# A Concentration Phenomenon for p-Laplacian Equation

Authors: Yansheng Zhong
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

It is proved that if the bounded function of coefficientQ
n in the following equation  -
div
⁡
{
|
∇
u
|
p
-
2
∇
u
}
+
V
(
x
)
|
u
|
p
-
2
u
=
Q
n
(
x
)
|
u
|
q
-
2
u
,
u
(
x
)
=
0
as
x
∈
∂
Ω
.
u
(
x
)
⟶
0
as
|
x
|
⟶
∞ is positive in a region contained in Ω and negative outside the region, the sets {
Q
n
>
0
} shrink to a point x
0
∈
Ω as n
→
∞, and then the sequence u
n generated by the nontrivial solution of the same equation, corresponding to Q
n, will concentrate at x
0 with respect to W
0
1
,
p
(
Ω
) and certain L
s
(
Ω
)-norms. In addition, if the sets {
Q
n
>
0
} shrink to finite points, the corresponding ground states {
u
n
} only concentrate at one of these points. These conclusions extend the results proved in  the work of Ackermann and Szulkin (2013) for case p
=
2.


---

## Body

## 1. Introduction

We study a new concentration phenomenon for the following p-Laplacian equations:(1)
-
div
⁡
{
|
∇
u
|
p
-
2
∇
u
}
+
V
(
x
)
|
u
|
p
-
2
u
=
Q
n
(
x
)
|
u
|
q
-
2
u
,
u
(
x
)
=
0
as
x
∈
∂
Ω
.
u
(
x
)
⟶
0
as
|
x
|
⟶
∞
,
where Ω
⊂
R
N is a smooth domain and V
⩾
0
(
∈
L
∞
(
Ω
)
), and p
<
q
<
p
*, where p
*
:
=
N
p
/
(
N
-
p
) if N
⩾
p and p
*
:
=
∞ if N
<
p. If Ω is unbounded, we assume additionally that σ
(
-
div
⁡
(
|
∇
·
|
p
-
2
∇
·
)
+
V
|
·
|
p
-
2
·
)
⊂
(
0
,
∞
).And an assumption ofQ
n is as follows.(
*
) The set {
x
∣
Q
n
(
x
)
>
0
} contained in the neighborhood of zero has positive measure, and |
Q
n
|
L
∞
(
Ω
)
⩽
C with the constant C is independent of n. Moreover, for each ϵ
>
0 there exist constants δ
ϵ
(
>
0
) and N
ϵ such that Q
n
⩽
-
δ
ϵ whenever x
∉
B
ϵ
(
0
) and n
⩾
N
ϵ.As it is known,u
≡
0 is the only solution to (1) if Q
n
(
x
)
⩽
0 for all x
∈
Ω. In addition, if Q
n
(
x
)
>
0 is based on a bounded set of positive measures, it is clear that there exists a solution u
≢
0 (see Theorem 1). Hence, without loss of generality, we assume that 0
∈
Ω and let Q
=
Q
n be such that Q
n
>
0 on the ball B
1
/
n
(
0
) and Q
n
<
0 on Ω
∖
B
2
/
n
(
0
) and u
n
≢
0 are the solutions to (1) associated with Q
n
(
x
). Accordingly, the question is what happens to u
n as n
→
∞. Furthermore, this phenomenon can be found in physics. For instance, considering the materials separately from Q positive or negative (see [1]), it corresponds to investigating the existence of bright (
Q
>
0
) or dark (
Q
<
0
) solitons.Equations of these types have been studied extensively in many monographs and lectures (e.g., [2–10] for p
=
2, [11–18] for general p). In [2], Byeon and Wang considered the standing wave solutions ψ
(
x
,
t
)
≡
exp
(
-
i
E
t
/
ħ
)
v
(
x
) for the nonlinear Schrödinger equation:
(2)
i
ħ
∂
ψ
∂
t
+
ħ
2
2
Δ
ψ
-
V
(
x
)
ψ
+
|
ψ
|
p
-
1
ψ
=
0
,
(
t
,
x
)
∈
R
×
R
N
.
Thus, they needed only to discuss the function v which satisfies
(3)
ħ
2
2
Δ
v
-
(
V
(
x
)
-
E
)
v
+
|
v
|
p
-
1
v
=
0
,
x
∈
R
N
,
and rewrote it in the following form:
(4)
ϵ
2
Δ
v
-
V
(
x
)
v
+
v
p
=
0
,
v
>
0
,
x
∈
R
N
lim
⁡
|
x
|
→
0
v
(
x
)
=
0
.
By a rescaling, it is transformed to
(5)
Δ
u
-
V
(
ϵ
x
)
u
+
u
p
=
0
,
u
>
0
,
x
∈
R
N
lim
⁡
|
x
|
→
0
u
(
x
)
=
0
.Let the zero setZ
≜
{
x
∈
R
N
∣
V
(
x
)
=
0
} and A be an isolated component of Z, and they distinguished three cases of A to prove the concentration as ϵ
→
0. And then, in [3] by replacing v
p with a fairly general class nonlinearity f
(
v
), they also obtained the concentration. Furthermore, in [4], Byeon and Jeanjean gave the almost optimal condition on f for the concentration. Recently, in [19], different from above with the linearity term V
(
ϵ
x
)
u, Ackermann and Szulkin considered the concentration phenomenon in the nonlinearity; that is, -
Δ
u
+
V
(
x
)
u
=
Q
n
(
x
)
|
u
|
p
-
2
u. In contrast, by following the similar strategy as in [19], we first show that the concentration phenomenon also occurs in the general p-Laplacian equation. It seems that this concentration phenomenon was unknown earlier, but to some extent, it answers the question mentioned above.This paper is organized as follows. In Section2, we prove that the solutions {
u
n
} to (1) concentrate at the origin in the W
0
1
,
p
(
Ω
) and the L
q-norm; in Section 3, concentration in the L
s-norms for different s is considered and Section 4 shows that the ground states only concentrate at one of these points when Q
n is positive in a neighbourhood of a finite number of points.


## 2. Concentration in theW
0
1
,
p
(
Ω
) and L
q
(
Ω
)

We begin with some notations.LetE
:
=
W
0
1
,
p
(
Ω
) and
(6)
∥
u
∥
:
=
(
∫
Ω
‍
(
|
∇
u
|
p
+
V
|
u
|
p
)
d
x
)
1
/
p
is an equivalent norm in E (due to σ
(
-
div
⁡
(
|
∇
·
|
p
-
2
∇
·
)
+
V
|
·
|
p
-
2
·
)
⊂
(
0
,
∞
)). Set
(7)
|
u
|
s
,
A
:
=
(
∫
A
‍
|
u
|
s
d
x
)
1
/
s
,
|
u
|
∞
,
A
=
esssu
p
A
|
u
|, and we abbreviate |
u
|
s
,
A to |
u
|
s sometimes. Moreover,
(8)
B
r
(
a
)
:
=
{
x
∈
R
n
:
|
x
-
a
|
<
r
}
denotes a ball.Here we offer the existence result for (1).Theorem 1.
Suppose thatQ
n satisfies the assumption (
*
) above and q
∈
(
p
,
p
*
); then for all sufficiently large n, there is a positive ground state solution u
n
∈
E to problem (1). Moreover, there exists a constant α
>
0 independent of n, such that ∥
u
n
∥
⩾
α.Proof.
As in [19], let J
n
(
v
)
=
∫
Ω
‍
Q
n
|
v
|
q
d
x and
(9)
s
n
:
=
inf
⁡
J
n
(
v
)
>
0
∥
v
∥
p
|
J
n
(
v
)
|
p
/
q
=
inf
⁡
J
n
(
v
)
>
0
∫
Ω
‍
|
∇
v
|
p
+
V
|
v
|
p
d
x
(
∫
Ω
‍
Q
n
|
v
|
q
d
x
)
p
/
q
.
Suppose that (
v
k
) is a minimizing sequence for s
n, normalized by J
n
(
v
k
)
=
1; then ∥
v
k
∥ is bounded. Hence, v
n
⇀
v in E and v
k
(
x
)
→
v
(
x
) a.e. in Ω (by choosing a subsequence). Note that Q
n
<
0 on |
x
|
>
1 for n large. The Rellich-Kondrachov Theorem and Fatous’s Lemma say that
(10)
s
n
=
lim
⁡
k
→
∞
∥
v
k
∥
p
=
lim
⁡
k
→
∞
∥
v
k
∥
p
(
∫
|
x
|
<
1
‍
Q
n
|
v
k
|
q
d
x
+
∫
|
x
|
>
1
‍
Q
n
|
v
k
|
q
d
x
)
p
/
q
⩾
∥
v
∥
p
J
n
(
v
)
p
/
q
⩾
s
n
.
Thus v is a minimizer.
And then, the lagrange multiple rule implies thatu
n
=
c
n
v
n is a solution to (1) for some appropriate constant c
n
>
0. Moreover, since v
n may be replaced by |
v
n
|, v
n
⩾
0 (and hence u
n
⩾
0). To show that u
n
>
0, we note that u
n satisfies
(11)
-
div
⁡
(
|
∇
v
|
p
-
2
∇
v
)
+
(
V
(
x
)
u
n
p
-
2
+
Q
n
-
(
x
)
u
n
(
x
)
q
-
2
)
v
=
Q
n
+
(
x
)
u
n
(
x
)
q
-
1
⩾
0
,
where Q
n
±
:
=
max
⁡
{
±
Q
n
(
x
)
,
0
}. Since V
(
x
)
u
n
p
-
2
+
Q
n
-
(
x
)
u
n
(
x
)
q
-
2
⩾
0, it follows from the strong maximum principle (see [20, 21]) that u
n
>
0.
Ifu
n
≠
0 is a solution to (1), then, via multiplying the equation by u
n, integrating by parts, and using the Sobolev inequality, one deduces that
(12)
∥
u
n
∥
p
=
∫
Ω
‍
Q
n
|
u
n
|
q
d
x
⩽
c
1
|
u
n
|
q
q
⩽
c
2
∥
u
n
∥
q
;
hence, ∥
u
n
∥
⩾
α for some α
>
0 and all large n.The next step is to consider the property of the nontrivial solution{
u
n
} to (1) and w
n
:
=
u
n
/
∥
u
n
∥.Lemma 2.
Consider(13)
∥
u
n
∥
⟶
∞
a
s
n
⟶
∞
.Proof.
We present an abridged version of the proof highlighting the main differences to that in [19]. It will be proved by contradiction. Assume u
n
⇀
u in E and u
n
→
u in L
loc
⁡
q
(
Ω
) after passing to a subsequence. Multiplying (1) (with u
=
u
n) by u
n, integrating by parts, and recalling that Q
n
<
0 for each ϵ
>
0 and n
⩾
N
ϵ, it holds that
(14)
limsup
⁡
n
→
∞
∥
u
n
∥
p
=
limsup
⁡
n
→
∞
∫
Ω
‍
Q
n
|
u
n
|
q
d
x
⩽
limsup
⁡
n
→
∞
∫
|
x
|
<
ϵ
‍
Q
n
|
u
n
|
q
d
x
⩽
c
∫
|
x
|
<
ϵ
‍
|
u
|
q
d
x
.Ifϵ
→
0, u
n
→
0 in E. It is a contradiction to ∥
u
n
∥
⩾
α
>
0 given in Theorem 1.Lemma 3.
Consider(15)
w
n
⇀
0
i
n
E
a
s
n
⟶
∞Proof.
We prove it by contradiction as well. We may assume thatw
n
⇀
w
(
≢
0
) in E. Multiplying (1) (with u
=
u
n) by u
n
/
∥
u
n
∥
p yields that
(16)
1
=
∥
w
n
∥
p
=
∥
u
n
∥
q
-
p
∫
Ω
‍
Q
n
|
w
n
|
q
d
x
.
Due to Lemma 2 with q
>
p, ∫
Ω
‍
Q
n
|
w
n
|
q
→
0.
On the other hand, we have for0
<
ϵ
<
ϵ
1
(17)
0
=
lim
⁡
n
→
∞
∫
Ω
‍
Q
n
|
w
n
|
q
d
x
=
lim
⁡
n
→
∞
(
∫
|
x
|
<
ϵ
‍
Q
n
|
w
n
|
q
d
x
+
∫
|
x
|
>
ϵ
‍
Q
n
|
w
n
|
q
d
x
)
⩽
lim
⁡
n
→
∞
(
∫
|
x
|
<
ϵ
‍
Q
n
|
w
n
|
q
d
x
+
∫
|
x
|
>
ϵ
1
‍
Q
n
|
w
n
|
q
d
x
)
⩽
c
∫
|
x
|
<
ϵ
‍
|
w
n
|
q
d
x
-
δ
ϵ
1
∫
|
x
|
>
ϵ
1
‍
|
w
n
|
q
d
x
.
We may choose small ϵ
1 such that the second integral on the right-hand side above is positive as w
≢
0. Then we get the contradiction as ϵ
→
0.
In the sequel, we study concentration of{
u
n
} as n
→
∞. Let ϵ
>
0 be given and χ
∈
C
∞
(
Ω
,
[
0,1
]
) be such that χ
(
x
)
=
0 for x
∈
B
ϵ
/
2
(
0
) and χ
(
x
)
=
1 for x
∉
B
ϵ
(
0
).
Multiplying (1) (with u
=
u
n) by χ
u
n we obtain
(18)
∫
Ω
‍
(
|
∇
u
n
|
p
-
2
∇
u
n
·
∇
(
χ
u
n
)
+
χ
V
u
n
p
)
d
x
=
∫
Ω
‍
χ
Q
n
|
u
n
|
q
d
x
,
namely,
(19)
∫
Ω
‍
χ
(
|
∇
u
n
|
p
+
V
u
n
p
)
d
x
-
∫
Ω
‍
χ
Q
n
|
u
n
|
q
d
x
=
-
∫
Ω
‍
|
∇
u
n
|
p
-
2
∇
u
n
·
∇
χ
·
u
n
d
x
.
Given ϵ
>
0, we have Q
n
⩽
-
δ
ϵ on suppχ, provided that n is large enough. Hence for all such n,
(20)
0
⩽
∫
Ω
∖
B
ϵ
(
0
)
‍
(
|
∇
u
n
|
p
+
V
u
n
p
)
d
x
+
δ
ϵ
∫
Ω
∖
B
ϵ
(
0
)
‍
|
u
n
|
q
d
x
⩽
∫
Ω
‍
χ
(
|
∇
u
n
|
p
+
V
u
n
p
)
d
x
-
∫
Ω
‍
χ
Q
n
|
u
n
|
q
d
x
=
-
∫
Ω
‍
|
∇
u
n
|
p
-
2
∇
u
n
·
∇
χ
·
u
n
d
x
⩽
d
ϵ
∫
B
ϵ
(
0
)
∖
B
ϵ
/
2
(
0
)
‍
|
u
n
|
|
∇
u
n
|
p
-
1
d
x
,
where d
ϵ is a constant independent of n. Since w
n
=
u
n
/
∥
u
n
∥
→
0 in L
loc
⁡
p
(
Ω
) according to Lemma 3, it follows from Hölder inequality that
(21)
∫
B
ϵ
(
0
)
∖
B
ϵ
/
2
(
0
)
‍
|
w
n
|
|
∇
w
n
|
p
-
1
d
x
⟶
0
.
So (20) implies
(22)
∫
Ω
∖
B
ϵ
(
0
)
‍
(
|
∇
w
n
|
p
+
V
w
n
p
)
d
x
+
δ
ϵ
∥
u
n
∥
q
-
p
∫
Ω
∖
B
ϵ
(
0
)
‍
|
w
n
|
q
d
x
=
0
.Theorem 4.
Suppose thatQ
n satisfies the assumption (
*
) and q
∈
(
p
,
p
*
). Let u
n be a nontrivial solution to (1) and put w
n
=
u
n
/
∥
u
n
∥. Then for every ϵ
>
0 they hold that
(23)
lim
⁡
n
→
∞
∫
Ω
∖
B
ϵ
(
0
)
‍
(
|
∇
w
n
|
p
+
V
w
n
p
)
d
x
=
0
,
(24)
lim
⁡
n
→
∞
∥
u
n
∥
q
-
p
∫
Ω
∖
B
ϵ
(
0
)
‍
|
w
n
|
q
d
x
=
0
.
Moreover,
(25)
lim
⁡
n
→
∞
∫
Ω
∖
B
ϵ
(
0
)
‍
(
|
∇
w
n
|
p
+
V
w
n
p
)
d
x
∫
Ω
‍
(
|
∇
w
n
|
p
+
V
w
n
p
)
d
x
=
0
,
lim
⁡
n
→
∞
∫
Ω
∖
B
ϵ
(
0
)
‍
|
w
n
|
q
d
x
∫
Ω
‍
|
w
n
|
q
d
x
=
0
.Proof.
(23) and (24) can be easily obtained by (22). Note that
(26)
∫
Ω
‍
(
|
∇
w
n
|
p
+
V
w
n
p
)
d
x
=
∥
w
n
∥
p
=
1
.
From (23), one concludes that
(27)
lim
⁡
n
→
∞
∫
Ω
∖
B
ϵ
(
0
)
‍
(
|
∇
u
n
|
p
+
V
u
n
p
)
d
x
∫
Ω
‍
(
|
∇
u
n
|
p
+
V
u
n
p
)
d
x
=
lim
⁡
n
→
∞
∫
Ω
∖
B
ϵ
(
0
)
‍
(
|
∇
w
n
|
p
+
V
w
n
p
)
d
x
∫
Ω
‍
(
|
∇
w
n
|
p
+
V
w
n
p
)
d
x
=
0
.
According to (16), we get
(28)
c
∥
u
n
∥
q
-
p
∫
Ω
‍
|
w
n
|
q
d
x
⩾
∥
u
n
∥
q
-
p
∫
Ω
‍
Q
n
|
w
n
|
q
d
x
=
∥
w
n
∥
p
=
1
.
This and (24) imply
(29)
lim
⁡
n
→
∞
∫
Ω
∖
B
ϵ
(
0
)
‍
|
w
n
|
q
d
x
∫
Ω
‍
|
w
n
|
q
d
x
=
lim
⁡
n
→
∞
∥
u
n
∥
q
-
p
∫
Ω
∖
B
ϵ
(
0
)
‍
|
w
n
|
q
d
x
∥
u
n
∥
q
-
p
∫
Ω
‍
|
w
n
|
q
d
x
=
0
.


## 3. Concentration in theL
s-Norm

The next is to consider the concentration in other norms.Theorem 5.
Letu
n denote a nontrivial solution to (1) for each n
∈
N. Suppose that the assumption (
*
) holds and there exists R, λ
>
0 such that V
⩾
λ whenever x
∈
Ω
∖
B
R
(
0
), and there exists ϵ
>
0 such that B
ϵ
(
0
)
¯
⊂
Ω; then one can get that(a)
∃
C, for all s
∈
[
1
,
∞
], n
∈
N, |
u
n
|
s
,
Ω
∖
B
ϵ
(
0
)
⩽
C;
(b)
ifδ
=
δ
ϵ
>
0 in (
*
) can be chosen independently of ϵ
(
>
0
), then lim
⁡
n
→
∞
|
u
n
|
s
,
Ω
∖
B
ϵ
(
0
)
=
0, for every s
∈
[
1
,
∞
];
(c)
for alls
(
⩾
1
)
∈
(
N
(
q
-
p
)
/
p
,
∞
], one has lim
⁡
n
→
∞
|
u
n
|
s
=
∞ and
(30)
lim
⁡
n
→
∞
|
u
n
|
s
,
Ω
∖
B
ϵ
(
0
)
|
u
n
|
s
=
0
;
(d)
ifN
(
q
-
p
)
/
p
⩾
1, then for s
=
N
(
q
-
p
)
/
p it holds that
(31)
liminf
⁡
n
→
∞
|
u
n
|
s
>
0
.
If the hypotheses in (b) are satisfied, then (30) also holds for this
s
.Proof.
There is clearly a positive classical solutionw to the equation
(32)
-
div
⁡
(
|
∇
u
|
p
-
2
∇
u
)
=
-
δ
ϵ
/
2
|
u
|
q
-
2
u
,
x
∈
R
n
∖
B
ϵ
/
2
(
0
)
¯
lim
⁡
|
x
|
→
ϵ
/
2
w
(
x
)
=
∞
,
lim
⁡
|
x
|
→
∞
w
(
x
)
=
0
.
In fact, by [22, 23], the radial solution u
p
(
x
)
=
u
p
(
|
x
|
) satisfies the ordinary differential equation
(33)
(
r
n
-
1
|
u
′
|
p
-
2
u
′
)
′
=
-
δ
ϵ
/
2
r
n
-
1
u
q
u
(
r
)
=
∞
as
r
⟶
ϵ
/
2
,
u
(
r
)
⟶
0
as
r
⟶
∞
.
Set z
n
=
w
-
u
n and
(34)
φ
n
(
x
)
:
=
(
q
-
1
)
∫
0
1
‍
|
s
w
(
x
)
+
(
1
-
s
)
u
n
(
x
)
|
q
-
2
d
s
⩾
0
,
ϕ
n
(
x
)
:
=
(
p
-
1
)
∫
0
1
‍
|
s
w
(
x
)
+
(
1
-
s
)
u
n
(
x
)
|
p
-
2
d
s
⩾
0
,
φ
n
(
x
)
z
n
=
(
q
-
1
)
∫
0
1
‍
|
s
w
(
x
)
+
(
1
-
s
)
u
n
(
x
)
|
q
-
2
(
w
-
u
n
)
d
s
=
∫
0
1
‍
d
d
s
(
|
s
w
+
(
1
-
s
)
u
n
|
p
-
2
(
s
w
+
(
1
-
s
)
u
n
)
)
d
s
=
w
q
-
1
-
|
u
n
|
q
-
2
u
n
,
ϕ
n
(
x
)
z
n
=
(
q
-
1
)
∫
0
1
‍
|
s
w
(
x
)
+
(
1
-
s
)
u
n
(
x
)
|
p
-
2
(
w
-
u
n
)
d
s
=
∫
0
1
‍
d
d
s
(
|
s
w
+
(
1
-
s
)
u
n
|
p
-
2
(
s
w
+
(
1
-
s
)
u
n
)
)
d
s
=
w
p
-
1
-
|
u
n
|
p
-
2
u
n
and hence from (
*
)
(35)
-
div
⁡
(
|
∇
w
|
p
-
2
∇
w
)
-
(
-
div
⁡
(
|
∇
u
n
|
p
-
2
∇
u
n
)
)
+
(
V
ϕ
n
(
x
)
-
Q
n
ψ
n
)
z
n
=
-
div
⁡
|
∇
w
|
p
-
2
+
V
|
w
|
p
-
2
w
-
Q
n
w
q
-
1
-
[
-
div
⁡
|
∇
u
|
p
-
2
∇
u
+
V
|
∇
u
|
p
-
2
∇
u
-
Q
n
|
u
n
|
q
-
2
u
n
]
=
-
div
⁡
|
∇
w
|
p
-
2
+
V
|
w
|
p
-
2
w
-
Q
n
w
q
-
1
⩾
-
div
⁡
|
∇
w
|
p
-
2
+
δ
ϵ
/
2
w
q
-
1
=
0
.
Note that V
ϕ
n
(
x
)
-
Q
n
φ
n
⩾
0 in Ω
∖
B
ϵ
/
2
(
0
)
¯ when n
⩾
N
ϵ
/
2. Due to the continuity of u
n and the fact that w
n
(
x
)
→
∞ as x
→
∂
B
ϵ
/
2
(
0
), there is r
∈
(
ϵ
/
2
,
ϵ
) such that z
n
⩾
0 on ∂
B
r
(
0
). Moreover, z
n
⩾
0 on ∂
Ω. If Ω is bounded, the maximum principle says that z
n
⩾
0 in Ω
∖
B
r
(
0
) (see [20, 21]). If Ω is unbounded, by virtue of w
(
x
) tending to 0 as |
x
|
→
∞ by construction, thus for any γ
>
0, we may pick R
~
>
0 such that z
n
⩾
-
γ in Ω
∖
B
R
~
(
0
). Moreover, applying regularity theory to u
n
∈
W
0
1
,
p
(
Ω
), we can get u
n
(
x
)
→
0 as |
x
|
→
∞. Now the same maximum principle is applied on Ω
∩
(
B
R
~
∖
B
r
(
0
)
¯
), which implies that z
n
⩾
-
γ in all of Ω
∖
B
r
(
0
). Letting γ
→
0, we obtain z
n
>
0 again. By analogy we obtain u
n
⩾
-
w (take z
n
:
=
w
+
u
n); hence
(36)
|
z
n
|
⩾
w
in
Ω
∖
B
ϵ
(
0
)
,
∀
n
⩾
N
ϵ
/
2
.
Hence (a) follows from above arguments with the fact that w is continuous in Ω
∖
B
ϵ
(
0
).
Next, the hypotheses in (b) imply that there isδ
>
0 such that Q
n
⩽
-
δ on Ω
∖
B
1
/
n
(
0
) for each n large enough. Let w
n be a positive solution to
(37)
-
div
⁡
(
|
∇
u
|
p
-
2
∇
u
)
=
-
δ
|
u
|
q
-
2
u
,
x
∈
R
n
∖
B
1
/
n
(
0
)
lim
⁡
|
x
|
→
1
/
n
w
n
(
x
)
=
+
∞
,
lim
⁡
|
x
|
→
∞
w
n
(
x
)
=
0
.
Then the sequence w
n is monotone decreasing, by using the maximum principle to w
n
⩾
w
n
+
1 on ∂
B
1
/
n
(
0
) for every n
∈
N. Therefore, w
n converges locally and uniformly to a nonnegative solution w to (37) on R
n
∖
{
0
}. It follows from our hypotheses on N and p that w is an entire solution to (37) by applying the argument as in [24]. And then, due to [25], w
≡
0. For another, the function w
n dominates the solution u
n on Ω
¯
∖
B
r
(
0
) for some r
∈
(
ϵ
/
2
,
ϵ
), as seen in the proof of (a). Thus, u
n also converges to 0 locally and uniformly in Ω
∖
B
r
(
0
); that is, lim
⁡
n
→
∞
|
u
n
|
s
,
Ω
∖
B
r
(
0
)
=
0.
For (c), we first consider the cases
(
⩾
1
)
∈
(
N
(
q
-
p
)
/
p
,
q
]. By interpolation inequality, we have the following estimate for solution u
n:
(38)
∥
u
n
∥
p
=
∫
Ω
‍
Q
n
|
u
n
|
q
d
x
⩽
c
1
|
u
n
|
q
q
⩽
c
1
|
u
n
|
s
q
θ
|
u
n
|
p
*
q
(
1
-
θ
)
⩽
c
2
|
u
n
|
s
q
θ
∥
u
n
∥
q
(
1
-
θ
)
.
Here c
1, c
2 are independent of n, and θ satisfies that
(39)
1
q
=
θ
s
+
1
-
θ
p
*
.
According to Lemma 2, it suffices to impose that q
(
1
-
θ
)
<
p or equivalent s
>
N
(
q
-
p
)
/
p. This and (a) prove the case s
∈
(
N
(
q
-
p
)
/
p
,
q
]. And then, (38) and (a) yield |
u
n
|
q
,
B
ϵ
(
0
)
→
∞; hence |
u
n
|
s
,
B
ϵ
(
0
)
→
∞ for every s
∈
(
q
,
∞
] as n
→
∞. Using (a) again we get (30).
Note that (38) implies (30) for s
=
N
(
q
-
p
)
/
p, so case (d) is easily followed.


## 4. Concentration at Several Points

Now we assume that the functionQ
n is positive in a neighbourhood of two distinct points x
1
,
x
2
∈
Ω (indeed, the following argument is also valid for any finite number of points in Ω). More precisely, we assume.(
*
*
)  Q
n
>
0 in a neighbourhood of {
x
1
}
∪
{
x
2
}, and there exists a constant C such that |
Q
n
|
L
∞
(
Ω
)
⩽
C for all n. Moreover, for each ϵ
>
0 there exist constants δ
ϵ
>
0 and N
ϵ such that Q
n
⩽
-
δ
ϵ whenever x
∉
B
ϵ
(
x
1
)
∪
B
ϵ
(
x
2
) and n
⩾
N
ϵ.As in Section2, we put J
n
(
u
)
=
∫
Ω
‍
Q
n
|
u
|
q
d
x:
(40)
s
n
:
=
inf
⁡
J
n
(
u
)
∥
u
∥
p
J
n
(
u
)
p
/
q
≡
inf
⁡
J
n
(
u
)
∫
Ω
‍
(
|
∇
u
|
p
+
V
|
u
|
p
)
d
x
(
∫
Ω
‍
Q
n
|
u
|
q
d
x
)
p
/
q
.Theorem 6.
SupposeQ
n satisfies (
*
*
) and q
∈
(
p
,
p
*
), and u
n is a ground state solution to (1). Then, for n large, u
n concentrates at x
1 or x
2. More precisely, for each ϵ
>
0 we have by passing to a subsequence
(41)
lim
⁡
n
→
∞
∫
Ω
∖
B
ϵ
(
x
j
)
‍
(
|
∇
u
|
p
+
V
u
n
p
)
d
x
∫
Ω
‍
(
|
∇
u
|
p
+
V
u
n
p
)
d
x
=
0
,
lim
⁡
n
→
∞
∫
Ω
∖
B
ϵ
(
x
j
)
‍
Q
n
u
n
p
d
x
∫
Ω
‍
Q
n
u
n
p
d
x
=
0
for j
=
1 or 2 (but not for j
=
1 and 2).Remark 7.
Note that, in view of the obvious modification of Theorem4, the limits in (41) are 0 if Ω
∖
B
ϵ
(
x
j
) is replaced by Ω
∖
B
ϵ
(
x
1
)
∪
B
ϵ
(
x
2
). So if j
=
1 in (41), then concentration occurs at x
1 and if j
=
2, it occurs at x
2.Proof.
As in [19], we may assume that J
n
(
u
n
)
=
∫
Ω
‍
Q
n
|
u
n
|
p
d
x
=
1 by renormalizing (u
n may not be a solution to (1), but we still have s
n
:
=
∥
u
n
∥
p
/
J
n
(
u
n
)
p
/
q). Let ξ
j
∈
C
0
∞
(
Ω
,
[
0,1
]
) be a function such that ξ
j
=
1 on B
ϵ
/
2
(
x
j
) and ξ
j
=
0 on Ω
∖
B
ϵ
(
x
j
), j
=
1,2, where ϵ is so small that B
ϵ
(
x
j
)
¯
⊂
Ω and B
ϵ
(
x
1
)
¯
∩
B
ϵ
(
x
2
)
¯
=
∅. Set v
n
:
=
ξ
1
u
n, w
n
:
=
ξ
2
u
n, and z
n
:
=
u
n
-
v
n
-
w
n. Since suppz
n
⊂
Ω
∖
(
B
ϵ
/
2
(
x
1
)
∪
B
ϵ
/
2
(
x
2
)
) and the conclusion of Theorem 4 remains valid after a modification, we have
(42)
∥
u
n
∥
p
=
∫
Ω
‍
(
|
∇
u
n
|
p
+
V
u
n
p
)
d
x
=
(
∫
Ω
‍
(
|
∇
v
n
|
p
+
V
u
n
p
)
d
x
hhlh
+
∫
Ω
‍
(
|
∇
w
n
|
p
+
V
w
n
p
)
d
x
)
(
1
+
o
(
1
)
)
=
(
∥
v
n
∥
p
+
∥
w
n
∥
p
)
(
1
+
o
(
1
)
)
,
1
=
J
n
(
u
n
)
=
∫
Ω
‍
Q
n
|
u
n
|
q
d
x
=
∫
Ω
‍
Q
n
|
v
n
|
q
d
x
+
∫
Ω
‍
Q
n
|
w
n
|
q
d
x
+
o
(
1
)
=
J
n
(
v
n
)
+
J
n
(
w
n
)
+
o
(
1
)
.
First, we assume that limsup
⁡
n
→
∞
J
n
(
v
n
)
⩾
0 and limsup
⁡
n
→
∞
J
n
(
w
n
)
⩾
0. By passing to a subsequence, we may assume that J
n
(
v
n
)
→
c
0
∈
[
0,1
] and J
n
(
w
n
)
→
1
-
c
0
∈
[
0,1
]. If c
0
∈
(
0,1
), recalling that q
>
p, we get a contradiction from the following inequality:
(43)
s
n
=
∥
u
n
∥
p
J
n
(
u
n
)
p
/
q
=
(
∥
v
n
∥
p
+
∥
w
n
∥
p
)
(
1
+
o
(
1
)
)
(
J
n
(
v
n
)
+
J
n
(
w
n
)
+
o
(
1
)
)
p
/
q
>
∥
v
n
∥
p
+
∥
w
n
∥
p
J
n
(
v
n
)
p
/
q
+
J
n
(
w
n
)
p
/
q
⩾
min
⁡
{
∥
v
n
∥
p
J
n
(
v
n
)
p
/
q
,
∥
w
n
∥
p
J
n
(
w
n
)
p
/
q
}
⩾
s
n
.
So c
0
=
0 or 1. If c
0
=
1 (say), then the second limit in (41) is 0 for j
=
1 because suppv
n
⊂
B
ϵ
(
x
1
). The first limit is 0 as well, since ∥
w
n
∥
p
/
∥
v
n
∥
p is otherwise bounded away from 0 for large n, and we obtain a contradiction again from
(44)
s
n
=
(
∥
v
n
∥
p
+
∥
w
n
∥
p
)
(
1
+
o
(
1
)
)
(
J
n
(
v
n
)
+
J
n
(
w
n
)
+
o
(
1
)
)
p
/
q
>
∥
v
n
∥
p
J
n
(
v
n
)
p
/
q
⩾
s
n
.
Finally, supposelimsup
⁡
n
→
∞
J
n
(
w
n
)
<
0 (the case limsup
⁡
n
→
∞
J
n
(
v
n
)
<
0 is of course analogous); it passes to a subsequence J
n
(
w
n
)
⩽
-
η for some η
>
0 when n is large enough. Then a contradiction (44) holds for such n because J
n
(
v
n
)
>
J
n
(
v
n
)
+
J
n
(
w
n
)
+
o
(
1
).



---

*Source: 148902-2014-07-20.xml*