N −LAPLACIAN EQUATIONS IN RN WITH CRITICAL
GROWTH
JOÃO MARCOS B. DO Ó
Abstract. We study the existence of nontrivial solutions to the following
problem:
u ∈ W 1,N (RN ), u ≥ 0 and
−div(| ∇u |N −2 ∇u) + a (x) | u |N −2 u = f (x, u) in RN (N ≥ 2),
where a is a continuous function which is coercive, i.e., a (x) → ∞ as
| x |→ ∞ and the nonlinearity f behaves like exp α | u |N/(N −1) when
| u |→ ∞.
1. Introduction
In this paper, we apply a mountain pass type argument to prove the
existence of nontrivial weak solutions to the following class of semilinear
elliptic problems in RN (N ≥ 2) , involving critical growth:
(1)
u ∈ W 1,N (RN ),
u≥0
and
∇u) + a (x) | u |N −2 u = f (x, u)
−div(| ∇u
|N −2
in RN ,
where a : RN → R is a continuous function satisfying a (x) ≥ a0 , ∀x ∈ RN ,
and such that a (x) → ∞ as | x |→ ∞. It is assumed that the nonlinearity
f : RN × R → R is also continuous and f (x, 0) ≡ 0. Thus,
u ≡ 0 is a solution
of (1) and has critical growth, i.e., f behaves like exp α | u |N/(N −1) when
| u |→ ∞.
For N = 2, problems of this type, that is, involving the Laplacian operator
and critical growth in the whole R2 , have been considered by Cao in [10] and
by Cao and Zhengjie in [11], under the decisive hypothesis that the function
1991 Mathematics Subject Classification. 35A15, 35J60.
Key words and phrases. Elliptic equations, p−Laplacian, critical growth, Mountain
Pass theorem, Trudinger-Moser inequality.
Work partially supported by CNPq/Brazil.
Received: June 10, 1997.
c
1996
Mancorp Publishing, Inc.
301
302
JOÃO MARCOS B. DO Ó
a be a constant. For that purpose, they used the concentration-compactness
principle of P. L. Lions.
Recently, Rabinowitz in [29], among other results, obtained a nontrivial
solution to the problem −∆u + a (x) u = f (x, u) in RN , under
the as
sumption that a is coercive and that the potential F (x, u) = 0u f (x, s)ds
is superquadratic and f (x, u) has subcritical growth, that is, | f (x, u) |≤
b1 + b2 | u |s , where s ∈ (1, (N + 2)/(N − 2)). This result was extended by
Costa [15] to a class of potentials F (x, u) which are nonquadratic at infinity. Miyagaki, in [24], has treated this problem for N ≥ 3, involving critical
Sobolev exponent, namely for f (x, u) = λ | u |q−1 u+ | u |p−1 u, where
1 < q < p ≤ (N − 2)/(N + 2) and λ > 0. In [3], this result was generalized
by Alves to the p−Laplacian operator.
In this paper, we complement the results mentioned above by establishing
sufficient conditions for the existence of nontrivial solutions to (1).
To treat
N variationally this class of problems, with f behaving like exp α | u | N −1
when | u |→ ∞, we introduce a Trudinger-Moser type inequality. On the
other hand, to overcome the lack of compactness that has arisen from the
critical growth and the unboundedness of the domain, we use some recent
ideas from [16, 19] together with a compact imbedding result essentially
given by the coerciveness of a (cf. [15]).
We would also like to mention that problems involving the Laplacian
operator with critical growth range in bounded domains of R2 have been
investigated, among others, by [2, 5, 6, 9, 16, 17, 22, 23, 30]. We refer to
[1, 19, 26] for semilinear problems with critical growth for the N −Laplacian
in bounded domains of RN .
Now we shall describe the conditions on the functions a and f. Namely,
for a we suppose that:
(a1 )
there exists a positive real number a0 such that a (x) ≥ a0 , ∀x ∈ RN ,
(a2 )
a (x) → ∞
as
| x |→ ∞.
On the other hand, motivated by a Trudinger-Moser type inequality (cf.
Lemma 1 below), we assume the following growth condition on the nonlinearity f (x, u),
(f1 )
the function f : RN × R → R is continuous and for all (x, u) ∈ RN × R,
N
| f (x, u) |≤ b1 | u |N −1 +b2 exp α0 | u | N −1 − SN −2 (α0 , u) ,
for some constants α0 , b1 , b2 > 0, where
SN −2 (α0 , u) =
N
−2
k=0
N
α0k
| u | N −1 k .
k!
Moreover, f is assumed to satisfy the following conditions:
(f2 )
there is a constant µ > N such that, for all x ∈ RN and u > 0,
0 ≤ µF (x, u) ≡ µ
u
0
f (x, t)dt ≤ uf (x, u),
N -LAPLACIAN EQUATIONS
(f3 )
303
there are constants R0 , M0 > 0 such that, for all x ∈ RN and u ≥ R0 ,
0 < F (x, u) ≤ M0 f (x, u);
(f4 )
N
lim uf (x, u) exp −α0 | u | N −1
u→+∞
subsets of RN .
≥ β0 > 0
uniformly on compact
As usual, W 1,N (RN ) denotes the Sobolev space of functions in LN (RN )
such that their weak derivatives are also in LN (RN ) with the norm
.
u N
W 1,N =
RN
(| ∇u |N + | u |N )dx,
and we consider the subspace E ⊂ W 1,N (RN ) given by
E = {u ∈ W 1,N RN :
endowed with the norm
u
.
N
E=
RN
RN
a(x) | u |N dx < ∞}
(| ∇u |N +a(x) | u |N )dx.
see that the Banach space E is a continuously
Since a(x) ≥ a0 > 0, we clearly
embedded in W 1,N RN and, moreover,
(2)
λ1 (N ) = inf
0=u∈E
u N
E
≥ a0 > 0.
u N
LN
The main result of this paper is the following
Theorem 1. Suppose (a1 ) − (a2 ) and (f1 ) − (f4 ) are satisfied. Furthermore,
assume that
(f5 )
lim supu→0+
N F (x, u)
< λ1 (N )
| u |N
unif ormly in x ∈ RN .
Then the problem (1) has a nontrivial weak solution u ∈ E.
Remark 1. The assumption (a2 ) implies that the Banach space E is compactly immersed in Lq if N ≤ q < ∞. We observe that this compact embedding result is used here only to prove that the Palais-Smale sequence obtained
by mountain pass type argument converges to a weak nontrivial solution.
Therefore, the same device can be applied when we have some assumption
which implies a compact embedding result as the cited above. For instance,
when the function a is a radially symmetric function, that is, a(x) = a(y) if
| x |=| y | (cf. [4, 18]).
304
JOÃO MARCOS B. DO Ó
2. A Trudinger-Moser inequality
Let Ω be a bounded domain in RN (N ≥ 2). The Trudinger-Moser inequality (cf. [25, 31]) asserts that
N
∀u ∈ W01,N (Ω) ,
exp(α | u | N −1 ) ∈ L1 (Ω),
∀α > 0
and that there exists a constant C(N ) which depends on N only, such that
u
sup
N
Ω
1,N ≤1
W
0
exp(α | u | N −1 ) ≤ C(N ) | Ω |,
if
α ≤ αN ,
1
−1
and wN −1 is the (N − 1)−dimensional
where | Ω |= Ω dx, αN = N wNN−1
measure of the (N − 1)−sphere.
Inspired of this inequality and based on the related results [7, 10, 11, 12,
13, 14], we get the following result.
Lemma 1. If N ≥ 2, α > 0 and u ∈ W 1,N RN , then
(3)
RN
N
exp α | u | N −1 − SN −2 (α, u) < ∞.
1
−1
≤ 1, u LN ≤ M < ∞ and α < αN = N wNN−1
,
Moreover, if ∇u N
LN
then there exists a constant C = C(N, M, α), which depends only on N, M
and α, such that
(4)
RN
N
exp α | u | N −1 − SN −2 (α, u) ≤ C (N, M, α) .
Proof. We may assume u ≥ 0, since we can replace u by | u | without causing any increase in the integral of the gradient. Since we shall use Schwarz
symmetrization method, we recall briefly some of theirs basic properties (cf.
[20, 27]). Let 1 ≤ p ≤ ∞ and u ∈ Lp (RN ) such that u ≥ 0. Thus, there is a
unique nonnegative function u∗ ∈ Lp (RN ), called the Schwarz symmetrization of u, such that it depends only on | x |, u∗ is a decreasing function of
| x |; for all λ > 0
| {x : u∗ (x) ≥ λ} |=| {x : u (x) ≥ λ} |
and there exists Rλ > 0 such that {x : u∗ ≥ λ} is the ball B (0, Rλ ) of radius
Rλ centered at origin. Moreover, suppose that G : [0, +∞) → [0, +∞) is a
continuous and increasing function such that G (0) = 0. Then, we have
∗
RN
G (u (x)) dx =
Further, if u ∈ W01,p RN
RN
RN
G (u (x)) dx.
then u∗ ∈ W01,p RN
| ∇u∗ |p (x) dx ≤
RN
and
| ∇u |p (x) dx.
N -LAPLACIAN EQUATIONS
Thus, we can write
RN
N
exp α | u | N −1
305
−SN −2 (α, u)]
=
RN
N
exp α | u∗ | N −1 − SN −2 (α, u∗ ) ,
and, for a real number r > 1 to be determined, we have
RN
=
|x|<r
+
≤
N
exp α | u∗ | N −1 − SN −2 (α, u∗ )
|x|≥r
|x|<r
N
exp α | u∗ | N −1 − SN −2 (α, u∗ )
N
exp α | u∗ | N −1 − SN −2 (α, u∗ )
N
exp α | u∗ | N −1 +
|x|≥r
N
exp α | u∗ | N −1 − SN −2 (α, u∗ ) .
Let us recall two elementary inequalities. Using the fact that the function
N
N
1
h : (0, +∞) → R given by h (t) = [(t + 1) N −1 − t N −1 − 1]/t N −1 is bounded,
we have a positive constant A = A(N ) such that
N
(5)
N
1
N
(u + v) N −1 ≤ u N −1 + Au N −1 v + v N −1 ,
∀u, v ≥ 0.
If γ and γ are positive real numbers such that γ + γ = 1, then for all ε > 0,
we have
uγ v γ ≤ εu + ε
(6)
− γγ
v,
∀u, v ≥ 0,
because g : [0, +∞) → R, given by g (t) = tγ − εt, is bounded.
Let v (x) = u∗ (x) − u∗ (rx0 ) where x0 is some fixed unit vector in RN .
Notice that v ∈ W01,N (B (0, r)) . Here, B(0, r) denotes the ball of radius r
centered at the origin of RN . Now, from (5) and (6), we have, respectively,
N
N
N
1
N
| u∗ | N −1 =| v + u∗ (rx0 ) | N −1 ≤ v N −1 + Av N −1 u∗ (rx0 ) + u∗ (rx0 ) N −1 ,
1
v N −1 u∗ (rx0 ) =
≤
N
v N −1
1 N
ε NN−1
v
+
A
N
u∗ (rx0 ) N −1
ε
A
1
1−N
N −1
N
N
u∗ (rx0 ) N −1 ,
and hence,
N
N
N
| u∗ | N −1 ≤ (1 + ε) v N −1 + K(ε, N )u∗ (rx0 ) N −1 ,
N
1
where K(ε, N ) = A N −1 ε 1−N + 1. Therefore,
N
∗ N −1 ≤
|x|≤r exp α | u |
N
exp K(ε, N )u∗ (rx0 ) N −1
N
N −1
,
|x|≤r exp α | (1 + ε) v |
306
JOÃO MARCOS B. DO Ó
which, in view of Trudinger-Moser inequality, implies,
(7)
N
exp α | u∗ | N −1 < ∞,
|x|≤r
∀u ∈ W 1,N RN ,
∀α > 0.
Furthermore, taking ε > 0 such that (1 + ε) α < αN , we obtain
|x|≤r
N
exp α | u∗ | N −1
≤ C (N )
(8)
N wN −1 rN
exp K(ε, N )u∗ (rx0 ) N −1
N

NMN
wN −1 rN
exp 
≤ C (N )
N
wN −1
1
N −1

K(ε, N ) 
N
r N −1
,
≤ 1 and u LN ≤ M , where in
for all u ∈ W 1,N RN such that ∇u N
LN
the last inequality we have used Radial Lemma A.IV in [8]:
∗
−1
| u (x) |≤| x |
|x|≥r
(9)
N
1
N
∀x = 0.
∗ N
|u |
|x|≥r
| x | N −1 k
dx = wN −1
=
∞
αk
+
k=N
Now, notice that the estimate
|x|≥r
u∗ LN (RN ) ,
exp α | u∗ | N −1 − SN −2 (u∗ )
αN −1
=
(N − 1)!
N
wN −1
On the other hand, we have
1
N
∞ N −1
t
r
N
t N −1 k
wN −1
N
N −1 k − N
k!
N
|x|≥r
| u∗ | N −1 k .
dt
N
rN − N −1 k ≤
wN −1 rN
N
r N −1 k
, ∀k ≥ N,
together with Radial Lemma, lead to
∞
αk
k=N
(10)
k!
N
|x|≥r
| u∗ | N −1 k
≤ wN −1 r
N
∞
αk
N
k=N
k!
1 N
wN −1
u∗ LN (RN )
r
NN−1 k
.
Finally, (7), (9) and (10) imply the existence of the integral in (3). Furthermore, in the case that α < αN and u LN ≤ M , if we choose r =
M
N
wN −1
1
N
, we have
|x|≥r
N
exp α | u∗ | N −1 − SN −2 (u) ≤ N M N exp (αN ) ,
which, in combination with (8), implies (4).
N -LAPLACIAN EQUATIONS
307
3. The variational formulation
First, we observe that since we are interested in obtaining nonnegative
solutions, it is convenient to define
f (x, u) = 0,
∀(x, u) ∈ RN × (−∞, 0].
From (f1 ), we obtain for all (x, u) ∈ RN × R,
N
| F (x, u) |≤ b3 exp α1 | u | N −1 − SN −2 (α1 , u) ,
(11)
1
N
for some constants
α1 ,b3 > 0. Thus, by lemma 1, we have F (x, u) ∈ L (R )
for all u ∈ W 1,N RN . Therefore, the functional I : E → R given by
1
I(u) =
u N
E −
N
F (x, u)dx
RN
is well defined. Furthermore, using standard arguments (cf. Theorem A.VI
in [8]) aswellas the fact that for any given strong convergent sequence (un)
in W 1,N RN there is a subsequence (unk ) and there exists h ∈ W 1,N RN
such that | unk (x) |≤ h(x) almost everywhere in RN , we see that I is a C 1
functional on E with
I (u)v =
(| ∇u |N −2 ∇u∇v+a(x) | u |N −2 uv)dx−
RN
f (x, u)vdx, ∀v ∈ E.
RN
Consequently, critical points of the functional I are precisely the weak solutions of problem (1). Here, like in [29, 24, 19], we are going to use a
Mountain-Pass Theorem without a compactness condition such as the one
of Palais-Smale type. This version of Mountain -Pass Theorem is a consequence of Ekeland’s variational principle (cf. [21]). In the next two lemmas we check that the functional I satisfies the geometric conditions of the
Mountain-Pass Theorem (cf. [28]).
Lemma 2.
that (a1 ), (f1 ) , (f2 ) and (f3 ) are satisfied. Then for any
Assume
1,N
N
u∈W
R −{0} with compact support and u ≥ 0, we have I (tu) → −∞
as t → ∞.
Proof. Let u ∈ W 1,N RN − {0} with compact support and u ≥ 0. By
(f2 ) and (f3 ) there are positive constants c, d such that
F (x, s) ≥ csµ − d,
Thus,
I (tu) ≤
∀x ∈ supp (u) , ∀s ∈ [0, +∞).
tN
µ
u N
E −ct
N
uµ + d | supp (u) |,
RN
which implies that I (tu) → −∞ as t → ∞, since µ > N.
Lemma 3. Suppose that (a1 ), (f1 ) and (f5 ) hold. Then there exist α, ρ > 0
such that
I (u) ≥ α if u E = ρ.
308
JOÃO MARCOS B. DO Ó
Proof.
From (f5 ), there exist ε, δ > 0 in such a way that | u |≤ δ implies
(λ1 (N ) − ε)
| u |N
N
for all x ∈ RN . On the other hand, for q > N , by (f1 ), there are positive
constants β, C = C(q, δ) such that | u |≥ δ implies
F (x, u) ≤
N
F (x, u) ≤ C | u |q exp β | u | N −1 − SN −2 (β, u)
for all x ∈ RN . These two estimates yield,
N (λ1 (N ) − ε)
| u |N +C | u |q exp β | u | N −1 − SN −2 (β, u) ,
N
N
for all (x, u) ∈ R × R. In what follows we make use of the inequality
F (x, u) ≤
(12)
RN
N
| u |q exp β | u | N −1 − SN −2 (β, u) ≤ C(β, N ) u qE ,
to be proved later, assuming that u E ≤ M holds, where M is sufficiently
small. Under the assumption we have just done, by means of (2) and the
continuous imbedding E 3→ LN (RN ), we achieve
1
(λ1 (N ) − ε)
q
u N
u N
E −
LN −C u E
N
N
(λ1 (N ) − ε)
1
q
) u N
≥ (1 −
E −C u E .
N
λ1 (N )
I (u) ≥
Thus, since ε > 0 and q > N , we may choose α, ρ > 0 such that I (u) ≥
α if u E = ρ.
Now, let us obtain inequality (12). As it has been done in the proof of
lemma
1, weN use
shall the method of symmetrization. Letting R(β, u) =
N
−1
exp β | u |
− SN −2 (β, u), we have
R(β, u) | u | dx =
RN
∗
RN
R(β, u∗ ) | u∗ |q dx
RN
and
q
∗ q
R(β, u ) | u | dx =
∗
|x|≤σ
∗ q
R(β, u ) | u | dx+
|x|≥σ
R(β, u∗ ) | u∗ |q dx,
where σ is a number to be determined later. Using the Hölder inequality,
we obtain
|x|≤σ
R(β, u∗ ) | u∗ |q dx ≤
≤
|x|≤σ [exp(β
|x|≤σ
N
| u∗ | N −1 )] | u∗ |q dx
exp(βr |
u∗
|
N
N −1
1 r
)
|x|≤σ
| u∗ |qs
1
s
,
where 1/r + 1/s = 1. Now, proceeding as in the proof of lemma 1, we obtain
|x|≤σ
N
exp(βr | u∗ | N −1 )dx ≤ C(β, N )
N -LAPLACIAN EQUATIONS
309
N
if u E ≤ M, where M is such that βrM N −1 < αN . Thus, using the
continuous imbedding E 3→ Lqs (RN ), we have
(13)
|x|≤σ
R(β, u∗ ) | u∗ |q dx ≤ C(β, N ) u qE .
On the other hand, the Radial Lemma leads to
N
|x|≥σ
≤
| u∗ | N −1 k | u∗ |q dx
1
N
N
wN −1
≤
1
N
N
wN −1
∗
u LN (RN )
NN−1 k |x|≥σ
NN−1 k 

u∗ LN (RN )
N
 wN −1
≤ wN −1 σ N 

1
N
LN (RN ) 
≤ C(N, M ) u qE ,
for all k ≥ N , if we choose σ r = M0
(N )1/N M .
|x|≥σ
We also have that
| u∗ |N | u∗ |q dx ≤
|x|≥σ
N
| x | N −1 kr
1 r

|x|≥σ
1
∗ qs
|u |
s
N
k
N −1
u qLsq (RN )


σr
dx
1

u∗
N
| x | N −1 k
|x|≥σ

λ1
| u∗ |q
N
wN −1
1
N
| u∗ |N r dx
where u LN (RN ) ≤ M0 =
1 r
|x|≥σ
| u∗ |qs dx
1
s
≤ u∗ N
u∗ qLqs (RN )
LN r (RN )
≤ C (N, M ) u∗ qE ,
if u∗ qE ≤ M, via the continuous imbedding E 3→ W 1,N (RN ) 3→ LN r (RN ).
Therefore,
(14)
|x|≥R
RN (β, u∗ ) | u∗ |q dx ≤ C(N, M ) exp(β) u qE .
Finally, the combination of estimates (13) and (14) leads to (12).
In order to get a more precise information about the minimax level obtained by the Mountain Pass Theorem, let us consider the following sequence
of nonnegative functions

N −1

 (log n) N
1 
1
r
n (x, r) = w− N
N
M
N −1  log( | x | )/(log n)


0
if
if
| x |≤ r/n
r/n ≤| x |≤ r
if
| x |≥ r.
n (·, r) ∈ W 1,N (RN ), the support of M
n (x, r), is the ball
Notice that: M
n (x, r) |N dx = 1 and
B[0, r] of radius r centered at zero, RN | ∇M
310
JOÃO MARCOS B. DO Ó
n (x, r) |N dx = O(1/ log n) as n → ∞. Moreover, let Mn (x, r) =
|M
n (x, r)/ M
n E . Thus, it is not difficult to see that
M
RN
N
−
1
N −1
MnN −1 (x, r) = wN −1
log n + dn , ∀ | x |≤ r/n,
(15)
where dn is a bounded sequence of nonnegative numbers.
Lemma 4. Suppose that (a1 ) and (f1 ) − (f5 ) hold true. Then there exists
Mn (·, r) such that
1 αN N −1
)
.
max{I(tMn ) : t ≥ 0} < (
N α0
Proof.
Let r be a fixed positive real number such that
1
β0 > N
r
(16)
N
α0
N −1
.
Suppose, by contradiction, that for all n we have
1 αN N −1
max{I(tMn ) : t ≥ 0} ≥ (
)
,
N α0
where Mn (x) = Mn (x, r). In view of Lemma 2, given n there exists tn > 0
such that
I (tn Mn ) = max{I (tMn ) : t ≥ 0}.
So,
tN
1 αN N −1
I (tn Mn ) = n −
(17)
F (x, tn Mn ) dx ≥ (
)
,
N
N α0
RN
and using the fact that F (x, u) ≥ 0, we obtain
αN N −1
tN
(18)
)
.
n ≥(
α0
d
Since at t = tn , we have I (tMn ) = 0, it follows that
dt
tN
n =
(19)
tn Mn f (x, tn Mn ) dx =
RN
|x|≤r
tn Mn f (x, tn Mn ) dx.
Now, using hypothesis (f5 ), given ε > 0 there exists Rε > 0 such that for all
u ≥ Rε and for all | x |≤ r,
N
uf (x, u) ≥ (β0 − ε) exp(α0 | u | N −1 ).
(20)
From (19) and (20), for large n, we obtain,
tN
n ≥ (β0 − ε)
r
|x|≤ n
wN −1
= (β0 − ε)
N
N
exp(α0 | tn Mn | N −1 )dx
N
r
n
N
−
1
N
N −1
exp(α0 tn N −1 wN −1
log n + α0 tn N −1 dn ).
Thus,
1 ≥ (β0 − ε)
N
wN −1 N
α0 N log n NN−1
tn
+ α0 tn N −1 dn − N log tn − N log n].
r exp[
N
αN
N -LAPLACIAN EQUATIONS
311
Therefore, the sequence tn is bounded, since otherwise, up to subsequences,
we would have
N
α0 N log n NN−1
tn
+ α0 tn N −1 dn − N log tn − N log n = +∞,
n→∞
αN
lim
which leads to a contradiction. Moreover, by (18) and
N
tN
n
N
wN −1 N
α0 tn N −1
≥ (β0 − ε)
− 1)N log n + α0 tn N −1 dn ]
r exp[(
N
αN
it follows that
tN
n →(
(21)
αN N −1
)
,
α0
as
n → ∞.
Now, in order to estimate (19) more precisely, we consider the sets
An = {x ∈ B[0, r] : tn Mn ≥ Rε } and Bn = B[0, r] − An .
From (19) and (20) we arrive at
tN
n
≥ (β0 − ε)
|x|≤r
− (β0 − ε)
exp(α0 | tn Mn |
N
N −1
)dx +
Bn
tn Mn f (x, tn Mn ) dx
N
Bn
exp(α0 | tn Mn | N −1 )dx.
Notice that Mn (x) → 0 and the characteristic functions χBn → 1 for almost
every x such that | x |≤ r. Therefore, the Lebesgue Dominated Convergence
Theorem implies
Bn
tn Mn f (x, tn Mn ) dx → 0 and
Bn
exp(α0
|
N
tn Mn | N −1 )dx →
wN −1 N
r as n → ∞.
N
αN N −1
,
Note also that, by (18), tN
n ≥ ( α0 )
N
|x|≤r
=
exp(α0 | tn Mn | N −1 )dx ≥
r
|x|≤ n
r
|x|≤ n
exp(αN | Mn |
N
N −1
)dx +
N
exp(αN | Mn | N −1 )dx =
N
|x|≤r
exp(αN | Mn | N −1 )dx
N
r
≤|x|≤r
n
r
|x|≤ n
exp(αN | Mn | N −1 )dx,
−
1
N −1
exp[αN ωN −1
log n + dn αN ]
ωN −1 rN
exp[N log n] exp[dn αN ]
N nN
ωN −1 rN
ωN −1 rN
exp[dn αN ] ≥
,
=
N
N
=
312
JOÃO MARCOS B. DO Ó
since dn ≥ 0. Using the change of variable τ = log rs /(ζn log n), with
n E , we have, by straightforward computation,
ζn = M
N
r/n≤|x|≤r
exp(αN | Mn | N −1 )dx
N
= wN −1 r ζn log n
ζn−1
0
N
exp[N log n(τ N −1 − ζn τ )]dτ → wN −1 rN as n → ∞
Finally, passing to limits, using (21) and the latter fact we obtain
(
αN N −1
ωN −1 rN
)
≥ (β0 − ε)
{exp[d0 αN ] − 1}
α0
N
+(β0 − ε)wN −1 rN ,
where d0 = lim inf n→∞ dn is a nonnegative number. Thus,
αN N −1
)
≥ (β0 − ε)wN −1 rN ,
(
α0
which implies
1 N N −1
,
β0 ≤ N
r
α0
contradicting (16).
4. Proof of Theorem 1.1
In view of lemmas 2 and 3 we can apply the Mountain-Pass Theorem to
obtain a sequence (un ) ⊂ E such that I(un ) → c > 0 and I (un ) → 0, that
is,
1
N
(22)
F (x, un )dx → c, as n → ∞,
u n E −
N
RN
(23)
|
RN
N −2
| ∇un |
N −2
∇un ∇v − a(x) | un |
un v −
RN
f (x, un )v |
≤ εn v E ,
for all v ∈ E, where εn → 0 as n → ∞. Furthermore, by lemma 4, the level
c is less than N1 ( ααN0 )N −1 ). From now on, we shall be working in order to
prove that (un ) converges to a weak nontrivial solution u of problem (1).
From (22), (23) and (f2 ),
µ
(µF (x, un ) − f (x, un )un )dx
C + εn un E ≥ ( − 1) un E −
N
RN
µ
≥ ( − 1) un N
E,
N
which implies that
un E ≤ C,
RN
f (x, un )un dx ≤ C,
RN
F (x, un )dx ≤ C.
Now, using the same argument as in Proposition 2.1 of [15], in view of
Sobolev’s Theorem together with conditions (a1 ) − (a2 ), the Banach space
E is compactly immersed in Lq if N ≤ q < ∞, (cf. [3]). Therefore, up to
N -LAPLACIAN EQUATIONS
313
subsequences, we have un : u weakly in E, un → u in Lq (RN ), ∀q ≥ N
and un (x) → u(x) almost everywhere in RN . Moreover, arguing as in lemma
4 of [19], we get


f (x, un ) → f (x, u) in L1 (B(0, R)),


| ∇un |N −2 ∇un →| ∇u |N −2 ∇u
(24)


weakly in LN/(N −1) (B(0, R)) N ,
for all R > 0. Therefore, by (23), passing to the limit,
RN
| ∇u |N −2 ∇u∇ϕ − a(x) | u |N −2 uϕ dx =
RN
f (x, u)ϕdx
for all ϕ ∈ C0∞ (RN ), that is, u is a weak solution of (1). Let us show that u
is nontrivial. Assume, by contradiction, that u ≡ 0. Using the Generalized
Lebesgue Dominated Convergence Theorem (cf. [20]), by (f3 ) and the first
result in (24), we conclude that F (x, un ) → 0 in L1 (B(0, R)), for all R > 0.
Thus, using (11), in view of Radial Lemma, we obtain F (x, un ) → 0 in
L1 (RN ). This together with (22) imply
(25)
lim
n→∞ RN
| ∇un |N = N c,
and hence given < > 0, we have ∇un N
≤ N c + <, for large n. Using c <
LN
1 αN N −1
(
)
and
choosing
q
>
1
sufficiently
close
to 1 and < sufficiently small,
N α0
N
we obtain qα0 ∇un LNN−1 < αN . Hence, by the same kind of argument as
it has been done in the proof of lemma 1, we conclude that
RN
N
exp α | un | N −1 − SN −2 (α, un ) ≤ C, ∀n,
which, in combination with the Hölder inequality and (f1 ), implies that
lim
n→∞ RN
| f (x, un ) |q dx = 0.
Therefore, from (23) with v = un , we achieve
lim
n→∞ RN
| ∇un |N = 0,
which contradicts (25) since c > 0. Thus, u is nontrivial and the proof of our
main result is complete.
References
[1] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with
critical growth for the N −Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17
(1990), 393–413.
[2] Adimurthi and S. L. Yadava, Multiplicity results for semilinear elliptic equations in
bounded domain of R2 involving critical exponent, Ann. Scuola Norm. Sup. Pisa Cl.
Sci. (4) 17 (1990), 481–504.
[3] C. O. Alves, Existência de Solução Positiva de Equações Elı́pticas Não-Lineares Variacionais em RN , Doct. Dissert., UnB, 1996.
314
JOÃO MARCOS B. DO Ó
[4] C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, Radially Symmetry Solutions
for a Class of Critical Exponent Elliptic Problems in RN , Electron. J. Differential
Equations, 7 (1996), 1–12.
[5] F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J.
Differetnial Equations, 70 (1987), 349–365.
[6] F. V. Atkinson and L. A. Peletier, Ground states and Dirichlet problems for −∆u =
f (u) in R2 Arch. Rational Mech. Anal. 96 (1986), 147-165.
[7] T. Aubin, Nonlinear Analysis on Manifolds, Monge-Ampère Equations, SpringerVerlag, New York, 1982, pp. 204.
[8] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I. Existence of ground
state, Arch. Rational Mech. Anal. 82 (1983), 313–346.
[9] L. Carleson and A. Chang, On the existence of an extremal function for an inequality
of J. Moser, Bull. Sci. Math. 110 (1986), 113–127.
[10] D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent
in R2 , Comm. Partial Differential Equations, 17 (1992), 407–435.
[11] D. M. Cao and Z. J. Zhang, Eigenfunctions of the nonlinear elliptic equation with
critical exponent in R2 , Acta Math. Sci. (English Ed.) 13 (1993), 74–88.
[12] Pascal Cherrier, Une inégalité de Sobolev sur les variétés Riemanniennes, Bull. Sci.
Math., 2me Série, 103 (1979), 353–374.
[13] Pascal Cherrier, Melleurs constants dans des inegalities relative aux espaces de
Sobolev, Bull. Sci. Math. 2me Série, 108 (1984), 225–262.
[14] Pascal Cherrier, Problèms de Neumann non linéaires sur les variétés riemanniennes,
J. Funct. Anal. 57 (1984), 154–206.
[15] David G. Costa, On a nonlinear elliptic problem in RN , Center for Mathematical
Sciences, Univ. Winconsin, CMS-Tech. Summary Rep. #93-13, 1993.
[16] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3
(1995), 139–153.
[17] D. G. de Figueiredo and B. Ruf, Existence and non-existence of radial solutions for
elliptic equations with critical growth in R2 , Comm. Pure Appl. Math. (to appear).
[18] D. C. de Morais Filho, João Marcos B. do Ó and M. A. S. Souto, A compactness
embedding result and applications to elliptic equations, preprint.
[19] João Marcos B. do Ó, Semilinear Dirichlet problems for the N −Laplacian in RN
with nonlinearities in critical growth range, Differential Integral Equations, 5 (1996),
967–979.
[20] O. Kavian, Introduction à la Théorie des Points Critiques et Applications aux
Problèmes Elliptiques, Springer-Verlag, Paris, 1993.
[21] J. Mawhim and M. Willem, Critical Point Theory and Hamiltonian Systems, SpringerVerlag, Berlin, 1989.
[22] J. L. McLeod and K. B. McLeod, Critical Sobolev exponents in two dimensions, Proc.
Roy. Soc. Edinburgh A, 109 (1988), 1–15.
[23] J. B. McLeod and L. A. Peletier, Observations on Moser’s Inequality, Arch. Rational
Mech. Anal. 106 (1989), 261–285.
[24] O. H. Miyagaki, On a class of semilinear elliptic problems in RN with critical growth,
Univ. Winconsin, CMS-Tech. Summary Rep. #94-112, 1994.
[25] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20
(1971), 1077–1092.
[26] R. Panda, On semilinear Neumann problems with critical growth for the n−Laplacian,
Nonlinear Anal. 26 (1996), 1347–1366.
[27] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton
Univ. Press, Princeton, 1951.
[28] P. H. Rabinowitz, Minimax Methods in Critical Point Theory With Applications to
Differential Equations, 65th CBMS Regional Conf. Math., 1986, pp. 100.
N -LAPLACIAN EQUATIONS
315
[29] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math.
Phys. 43 (1992), 270–291.
[30] M.C. Shaw, Eigenfunctions of the nonlinear equation ∆u + νf (x, u) = 0 in R2 , Pacific
J. Math. 129 (1987), 349–356.
[31] N. S. Trudinger, On the imbedding into Orlicz spaces and some applications, J. Math.
Mech. 17 (1967), 473–484.
Departamento de Matemática
Universidade Federal da Paraı́ba
58059.900 João Pessoa, Pb BRAZIL
E-mail address: jmbo@mat.ufpb.br