Existence and multiple solutions for a critical quasilinear
equation with singular potentials∗
Shaowei Chen a
Zhi-Qiang Wang b,c
a
c
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, P. R. China
b
Center for Applied Mathemtics, Tianjin University, Tianjin, 300072, P. R. China
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA
Abstract: We study the following quasilinear elliptic equations
−∆p u + V (x)|u|p−2 u = K(x)|u|q−2 u in RN
where 1 < p < N and q = p(N − ps/b)/(N − p) with constants b and s such that b < p, b 6= 0,
0 < sb < 1. This exponent q behaves like a critical exponent due to the presence of the potentials
even though p < q < p∗ = NpN
the Sobolev critical exponent. The potential functions V and K are
−p
locally bounded functions and satisfy that there exist positive constants L, C1 , C2 , D1 and D2 such
that C1 ≤ |x|b V (x) ≤ C2 and D1 ≤ |x|s K(x) ≤ D2 for |x| ≥ L. We prove that below some energy
threshold, the Palais-Smale condition holds for the functional corresponding to this equation. And we
1−b/p
show that the finite energy solutions of this equation have exponential decay like e−γ|x|
at infinity.
If V has a critical frequency, i.e., V −1 (0) has a non-empty interior, we prove that
−∆p u + λV (x)|u|p−2 u = K(x)|u|q−2 u in RN
has more and more solutions as λ → +∞.
Key words: quasilinear elliptic equation; critical exponent; decaying or coercive potentials; exponential
decay.
2000 Mathematics Subject Classification: 35J20, 35J60
1
Introduction and statement of results
In this paper, we consider the following quasilinear elliptic equation
−∆p u + V (x)|u|p−2 u = K(x)|u|q−2 u in RN , u(x) → 0, as |x| → ∞.
(1.1)
where N ≥ 2, 1 < p < N and ∆p u = div(|∇u|p−2 ∇u). The exponent
q=
p(N − ps
b )
N −p
(1.2)
with the real numbers b and s satisfying
b < p, b 6= 0, 0 <
s
< 1.
b
(1.3)
By this definition, it is easy to see p < q < p∗ := pN/(N − p) with p∗ the Sobolev critical exponent.
Even though q < p∗ we will see that q behaves like a critical exponent due to the effects of the potential
functions V and K which may have singular asymptotic near infinity.
For the potential functions V and K, we assume
∗ E-mail
address: [email protected] (S. Chen) and [email protected] (Z.-Q. Wang).
1
(A1 ). V and K are locally bounded nonnegative functions in RN .
(A2 ). There exist positive constants L, C1 , C2 , D1 and D2 such that, for |x| ≥ L,
C1 ≤ |x|b V (x) ≤ C2 , D1 ≤ |x|s K(x) ≤ D2 .
(1.4)
When 0 < b < p, V and K decay to zero at infinity and when b < 0, V and K are coercive.
Eq.(1.1) arises in various applications, such as superfluidity, plasticity, population genetics, chemical reactor theory, and the study of standing wave solutions of certain nonlinear Schrödinger equations.
Therefore, it has received growing attention in recent years (we referee [6, 9, 12, 20, 21, 25, 28, 29] for
more references therein).
There have been mathematical work for related problems in recent years. When p = 2, b > 0 and
2(N − 2s/b)/(N − 2) < q < 2∗ , Ambrosetti, Felli and Malchiodi [5] proved that (1.1) has a nonzero
solution under the assumptions (A1 ) and (A2 ). In fact, it was proved in [5] that there is a compact
embedding from a weighted Sobolev space into a weighted Lq space, which provides the compactness of
the variational formulation. However, the case q = 2(N − 2s/b)/(N − 2) was left open in [5] since for
q = 2(N − 2s/b)/(N − 2) the compact embedding breaks down in general and in a sense it is a critical
exponent problem as this q appears on the boundary of the embedding range. For general p ∈ (1, N ),
Lyberopoulos proved in [15] (see also [16]) that when b > 0, 1 < p < N and p(N − ps/b)/(N − p) <
q < p∗ , (1.1) has a nonzero solution if V and K satisfy (A1 ) and (A2 ). When V and K are radially
symmetric functions, existence of solutions for (1.1) can be obtained through some compact embedding
theorems of weighted Sobolev spaces. Again the critical case q = p(N − ps/b)/(N − p) was not treated
in [15, 16], We also mention [19, 20, 22, 23, 24, 25] in which cases, certain compact embedding was
proved so the Euler-Lagrange functional corresponding to (1.1) satisfies the Palais-Smale condition and
the nonzero solution of (1.1) can be obtained through the standard critical point theorems.
When b and s satisfy (1.3), q = p(N − ps/b)/(N − p) and V and K satisfy (A1 ) and (A2 ), the
functional corresponding to (1.1) does not satisfy the Palais-Smale condition in general. From this point
of view, q = p(N − ps/b)/(N − p) should be seen as a kind of critical exponent for Eq.(1.1). For the
case p = 2, in [11], through a transformation, Chen obtained an equivalent equation for (1.1) and then,
using the concentration-compactness principle, he got a global compactness theorem for this equivalent
equation. This theorem reveals why the Palais-Smale condition does not holds. Using this theorem, he
obtained a nonzero solution for (1.1) under some additional conditions. However, the method in [11] fails
for the general case p 6= 2, and it seems difficult to find a similar transform for (1.1) as the case p = 2. In
this paper, we develop a new approach, which is unified for all case p > 1, for the existence of solutions
of (1.1). We investigate the compactness first and find that below some energy threshold, the functional
corresponding to (1.1) satisfies the Palais-Smale condition (Theorem 3.3). As a consequence, we provide
a sufficient condition to insure the existence of a minimizer for the infimum (3.1) (Corollary 3.4), which
corresponds to a nonzero solution of (1.1). Here is an example for Eq. (1.1) with V and K satisfying
(A1 ) and (A2 )
−∆p u +
4 + arctan |x| q−2
100 + sin(x1 x42 ) p−2
|u| u =
|u| u in RN .
b
(1 + |x|)
(1 + |x|)s
(1.5)
In the second part of the paper we study the following equation with a parameter λ > 0
−∆p u + λV (x)|u|p−2 u = K(x)|u|q−2 u in RN , |u(x)| → 0, as |x| → ∞
where V has a critical frequency in the sense that the interior of the set Z = V −1 (0) is non-empty. This
is a steep potential well problem with λ controlling the depth of the potential. We prove that for any
m ∈ N, there exists Λm > 0 such that when λ ≥ Λm , this equation has at least m solutions (Theorem
4.2) and for fixed m ∈ N, as λ → ∞, these solutions concentrate on Z (Theorem 4.3). These results
generalize the classical work of Beyon, Bartsch, Pankov and Wang for semilinear Schrödinger equations
(see [7, 8, 10, 13]) to the quasilinear Schrödinger equations. Finally, we prove that the finite energy
1−b/p
solutions for (1.1) have e−γ|x|
decay at infinity, where γ > 0 (Theorem 5.1).
2
Notation. B(a, r) denotes the open ball of radius r and centered at a in RN . For a Banach space E,
we denote the dual space of E by E 0 , and denote the strong and the weak convergences in E by →
and *, respectively. For ϕ ∈ C 1 (E; R), we denote the Fréchet derivative of ϕ at u by ϕ0 (u). The
N
Gateaux derivative of ϕ is denoted by hϕ0 (u), vi, ∀u, v ∈ E. L∞
loc (R ) denotes the space of local bounded
N
N
∞
functions in R . Let Ω be a domain in R . C0 (Ω) denotes the space of smooth functions with compact
supports in Ω.
2
Variational structure for Eq. (1.1)
In this section, we shall prove that under the assumptions (A1 ) and (A2 ), Eq.(1.1) has a variational
structure.
Lemma 2.1. Suppose that p, b, s and q satisfy (1.2) and (1.3) and V and K satisfy (A1 ) and (A2 ). Then
there exists C > 0 such that the following inequality holds
Z
Z
p/q
Z
q
p
K(x)|u| dx
≤C
|∇u| dx +
V (x)|u|p dx , ∀u ∈ C0∞ (RN ).
(2.1)
RN
RN
Proof. Let
RN
χ(x) :=
1, |x| ≤ L
0, |x| > L,
V1 = V + χ and K1 = K + χ. By (A1 ) and (A2 ), there exist positive constants C 00 and C 0 such that for
all x ∈ RN ,
C 0 (1 + |x|)−b ≤ V1 (x) ≤ C 00 (1 + |x|)−b , C 0 (1 + |x|)−s ≤ K1 (x) ≤ C 00 (1 + |x|)−s .
(2.2)
By (2.2), the Hölder and the Sobolev inequalities (see, e.g., [27, Theorem 1.8]), we have, for every u ∈
C0∞ (RN ),
Z
1
(
K1 (x)|u|q dx) q
RN
Z
1
|u|q
≤ C(
dx) q (here we used (2.2))
s
RN (1 + |x|)
ps
Z
ps
1
|u| b
= C(
· |u|q− b dx) q
s
N (1 + |x|)
ZR
Z
∗
s
1
s
|u|p
s
ps
qb (
≤ C(
dx)
|u|p dx) q (1− b ) (here we used (1 − )−1 (q − ) = p∗ )
b
b
b
N (1 + |x|)
N
ZR
ZR
p∗
s
s
|u|p
≤ C(
dx) qb (
|∇u|p dx) qp (1− b ) (here we used the Sobolev inequality)
b
RN (1 + |x|)
RN
Z
Z
ps
1 ps
1
|u|p
p∗
s
ps
p
p · qb (
p ·(1− qb ) (here we used
= C(
|∇u|
dx)
dx)
(1 − ) = 1 − )
b
q
b
bq
N (1 + |x|)
N
ZR
ZR
ps
1 ps
1
·
·(1−
)
qb ,
≤ C(
V1 (x)|u|p dx) p qb (
|∇u|p dx) p
(here we used (2.2))
(2.3)
RN
RN
where C is a positive constant independent of u. It follows that there exists a constant C > 0 such that
Z
Z
Z
Z
(
K|u|q dx)1/q ≤ (
K1 |u|q dx)1/q ≤ C(
|∇u|p dx)1/p + C(
V1 |u|p dx)1/p .
(2.4)
RN
RN
RN
RN
Moreover, by the Sobolev inequality, we have, for any u ∈ C0∞ (RN ),
Z
Z
Z
Z
p∗
p
p
N (1− pp∗ )
p
p∗ p ∗
∗ −p 1− p∗
p
χ(x)|u| dx ≤ (
χ
)
(
|u| ) ≤ CL
(
RN
RN
RN
3
RN
|∇u|p )
(2.5)
where C = C(N, p) > 0 is a constant depend only on N and p. Inequality (2.1) follows from (2.4) and
(2.5) immediately.
2
Let E be the completion of C0∞ (RN ) with respect to the norm
Z
Z
||u||E = (
|∇u|p dx +
V (x)|u|p dx)1/p .
RN
RN
From [1], it is easy to verify that, under this norm, E is a uniformly convex Banach space. By Lemma 2.1,
the functional
Z
1
1
K(x)|u|q dx, u ∈ E
(2.6)
Φ(u) = ||u||pE −
p
q RN
is well defined in E. And it is easy to check that Φ is a C 1 functional and the derivative of Φ is given by
Z
Z
Z
hΦ0 (u), vi =
|∇u|p−2 ∇u∇vdx +
V (x)|u|p−2 uvdx −
K(x)|u|q−2 uvdx, ∀u, v ∈ E.
RN
RN
RN
It follows that the critical points of Φ are weak solutions of (1.1).
3
Palais-Smale condition for the functional Φ
Let
R
S=
RN
inf
u∈E\{0}
R
|∇u|p dx + RN V (x)|u|p dx
R
.
( RN K(x)|u|q dx)p/q
For R ≥ L, let (B(0, R))c = {x ∈ RN | |x| > R} and
R
R
|∇u|p dx + (B(0,R))c V (x)|u|p dx
(B(0,R))c
R
SR :=
inf
.
u∈C0∞ ((B(0,R))c )\{0}
( (B(0,R))c K(x)|u|q dx)p/q
(3.1)
(3.2)
Let
S∞ := lim SR .
(3.3)
R→∞
Lemma 3.1. Suppose that p, b, s and q satisfy (1.2) and (1.3) and V and K satisfy (A1 ) and (A2 ). Then
1− ps
qb
S∞ ≥ C1
where
−p
q
D2
max
n qb
ps
, (1 −
ps −1 o−1
)
· A,
qb
(3.4)
R
ps R
ps
( (B(0,L))c |∇u|p dx) qb ( (B(0,L))c |x|−b |u|p dx)1− qb
R
A=
inf
.
u∈C0∞ ((B(0,L))c )\{0}
( (B(0,L))c |x|−s |u|q dx)p/q
Proof. By assumptions (A1 ) and (A2 ), for |x| ≥ L,
V (x) ≥
C1
D2
, and K(x) ≤
.
b
|x|
|x|s
Together with (3.2) and (3.3), this implies that
R
S∞ ≥ lim
inf
R→∞ u∈C0∞ ((B(0,R))c )\{0}
R
|∇u|p dx + C1 (B(0,R))c |x|−b |u|p dx
R
.
(D2 (B(0,R))c |x|−s |u|q dx)p/q
(B(0,R))c
4
(3.5)
From (2.3), we deduce that for R ≥ L and u ∈ C0∞ ((B(0, R))c ),
Z
Z
qb
ps −1
p
|∇u| dx + (1 − )
C1 |x|−b |u|p dx
ps (B(0,R))c
qb
(B(0,R))c
Z
Z
ps
ps
1− ps
|∇u|p dx) qb (
≥ C1 qb (
|x|−b |u|p dx)1− qb
(B(0,R))c
(B(0,R))c
Z
1− ps
≥ AR C1 qb (
|x|−s |u|q dx)p/q
(3.6)
(B(0,R))c
where
R
ps
ps R
( (B(0,R))c |∇u|p dx) qb ( (B(0,R))c |x|−b |u|p dx)1− qb
R
≥ A.
AR =
inf
u∈C0∞ ((B(0,R))c )\{0}
( (B(0,R))c |x|−s |u|q dx)p/q
From (3.6) and AR ≥ A, we have that, for any R ≥ L and u ∈ C0∞ ((B(0, R))c ),
Z
n qb
o Z
ps
max
, (1 − )−1 · (
|∇u|p dx +
C1 |x|−b |u|p dx)
ps
qb
c
c
(B(0,R))
(B(0,R))
Z
1− ps
qb
≥ AC1
(
|x|−s |u|q dx)p/q .
(B(0,R))c
2
Inequality (3.4) follows from this inequality and (3.5).
Definition 3.2. Let φ ∈ C 1 (E; R). A sequence {un } ⊂ E is called a Palais-Smale sequence at level c
((P S)c sequence for short) for φ, if φ(un ) → c and ||φ0 (un )||E 0 → 0 as n → ∞. φ is called satisfying
(P S)c condition if every (P S)c sequence of φ contains a convergent subsequence in E.
q
q−p
, the functional Φ, defined by (2.6), satisfies the (P S)c condition.
Theorem 3.3. For c < ( p1 − 1q )S∞
Proof. Let {un } ⊂ E be a (P S)c sequence for Φ, i.e., Φ(un ) → c and Φ0 (un ) → 0 in E 0 . Then
q
pqc
||un ||pE = ( − 1)−1 (qΦ(un ) − hΦ0 (un ), un i) =
+ o(1) · ||un ||E ,
p
q−p
where o(1) denotes the infinitesimal depending only on n, i.e., o(1) → 0 as n → ∞. It follows that
sup ||un ||E < +∞.
(3.7)
n
Then
Z
RN
1 1
1
1 1
K(x)|un |q dx = ( − )−1 (Φ(un ) − hΦ0 (un ), un i) = ( − )−1 c + o(1).
p q
p
p q
q
q−p
Together with the assumption c < ( p1 − 1q )S∞
and the fact that limR→∞ SR = S∞ , this implies that
there exist positive constants R0 , δ0 and m0 such that, for n ≥ m0 and R ≥ R0 ,
Z
q−p
−1
1 − SR (
K(x)|un |q dx) q ≥ 2δ0 .
(3.8)
RN
Let η ∈ C ∞ [0, +∞) be such that
0 ≤ η ≤ 1; η(r) = 0, r ∈ [0, 1]; η(r) = 1, r ∈ [2, +∞); 0 ≤ η 0 (r) ≤ 2.
For R > 0 and x ∈ RN , let ηR (x) = η(|x|/R). Then
|∇ηR (x)| ≤ 2/R, ∀x ∈ RN .
5
(3.9)
By (1.4) and (3.9), for R ≥ R0 ,
Z
Z
Z
p
p2
p p
p
p
|∇(ηR un )| dx ≤ C
ηR |∇un | dx + C
|un |p |∇ηR
| dx
N
N
N
R
ZR
Z R
C
≤ C
|∇un |p dx + p
|un |p dx
R R≤|x|≤2R
RN
Z
Z
C
≤ C
|∇un |p dx + p−b
V (x)|un |p dx
R
N
R≤|x|≤2R
R
≤ C||un ||pE . (here we used p − b > 0).
(3.10)
Together with (3.7) and (3.10), this yields
p
sup{||ηR
un ||E | R ≥ R0 , n ≥ 1} < +∞.
p
un i = o(1), i.e.,
Then, by Φ0 (un ) → 0 in E 0 , we obtain hΦ0 (un ), ηR
Z
Z
p
p−2
|∇un | ∇un ∇(ηR un )dx +
V (x)|ηR un |p
RN
RN
Z
p
=
K(x)|un |q−2 un · (ηR
un )dx + o(1).
(3.11)
RN
By the Hölder inequality, for any > 0, there exists C > 0 such that
p−1
p
|un ηR
|∇un |p−2 ∇un ∇ηR | ≤ ηR
|∇un |p + C |un |p |∇ηR |p .
Using this inequality with = δ0 /2p, we get that there exists positive constant C > 0 such that
Z
Z
Z
p
p
p−1
|∇un |p−2 ∇un ∇(ηR
un )dx =
|∇un |p ηR
dx + p
un ηR
|∇un |p−2 ∇un ∇ηR dx
N
N
N
R
R
R
Z
Z
δ0
p p
≥ (1 − )
|∇un | ηR dx − C
|un |p |∇ηR |p dx (3.12)
2 RN
RN
By the mean value theorem, we have
|ηR ∇un |p − |∇(ηR un )|p
= p((1 − θ)|ηR ∇un | + θ|∇(ηR un )|)p−1 (|ηR ∇un | − |∇(ηR un )|)
≥
−p(|ηR ∇un | + θ|un ∇ηR |)p−1 |un ∇ηR |,
(3.13)
where 0 < θ = θ(x) < 1, x ∈ RN . And by the Hölder inequality, for any > 0, there exists C > 0 such
that
(|ηR ∇un | + θ|un ∇ηR |)p−1 |un ∇ηR |
≤
(|ηR ∇un | + θ|un ∇ηR |)p + C |un ∇ηR |p
≤
Cp |ηR ∇un |p + (Cp + C )|un ∇ηR |p
(3.14)
where Cp > 0 depends only on p. Choosing = δ0 /2pcp in (3.14) and combining (3.13) with (3.14), we
get
|ηR ∇un |p ≥
δ0
|∇(ηR un )|p − C|un ∇ηR |p .
2
Combining (3.12) with (3.15) leads to
Z
Z
p
p−2
|∇un | ∇un ∇(ηR un )dx ≥ (1 − δ0 )
RN
C
|∇(ηR un )| dx − p
R
p
RN
(3.15)
Z
|un |p dx,
where C is a positive constant which is independent of n and R. From (1.4) and (3.7), we have
Z
Z
1
C
C
p
|un | dx ≤ p−b
V (x)|un |p dx ≤ p−b .
Rp R≤|x|≤2R
R
R
R≤|x|≤2R
6
(3.16)
R≤|x|≤2R
(3.17)
Combining (3.16) with (3.17) yields that there exists C > 0 which is independent of n and R such that
Z
Z
C
p
p−2
(3.18)
|∇un | ∇un ∇(ηR un )dx ≥ (1 − δ0 )
|∇(ηR un )|p dx − p−b .
R
RN
RN
Together with (3.11), this implies
Z
Z
p
|∇un |p−2 ∇un ∇(ηR
un )dx +
RN
RN
V (x)|ηR un |p ≥ (1 − δ0 )||ηR un ||pE −
C
.
Rp−b
Combining (3.19), (3.11), (3.8) with the following inequality
Z
Z
Z
q−p
p
p
q−2
q q
K(x)|un | un · (ηR un )dx ≤ (
K(x)|un | )
(
K(x)|ηR un |q ) q
N
N
R
|x|≥R
R
Z
q−p
−1
≤ SR
(
K(x)|un |q ) q ||ηR un ||pE ,
(3.19)
(3.20)
|x|≥R
we have that, for R ≥ R0 and n ≥ m0 ,
δ0 ||ηR un ||pE ≤
C
+ o(1).
Rp−b
(3.21)
Together with (3.1), this yields
Z
δ0 S(
p
K(x)|ηR un |q dx) q ≤
RN
C
+ o(1).
Rp−b
(3.22)
Since E is a reflexive Banach space and {||un ||E } is bounded in E, we deduce that there exists u0 ∈ E
such that, up to a subsequence, un * u0 in E. It is easy to verify that u0 is a critical point of Φ.
By (3.22), we have
Z
Z
q
|
K(x)|un | dx −
K(x)|u0 |q dx|
RN
RN
Z
Z
≤ |
K(x)|(1 − ηR )un |q dx −
K(x)|(1 − ηR )u0 |q dx|
N
N
R
Z
Z R
q
+|
K(x)|ηR un | dx| +
K(x)|ηR u0 |q dx
RN
RN
Z
Z
q
≤ |
K(x)|(1 − ηR )un | dx −
K(x)|(1 − ηR )u0 |q dx|
RN
RN
Z
C
+ q (p−b) +
K(x)|ηR u0 |q dx + o(1).
(3.23)
p
N
R
R
Since the imbedding E ,→ Lploc (RN ) is compact, we have
Z
Z
lim |
K(x)|(1 − ηR )un |q dx −
K(x)|(1 − ηR )u0 |q dx| = 0.
n→∞
RN
RN
Then, by (3.23), we obtain that
Z
Z
q
lim sup |
K(x)|un | dx −
n→∞
RN
K(x)|u0 | dx| ≤
RN
q
Together with CR− p (p−b) +
C
q
q
R p (p−b)
Z
+
K(x)|ηR u0 |q dx.
RN
K(x)|ηR u0 |q dx → 0 as R → ∞, this implies
Z
Z
lim sup |
K(x)|un |q dx −
K(x)|u0 |q dx| = 0.
n→∞
R
RN
RN
RN
7
(3.24)
By hΦ0 (un ), un i = o(1) and hΦ0 (u0 ), u0 i = 0, we have
Z
Z
p
p
q
||un ||E =
K(x)|un | dx + o(1), ||u0 ||E =
RN
K(x)|u0 |q dx.
RN
Together with (3.24), we get limn→∞ ||un ||E = ||u0 ||E . Finally, from the fact that E is a uniformly
convex Banach space, we deduce that un → u0 in E. This completes the proof.
2
Theorem 3.4. If S < S∞ , then the infimum S can be achieved.
R
Proof. Let {un } be a minimizing sequence. We assume RN K(x)|un |q dx = 1, n = 1, 2, · · · . Otherwise,
R
we choose un /( RN K(x)|un |q dx)1/q as the minimizing sequence. Then,
Z
p
||un ||E → S,
K(x)|un |q dx = 1, n = 1, 2, · · · .
(3.25)
RN
From the Ekeland variational principle (see, e.g., [27]) and the Lagrange multiplier rule, we have that
Z
Z
Z
|∇un |p−2 ∇un ∇ϕdx +
V (x)|un |p−2 un ϕdx − S
K(x)|un |q−2 un ϕdx → 0
(3.26)
RN
RN
RN
1
holds uniformly for ϕ ∈ E with ||ϕ||E ≤ 1. Let vn = S q−p un . Then, from (3.25) and (3.26), we deduce
q
that {vn } is a (P S)c sequence for Φ with c = ( p1 − 1q )S q−p . Since S < S∞ , Theorem 3.3 implies that
1
there exists v ∈ E such that, up to a subsequence, vn → v in E. It follows that un → u = S − q−p v.
Hence, the infimum S can be achieved by u.
2
4
Eq. (1.1) with a steep potential well and a critical frequency
In this section, we assume that V and K satisfies the additional assumptions
(V). There exists a closed subset Z ⊂ {x ∈ RN | |x| < L} with nonempty interior Ω = intZ such that
V (x) = 0 for x ∈ Z.
(K). inf |x|≤L K(x) > 0.
Consider the equation
−∆p u + λV (x)|u|p−2 u = K(x)|u|q−2 u,
u ∈ Eλ
(4.1)
where λ is a positive parameter and Eλ is the Banach space E defined in Section 2 with V replaced by
λV . The norm of Eλ is denoted by || · ||λ . The functional corresponding to this equation is denoted by Φλ
λ
which is the functional defined by (2.6) with ||u||pE replaced by ||u||pλ . Let Sλ and SR
be S and SR with
λ
λ
V replaced by λV and let S∞ = limR→∞ SR . From (3.4), we have
n qb
o−1
ps
−p
ps
λ
, (1 − )−1
· A.
S∞
≥ (λC1 )1− qb D2 q max
ps
qb
This implies
λ
lim S∞
= +∞.
λ→+∞
(4.2)
On the other hand, choose a function v0 ∈ C0∞ (RN ) satisfying v0 6= 0 and the support of v0 is contained
in Z. Then by (V) and (K), we get that, for every λ > 0,
R
p
N |∇v0 | dx
Sλ ≤ R R
.
( RN K(x)|v0 |q dx)p/q
Together with (4.2) and Theorem 3.4, this leads to the following corollary
8
Corollary 4.1. Suppose that (A1 ), (A2 ), (V) and (K) hold. Then, there exists Λ∗ > 0, such that, for
λ ≥ Λ∗ , the infimum Sλ can be achieved.
Theorem 4.2. Suppose that (A1 ), (A2 ), (V) and (K) hold. Then, for every integer m there exists
λ
Λm ≥ 1 such that (4.1) with λ ≥ Λm has at least m distinct pairs ±v1λ , · · · , ±vm
of weak solutions.
Moreover, there exist positive constants a, bm (independent of λ) such that
a ≤ ||vjλ ||λ ≤ bm , 1 ≤ j ≤ m
(4.3)
hold for all λ ≥ Λm .
|∇u|p dx < +∞}. By (V), W01,p (Ω) ⊂ Eλ for all λ > 0. Let
Z
Z
1
1
Φ0 (u) =
|∇u|p dx −
K(x)|u|q dx, u ∈ W01,p (Ω).
p Ω
q Ω
Proof. Let W01,p (Ω) = {u |
R
Ω
If 0 6= u ∈ Eλ is a critical point of Φλ . By hΦ0λ (u), ui = 0, S ≤ Sλ (λ ≥ 1) and Lemma 2.1, we get that,
for λ ≥ 1,
Z
||u||pλ =
−q
RN
q
K(x)|u|p dx ≤ Sλ p ||u||qλ ≤ S − p ||u||qλ .
It follows that for λ ≥ 1,
||u||λ ≥ S q/p(q−p) := a.
(4.4)
Let {en } ⊂ W01,p (Ω) be a linearly independent sequence and Ej := span{e1 , · · · , ej }. By (V),
Φλ (u) = Φ0 (u), ∀u ∈ W01,p (Ω). Then, (K) and q > p imply that there exists an increasing sequence of
positive numbers {Rj } (independent of λ) such that
Φλ (u) ≤ 0, ∀u ∈ Ej , ||u||λ ≥ Rj .
Let Dj = {u | u ∈ Ej , ||u||λ ≤ Rj } and ∂Dj be the boundary of Dj in Ej . Let Σ be the class of closed
symmetric subsets of Eλ and γ(Y ) be the genus of Y for Y ∈ Σ (see [17] for its definition). Define for
j = 1, 2, · · · ,
Gj = {h | h ∈ C(Dj , Eλ ), h is odd, h|∂Dj = id},
e j = {h | h ∈ C(Dj , W 1,p (Ω)), h is odd, h|∂D = id},
G
j
0
Γj = {h(Dk \ Y ) | h ∈ Gk , k ≥ j, Y ∈ Σ, γ(Y ) ≤ k − j},
e j = {h(Dk \ Y ) | h ∈ G
e k , k ≥ j, Y ∈ Σ, γ(Y ) ≤ k − j},
Γ
e j ⊂ Γj for all j. Define for j = 1, 2, · · · ,
where F denotes the closure of a set F. It follows that Γ
cλj = inf sup Φλ (u), c̃j = inf sup Φ0 (u).
B∈Γj u∈B
ej u∈B
B∈Γ
We have
cλ1 ≤ cλ2 ≤ · · · ,
e j ⊂ Γj , we obtain
And by Γ
c̃1 ≤ c̃2 ≤ · · · .
cλj ≤ c̃j , ∀j.
It easy to verify that Φλ and Φ0 satisfy the (I10 ) and (I20 ) assumptions of [17, Theorem 9.12]. By (4.2), for
λ q/(q−p)
any m, there exists Λm > 1 such that for λ ≥ Λm , c̃m ≤ ( p1 − 1q )(S∞
)
. Then, by [17, Theorem
λ
λ
9.12] and Theorem 3.3, for λ ≥ Λm , c1 , · · · , cm are critical values of Φλ . Let vjλ , i = 1, · · · , m be critical
points of Φλ with Φλ (vjλ ) = cλj . Then ||vjλ ||λ = ( p1 − 1q )−1 (cλj )1/p ≤ ( p1 − 1q )−1 (c̃m )1/p := bm . Together
with (4.4), this implies (4.3).
2
9
Theorem 4.3. Suppose (A1 ), (A2 ), (V) and (K) hold. Let un ∈ Eλn , n ∈ N be a non-zero weak
solution of (4.1) for some sequence λn → +∞ and suppose that
sup ||un ||λn < ∞
(4.5)
n∈N
holds. Then there exists 0 6= ū ∈ E such that un → ū along a subsequence strongly in E. Moreover, ū is
a weak solution of the equation
−∆p u = K(x)|u|q−2 u, for x ∈ Ω = intZ
(4.6)
and ū = 0 a.e. in RN \ Z.
Proof. We divide the proof into several steps.
1). From (4.5), we deduce that {un } is bounded in E. Then there exists ū ∈ E such that, up to a
subsequence, un * ū, and by the compact embedding E ,→ Lθloc (RN ) (see for example, [1]), un → ū in
Lθloc (RN ) for all p ≤ θ < p∗ . It follows that ū is a weak solution of (4.6).
2). For k ∈ N, let Ak =R {x ∈ RN | |x|
R ≤ k, V (x) ≥ 1/k}. From the compact embedding
E ,→ Lploc (RN ), we have that Ak |un |p dx → Ak |ū|p dx, n → ∞. Then by
Z
Z
k
||un ||pλn → 0, n → ∞,
|un |p dx ≤ k
V (x)|un |p dx ≤
λ
n
Ak
Ak
R
p
N
we get Ak |ū| dx = 0. Therefore, ū = 0 a.e. in ∪∞
k=1 Ak = R \ Z.
λn
3). From λn → ∞ and (4.2), we have that S∞ → +∞ as n → ∞. Then, following the same
argument as (3.24), we obtain
Z
Z
lim sup |
K(x)|un |q dx −
K(x)|ū|q dx| = 0.
(4.7)
n→∞
||un ||pλn
RN
RN
R
Together with
= RN K(x)|un | dx and (4.4), this implies RN K(x)|ū|q dx ≥ S q/q−p . Hence
ū 6= 0.
4). Since un is a weak solution of (4.1), we have
Z
Z
Z
p−2
p−2
|∇un | ∇un ∇ūdx + λn
V (x)|un | un ūdx =
K(x)|un |q−2 un ūdx.
q
R
RN
RN
RN
N
By ū = 0 a.e. in R \ Z and V = 0 in Z, we get
Z
Z
p−2
|∇un | ∇un ∇ūdx =
RN
K(x)|un |q−2 un ūdx.
RN
Sending n into infinity, we obtain
Z
|∇ū|p dx =
RN
Z
K(x)|ū|q dx.
(4.8)
RN
R
R
5). By λn → +∞, λn RN V (x)|un |p dx ≤ ||un ||pλn and (4.5), we get that RN V (x)|un |p dx → 0.
R
R
Therefore, to prove un → ū in E, it suffices to prove RN |∇un |p dx → RN |∇ū|p dx. By ||un ||pλn =
R
K(x)|un |q dx, we have
RN
Z
Z
|∇un |p dx ≤
RN
Then, (4.7) and (4.8) imply that
Z
Z
lim sup
|∇un |p dx ≤ lim sup
n→∞
n→∞
RN
Together with lim inf n→∞
R
RN
K(x)|un |q dx.
RN
K(x)|un |q dx =
RN
Z
K(x)|ū|q dx =
RN
Z
|∇ū|p dx.
RN
R
|∇un |p dx ≥ RN |∇ū|p dx, this yields
Z
Z
lim
|∇un |p dx =
|∇ū|p dx.
n→∞
RN
RN
2
This completes the proof.
10
5
Exponential decay of solutions of Eq. (1.1)
The main result of this section is the following theorem
Theorem 5.1. Suppose V and K satisfy (A1 ) and (A2 ). If u ∈ E is a weak solution of (1.1), then there
exist C > 0 and γ > 0 such that
1− b
p
|u(x)| ≤ Ce−γ|x|
, x ∈ RN .
(5.1)
To prove this theorem, we need some lemmas.
Lemma 5.2. Suppose V and K satisfy (A1 ) and (A2 ). If u ∈ E is a weak solution of (1.1), then there
exist α > max{|s| + q|b|(N − p)/p2 , p|s|/(q − p)}, C > 0, and R0 > 0 such that for R > R0 ,
Z
K|u|q dx ≤ CR−α .
(5.2)
|x|≥R
∞
Proof. Let φ ∈ C (R) be such that
0 ≤ φ ≤ 1; φ(r) = 0, r ∈ (−∞, 0]; φ(r) = 1, r ∈ [1, +∞); 0 ≤ φ0 (r) ≤ 2.
For R0 > L and n ∈ N, let ηn (x) = φ(|x|/2n R0 ), x ∈ RN . Then supp |∇ηn |, the support of |∇ηn |, is
contained in {x ∈ RN | 2n R0 ≤ |x| ≤ 2n+1 R0 }. Using ηnp u as a test function on (1.1), we obtain
Z
Z
Z
p−2
p
p
|∇u| ∇u∇(ηn u)dx +
V (x)|ηn u| =
K(x)|u|q−2 u · (ηnp u)dx.
(5.3)
RN
RN
RN
As the proof of (3.16), we have
Z
Z
Z
1
C
|∇u|p−2 ∇u∇(ηnp u)dx ≥
|∇(ηn u)|p dx − n 0 p
|u|p dx,
2 RN
(2 R ) 2n R0 ≤|x|≤2n+1 R0
RN
(5.4)
where C > 0 is independentRof n and R0
From u ∈ E, we have RN K(x)|u|q dx < +∞. It follows that there exists R0 > 0 such that, for
0
R > R0 and n ≥ 1,
Z
q−p
1
S −1 (
K(x)|u|q dx) q ≤ .
(5.5)
4
|x|≥2n R0
And the similar proof for (3.20) yields
Z
Z
K(x)|u|q−2 u · (ηnp u)dx ≤ S −1 (
K(x)|u|q )
|x|≥2n R0
RN
Combining (5.3)-(5.6), we obtain
Z
V (x)|u|p dx
Z
≤
|x|≥2n+1 R0
p
q−p
q
||ηn u||pE
(5.6)
Z
|∇(ηn u)| dx +
V (x)|ηn u|p dx
RN
Z
C
|u|p dx.
(2n R0 )p 2n R0 ≤|x|≤2n+1 R0
RN
≤
(5.7)
From (A2 ), there exists C > 0 which is independent of R0 and n, such that for R0 > R0 and n ≥ 1,
Z
Z
|u|p dx ≤ C(2n R0 )b
V (x)|u|p dx.
(5.8)
2n R0 ≤|x|≤2n+1 R0
Let an =
R
|x|≥2n R0
2n R0 ≤|x|≤2n+1 R0
V (x)|u|p dx. By (5.7) and (5.8), we get that
C
an+1
≤ n 0 p−b .
an
(2 R )
11
(5.9)
where C > 0 is independent of n and R0 > R0 .
Let n0 ∈ N be such that
n0 ≥
1
max{|s| + q|b|(N − p)/p2 , p|s|/(q − p)} + 2
p−b
and α = (p − b)(n0 − 1). Then α > max{|s| + q|b|(N − p)/p2 , p|s|/(q − p)}. By (5.9), we obtain
an0 ≤ C (n0 −1) 2−
n0 (n0 −1)
(p−b)
2
(R0 )−α a1 ≤ C(R0 )−α .
where C > 0 is independent of n and R0 > R0 . Combining (5.8), (5.7) with (5.10), we obtain
Z
Z
C
|∇(ηn0 u)|p dx +
V (x)|ηn0 u|p dx ≤
(R0 )α
RN
RN
(5.10)
(5.11)
where C > 0 is independent of n and R0 > R0 . Let R = 2n0 +1 R0 . Then, (5.2) follows from (5.11) and
the inequality (2.1).
2
Lemma 5.3. Suppose that V and K satisfy (A1 ) and (A2 ). If u ∈ E is a weak solution of (1.1), then
there exists C > 0 such that, for |x| > R0 + 2,
|u(x)| ≤ C|x|−(α−s)/q .
(5.12)
where α and R0 are the constants appearing in Lemma 5.2.
Proof. For x ∈ RN , let u± (x) = max{±u(x), 0}. Then, u+ and u− satisfy the inequality
−∆p v + V (x)v p−1 ≤ K(x)v q−1 in RN .
(5.13)
To prove this lemma, it suffices to prove that u± satisfy (5.12).
Take G(s) = sβ if s > 0, and zero otherwise, and put
Z s
G0 (t)p dt = β p spβ−p+1 /(pβ − p + 1).
F (s) =
0
It is easy to verify that
sp−1 F (s) ≤ sp G0 (s)p ≤ β p G(s)p , if β ≥ 1.
(5.14)
Take φ ∈ C ∞ [0, +∞) such that
0 ≤ φ ≤ 1; φ(r) = 1, r ∈ [0, 1]; φ(r) = 0, r ∈ [2, +∞); −2 ≤ φ0 (r) ≤ 0.
For |x0 | > R0 + 2, let η(x) = φ(|x − x0 |), x ∈ RN . Using η p F (v) as a test function on the inequality
(5.13), we obtain
Z
Z
Z
|∇v|p G0 (v)p η p dx +
V |v|p−2 v · η p F (v)dx + p
|∇v|p−2 F (v)η p−1 (∇v∇η)dx
RN
RN
RN
Z
≤
Kv q−1 · η p F (v)dx.
(5.15)
RN
p
p
Using the Hölder inequality: For a, b ≥ 0, ab ≤ a p−1 + p−1 ( p−1
)−(p−1) bp and (5.14), we get that
|∇v|p−2 F (v)η p−1 (∇v∇η)
≤ |∇v|p η p v −1 F (v) + C0 v p−1 F (v)|∇η|p
≤ |∇v|p η p G0 (v)p + C0 β p |∇η|p G(v)p ,
12
p
where C0 = p−1 ( p−1
)−(p−1) . It follows that
Z
≤
≤
|∇v|p−2 F (v)η p−1 (∇v∇η)dx
N
R
Z
Z
|η∇G(v)|p dx + C0 β p
|∇η|p G(v)p dx
RN
RN
Z
Z
p
p
p
|η∇G(v)| dx + C β (
|G(v)|q dx) q .
RN
(5.16)
B(x0 ,2)
where C > 0 depends only on N, and p. Using the similar proof for (3.16), we have
Z
Z
Z
1
p 0
p p
p
|∇v| G (v) η dx ≥
|∇(ηG(v))| dx − C
|G(v)|p |∇η|p dx
2 RN
RN
RN
Z
Z
p
1
|∇(ηG(v))|p dx − C(
|G(v)|q dx) q
≥
2 RN
B(x0 ,2)
where C > 0 depends only on p and N . Moreover, by (5.14), we get that
Z
Z
Kv q−1 · η p F (v)dx =
Kv q−p v p−1 · η p F (v)dx
RN
RN
Z
p
≤ β
K|v|q−p (ηG(v))p dx
supp η
Z
Z
q
q−p
p
q
q−p
q
≤ β (
K
|v| dx)
(
supp η
p
|G(v)|q dx) q .
(5.17)
(5.18)
B(x0 ,2)
Note that |x0 | − 2 > R0 and supp η is contained in {x | |x| > |x0 | − 2}. From (5.2) and the inequality
(2.1), we have that (choose R = |x0 | − 2)
Z
Z
q
K|v| dx ≤
K|v|q dx ≤ C|x0 |−α .
(5.19)
|x|>|x0 |−2
supp η
By (A2 ), there exists C > 0 independent of x0 , such that,
K(x) ≤ C|x0 |−s , ∀x ∈ B(x0 , 2).
Together with (5.19) and the fact that − pq s −
Z
(
q
q−p
q α
K q−p |v|q dx)
q−p
q
(5.20)
< 0, this yields
p
≤ C|x0 |− q s−
q−p
q α
≤C
(5.21)
supp η
where C > 0 is independent of x0 . Then, by (5.18), we obtain
Z
Z
p
Kv q−1 · η p F (v)dx ≤ Cβ p (
|G(v)|q dx) q .
RN
(5.22)
B(x0 ,2)
Choosing = 1/4p in (5.16) and combining (5.15), (5.16), (5.17) with (5.22), we obtain
Z
Z
p
|∇(ηG(v))|p dx ≤ C(1 + β p )(
|G(v)|q dx) q .
RN
B(x0 ,2)
Then, using the Sobolev inequality, we get
Z
Z
p
∗
(
|G(v)|p dx) p∗ ≤ C(1 + β p )(
B(x0 ,1)
B(x0 ,2)
13
p
|G(v)|q dx) q ,
where C > 0 is independent of x0 and β. From this, the standard Moser’s iteration (see [14, Theorem
8.18]) shows that for x ∈ B(x0 , 1/2),
Z
1
v(x) ≤ C(
|v|q dx) q
B(x0 ,2)
where C > 0 is independent of x0 . By (A2 ), K(x) ≥ C|x0 |−s , ∀x ∈ B(x0 , 2). Then by (5.19) and
(5.20), we deduce that
Z
−α+s
1
(
|v|q dx) q ≤ C|x0 | q
B(x0 ,2)
where C > 0 is independent of x0 . The result of this lemma follows form the above two inequalities. 2
Lemma 5.4. Suppose V satisfies (A1 ) and (A2 ). There exists R∗ > L such that, if u ∈ E is a nonnegative and local bounded function satisfies the inequality:
1
−∆p u + V (x)up−1 ≤ 0, |x| > R∗
2
(5.23)
then there exist γ > 0 and C > 0 such that
u(x) ≤ Ce−γ|x|
Proof. For > 0 and r = |x|, let w (r) = e−r
∆p w (|x|)
1− b
p
1− b
p
(5.24)
. Direct computations shows that
N −1 0 = |w0 (r)|p−2 (p − 1)w00 (r) +
w (r)
r
= p B1 r−b e−(p−1)r
where
, |x| > R∗ .
1− b
p
b
+ p−1 B2 r−b−(1− p ) e−(p−1)r
1− b
p
.
b
b
b
B1 = (p − 1)(1 − )p , B2 = (1 − )p−1 (b − − N + 1).
p
p
p
Then by (A2 ), we can choose R∗ > L and a sufficiently small 0 > 0 such that for |x| ≥ R∗ ,
1
−∆p w0 + V (x)wp−1
≥ 0.
0
2
(5.25)
Since u is a local bounded function, we can choose λ > 0 such that
ū := λw0 > u for |x| = R∗ .
It follows that (ū − u)− ∈ E. By (5.23) and (5.25), we have
1
−∆p ū + ∆p u + V (ūp−1 − up−1 ) ≥ 0.
2
Multiplying both sides of this inequality by (ū − u)− and integrating in {x | |x| > R∗ }, we obtain
Z
Z
1
p−2
p−2
(|∇ū| ∇ū − |∇u| ∇u)(∇ū − ∇u)dx +
V (x)(|ū|p−2 ū − |u|p−2 u)(ū − u)dx ≤ 0 (5.26)
2 Ω
Ω
where Ω = {x | ū(x) − u(x) < 0, |x| > R∗ }. From [18], there exists c > 0 such that for any x, y ∈ Rm
(m ≥ 1),
(
c|x − y|p , p ≥ 2
(|x|p−2 x − |y|p−2 y) · (x − y) ≥
|x−y|2
c (|x|+|y|)
p−2 , 1 < p < 2.
By this inequality and (5.26), we deduce that Ω has zero measure. The result of this lemma follows.
14
2
Proof of Theorem 5.1. Since V and K are local bounded functions and q < p∗ , by the classical regular
N
theory of quasilinear elliptic equation (see for example [26]), we have u ∈ L∞
loc (R ).
By assumption, u satisfies
−∆p u + (V − K|u|q−p )|u|p−2 u = 0 in RN .
From the definition of α (see Lemma 5.2), we have
α−s
(q − p) + s > b.
q
Together with Lemma 5.3, this implies that there exists R∗∗ > R∗ such that V − K|u|q−p > V /2 if
|x| ≥ R∗∗ . It follows that u+ satisfies
1
−∆p u+ + V (x)(u+ )p−1 ≤ 0 if |x| ≥ R∗∗ .
2
1− b
p
Then, Lemma 5.4 implies that there exist C > 0 and γ > 0 such that u+ (x) ≤ Ce−γ|x|
−
if |x| ≥ R∗∗ .
1− b
p
−γ|x|
Similarly, we can prove that u (x) ≤ Ce
if |x| ≥ R∗∗ . The result of this theorem follows from
N
these two inequalities and fact that u ∈ L∞
(R
).
2
loc
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Stud. 6 (2006) 491-509.
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