Electronic Journal of Differential Equations, Vol. 2010(2010), No. 96, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE AND CONCENTRATION OF SOLUTIONS FOR A
p-LAPLACE EQUATION WITH POTENTIALS IN RN
MINGZHU WU, ZUODONG YANG
Abstract. We study the p-Laplace equation with Potentials
− div(|∇u|p−2 ∇u) + λV (x)|u|p−2 u = |u|q−2 u,
u ∈
x ∈ RN where 2 ≤ p, p < q < p∗ . Using a concentrationcompactness principle from critical point theory, we obtain existence, multiplicity solutions, and concentration of solutions.
W 1,p (RN ),
1. Introduction
This article concerns the existence and the multiplicity of decaying solutions for
the equation
− div(|∇u|p−2 ∇u) + V (x)|u|p−2 u = |u|q−2 u,
x ∈ RN
(1.1)
− div(|∇u|p−2 ∇u) + λV (x)|u|p−2 u = |u|q−2 u,
x ∈ RN ,
(1.2)
−εp div(|∇u|p−2 ∇u) + V (x)|u|p−2 u = |u|q−2 u,
x ∈ RN
(1.3)
and for the related equations
and
respectively as λ → ∞, and ε → 0. We also consider concentration of solutions as
λ → ∞ or ε → 0.
We assume throughout that V and p, q satisfy the following conditions:
(V1) V ∈ C(RN ) and V is bounded.
(V2) There exists b > 0 such that the set {x ∈ RN : V (x) < b} is nonempty and
has finite measure.
∗
(P1) 2 ≤ p, p < q < p∗ where p∗ = NpN
−p if N > p and p = ∞ if 1 ≤ N ≤ p.
−1
Note that if εp = λ−1 , then u is a solution of (1.2) if and only if v = λ q−p u is
a solution of (1.3), hence as far as the existence and the number of solutions are
concerned, these two problems are equivalent.
2000 Mathematics Subject Classification. 35J25, 35J60.
Key words and phrases. Potentials; critical point theory; concentration; existence;
concentration-compactness; p-Laplace.
c
2010
Texas State University - San Marcos.
Submitted January 22, 2010. Published July 15, 2010.
Supported by grants 10871060 from the National Natural Science Foundation of China,
and 08KJB110005 from the Natural Science Foundation of the Jiangsu Higher Education
Institutions of China.
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M. WU, Z. YANG
EJDE-2010/96
kukp will denote the usual Lp (RN ) norm and V ± (x) = max {±V (x), 0}. Bρ and
Sρ will respectively denote the open ball and the sphere of radius ρ and center at
the origin.
It is well known that the functional
Z
Z
1
1
p
p
Φ(u) =
(|∇u| + V (x)|u| )dx −
|u|q dx
p RN
q RN
is of class C 1 in the Sobolev space
E = {u ∈ W
1,p
N
p
Z
(R ) : kuk =
(|∇u|p + V + (x)|u|p )dx < ∞}
(1.4)
RN
and critical points of Φ correspond to solutions u of (1.1). Moreover, u(x) → 0 as
|x| → ∞. It is easy to see that if
R
(|∇u|p + V (x)|u|p )dx
RN
M = inf
(1.5)
kukpq
u∈E\{0}
1
is attained at some u and M is positive, then u = M q−p u/kukq is a solution of
(1.1) and u(x) → 0 as |x| → ∞. Such u is called a ground state. Because we have
Poincaré inequality
Z
Z
p
|u| dx ≤ C
|∇u|p dx, 1 ≤ p < +∞, u ∈ W01,p (Ω)
Ω
Ω
so E is continuously embedded in W 1,p (RN ).
Recently, there have been numerous works for the eigenvalue problem
− div(|∇u|p−2 ∇u) = V (x)|u|p−2 u
u ∈ D01,p (Ω),
(1.6)
u 6= 0
where Ω ⊆ RN . We can see [3, 16, 24, 25] for different approaches. Szulkin and
Willem [25] generalized several earlier results concerning the existence of an infinite
sequence of eigenvalues.
Consider the quasilinear elliptic equation
− div(|∇u|p−2 ∇u) + λ|u|p−2 u = f (x, u),
u∈
W01,p (Ω),
u 6= 0
in Ω
(1.7)
where 1 < p < N , N ≥ 3, λ is a parameter, Ω is an unbounded domain in RN .
Existence of solutions to (1.7) has been investigation in the previoius decade, see for
example [12, 15, 21, 22, 27, 28]. Because of the unboundedness of the domain, the
Sobolev compact embedding do not hold. There are some methods to overcome this
difficulty. In [28], the authors used the concentration-compactness principle posed
by Lions and the mountain pass lemma to solve problem (1.3). In [27], the author
use that the projection u 7→ f (x, u) is weak continuous in W01,p (Ω) to consider
the problem. In [8, 9], the authors study the problem in symmetric Sobolev spaces
which possess Sobolev compact embedding. By the result and a min-max procedure
formulated by Bahri and Li [5], they considered the existence of positive solutions
of
− div(|∇u|p−2 ∇u) + up−1 = q(x)uα in RN ,
where q(x) satisfies certain conditions.
When p = 2, problem (1.1) has been studied in [1, 4, 6, 7, 14, 17, 17, 19]. In
[20], a quasilinear problem in bounded domains was considered with Hardy type
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potentials. To the best of our knowledge, there is very little work on the case p 6= 2
for problem (1.1).
From its first appearance in the work by Lions [21, 22], the concentrationcompactness principle in calculus of variations has been widely used and by many
authors. In fact, one should refer to the two concentration-compactness principles,
as “escape to infinity” and “concentration around points” as treated separately,
originally. This seemingly harmless dichotomy however often leads to rather cumbersome and tricky calculations. To get rid of these difficulties, some authors have
developed variants that encompass both possible loss of compactness in a whole;
see for instance Ben-Naoum et al. [10] and Bianchi et al. [11] which seem to be
the first works in this direction. When using the original principle or its variants,
it is necessary beforehand to discover the so-called limiting problems that are responsible for non-compactness. Often, these are related to the invariance of the
considered functional and constraint under a non-compact group; translations and
dilations being the two most studied.
Motivated by the results in [6, 11, 13, 14, 17, 18, 19, 20, 22, 26, 27], we obtain
the existence and the multiplicity of solutions in Theorems 3.1–3.3 by using critical
point theory. By Theorems 3.4 and 3.5, we can obtain the concentration of solutions.
This paper is organized as follows. In Section 2, we state some condition and
many lemmas which we need in the proof of the main Theorem. In Section 3, we
give the proof of the main result of the paper.
2. Preliminaries
Lemma 2.1. Let Ω ⊆ R be an open subset. (un ) ⊆ W01,p (Ω) be a sequence such
that un * u in W01,p (Ω) and p ≥ 2. Then
Z
Z
Z
p
p
lim
|∇un | dx ≥ lim
|∇un − ∇u| dx +
|∇u|p dx.
N
n→∞
n→∞
Ω
Ω
Ω
Proof. When p = 2 from Lieb Lemma we have
Z
Z
Z
2
2
lim
|∇un | dx = lim
|∇un − ∇u| dx +
|∇u|2 dx.
n→∞
n→∞
Ω
Ω
Ω
p
For 3 ≥ p > 2, using the lower semi-continuity of the L -norm with respect to the
weak convergence and un * u in W 1,p (Ω), we deduce
lim h|∇un |p−2 ∇un , ∇un i ≥ h|∇u|p−2 ∇u, ∇ui
n→∞
and
lim h|∇un − ∇u|p−2 (∇un − ∇u), ∇un − ∇ui
n→∞
= 0 ≥ lim h|∇un − ∇u|p−2 (∇u − ∇u), ∇u − ∇ui.
n→∞
So
lim h|∇un − ∇u|p−2 ∇un , ∇un i ≥ lim h|∇un − ∇u|p−2 ∇un , ∇ui
n→∞
n→∞
= lim h|∇un − ∇u|p−2 ∇u, ∇un i
n→∞
= lim h|∇un − ∇u|p−2 ∇u, ∇ui.
n→∞
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M. WU, Z. YANG
EJDE-2010/96
Then
Z
(|∇un |p − |∇u|p )dx
Z
Z
p−2
2
2
= lim
|∇un | (|∇un | − |∇u| )dx + lim
(|∇un |p−2 − |∇u|p−2 )|∇u|2 dx
n→∞ Ω
n→∞ Ω
Z
= lim
(|∇un |p−2 + |∇u|p−2 )(|∇un |2 − |∇u|2 )dx
n→∞ Ω
Z
+ lim
(|∇un |p−2 |∇u|2 − |∇u|p−2 |∇un |2 )dx.
lim
n→∞
Ω
n→∞
Ω
From un * u in W 1,p (Ω),
Z
lim
(|∇un |p−2 |∇u|2 − |∇u|p−2 |∇un |2 )dx = 0.
n→∞
Ω
So
Z
n→∞
Z
(|∇un |p − |∇u|p )dx = lim
lim
n→∞
Ω
(|∇un |p−2 + |∇u|p−2 )(|∇un |2 − |∇u|2 )dx
Ω
Z
|∇un − ∇u|p−2 (|∇un |2 − |∇u|2 ).
≥ lim
n→∞
Ω
So we have
lim h|∇un |p−2 ∇un , ∇un i + h|∇un − ∇u|p−2 ∇u, ∇un i + h|∇un − ∇u|p−2 ∇un , ∇ui
n→∞
≥ lim h|∇un − ∇u|p−2 ∇un , ∇un i
n→∞
+ lim h|∇un − ∇u|p−2 ∇u, ∇ui + h|∇u|p−2 ∇u, ∇ui.
n→∞
Then
lim h|∇un |p−2 ∇un , ∇un i
n→∞
≥ lim h|∇un − ∇u|p−2 ∇un − ∇u, ∇un − ∇ui + h|∇u|p−2 ∇u, ∇ui.
n→∞
and
Z
lim
n→∞
|∇un |p dx ≥ lim
Z
n→∞
Ω
|∇un − ∇u|p dx +
Ω
Z
|∇u|p dx.
Ω
For p > 3, there exist a k ∈ N that 0 < p − k ≤ 1. Then, we only need to prove
the inequality
Z
Z
lim
(|∇un |p − |∇u|p )dx ≥ lim
|∇un − ∇u|p−k (|∇un |k − |∇u|k ).
n→∞
n→∞
Ω
Ω
The proof of this iequality is similar to the above, so we omit it. Therefore, the
lemma is proved.
Let Vb (x) = max {V (x), b} and
R
Mb =
inf
u∈E\{0}
RN
(|∇u|p + Vb (x)|u|p )dx
.
kukpq
(2.1)
Denote the spectrum of −∆p +V in Lp (RN ) by σ(−∆p +V ) and recall the definition
(1.5) of M .
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EXISTENCE AND CONCENTRATION OF SOLUTIONS
5
Lemma 2.2. Suppose (V1), (V2), (P1) are satisfied and σ(−∆p + V ) ⊂ (0, ∞). If
M < Mb , then each minimizing sequence for M has a convergent subsequence. So
in particular, M is attained at some u ∈ E \ {0}.
Proof. Let {um } be a minimizing sequence. We may assume kum kq = 1. Since
V < 0 on a set of finite measure, {um } is bounded in the norm of E given by (1.4).
Passing to a subsequence we may assume um * u in E and by the continuity of
the embedding E ,→ W 1,p (RN ), um → u in Lploc (RN ), Lqloc (RN ) and a.e. in RN .
Let um = vm + u. Then by Lemma 2.1, we have
Z
lim
(|∇um |p + V (x)|um |p )dx
m→∞ RN
Z
Z
(2.2)
≥ lim
(|∇vm |p + V (x)|vm |p )dx +
(|∇u|p + V (x)|u|p )dx,
m→∞
RN
RN
by the Lieb Lemma,
Z
|um |p dx = lim
lim
m→∞
Z
m→∞
RN
|vm |p dx +
RN
Z
|u|p dx.
Moreover, by (V2) and since vm * 0 as m → ∞,
Z
lim
(V (x) − Vb (x))|vm |p dx → 0.
m→∞
(2.3)
RN
(2.4)
RN
Using (2.2)-(2.4) and the definitions of M , Mb , we obtain
Z
Z
p
p
(|∇u| + V (x)|u| )dx + lim
(|∇vm |p + V (x)|vm |p )dx
m→∞
RN
≤ M lim
m→∞
RN
kum kpq
p
= M lim (kukqq + kvm kqq ) q
m→∞
≤ M lim (kukpq + kvm kpq )
Z
Z m→∞
(|∇vm |p + Vb (x)|vm |p )dx
≤
(|∇u|p + V (x)|u|p )dx + M Mb−1 lim
m→∞ RN
RN
Z
Z
−1
p
p
≤
(|∇u| + V (x)|u| )dx + M Mb lim
(|∇vm |p + V (x)|vm |p )dx.
RN
M Mb−1
m→∞
R
−1
RN
p
Since
< 1 and RN V (x)|vm | dx → 0 as m → ∞, it follows that vm → 0
and therefore um → u as m → ∞. It is clear that u 6= 0.
From the above lemma it follows that if σ(−∆p +V ) ⊂ (0, ∞) and M < Mb , then
there exists a ground state solution of (1.1). Recall that {um } is called a PalaisSmale sequence at the level c (a (P S)c -sequence) if Φ0 (um ) → 0 and Φ(um ) → c.
If each (P S)c -sequence has a convergent subsequence, then Φ is said to satisfy the
(P S)c -condition.
Lemma 2.3. If (V1), (V2), (P1) hold, then Φ satisfies (P S)c for all
q
1 1
c < ( − )Mb(q−p) .
p q
Proof. Let {um } be a (P S)c -sequence with c satisfying the inequality above. First
we show that {um } is bounded. We have
1
1 1
2c + dkum k ≥ Φ(um ) − hΦ0 (um ), um i = ( − )kum kqq
(2.5)
p
p q
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M. WU, Z. YANG
EJDE-2010/96
and
1
2c + dkum k ≥ Φ(um ) − hΦ0 (um ), um i
q
Z
(2.6)
1 1
1 1
= ( − )kum kp − ( − )
V − (x)|um |p dx
p q
p q RN
for some constants d > 0. Suppose kum k → ∞ as m → ∞ and let wm =
um /kum k. Dividing (2.5) by kum kq we see that wm → 0 in Lq (RN ) as m → ∞
and
therefore wm * 0 in E as m → ∞ after passing to a subsequence. Hence
R
−
V
(x)|wm |p dx → 0 as m → ∞. So dividing (2.6) by kum kp , it follows that
RN
wm → 0 in E as m → ∞, a contradiction. Thus {um } is bounded.
As in the preceding proof, we may assume um * u in E and um → u in Lploc (RN ).
Set um = vm + u. Since Φ0 (u) = 0 and
1 1
1
Φ(u) = Φ(u) − hΦ0 (u), ui = ( − )kukqq ≥ 0,
p
p q
it follows from (2.2), (2.3) that
lim (|kvm kp − kvm kqq |) ≤ lim (|kum kp − kum kqq | + |kukp − kukqq |) = 0
m→∞
m→∞
so
lim (kvm kp − kvm kqq ) = 0
(2.7)
m→∞
and
c = lim Φ(um ) ≥ lim (Φ(vm ) + Φ(u)) ≥ lim Φ(vm ).
m→∞
By (2.7), we have
Z
lim
m→∞
m→∞
p
m→∞
Z
p
(|∇vm | + V (x)|vm | )dx = lim
m→∞
RN
|vm |q dx = γ
(2.8)
(2.9)
RN
possibly after passing to a subsequence, and therefore it follows from (2.8) that
1 1
(2.10)
c ≥ ( − )γ.
p q
By (2.4),
Z
lim
m→∞
(|∇vm |p + Vb (x)|vm |p )dx = lim
Z
m→∞
RN
(|∇vm |p + V (x)|vm |p )dx = γ .
RN
On the other hand,
kvm kpq
≤
Mb− lim
m→∞
Z
(|∇vm |p + Vb (x)|vm |p )dx;
RN
p
therefore, γ q ≤ Mb− γ. Combining this with (2.10), we see that either γ = 0, or
q
1 1
c ≥ ( − )Mb(q−p)
p q
hence γ must be 0 by the assumption on c. So according to (2.9), we have
Z
Z
p
+
p
lim
(|∇vm | + V (x)|vm | )dx = lim
(|∇vm |p + V (x)|vm |p )dx = 0.
m→∞
RN
m→∞
RN
Therefore, vm → 0 and um → u in E as m → ∞.
Next we recall a usual critical point theory which will be used in the below
Theorem. Here γ(A) is the Krasnoselskii genus of A.
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EXISTENCE AND CONCENTRATION OF SOLUTIONS
7
Theorem 2.4. Suppose E ∈ C 1 (M ) is an even functional on a complete symmetric
C 1,1 -manifold M ⊂ V \{0} in some Banach space V . Also suppose E satisfies (P S)
and is bounded below on M . Let γ
e(M ) = sup{γ(K); K ⊂ M and symmetric}. Then
the functional E possesses at least γ
e(M ) ≤ ∞ pairs of critical points.
3. Proof of Main Theorems
Theorem 3.1. Suppose Assumptions (V1), (P1) are satisfied, σ(−∆p + V ) ⊂
(0, ∞), supx∈RN V (x) = b > 0 and the measure of the set {x ∈ RN : V (x) < b − ε}
is finite for all ε > 0. Then the infimum in (1.5) is attained at some u ≥ 0. If
V ≥ 0, then u > 0 in RN .
Proof. Since V + is bounded, E = W 1,p (RN ) here. Let ub be the radially symmetric
positive solution of the equation
− div(|∇u|p−2 ∇u) + b|u|p−2 u = |u|q−2 u,
x ∈ RN .
It is well known that such ub exists, is unique and minimizes
R
(|∇u|p + b|u|p )dx
RN
Nb = inf
kukpq
u∈E\{0}
(3.1)
(see [12]). So if V ≡ b, the proof is complete. Otherwise we may assume without
loss of generality that V (0) < b. Then
R
(|∇u|p + V (x)|u|p )dx
RN
M = inf
kukpq
u∈E\{0}
R
p
p
N (|∇ub | + V (x)|ub | )dx
≤ R
p
kub kq
R
p
p
N (|∇ub | + b|ub | )dx
< R
p
kub kq
= N b = Mb ,
where the last equality follows from the fact that Vb = b. To apply Lemma 2.2 we
need to show that M < Mb−ε for some ε > 0. A simple computation shows that if
λ > 0, then Nλb is attained at
1
1
uλb (x) = λ (q−p) ub (λ p x)
and Nλb = λr Nb ,
where r = 1 − Np + Nq .
Choosing λ = (b − ε)/b we see that Nb−ε < Nb and Nb−ε → Nb as ε → 0. So for
ε small enough we have
R
(|∇u|p + (b − ε)|u|p )dx
RN
M < Nb−ε = inf
kukpq
u∈E\{0}
R
(3.2)
(|∇u|p + Vb−ε (x)|u|p )dx
RN
≤ inf
kukpq
u∈E\{0}
= Mb−ε .
Hence M is attained at some u. If u is replaced by |u|, the expression on the
right-hand side of (1.5) does not change, we may assume u ≥ 0. By the maximum
principle, if V ≥ 0, then u > 0 in RN .
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M. WU, Z. YANG
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Theorem 3.2. Suppose V ≥ 0 and (V1), (V2), (P1) are satisfied. Then there
exists Λ > 0 such that for each λ ≥ Λ the infimum in (1.5) is attained at some
uλ > 0. Here V (x) replaced by λV (x).
Proof. Here V = V + . Let b be as in (V2) and
R
(|∇u|p + λV (x)|u|p )dx
λ
RN
M = inf
,
kukpq
u∈E\{0}
R
(|∇u|p + λVb (x)|u|p )dx
RN
.
Mbλ = inf
kukpq
u∈E\{0}
(3.3)
It suffices to show that M λ < Mbλ for all λ large enough. We may assume V (0) < b
and choose ε, δ > 0 so that V (x) < b − ε whenever |x| < 2δ. Let ϕ ∈ C0∞ (RN , [0, 1])
be a function such that ϕ(x) = 1 for |x| ≤ δ and ϕ(x) = 0 for |x| ≥ 2δ. Set
1
1
wλb (x) = ϕ(x)uλb (x) = λ q−p ub (λ p x)ϕ(x), where ub is as in the proof of Theorem
3.1. Then for all sufficiently large λ and some C0 > 0,
R
p
p
N (|∇wλb | + λV (x)|wλb | )dx
Mλ ≤ R
p
kwλb kq
R
p
p
N (|∇wλb | + λ(b − ε)|wλb | )dx
≤ R
p
kwλb kq
R
R
(|∇ub |p + λb|ub |p )dx − ε RN |ub |p dx
r
RN
≤λ (
+ ε)
kub kpq
≤ λr (Nb − C0 ε)
where Nb is defined in (3.1) and r in (3.2). Using (3.2) and (3.3) we also see that
R
Mbλ
≥
inf
u∈E\{0}
RN
(|∇u|p + λb|u|p )dx
= Nλb = λr Nb ,
kukpq
(3.4)
hence M λ < Mbλ . By the argument at the end of the proof of Theorem 3.1, the
infimum is attained at some uλ > 0.
Next we consider the existence of multiple solutions under the hypothesis that
V −1 (0) has nonempty interior.
Theorem 3.3. Suppose V ≥ 0, V −1 (0) has nonempty interior and (V1), (V2),
(P1) are satisfied. For each k ≥ 1 there exists Λk > 0 such that if λ ≥ Λk , then
(1.2) has at the least k pairs of nontrivial solutions in E.
Proof. For a fixed k we can find ϕ1 ,. . . ,ϕk ∈ C0∞ (RN ) such that supp ϕj , 1 ≤ j ≤ k,
is contained in the interior of V −1 (0) and supp ϕi ∩ supp ϕj = ∅ whenever i 6= j.
Let
Fk = span {ϕ1 , . . . , ϕk }.
Since V ≥ 0, Φ(u) = p1 kukp − 1q kukqq and therefore there exist α, ρ > 0 such that
Φ|Sρ ≥ α. Denote the set of all symmetric (in the sense that −A = A) and closed
subsets of E by Σ, for each A ∈ Σ let γ(A) be the Krasnoselski genus and
i(A) = min γ(h(A) ∩ Sρ )
h∈Γ
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EXISTENCE AND CONCENTRATION OF SOLUTIONS
9
where Γ is the set of all odd homeomorphisms h ∈ C(E, E). Then i is a version of
Benci’s pseudoindex. Let
Z
Z
1
1
Φλ (u) =
(|∇u|p + λV (x)|u|p )dx −
|u|q dx, λ ≥ 1
p RN
q RN
and
cj = inf sup Φλ (u), 1 ≤ j ≤ k.
i(A)≥j u∈A
Since Φλ (u) ≥ Φ(u) ≥ α for all u ∈ Sρ and since i(Fk ) = dim Fk = k,
α ≤ c1 ≤ · · · ≤ ck ≤ sup Φλ (u) = C.
u∈Fk
It is clear that C depends on k but not on λ. As in (3.4), we have
Mbλ ≥ Nλb = λr Nb
q
where r > 0, and therefore Mbλ → ∞. Hence C < ( p1 − 1q )(Mbλ ) (q−p) whenever λ
is large enough and it follows from Lemma 2.3 that for such λ the Palais-Smale
condition is satisfied at all levels c ≤ C. By the usual critical point theory Theorem
2.4, all cj are critical levels and Φλ has at least k pairs of nontrivial critical points.
Theorem 3.4. Suppose (V1), (V2), (P1) are satisfied and V −1 (0) has nonempty
interior Ω. Let um ∈ E be a solution of the equation
− div(|∇u|p−2 ∇u) + λm V (x)|u|p−2 u = |u|q−2 u,
x ∈ RN .
(3.5)
If λm → ∞ and kum kλm ≤ C for some C > 0, then, up to a subsequence, um → u
in Lq (RN ), where u is a weak solution of the equation
− div(|∇u|p−2 ∇u) = |u|q−2 u,
N
and u = 0 a.e. in R \ V
−1
x ∈ Ω,
(3.6)
(0). If moreover V ≥ 0, then um → u in E as m → ∞.
Proof. Since λm ≥ 1, kum k ≤ kum kλm ≤ C. Passing to a subsequence, um * u in
E and um → u in Lqloc (RN ) as m → ∞. Since hΦλm 0 (um ), ϕi = 0, we see that
Z
Z
lim
V (x)|um |p−2 um ϕdx = 0,
V (x)|u|p−2 uϕdx = 0
m→∞
RN
C0∞ (RN ).
RN
and for all ϕ ∈
Therefore, u = 0 a.e. in RN \ V −1 (0).
We claim that um → u in Lq (RN ) as m → ∞. Assuming the contrary, it follows
from Lion vanishing lemma that
Z
|um − u|p dx ≥ γ
Bρ (xm )
for some {xm } ⊂ RN , ρ, γ > 0 and almost all m, where Bρ (x) denotes the open
ball of radius ρ and center x.
Since um → u in Lqloc (RN ), |xm | → ∞. Therefore, the measure of the set
Bρ (xm ) ∩ {x ∈ RN : V (x) < b} tends to 0 and
Z
lim kum kpλm ≥ lim λm b
|um |p dx
m→∞
m→∞
Bρ (xm )∩{V ≥b}
Z
= lim λm b(
m→∞
which is a contradiction.
Bρ (xm )
|um − u|p dx) = ∞,
10
M. WU, Z. YANG
EJDE-2010/96
Let now V ≥ 0. Since um satisfies (3.5), hΦ0λm (um ), ui = 0 and u = 0 whenever
V > 0, it follows that
kum kp ≤ kum kpλm = kum kqq
and
kukp = kukpλm = kukqq .
Hence lim supm→∞ kum kp ≤ kukqq = kukp ; therefore, um → u in E as m → ∞.
Theorem 3.5. Suppose (V1), (V2), (P1) are satisfied and V −1 (0) has nonempty
interior, V ≥ 0, um ∈ E is a solution of (3.5), λm → ∞ and Φλm (um ) is bounded
and bounded away from 0. Then the conclusion of Theorem 3.4 is satisfied and
u 6= 0.
Proof. We have
Φλm (um ) =
1
1
kum kpλm − kum kqq
p
q
and
1
1 1
Φλm (um ) = Φλm (um ) − hΦ0λm (um ), um i = ( − )kum kqq
p
p q
Hence kum kq , and therefore also kum kλm is bounded. So the conclusion of Theorem
3.4 holds. Moreover, as kum kq is bounded away from 0, u 6= 0.
As a consequence of this corollary, if k is fixed, then any sequence of solutions
um of (1.2) with λ = λm → ∞ obtained in Theorem 3.3 contains a subsequence
concentrating at some u 6= 0. Moreover, it is possible to obtain a positive solution
for each λ, either via Theorem 3.1 or by the mountain pass theorem. It follows
that each sequence {um } of such solutions with λm → ∞ has a subsequence concentrating at some u which is positive in Ω. Corresponding to um are solutions
−p/(q−p)
p/(q−p)
vm → u.
um of (1.3), where εpm = λ−1
vm = εm
m . Then vm → 0 and εm
subsection*Remark In the proof of Lemmas 2.2 and 2.3 and Theorems 3.1–3.3,
the condition (V1) can be replaced by
(V1’) v ∈ L1loc (RN ) and V − = max {−V, 0} ∈ Lq (RN ), where q = N/p if N ≥
p + 1, q > 1 if N = p and q = 1 if N < p.
Meanwhile in Theorems 3.4 and 3.5 we also need V ∈ Lqloc (RN ).
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Mingzhu Wu
Institute of Mathematics, School of Mathematical Science, Nanjing Normal University,
Jiangsu Nanjing 210046, China
E-mail address: wumingzhu 2010@163.com
Zuodong Yang
Institute of Mathematics, School of Mathematical Science, Nanjing Normal University,
Jiangsu Nanjing 210046, China.
College of Zhongbei, Nanjing Normal University, Jiangsu Nanjing 210046, China
E-mail address: zdyang jin@263.net