
γ(x) = min
From (14)
for
we have that:
|s| ≥ δ
F (x, δ) F (x, −δ)
,
δθ
δθ
Then, we have:
Z
Z
ε
F (x, u) dx ≥
γ(x)|u| dx −
p
Ω
{x∈Ω:|u(x)|≥δ}
It follows that
J
θ
Z
|u|p v0 (x) dx
{x∈Ω:|u(x)|<δ}
veries the estimate:
p
Z
J(u) ≤ C1 kuk + (C2 − ε)kuk −
γ(x)|u|θ dx
(18)
Ω
We claim that we have the continuous embedding:
W 1,p (Ω; v0 , v1 ) ⊂ Lp (Ω; γ(x))
Indeed, from condition (11), we obtain that
|F (x, t)| ≤ f0 (x)|t| +
f1 (x) q
|t|
q
Hence from (12),
F (x, ±δ) ≤ Cw(x) ∀ x ∈ Ω
As a consequence,
γ(x) ≤ Cw(x) ∀ x ∈ Ω
13
(19)
Then, using Hölder inequality, we have that:
Z
θ/q Z
(q−θ)/q
|u| w(x) dx
w(x) dx
Z
θ
q
|u| γ(x) dx ≤ C
Ω
Since
Ω
w ∈ L1 (Ω),
Ω
and using the Sobolev embedding (5), we conclude that:
Z
|u|θ γ(x) dx ≤ Ckukθ
Ω
Then the continuous embedding (19) holds. In the nite dimensional subspace
X1 ,
the norm
k · kLp (Ω;γ)
is equivalent to the norm
k · k:
Kkuk ≤ kukLp (Ω;γ) ∀u ∈ X1
It follows from (18), that if
u ∈ S,
then
0 ≤ C1 kuk + (C2 − ε)kukp − K θ kukθ
Since
θ > p,
if we choose
δ < C2 ,
it follows that
S
is bounded.
Proof of Theorem 6.1
J is even. J veries also
(P S) condition. Proposition
5.2 implies that there exist constants α, ρ > 0 such that J(u) ≥ α if kuk1,p = ρ.
Finally from the previous lemma {v ∈ X1 : J(u) ≥ 0} is bounded whenever X1
1,p
is a nite dimensional subspace of W0 (Ω; v0 , v1 ). Then theorem 6.2 can be
applied to the functional J . This ends the proof.
Since the functions
that
J(0) = 0.
f
and
a
are odd, the functional
By proposition 5.1,
J
satises the
ACKNOWLEDGEMENT
The authors thank especially Prof. C. Lederman for her careful reading of
the of the Ph. D. thesis of Dr. Pablo De Nápoli, and for her fruitful remarks.
References
[1] R. Adams.
Sobolev Spaces.
Academic Press, (1975).
[2] A. Ambrosetti, P. Rabinowitz.
Theory and Applications.
Dual Variational Methods in Critical Point
Journal Functional Analysis, (14), pp. 349381,
(1973).
[3] P. Amster, M.C. Mariani.
nonparametric surfaces.
The prescribed mean curvature equation for
To appear in Nonlinear Analysis.
Existence and regularity of weak solutions to the prescribed mean curvature equation for a
nonparametric surface. To appear in Abstract and Applied Analysis.
[4] P. Amster, M. Cassinelli, M.C. Mariani, D.F. Rial.
14
[5] H. Brezis.
Analyse fonctionnelle.
Masson, Paris, (1983).
[6] P. De Nápoli, M.C. Mariani. Three solutions of some quasilinear equations
in
Rn
near resonance. Electron. J. Di. Eqns., Conf. 06, 2001, pp. 131-140.
Mountain Pass Solutions to Equations of pLaplacian type. Preprint, 2001.
[7] P. De Nápoli, M.C, Mariani.
Quasilinear Elliptic Systems of Resonant Type
and Nonlinear Eigenvalue Problems. To appear in Abstract and Applied
[8] P. De Nápoli, M.C, Mariani.
Analysis.
[9] T. Kilpeläinen.
Weighted Sobolev Spaces and Capacity. Annales Academiae
Scientiarum Fennicae, pp. 19:95113, (1994).
[10] G. Dinca, P. Jebelean, J. Mawhin.
for p-Laplacian.
A result of Ambrosetti-Rabinowitz type
Corduneanu, C. (ed.). Qualitative problems for dieren-
tial equations and control theory. Singapore: World Scientic. pp. 231-242
(1995).
[11] J.-P. Gossez. Some Remarks on the Antimaximum Principle. Revista de la
Unión Matemática Argentina- vol. 41, 1, pp. 79-84 (1998)
Semilinear Elliptic Problems in Unbounded Domains: Solutions
in Weighted Sobolev Spaces. Preprint 21 A 95. Freie Universität Berlin.
[12] K. Püger.
[13] João Marcos B. do Ó.
p-Laplacian in Rn .
Solutions to Perturbed Eigenvalue Problems of the
Electronic Journal of Di. Eq., Volumen 1997, Nro. 1,
pp. 115, (1997).
Minimax Methods in Critical Point Theory with Applications to Dierential Equations. Expository Lectures from the CBMS Re-
[14] P. Rabinowitz.
gional Conference held at the University of Miami. American Mathematical
Society, (1984).
[15] M. Willem.
Minimax Theorems.
Birkhäuser, Boston, (1996).
Pablo L. De Nápoli (e-mail: pdenapo@dm.uba.ar)
M. Cristina Mariani (*) (e-mail: mcmarian@dm.uba.ar)
Universidad de Buenos Aires
FCEyN - Departamento de Matemática
Ciudad Universitaria, Pabellón I
(1428) Buenos Aires, Argentina
(*) Conicet
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