Electronic Journal of Differential Equations, Vol. 1996(1996), No. 09, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp (login: ftp) 147.26.103.110 or 129.120.3.113
ON ELLIPTIC EQUATIONS IN RN WITH
CRITICAL EXPONENTS ∗
C.O. Alves, J.V. Goncalves, & O.H. Miyagaki
Abstract
In this note we use variational arguments –namely Ekeland’s Principle
and the Mountain Pass Theorem– to study the equation
∗
−∆u + a(x)u = λuq + u2
−1
in RN .
The main concern is overcoming compactness difficulties due both to the
unboundedness of the domain RN , and the presence of the critical exponent 2∗ = 2N/(N − 2).
1
Introduction
In this note we use variational methods to explore existence of weak solutions
for the problem

∗
+ a(x)u = Rλuq + u2 −1 in RN
 −∆u
R
(∗)
a(x)u2 < ∞,
|∇u|2 < ∞

u ≥ 0, u 6≡ 0
∗
where a is a nonnegative L∞
loc function, λ ≥ 0, 0 < q ≤ 1 and 2 is the critical
exponent, 2∗ = 2N/(N − 2), for N ≥ 3.
This problem has been explored by many authors including Brézis & Nirenberg [6], Ambrosetti-Brézis & Cerami [1], Guedda & Veron [9] (see also their
references) for the case of elliptic equations in bounded domains. As far as
unbounded domains are concerned we recall the work by Benci & Cerami [12],
Noussair-Swanson & Jianfu [3], Jianfu & Xiping [14], Egnell [7], Azorero &
Alonso [4], Miyagaki [10].
∗ 1991 Mathematics Subject Classifications: 35J20, 35K20.
Key words and phrases: Elliptic equations, unbounded domains, critical exponents,
variational methods.
c
1996
Southwest Texas State University and University of North Texas.
Submitted: August 7, 1996. Published October 22, 1996.
Partially supported by CNPq/Brasil.
1
Elliptic Equations in RN
2
EJDE–1996/09
In this work we shall assume the following condition on a.
Z
1
c
(a1 )
< ∞.
a(x) > 0, x ∈ BR
and
o
c
a
BR
o
Our main results are the following:
Theorem 1 Let 0 < q < 1 and assume (a1 ). Then there exists λ∗ > 0 such
that (∗) has a solution for 0 < λ < λ∗ , N ≥ 3.
Theorem 2 Let N ≥ 4 and q = 1. Assume (a1 ) and a(x) = 0, x ∈ B2r0
for some r0 ∈ (0, R0 /2). Then there is λ∗ > 0 such that (∗) has a solution for
0 < λ < λ∗ .
These theorems complement the results in [10] in the sense that here the
function a is allowed to vanish on a ball BRo . Actually in Theorem 2, we
require a to vanish on B2r0 . In addition we consider the case 0 < q ≤ 1 while
in [10], a > 0 is continuous and q ∈ (1, 2∗ ).
2
Preliminaries
Let
Z
1,2
2
E= u∈D |
au < ∞
with inner product and norm given by
Z
Z
hu, vi = (∇u∇v + auv) , kuk2 =
|∇u|2 + au2 .
Recall
that D1,2 is the closure of Co∞ with respect to the gradient norm kuk21 =
R
|∇u|2 . Moreover
n
o
∗
D1,2 = u ∈ L2 | ∂i u ∈ L2
and the norm
0
kuk ≡ |u|L2∗ + |∇u|L2
∗
is equivalent to the D1,2 norm. In addition D1,2 → L2 .
The following lemma is a variant of a result by Willem & Omana [13] and
by Costa [2].
Lemma 1 Assume (a1 ). Then E → Ls for 1 ≤ s ≤ 2∗ and E ,→ Ls for
1 ≤ s < 2∗ .
EJDE–1996/09
C.O. Alves, J.V. Goncalves, & O.H. Miyagaki
3
We shall look for the critical points of the functional
Z
Z
Z
∗
1
1
1
q+1
2
2
I(u) =
|∇u| + a|u| −
λu+ − ∗ u2+
2
q+1
2
in the Hilbert space E.
Using standard techniques we can show that I ∈ C 1 (E, R), and that its
derivative is given by
Z
Z
Z
q
2∗ −1
0
hI (u), vi = (∇u∇v + auv) − λ u+ v − u+
v.
Therefore, the critical points of I are the weak solutions of (∗).
The following auxiliary result concerns the geometry of I.
Lemma 2 If a satisfies (a1 ) and if 0 < q ≤ 1 then there exists λ∗ > 0 such that
if 0 < λ < λ∗ then
I(u) ≥ r, kuk = ρ, for some r, ρ > 0
(i)
If in addition φ ≥ 0, φ 6≡ 0, and φ ∈ E then
(ii)
I(tφ) → −∞ as t → ∞
(iii)
I(tφ) < 0, for small t > 0, and 0 < q < 1.
3
Proofs
For the sake of completeness, we present a proof of Lemma 3, which is based on
the proof in [2].
Proof of Lemma 3.
Z
c
BR
Z
|u| =
c
BR
At first let R > Ro . Then we have
a1/2 |u|
≤
a1/2
Z
c
BR
1
a
!1/2
!1/2
Z
2
a|u|
c
BR
≤ Ckuk
which shows that
|u|L1 ≤ Ckuk, u ∈ E.
Now using the interpolation inequality
1−α
|u|s ≤ |u|α
, α+
1 |u|r
∗
1−α
1
= , 1 ≤ s ≤ r ≤ 2∗ , 0 ≤ α ≤ 1
r
s
and the embedding E → L2 , we infer that E → Ls , 1 ≤ s ≤ 2∗ .
Elliptic Equations in RN
4
EJDE–1996/09
On the other hand, for sufficiently large R > 0 we have
Z
1
< .
c
a
BR
So if un * 0 in E, then for large n
Z
Z
|un | ≤ C
c
BR
c
BR
1
≤ .
a
Using compact Sobolev embeddings we also have
un → 0 in L1 (BR )
so that un → 0 in L1 . Using again the interpolation inequality stated above,
one concludes the proof of Lemma 3.
Proof of Lemma 4.
Verification of (i).
I(u) ≥
≥
From the continuous embedding in Lemma 3, we have
∗
−2 /2
∗
λcq+1
1
2
kukq+1 − S 2∗ kuk2
2 kuk −
q+1
−2∗ /2
∗
λcq+1
kukq+1 12 kuk2−(q+1) − q+1
− S 2∗ kuk2 −(q+1)
∗
where S is the best constant for the embedding D1,2 → L2 , that is
)
( R
|∇u|2
1,2
S = inf
R
2/2∗ | u ∈ D , u 6≡ 0 .
|u|2∗
Letting
∗
1
S −2 /2 2∗ −(q+1)
Q(t) ≡ t2−(q+1) −
t
, t ≥ 0,
2
2∗
there is ρ > 0 such that
max Q(t) = Q(ρ) > 0.
t≥0
∗
Taking kuk = ρ and λ =
Verification of (ii).
q+1
cq+1 Q(ρ)
we get (i).
Taking φ 6≡ 0, φ ≥ 0, φ ∈ E we have
t2
tq+1
I(tφ) = kφk2 −
λ
2
q+1
which gives (ii).
Z
∗
q+1
φ
t2
− ∗
2
Z
∗
φ2
EJDE–1996/09
C.O. Alves, J.V. Goncalves, & O.H. Miyagaki
5
Verification of (iii). It is clear from the expression of I(tφ) above taking into
account that 0 < q < 1.
Proof of Theorem 1. By the proof of lemma 4, I is bounded from below on
Bρ . By the Ekeland Principle [8], there exists u ∈ Bρ such that
I(u ) ≤ inf I + B¯ρ
and
I(u ) < I(u) + ku − u k, u 6≡ u .
Now since 0 < q < 1 it follows that
I(tφ) < 0, for small t > 0, φ 6≡ 0, and φ ∈ Co∞ .
Again by Lemma 4
inf I ≥ r > 0 and inf I < 0.
∂Bρ
Bρ
Choose > 0 such that
0 < < inf I − inf I.
∂Bρ
Bρ
Hence
I(u ) < inf I
∂Bρ
so that
u ∈ Bρ .
Hence letting
F (u) ≡ I(u) + ku − u k
we notice that u is a point of minimum of F on Bρ and so
I(u + δv) − I(u )
+ kvk ≥ 0
δ
which by passing to the limit as δ → 0 gives that
hI 0 (u ), vi + kvk ≥ 0
and hence kI 0 (u )k ≤ . Therefore, there is a sequence un ∈ Bρ such that
I(un ) → c∗ ≡ inf I < 0 and I 0 (un ) → 0.
Bρ
Since of course un is bounded,
un * u∗ in E
Elliptic Equations in RN
6
and
EJDE–1996/09
un → u∗ a.e. in RN .
Now passing to the limit in
Z
Z
Z
2∗ −1
o(1) = (∇un ∇φ + aun φ) − λ uqn+ φ − un+
φ, φ ∈ E
we infer that I 0 (u∗ ) = 0 showing that u∗ is a solution of problem (∗).
In order to show that u∗ 6≡ 0, we follow the arguments in [6]. Assume that
∗
u ≡ 0 and that
kun k2 → ` ≥ 0.
Using I 0 (un ) → 0 we have
Z
kun k2 −
so that
R
∗
u2n+ = o(1)
∗
u2n+ → ` and from the expression
Z
Z
∗
1
1
1
q+1
∗
−
un+ +
u2n+
c + o(1) = λ
2 q+1
N
we infer that
c∗ =
`
N
which is impossible.
Proof of Theorem 2. By Lemma 4 and the Mountain Pass Theorem, there
exists a sequence un in E such that
I(un ) → c and I 0 (un ) → 0
where
c = inf max I(γ(t)), c ≥ r
γ∈Γ 0≤t≤1
and
Γ = {γ ∈ C([0, 1], X) | γ(0) = 0, γ(1) = e}
where e ∈ E satisfies I(e) ≤ 0.
Claim. There is e ≡ eλ such that 0 < c < N1 S 2 , 0 < λ < λ∗ .
From the expression
∗
Z
2∗
2
0
∗
2
hI (un ), un i − 2 I(un ) = 1 −
kun k + λ
−1
u2n+
2
2
N
one shows, by taking λ∗ > 0 smaller than the one found in lemma 4, that un is
bounded. So that, passing to subsequences,
un * u in E and un → u a.e. in RN
for some u ∈ E.
EJDE–1996/09
C.O. Alves, J.V. Goncalves, & O.H. Miyagaki
7
Remark. I(un+ ) → c and I 0 (un+ ) → 0.
Indeed, un− is also bounded so that
Z
o(1) = hI 0 (un ), un− i = (|∇un− |2 + au2n− ) .
Moreover, if φ ∈ E then
Z
hI 0 (un+ ), φi
= hun+ , φi − λ
Z
un+ φ −
∗
2 −1
un+
φ
= hI 0 (un ), φi − hun− , φi
so that, I 0 (un+ ) → 0. On the other hand,
Z
1
I(un ) −
(|∇un− |2 + au2n− )
2
Z
Z
Z
∗
1
λ
1
=
(|∇un+ |2 + au2n+ ) −
u2n+ − ∗ u2n+
2
2
2
= I(un+ ) ,
which gives I(un+ ) → c. So we may assume that un ≥ 0 and thus u ≥ 0.
Now as in the proof of Theorem 1 one shows that u satisfies the equation in
(∗). Again arguing as in [6] we assume that u ≡ 0. Then
kun k2 → ` for some ` ≥ 0
and using the facts that
I(un ) → c, I 0 (un ) → 0
we infer that c ≥
1 N/2
,
NS
contradicting 0 < c <
1 N/2
NS
given by the Claim.
Proof of the Claim. (Arguments adapted from [6].)
Consider the cut-off function φ ∈ Co∞ such that
φ ≡ 1 on Br0 , φ ≡ 0 on RN \B2r0 .
Now consider the function
(N −2)/4
w (x) =
which satisfies
[N (N − 2)]
( + |x|2 )(N −2)/2
∗
−∆w = w2
−1
,
x ∈ RN , > 0
in RN .
It is well known (see e.g. Talenti [5], Aubin [11] ) that
∗
kw k21 = |w |22∗ = S N/2 .
Elliptic Equations in RN
8
EJDE–1996/09
Let
ψ = φw
and let v ∈
Co∞
given by
v = R
ψ
ψ 2∗
1/2∗ .
Now it can be shown (see e.g. [6], [10]) that X ≡
R
|∇v |2 satisfies
X ≤ S + O((N −2)/2 ).
Moreover there is some t > 0 such that
max I(tv ) = I(t v )
t≥0
and
d
I(tv )|t=t = 0.
dt
which gives
∗
0 < t < X1/(2
−2)
≡ t0 .
Notice that a = 0 on B2r0 and v = 0 on R \B2r0 . Moreover t ≥ d0 ≡ d0 (r0 )
for some d0 > 0. Otherwise since X is bounded, if t → 0, then I(t v ) → 0
contradicting
I(t v ) = max I(tv ) ≥ r > 0
N
t≥0
given by lemma 4 (i). On the other hand
Z
Z
∗
t2
t2
λt2
I(tv ) =
|∇v |2 − ∗ −
v2
2
2
2
2
Z
∗ t 2∗ −2 t2
λt2
≤
t0
− ∗ −
v2 .
2
2
2 B2r0 Now recalling that as a function of t,
2
∗ t 2∗ −2 t2
t
− ∗
2 0
2
increases on the interval (0, t0 ) we get
Z
1
1
λt2
2∗
I(t v ) ≤ t0
−
−
v2
2 2∗
2 B2r0 Z
1 2∗ λd20
≤
t0 −
v2
N
2 B2r0 i2∗ /(2∗ −2) λd2 Z
1 h
≤
S + O (N −2)/2
− 0
v2
N
2 B2r0 iN/2 λd2 Z
1 h
(N −2)/2
=
S +O − 0
v2 .
N
2 B2r0 EJDE–1996/09
C.O. Alves, J.V. Goncalves, & O.H. Miyagaki
9
Using the inequality
(b + c)α ≤ bα + α(b + c)α−1 c b, c ≥ 0, α ≥ 1
with b = S, c = O (N −2)/2 and α = N/2 we get
Z
1
v2 .
I(t v ) ≤ S N/2 + O((N −2)/2 ) − c0 λ
N
B2r0
Therefore,
1
I(t v ) ≤ S N/2 + (N −2)/2
N
(
)
Z
M − c0 λ(2−N )/2
v2
,
B2r0
where c0 = d20 /2 and M is a positive constant.
We shall show that
(
)
Z
(N −2)/2
(N −2)/2
2
M − c0 λ
v < 0,
for small > 0 .
B2r0
So that
I(t v ) <
and hence
0<c<
Noticing that
Z
1 N/2
S
N
1 N/2
S
.
N
∗
d1 ≤
ψ2 ≤ d2 , for some d1 , d2 > 0,
B2r0
(see [6]), it follows by a change of variables that
(
)
Z r0 −1/2
1 N/2
sN −1 ds
(N −2)/2
(4−N )/2
+
M − c0 λ
.
I(t v ) ≤ S
N
(1 + s2 )N −2
0
We are going to consider separately the cases N = 4 and N ≥ 5.
Case N = 4. We have
I(t v )
≤
≤
Now since
(
)
Z r0 −1/2
1 2
s3 ds
S + M − c0 λ
4
(1 + s2 )2
0
n
o
1 2
S + M − c0 λ ln(r0 −1/2 ) .
4
c0 λ ln(r0 −1/2 ) → ∞ as → 0
we infer that
I(t v ) <
1 2
S , for small > 0.
4
Elliptic Equations in RN
10
EJDE–1996/09
Case N ≥ 5. Noticing that
Z
r0 −1/2
c0 λ(4−N )/2
0
sN −1 ds
→ ∞ as → 0
(1 + s2 )N −2
we infer that
1 N/2
S
for small > 0 ,
N
which concludes the proof of this claim.
I(t v ) <
References
[1] A. Ambrosetti, H. Brézis & G. Cerami, Combined effects of concave and
convex nonlinearities in some elliptic problems, Preprint.
[2] D. Costa, On a nonlinear elliptic problem in RN , Preprint.
[3] E. S. Noussair, C. A. Swanson & Y. Jianfu, Positive finite energy solutions
of critical semilinear elliptic problems, Canadian J. Math. 44(1992) 10141029.
[4] J. G. Azorero & I. P. Alonzo, Multiplicity of solutions for elliptic problems
with critical exponents with a nonsymmetric term, Trans. Amer. Math.
Society 323(1991) 877-895.
[5] G. Talenti, Best constant in Sobolev inequality, Ann. Math. 110 (1976)
353-372.
[6] H. Brézis & L.Nirenberg, Some Variational problems with lack of compactness. Proc. Symp. Pure Math. (AMS) 45(1986) 165-201.
[7] H. Egnell, Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents, Arch. Rat. Mech. Anal. 104(1988) 57-77.
[8] I. Ekeland, On the variational principle. J. Math. Anal. App. 47(1974)
324-353.
[9] M. Guedda & L. Veron, Quasilinear elliptic equations involving critical
Sobolev exponents, Non. Anal. 12(1989) 879-902.
[10] O. Miyagaki, On a class of semilinear elliptic problems in IRn with critical
growth, CMS Technical Report, Univ. Wisconsin (1994).
[11] T. Aubin, Problemes isoperimetriques et espaces de Sobolev, J. Diff. Geometry 11(1976) 573-598.
[12] V. Benci & G. Cerami, Existence of positive solutions of the equation −∆u+
a(x)u = u(n+2)/(n−2) in Rn , J. Funct. Anal. 88(1990) 90-117.
EJDE–1996/09
C.O. Alves, J.V. Goncalves, & O.H. Miyagaki
11
[13] W. Omana & M. Willem, Homoclinic orbits for a class of Hamiltonian
systems, Diff. Int. Equations 5(1992) 1115-1120.
[14] Y. Jianfu & Z. Xiping, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains, Acta Math.
Sci. 7(1987) 341-359.
C.O. Alves
Dep. Mat. Univ. Fed. Paraiba
58109-970 - Campina Grande(PB), Brasil
E-mail coalves@dme.ufpb.br
J.V. Goncalves
Dep. Mat. Univ. Brası́lia
70.910-900 Brasilia(DF), Brasil
E-mail jv@mat.unb.br
O.H. Miyagaki
Dep. Mat. Univ. Fed. Viçosa
36570-000 Viçosa(MG), Brasil
E-mail olimpio@mail.ufv.br