Differential and Integral Equations
Volume 13 (4–6) April–June 2000, pp. 595–612
SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION
WITH CRITICAL SOBOLEV EXPONENT AND
PERTURBATIONS ON RN
Huan-Song Zhou
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences
P.O. Box 71010, Wuhan 430071, P. R. China
(Submitted by: G. Da Prato)
Abstract. We consider the following quasilinear elliptic problem
−div |∇u|p−2 ∇u) + c|u|p−2 u = |u|p
u∈
where c > 0, p∗ =
W 1,p (RN ),
Np
,
N −p
h(x) ∈ W
∗
−2 u
+ f (x, u) + h(x)
(*)
N > p ≥ 2,
p
−1, p−1
(RN ) (i.e., the dual space of W 1,p (RN )), f (x, 0) =
0 and f (x, u) is a lower-order perturbation of |u|p
∗
−2 u
in the sense that limu→∞
f (x,u)
∗
|u|p −2 u
=
0. It is well known that (*) has only a trivial solution if f (x, u) ≡ h(x) ≡ 0 by a Pohozaev
type identity, but (*) has a nontrivial solution if there is a subcritical perturbation, e.g.,
h(x) ≡ 0 and f (x, u) ≡ 0. In this paper, we prove that (*) has at least two distinct solutions
if there are two perturbations, i.e., f (x, u) ≡ 0 and h(x) ≡ 0 (inhomogeneous term) with
h small enough.
1. Introduction and the main results. In this paper, we study the existence
of solutions to the following quasilinear elliptic problem
∗
−div |∇u|p−2 ∇u) + c|u|p−2 u = |u|p
u ∈ W 1,p (RN ),
−2
N > p > 2,
u + f (x, u) + h(x)
(1.1)
p
p
where c > 0, p∗ = NN−p
, h(x) ∈ W −1,p (RN ) the dual space of W 1,p (RN ), p = p−1
),
∗
f (x, 0) = 0 and f (x, u) is a lower-order perturbation of |u|p −2 u in the sense that
lim
u→∞
f (x, u)
= 0.
|u|p∗ −2 u
This kind of problem occurs in many branches, see e.g., [2] [15] and the references
therein.
Received for publication October 1998.
AMS Subject Classifications: 35J60, 35J65.
595
596
HUAN-SONG ZHOU
If f (x, u) ≡ h(x) ≡ 0, the existence and nonexistence results for problem (1.1)
on bounded domains was studied by [9, 11] and [5] (for p = 2), etc. For the entire
space RN , by using the Gauss formula and the same ideas as in [11] and [1], we can
get the following Pohozaev type identity
∗
cp∗
|∇u|p dx =
|u|p dx −
|u|p dx.
(1.2)
p
N
N
N
R
R
R
On the other hand, if u is a solution to (1.1) with f (x, u) ≡ h(x) ≡ 0 we mean that
∗
|∇u|p dx =
|u|p dx − c
|u|p dx.
(1.3)
RN
RN
RN
So, it is easy to see from (1.2) and (1.3) that problem (1.1) has no nontrivial solution
if c > 0 and f (x, u) ≡ h(x) ≡ 0.
However, if h(x) ≡ 0 but f (x, u) ≡ 0, by using concentration-compactness principle (see [13], [14]), [22] proved that (1.1) has a nontrivial solution. A natural
question is whether (1.1) has more than one solutions if f (x, u) ≡ 0 and h(x) ≡ 0?
Recently, there are many results related to this kind of problems. For example, Li in
[12] proved that problem (1.1) with two perturbations has always a week solution.
In [7], problem (1.1) in a bounded domain was studied and the existence of two
solutions has obtained. Two solutions of the subcritical case of (1.1) on RN has
also been considered in [6], but in which
f (x, u) =
m
Qi (x)|u|qi −2 u.
i=1
There are also many works on (1.1) with p = 2, see, e.g., [15, 2, 8, 19] and the
references therein.
The aim of this paper is to use the variational methods to show that the more
general problem (1.1) has also at least distinct two solutions. The main ideas of
this paper are due to [6]. But here we encounter serious difficulties caused by the
Sobolev exponent and the subcritical perturbation f (x, u) is not a polynomial.
It is well known that the solutions of (1.1) are the critical points of the following
variational functional defined on W 1,p (RN ):
∗
1
1
I(u) =
|u|p − F (x, u) − hu,
(1.4)
(|∇u|p + c|u|p ) − ∗
p
p
u
where F (x, u) = 0 f (x, u)ds and from now on we omit “dx” and “RN ” in all
integration if there is no other indications.
Now we give a sketch of how to look for two distinct critical points of (1.4).
First, we consider a minimization of I(u) constrained in a neighborhood of zero, by
SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION
597
using Ekeland’s variational principle [10], we can find a critical point of I(u) which
achieves its level (i.e., the local minimum of I(u)), moreover, this local minimum
is negative, see Theorem 1.1. Next, around “zero”, by the mountain pass theorem
without (PS) condition [5], we can also get a critical point of I(u) and its level is
positive, (see Theorem 1.4). Because these two critical points are in different levels,
they must be distinct if both of them achieve their levels. Unfortunately, it seems
hard to prove that the critical point obtained by the mountain pass theorem [5]
achieves its level. In this paper, by establishing some new comparisons between the
levels of the above two critical points and then we can prove the two solutions are
distinct.
Throughout this paper, we denote the dual space of W 1,p (RN ) by W −1,p (RN )
and its norm by · ∗ ; the norms of Lp (RN ) and W 1,p (RN ) are denoted by · p
and · respectively, that is,
1
up = [
|u|p dx] p , if u ∈ Lp (RN );
RN
u = [
1
(|∇u|p + c|u|p )dx] p , if u ∈ W 1,p (RN ).
RN
The conditions posed on f (x, t) are as follows:
(f 1) f (x, t) ∈ C(RN × R).
is continuous with respect to t ∈ R1 .
(f 2) ∂(f (x,t)t)
∂t
(f 3)
f (x, t)
lim
= 0 uniformly in x ∈ RN ;
|t|→0 |t|p−2 t
f (x, t)
= 0 uniformly in x ∈ RN .
∗
|t|→∞ |t|p −2 t
lim
(f 4) There exists θ > 0 if N ≥ p2 ; or θ >
all x ∈ RN , t ≥ 0,
p(p2 −N )
(N −p)(p−1)
0 ≤ (p + θ)F (x, t) ≡ (p + θ)
if p < N < p2 such that for
t
f (x, s)ds ≤ tf (x, t).
0
(f 5) lim|x|→+∞ f (x, t) = f¯(t) uniformly for |t| bounded; f (x, t) ≥ f¯(t) ≥ 0,
¯
(t)
∀x ∈ RN and |t|fp−2
t is nondecreasing in t ≥ 0.
N
(f 6) meas {x ∈ R : f (x, t) > f¯(t)} > 0.
Corresponding to (1.4), for u ∈ W 1,p (RN ) we define that
∗
1
1
I ∞ (u) =
|u|p − F̄ (u);
(|∇u|p + c|u|p ) − ∗
p
p
(1.5)
598
HUAN-SONG ZHOU
1
1
p
p
p∗
|u| − F (x, u),
I0 (u) =
(|∇u| + c|u| ) − ∗
p
p
u
u
where F̄ (u) = 0 f¯(s)ds, F (x, u) = 0 f (x, s)ds.
Furthermore, we set
Λ = {u ∈ W 1,p (RN ) \ {0} : < I ∞ (u), u >= 0},
inf{I ∞ (u) : u ∈ Λ}, if Λ = φ
J∞ =
+∞,
if Λ = φ.
(1.6)
(1.7)
(1.8)
Remark 1.1. By a result of [22], if (f 1)–(f 5) hold, then I ∞ satisfies (P S)c conN
N
dition for any c ∈ (0, min{J ∞ , N1 S p }). Moreover, if J ∞ < N1 S p , then J ∞ is
achieved by some ū ∈ W 1,p (RN ) such that
J ∞ = I ∞ (ū) = sup I ∞ (tū),
(1.9)
t≥0
where S = inf{∇upp : u ∈ W 1,p (RN ), up∗ = 1}.
Our main results are that:
Theorem 1.1. If (f 1), (f 3) hold, then there exist R > 0 (small), C = C(R, N, p, f )
> 0 and u1 ∈ W 1,p (RN ) such that for any h ∈ W −1,p (RN ) with h∗ < C we have
∆
I(u1 ) = c1 := c1 (R)= inf I(u),
(1.10)
u∈B R
where B R = {u ∈ W 1,p (RN ) : u ≤ R} and I(u) is defined by (1.4). Furthermore,
u1 is a solution of (1.1) and c1 ≤ 0, c1 → 0 as h∗ → 0.
This theorem was essentially shown in [12], but in [12] we do not know if u1
achieves the local minimum c1 . In this paper we will prove it in Section 1.
Theorem 1.2. Let un ⊂ W 1,p (RN ) be a bounded (P S)c sequence of I(u) and for
n
some u ∈ W 1,p (RN ), un u weakly in W 1,p (RN ), then u is a weak solution of
(1.1). Moreover, either un → u in W 1,p (RN ) i.e. Ig (u0 ) = c; or, c ≥ I(u) + J ∞ .
This theorem is essentially a weaker version of the global compactness results,
see[17] [20] for bounded domains, [3] [21] for p = 2.
Theorem 1.3. Let I and I0 be given by (1.4) (1.6), respectively. Then if (f 1)−(f 6)
hold, there is a v0 ∈ W 1,p (RN ) such that
i)
1 N
sup I0 (tv0 ) < min{J ∞ , S p }.
(1.11)
N
t≥0
ii) There exist t̄ > 0 such that
I0 (tv0 ) < 0 if t ≥ t̄; I(tv0 ) < 0 if t ≥ t̄ and h∗ ≤ 1.
(1.12)
SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION
599
Furthermore, if we denote
Γ = {γ ∈ C([0, 1], W 1,p (RN )) : γ(0) = 0, γ(1) = t̄v0 },
(1.13)
c2 = inf max I(u).
(1.14)
γ∈Γ u∈γ
Thus, there is a positive constant C such that
c2 < c1 + min{J ∞ ,
1 Np
S },
N
(1.15)
if h∗ ≤ C, where c1 < 0 is given by (1.10)
Theorem 1.4. If (f 1)–(f 6) are satisfied and h(x) ∈ W −1,p (RN ), then, there is
a positive constant C = C(N, P, f ) such that (1.1) possesses at least two distinct
solutions for all h(x) ∈ W −1,p (RN ) with h ≤ C.
2. Some Lemmas and the Proof of Theorem 1.1. The first lemma is a
generalization of Brezis-Lieb Lemma [4], it was shown in [21] for p = 2 and in [6]
for general p > 1, by using the mean theorem and Strauss Lemma [16]
Lemma 2.1. Let {un } ⊂ W 1,p (RN ) be a bounded sequence such that for some
u0 ∈ W 1,p (RN ),
n
n
un u0 weakly in W 1,p (RN ), un → u0 a.e. in RN
and f (x, t) satisfies (f 1) and
f (x, t)
| ≤ M for some M > 0
t→0 tp−1
f (x, t)
lim | | = 0 uniformly in x ∈ RN , p < ≤ p∗ .
t
|t|→∞
lim |
Then,
i)
lim [
F (x, un )dx −
F (x, u0 )dx −
RN
u
where F (x, u) = 0 f (x, s)ds.
ii) If
n→∞
RN
F (x, un − u0 )dx] = 0,
RN
d
(f (x, t)t)
| ≤ M for some M > 0;
tp−1
d
(f (x, t)t)
lim | dt p∗ −1 | = 0 uniformly in x ∈ RN .
t
|t|→∞
lim | dt
t→0
Then
lim [
n→∞
RN
f (x, un )un dx −
f (x, u0 )u0 dx −
RN
f (x, un − u0 )(un − u0 )dx] = 0.
RN
600
HUAN-SONG ZHOU
Lemma 2.2. Assume (f1 )–(f3 ), (f5 ) hold and{vn } ⊂ W 1,p (RN ) is a sequence with
vn 0 weakly in W 1,p (RN ).
(2.1)
Then,
lim [I0 (vn ) − I ∞ (vn )] = 0;
(2.2)
n→∞
lim [I0 (vn ), vn n→∞
− I
∞
(vn ), vn ] = 0.
(2.3)
Proof. This lemma is essentially due to [6]. For the sake of completeness, we give
the proof here. By the definitions of (1.5) and (1.6), we see that
∞
I0 (vn ) = I (vn ) + [F̄ (vn ) − F (x, vn )].
(2.4)
It is clear that to prove (2.2) we need only show that
lim
|F (vn ) − F (x, vn )|dx = 0.
(2.5)
n→∞
In fact, from (2.1) and Sobolev’s embedding, we may assume that
vn → 0 strongly in Lqloc (RN ), for p ≤ q < p∗ .
n
(2.6)
Then, for any δ > 0, R > 0,
|F (x, vn ) − F̄ (vn )| ≤
|F (x, vn )| +
|F̄ (vn )|
RN
BR
BR
+[
+
+
]|F (x, vn ) − F̄ (vn )|.
|x|≥R
n |<δ −1
|x|≥R
{ |v
}
|<δ
{ δ≤|v
n
However,
}
{ |v
|x|≥R
n |≥δ −1
}
|x|≥R
{ |v
}
|<δ
|F (x, vn ) − F̄ (vn )| ≤ ε1 (δ)
0≤t<δ
|x|≥R
|F (x,t)−F̄ (t)|
|t|p
|x|≥R
{ δ≤|v |<δ−1
n
where ε(R) =
sup
|x|≥R
δ≤|t|<δ −1
}
|F (x, vn ) − F̄ (vn )| ≤ ε(R)δ −p
|vn |p ,
(2.9)
RN
|F (x, t) − F̄ (t)| −→ 0 as R → +∞ and δ > 0 fixed, by (f 5).
|x|≥R
{ |v |≥δ−1
n
|x|≥R
|t|≥δ −1
(2.8)
−→ 0 as δ → 0+ , by (f 3).
where ε2 (δ) = sup
|vn |p ,
RN
n
where ε1 (δ) = sup
(2.7)
}
|F (x, vn ) − F̄ (vn )| ≤ ε2 (δ)
|F (x,t)−F̄ (t)|
|t|p∗
∗
|vn |p ,
(2.10)
RN
−→ 0 as δ → 0, by (f 3).
It follows from (2.7)–(2.10) that (2.5) holds. Thus (2.2) is proved by using (2.4)
and (2.5). By the same way we can prove (2.3).
SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION
601
Lemma 2.3. Let α > 0, β > 0, γ > 0, then
∞ rα dr
i) 0 (1+r
β )γ < +∞, if β − 1 ≤ α < βγ − 1;
ii) For R > 0 fixed, there are K1 > 0, K2 > 0, K3 > 0 (which depend only on
R) such that for ε → 0+ ,
⎧
if α > βγ − 1;
⎪
R
⎨ K1 ,
α
r dr
= K2 |nε| + O(1), if α = βγ − 1;
β γ
⎪
α+1
0 (ε + r )
⎩
K3 ε β −γ ,
if β ≤ α < βγ − 1.
Proof. i) It is obvious by noticing that
∞
∞
rα dr
1 1 rα−β+1 drβ
=
+
rα−βγ dr.
β )γ
β )γ
(1
+
r
β
(1
+
r
0
0
1
ii) If α > βγ − 1, it is easy to see that
R
R
rα dr
1
lim
=
rα−βγ dr =
Rα−βγ+1 = K1 .
1
+
α
−
βγ
ε→0+ 0 (ε + r β )γ
0
If α = βγ − 1, since
0
R
rα dr
=
(ε + rβ )γ
−1
β
Rε
0
ρα
dρ, lim
(1 + ρβ )γ
ε→0+
Rε− β1
0
ρα
dρ
(1+ρβ )γ
Rε− β1
0
and
−1
β
Rε
1
1
dρ = n|1 + Rε− β | + O(1), as ε → 0+ .
1+ρ
0
Then,
= 1,
1
1+ρ dρ
R
rα dr
= K2 | ln ε| + O(1), as ε → 0+ .
(ε + rβ )γ
0
If β − 1 < α < βγ − 1, by the argument of i),
R
0
α+1
rα dr
= ε β −γ
(ε + rβ )γ
−1
β
Rε
0
α+1
ρα dρ
= K3 ε β −γ .
(1 + ρβ )γ
Proof of Theorem 1.1. By using Ekeland’s variational principle and the second
concentration-compactness Lemma [14], we can prove that, see [12] for the details,
there exist {un } ⊂ B̄R0 and u1 ∈ B̄R0 such that
n
I(un ) −→ c1 ,
I (un ) −→ 0 in W −1,p (RN );
n
n
n
un u1 weakly in W 1,p (RN ), un → u1 a.e. in RN ,
n
∇un −→ ∇u1 a.e. in R .
N
(2.11)
(2.12)
(2.13)
602
HUAN-SONG ZHOU
Moreover, u1 is a weak solution of (1.1), that is,
< I (u1 ), ϕ >= 0, ∀ϕ ∈ W 1,p (RN ).
(2.14)
Since, by (2.11) and by (f 4),
I(un ) =
∗
1
1
1
1
[ f (x, un )un − F (x, un )] + ( − ∗ ) |un |p + ( − 1) hun ,
p
p p
p
1
f (x, un )un − F (x, un ) ≥ 0.
p
Then, (2.12) and Fatou’s lemma give that
c1 = lim inf I(un )
n→+∞
1
1
1
1
p∗
≥ [ f (x, u1 )u1 − F (x, u1 )] + ( − ∗ ) |u1 | + ( − 1) hu1 .
p
p p
p
(2.15)
(2.16)
On the other hand, by the definition of c1 and u1 ∈ B̄R0 , then I(u1 ) ≥ c1 , i.e.,
1
I(u1 ) =
p
1
|∇u1 | + c|u1 | − ∗
p
p
p
p∗
|u1 |
−
F (x, u1 ) −
hu1 ≥ c1 .
(2.17)
Taking ϕ = u1 in (2.14) and substitute it into (2.17), then
I(u1 ) =
1
[ f (x, u1 )u1 − F (x, u1 )]
p
∗
1
1
1
+ ( − ∗ ) |u1 |p + ( − 1) hu1 ≥ c1 .
p p
p
(2.18)
Combining (2.16) and (2.18), (1.10) holds.
Since 0 ∈ B̄R0 , c1 ≤ I(0) = 0. By (f 2) and (f 3), we know that for any ε > 0,
there is a Cε > 0 such that
∗
|f (x, t)| ≤ ε|t|p−1 + Cε |t|p
−1
.
(2.19)
Letting ε = 2c , then by (1.4) (2.19) and Sobolev embedding and Young’s inequality,
we have
∗
1
I(u) ≥
up − Cup − h∗ u
2p
(2.20)
p
∗
1
≥ ( − η)up − Cup − Cη h∗p−1 .
2p
SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION
603
1
Now, if we choose η = 4p
and u ∈ B̄R0 for R0 > 0 small enough, it is clear that
I(u) ≥ 0 for u ∈ B̄R0 and h∗ small enough, i.e. c1 ≥ 0 as h∗ small, this implies
that c1 → 0 as h∗ → 0.
3. Proofs of Theorem 1.2 to Theorem 1.4.
Proof of Theorem 1.2. By the assumptions of this theorem we may assume that
there is an u0 ∈ W 1,p (RN ) such that
n
un −→ u0 weakly in W 1,p (RN ),
(3.1)
n
< I (un ), un >−→ 0;
(3.2)
I (un ) −→ c.
(3.3)
n
Then, by the results of [12] (see Section 2),
n
∇un −→ ∇u0 a.e. in RN ;
(3.4)
< I (u0 ), ϕ >= 0, ∀ϕ ∈ W 1,p (RN ).
(3.5)
Letting vn = un − u0 and by (3.1) (3.4) that
n
n
vn 0 weakly in W 1,p (RN ); ∇vn −→ 0 a.e. in RN .
Since (f 1) (f 2), it follows from Lemma 2.1 and the definition of I0 (see (1.6)) that
o(1) + c = I(un ) = I(u0 ) + I(vn ) = I(u0 ) + I0 (vn );
o(1) =< I (un ), un > =< I (u0 ), u0 > + < I (vn ), vn > +o(1)
=< I0 (vn ), vn > +o(1).
(3.6)
(3.7)
Now, there are two cases to be considered.
n
n
Case 1: If vn −→ 0 strongly in W 1,p (RN ), that is, un −→ u0 strongly in
1,p
W (RN ) and I(u0 ) = limn→+∞ I(un ).
n
Case 2: If vn → 0, we need to show that c ≥ I(u0 ) + J ∞ . In fact, in this case,
we may assume that for some > 0,
n
vn −→ .
(3.8)
Since (3.6) (3.7) and Lemma 2.2, we see that
c = I(u0 ) + I ∞ (vn ) + o(1),
<
I0 (vn ), vn
>=< I
∞
(vn ), vn >= o(1),
(3.9)
(3.10)
604
HUAN-SONG ZHOU
Let
µn =< I
∞
|∇vn | + c|vn | −
p
(vn ), vn >=
p∗
|vn |
p
f¯(vn )vn ,
−
(3.11)
and (3.10) implies that µn → 0 as n → +∞. Furthermore, by (2.19), (3.8) and
(3.10) we know that for some α > 0,
lim
n
∗
|vn |p ≥ α > 0.
(3.12)
By (3.9), to prove c ≥ I(u0 ) + J ∞ we need only to show that
I ∞ (vn ) ≥ J ∞ + o(1).
(3.13)
If vn ∈ Λ (See (1.7)), by the definition of J ∞ , (3.13) holds. In general, for t > 0,
since
∞
p
p
p
p∗
p∗
< I (tvn ), tvn >= t
(|∇vn | + c|vn | ) − t
|vn | − f¯(tvn )tvn .
If we can find a sequence {tn } with tn > 0, tn → 1 as n → +∞ and
< I ∞ (tn vn ), tn vn >= 0, i.e. tn vn ∈ Λ.
(3.14)
I ∞ (vn ) = I ∞ (tn vn ) + ◦(1) ≥ J ∞ + ◦(1),
(3.15)
Then,
this gives (3.13).
To prove (3.14), we let
g(t) = t
p
(|∇vn | + c|vn | ) − t
p
p
p∗
p∗
|vn |
−
f¯(tvn )tvn .
(3.16)
Since for any ε ∈ (0, 1), there is S0 > 0 such that
∗
f¯(s) ≥ −ε|s|p −1 ,
∀s ≥ S0 .
Then, for any fixed n and if t > 0 large enough, (3.12) and (3.16) give that
g(t) ≤ tp
∗
(|∇vn |p + c|vn |p ) − tp
∗
∗
|vn |p + εtp
∗
|vn |p < 0.
(3.17)
605
SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION
On the other hand, by (2.19) and (3.16) we see that for any ε ∈ (0, c), there exists
Cε > 0 such that
∗
∗
∗
∗
g(t) ≥ tp (|∇vn |p + c|vn |p ) − tp
|vn |p − εtp |vn |p − Cε tp
|vn |p
∗
∗
ε
≥ tp (1 − )vn p − (1 + Cε )tp vn pp∗
c
∗
∗
ε
p
≥ t (1 − )vn p − S(1 + Cε )tp vn p , by Sobolev inequality
c
> 0, as t → 0.
(3.18)
So, by (3.17) and (3.18), the mean theorem implies that for any fixed n, we can find
a tn > 0 such that g(tn ) = 0, that is,
∗
∗
< I ∞ (tn vn ), tn vn >= tpn (|∇vn |p + c|vn |p ) − tpn
|vn |p − tn f¯(tn vn )vn = 0.
(3.19)
Now, we may assume that
tn −→ a > 0.
(3.20)
and we claim that a = 1. In fact, by (3.19),
p
p
p∗ −p
p∗
1−p
lim
(|∇vn | +c|vn | )−a
|vn | = a
f¯(avn )vn . (3.21)
lim
lim
n→+∞
n→+∞
n→+∞
Combining (3.10) (3.21) and noticing that (f 5),
p∗ −p
p∗
1−p
¯
(1 − a
|vn | = − lim [ f (vn )vn − a
) lim
n→+∞
n→+∞
¯
f (vn )
f¯(avn )
p
−
]|vn | =
= − lim [
n→+∞ |vn |p−2 vn
|avn |p−2 avn
∗
f¯(avn )vn ]
≥ 0,
if a ≥ 1,
≤ 0,
if a ≤ 1,
∗
this implies that (1 − ap −p ) limn→+∞ |vn |p ≡ 0, that is, a = 1 by (3.12). So, by
(3.14) and (3.15) that (3.13) holds, then Case 2 is proved. We turn now to prove Theorem 1.3.
To this end, we first introduce that for any ε > 0,
uε (x) =
bε
N −p
p2
,
(3.22)
∇uε pp = uε pp∗ = S p ,
(3.23)
p
(ε + |x| p−1 )
N −p
p
where b > 0 is a constant. By the result of [18],
∗
N
606
HUAN-SONG ZHOU
where S is given by Remark 1.1.
Next, for R > 0, we choose ϕ(x) ∈ C0∞ such that ϕ(x) ≡ 1 if |x| ≤
if |x| ≥ R and let
ψε (x) = ϕ(x)uε (x),
R
2;
ϕ(x) ≡ 0
(3.24)
then we have the following estimates:
|∇ψε |p dx =
|ϕ∇uε + uε ∇ϕ|p dx
RN
p
=
|∇uε | dx +
[|ϕ∇uε + uε ∇ϕ|p − |∇uε |p ]dx
RN
N
= S p + O(ε
N −p
p
), by (3.23);
∗
∗
|ψε |p dx =
|uε |p dx +
RN
N
p
∗
[|ϕuε |p − |uε |p ]dx
(3.26)
N
p
+ O(ε ), by (3.23)
|ψε | dx =
∗
R
2 ≤|x|≤R
RN
=S
(3.27)
p
RN
(3.25)
R
2 ≤|x|≤R
RN
|uε | dx +
p
|x|≤ R
2
= ωN −1 bε
N −p
p
0
R
2 ≤|x|≤R
R
2
|ϕuε |p dx
rN −1
[ε + r
p
p−1
]N −p
⎧
N −p
N −p
p
+ O(ε p ),
⎪
⎨ K1 ε
= K2 εp−1 |nε| + O(εp−1 ),
⎪
⎩
N −p
K3 εp−1 + O(ε p ),
dr + O(ε
N −p
p
)
if p < N < p2 ;
if N = p2 ;
if N > p2 ≥
√
2;
(3.28)
by Lemma 2.3 ii), where ωN −1 is the measure of a unit ball in RN .
Furthermore, for I0 defined by (1.6), we have the following lemma.
Lemma 3.1. For any ε ∈ (0, 1), let ψε (x) be given by (3.24), if (f 1)–(f 3) hold,
then there exists tε > 0 such that
I0 (tε ψε ) = sup I0 (tψε ).
(3.29)
t≥0
Moreover, there are two positive constants K1 , K2 (independent on ε) such that
0 < K1 ≤ tε ≤ K2 < +∞.
(3.30)
SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION
607
Proof. For any fixed ε > 0, by the definition of I0 , it is easy to see that I0 (tψε ) > 0
for t > 0 small enough and I0 (tψε ) < 0 for t > 0 large enough, so (3.29) holds. By
Sobolev’s inequality, we know that
∗
||tε ψε ||pp∗ ≤ Ctε ψε p = C(∇ψε pp + ψε pp )tpε ,
that is,
∗
tpε
−p
≤C
∇ψε pp + ψε pp
∗
ψε pp∗
≤ A < +∞, ∀ε ∈ (0, 1),
(3.31)
by using (3.25) (3.26) and (3.28), where A does not depend on ε.
Now, we may assume that
tε −→ as ε → 0.
(3.32)
Obviously, 0 ≤ < +∞. But we claim that there exists α > 0 such that
≥ α > 0.
In fact, by (3.29),
d
dt |t=tε I0 (tψε )
(3.33)
= 0, that is
(|∇ψε |p + c|ψε |p )
∗
f (x, tε ψε )ψε
p
−p
− tε
− p−1 = 0,
∗
p
|ψε |
|ψε |p∗
tε
by letting ε → 0 in (3.34) and using (3.25)–(3.28) (3.32), we have
f (x, tε ψε )ψε
p∗ −p
1−
≤ lim | p−1 |.
ε→0 tε
|ψε |p∗
(3.34)
(3.35)
On the other hand, by (2.19) with ε = 1, there is C > 0 such that
∗
1 − p
∗
|ψε |p + tp−1
|ψε |p
ε
|
∗
p−1
ε→0
|ψε |p
tε
∗
∗
tpε −p C |ψε |p + |ψε |p
= lim |
|
ε→0
|ψε |p∗
∗
−p
≤ lim |
= Cp
∗
∗
Ctpε
−p
−1
, by using (3.25)–(3.28) (3.32),
∗
i.e. (1 + C)p −p ≥ 1,then p −p ≥ C̄, hence (3.33) holds and (3.30) is proved by
noticing (3.31) (3.33).
Proof of Theorem 1.3. We prove part i) by the following two cases.
Case 1): J ∞ ≤
N
1
p
NS
. In this case, we may choose v0 = ū (ū given by (1.9)).
608
HUAN-SONG ZHOU
In fact, by (f 3) and the definition of I0 (see (1.6)), we see that there is t0 > 0
such that
sup I0 (tū) = I0 (t0 ū).
(3.36)
t≥0
Since (f 6), we may assume that (if necessary, replacing ū(x) by ū(x + y) for an
appropriate y ∈ RN ):
I0 (t0 ū) < I ∞ (t0 ū).
(3.37)
Thus, it follows from (1.9) (3.36) and (3.37) that
sup I0 (tū) = I0 (t0 ū) < I ∞ (t0 ū) ≤ sup I ∞ (tū) = J ∞ ,
t≥0
t≥0
then by using J ∞ <
∞
N
1
p
NS
, we know (1.11) is true.
N
1
p
NS
Case 2): J >
. In this case, we take v0 = tε ψε , where ψε and tε are
given by (3.24) and Lemma 3.1, respectively.
Since (f 5), I0 (tε ψε ) ≤ I ∞ (tε ψε ) and by Lemma 3.1 that
sup I0 (tψε ) = I0 (tε ψε ) ≤ I ∞ (tε ψε ).
t≥0
Then (1.11) is proved if we know that
I ∞ (tε ψε ) <
1 Np
S .
N
(3.38)
We turn now to prove (3.38). By the definition of I ∞ and (3.25)–(3.28),
tp
I (tε ψε ) = ε
p
∞
∗
tp
(|∇ψε | + c|ψε | ) − ε∗
p
p
p
p∗
|ψε |
−
F̄ (tε ψε )
∗
N −p
N
N
tpε
tp
− ε∗ )S p + O(ε p ) + O(ε p ) − F̄ (tε ψε )
p
p
⎧
N −p
N −p
p
+ O(ε p ),
if p < N < p2
⎪
⎨ C1 ε
p−1
p−1
+ C2 ε |nε| + O(ε ), if p2 = N
⎪
√
⎩
N −p
C3 εp−1 + O(ε p ),
if N > p2 ≥ 2
1 N
≤ S p − F̄ (tε ψε )
N
⎧
N −p
N −p
p
+ O(ε p ),
if p < N < p2
⎪
⎨ C1 ε
p−1
p−1
+ C2 ε |nε| + O(ε ), if p2 = N
⎪
√
⎩
N −p
C3 εp−1 + O(ε p ),
if N > p2 ≥ 2.
=(
(3.39)
609
SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION
On the other hand, by (f 4) (f 5), there is C > 0 such that
F̄ (t) ≥ Ctp+θ , if t ≥ 1.
(3.40)
Choosing ε > 0 small enough such that
Rε−
p−1
p
≥ (ε−
p−1
p
K − 1)
p−1
p
,
(3.41)
p
p
where K = b N −p K1N −p , K1 given by (3.30). Then,
btε ε−
(1 + r
N −p p−1
p · p
p
p−1
)
≥ 1 if 0 < r ≤ (ε−
N −p
p
p−1
p
K − 1)
p−1
p
.
(3.42)
By using (3.40) (3.41)) and (3.42) we have
F̄ (tε ψε )dx = ε
p−1
p N
p−1
p
F̄ (
RN
≥ε
−
Rε
p−1
p N
(ε
p−1
p
p−1
K−1) p
btε ε
F̄ (
≥ Cε
(N −p)(p−1)
θ
p2
−
(ε
p−1
p
p
K−1)
p
p−1
N −p 1−p
p · p
(1 + r p−1 )
0
p−1−
N −p 1−p
p · p
(1 + r
0
−
btε ε
N −p
p
p−1
p
)
)rN −1 dr, by (3.41)
rN −1 dr
p
(1 + r p−1 )
0
)rN −1 dr
N −p
p
N −p
p (p+θ)
by (3.40) (3.42).
It follows from (3.43) and Lemma 2.3 i) that
2
−N )
a): If p < N < p2 , θ > (Np(p
−p)(p−1) and α = (p − 1) −
then
ε
− N p−p
−
(ε
p−1
p
K−1)
p−1
p
F̄ (tε ψε ) ≥ Cεα
−
N −p
p
< 0,
rN −1 dr
(1 + r p−1 )
0
,
(N −p)(p−1)
θ
p2
p
RN
(3.43)
N −p
p (p+θ)
(3.44)
−→ +∞, as ε −→ 0;
b): If N = p2 , θ > 0, then
(εp−1 |nε|)−1
F̄ (tε ψε )dx
RN
≥ Cε
(p−1)2
− p
|nε|−1
−
(ε
p−1
p
0
−→ +∞, as ε −→ 0;
K−1)
p−1
p
rN −1 dr
(1 + r
p
p−1
)(p−1)(p+θ)
(3.45)
610
HUAN-SONG ZHOU
c): If N > p2 , θ > 0, then
ε
1−p
F̄ (tε ψε )dx
RN
−
≥ Cε
(N −p)(p−1)
θ
p2
−
(ε
p−1
p
K−1)
p−1
p
rN −1 dr
(1 + r
0
p
p−1
)
(3.46)
N −p
p (p+θ)
−→ +∞, as ε −→ 0.
Combining (3.39) (3.44) (3.45) (3.46)), we see that (3.38) is true, so Case 2) and
then part i) is proved.
Finally, we prove part ii). Indeed, by (1.11) there exists an ε0 > 0 such that
sup I0 (tv0 ) < min{J ∞ ,
t≥0
1 Np
S } − ε0 .
N
For this ε0 > 0, by Theorem 1.1, there is M > 0 such that |c1 | <
Let u ∈ γ0 = {tt̄v0 : 0 ≤ t ≤ 1} and ||h||∗ <
|I(u) − I0 (u)| = |
ε0
2t̄||v0 || ,
(3.47)
ε0
2
then
hu dx| ≤ t̄|
as ||h||∗ ≤ M .
hv0 dx| ≤ t̄||v0 ||||h||∗ <
ε0
.
2
ε0
Taking C = min{M, 2t̄||v
}, then if ||h||∗ ≤ C we have
0 ||
|c1 | <
ε0
ε0
and I(u) < I0 (u) + ,
2
2
∀u ∈ γ0 .
(3.48)
Now by (3.47) (3.48), for c2 given by (1.14), we have
c2 ≤ sup I(u) ≤ sup I0 (u) +
u∈γ0
u∈γ0
ε0
ε0
≤ sup I0 (tv0 ) +
2
2
t≥0
1 N
ε0
< min {J ∞ , S p } −
< J ∞ + c1 .
N
2
this means (1.15), and the proof of Lemma 3.1 is completed.
Proof of Theorem 1.4. Combining Theorem 1.1 and (1.15), there exists M1 > 0
such that (1.1) has a solution u1 ∈ W 1,p (RN ) with c1 = I(u1 ) < 0 and (1.15) holds.
On the other hand, by (2.20) we know that there exist ρ > 0, M2 > 0 such that
I(u)|∂Bρ > 0, if h∗ ≤ M2 .
Then by the definition (1.14), c2 > 0.
SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION
611
Noticing (1.12)–(1.14), it follows from the mountain pass theorem without (P S)
Condition [5], we can find {un } ⊂ W 1,p (RN ) such that
n
I(un ) −→ c2 > 0;
I (un ) −→ 0,
n
and {un } is bounded in W 1,p (RN ) by (f 4). Then we may assume that there is some
u2 ∈ W 1,p (RN ) such that
n
un u2 weakly in W 1,p (RN ).
By Theorem 1.2, we know that u2 is also a weak solution of (1.1) and
either c2 = I(u2 );
or
c2 ≥ I(u2 ) + J ∞ .
Letting ||h||∗ ≤ min{M1 , M2 }, then u1 , u2 are solutions of (1.1), moreover,
c1 = I(u1 ) < 0; c2 = I(u2 ) > 0 or c2 ≥ I(u2 ) + J ∞ .
Thus, it is clear that u1 ≡ u2 by using (1.15).
Acknowledgment. This work was finished while the author visited EPFL-DMA,
he would like to express his sincere gratitude for the support of Swiss NSF and the
hospitality of EPFL-DMA during 1996-1997. This work was supported partially by
NSFC.
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