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Hindawi Publishing Corporation
Boundary Value Problems
Volume 2009, Article ID 185319, 19 pages
doi:10.1155/2009/185319
Research Article
A Class of p-q-Laplacian Type Equation with
Potentials Eigenvalue Problem in R
N
Mingzhu Wu
1
and Zuodong Yang
1, 2
1
School of Mathematics Science, Institute of Mathematics, Nanjing Normal University, Jiangsu,
Nanjing 210097, China
2
College of Zhongbei, Nanjing Normal University, Jiangsu, Nanjing 210046, China
Correspondence should be addressed to Zuodong Yang, zdyang
Received 20 October 2009; Accepted 6 December 2009
Recommended by Wenming Zou
The nonlinear elliptic eigenvalue problem −div|∇u|
p−2
∇u − div|∇u|
q−2
∇uλax|u|
p−2
u
λbx|u|
q−2
u fx, u,u ∈ W
1,p
∩ W
1,q
R
N
,where2≤ q ≤ p<Nand ax ∈ L
N/p
R
N
,bx ∈
L
N/q
R
N
,ax,bx > 0 are studied. The key ingredient is a special constrained minimization
method.
Copyright q 2009 M. Wu and Z. Yang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
In this paper, we are interested in finding nontrivial weak solutions for the nonlinear
eigenvalue problem
− div
|
∇u
|
p−2
∇u
− div
|
∇u
|
q−2
∇u
a
x
|
u
|
p−2
u b
x
|
u
|
q−2
u f
x, u
,
u ∈ W
1,p
∩ W
1,q
R
N
,u
/
0,
1.1
where 2 ≤ q ≤ p<Nand ax ∈ L
N/p
R
N
, bx ∈ L
N/q
R
N
, ax,bx > 0,
inf ax, inf bx
/
0, fx, u satisfy the following conditions:
A f ∈ CR
N
× R, R, lim
t → 0
fx, t/|t|
p−1
0, and lim
|t|→∞
fx, t/|t|
p−1p
2
/N
0
uniformly in x ∈ R
N
,
B lim
|x|→∞
fx, tft uniformly for t in bounded subsets of R.
2 Boundary Value Problems
Remark 1.1. We can see if ax ∈ L
N/p
R
N
, bx ∈ L
N/q
R
N
, then
R
N
a
x
|
u
|
p
dx <
R
N
a
x
N/p
1−p/p
∗
R
N
u
p
∗
p/p
∗
< ∞,
R
N
b
x
|
u
|
q
dx <
R
N
b
x
N/q
1−q/q
∗
R
N
u
q
∗
q/q
∗
< ∞,
1.2
where p
∗
Np/N − p and q
∗
Nq/N − q.
Problem 1.1 comes, for example, from a general reaction-diffusion system:
u
t
div
D
u
∇u
c
x, u
, 1.3
where Du|∇u|
p−2
|∇u|
q−2
. This system has a wide range of applications in physics and
related sciences such as biophysics, plasma physics, and chemical reaction design. In such
applications, the function u describes a concentration, the first term on the right-hand side of
1.3 corresponds to the diffusion with a diffusion coefficient Du; whereas the second one
is the reaction and relates to source and loss processes. Typically, in chemical and biological
applications, the reaction term cx, u is a polynomial of u with variable coefficients.
When p q 2, problem 1.1 is a normal Schrodinger equation which has been
extensively studied, for example, 1–8. The authors used many different methods to study
the equation. In 8, the authors established some embedding results of weighted Sobolev
spaces of radially symmetric functions which are used to obtain ground state solutions. In
6, the authors studied the equation depending upon the local behavior of V near its global
minimum. In 3, the authors used spectral properties of the Schrodinger operator to study
nonlinear Schrodinger equations with steep potential well. In 9, the author imposed on
functions k and K conditions ensuring that this problem can be written in a variational form.
We know that W
1,p
R
N
is not a Hilbert space for 1 <p<N, except for p 2. The space
W
1,p
R
N
with p
/
2 does not satisfy the Lieb lemma e.g., see 9. And R
N
results in the loss
of compactness. So there are many difficulties to study equation 1.1 of p q
/
2 by the usual
methods. There seems to be little work on the case p q
/
2 for problem 1.1 ,tothebestof
our knowledge. In this paper, we overcome these difficulties and study 1.1 of p ≥ q ≥ 2.
Recently, when p q, axbx, and fx, u0 then the problem is the following
eigenvalue problem has been studied by many authors:
− div
|
∇u
|
p−2
∇u
V
x
|
u
|
p−2
u,
u ∈ D
1,p
0
Ω
,u
/
0,
1.4
where Ω ⊆ R
N
. We can see 10–13.In13, Szulkin and Willem generalized several earlier
results concerning the existence of an infinite sequence of eigenvalues.
Boundary Value Problems 3
When p q and ax,bx is constant then the problem is the following quasilinear
elliptic equation:
− div
|
∇u
|
p−2
∇u
λ
|
u
|
p−2
u f
x, u
, in Ω,
u ∈ W
1,p
0
Ω
,u
/
0,
1.5
where 1 <p<N, N ≥ 3, λ is a parameter, Ω is an unbounded domain in R
N
. There are many
results about it we can see 14–18. Because of the unboundedness of the domain, the Sobolev
compact embedding does not hold. There are many methods to overcome the difficulty. In
15, the authors used the concentration-compactness principle posed by P. L. Lions and the
mountain pass lemma to solve problem 1.5 .In17, 18, the authors studied the problem in
symmetric Sobolev spaces which possess Sobolev compact embedding. By the result and a
min-max procedure formulated by Bahri and Li 16, they considered the existence of positive
solutions of
− div
|
∇u
|
p−2
∇u
u
p−1
q
x
u
α
in R
N
, 1.6
where qx satisfies some conditions. We can see if λ is function, then it cannot easily be
proved by the above methods.
When ax,bx is positive constant, He and Li used the mountain pass theorem and
concentration-compactness principle to study the following elliptic problem in 19:
− div
|
∇u
|
p−2
∇u
− div
|
∇u
|
q−2
∇u
m
|
u
|
p−2
u n
|
u
|
q−2
u f
x, u
in R
N
,
u ∈ W
1,p
∩ W
1,q
R
N
,
1.7
where m, n > 0, N ≥ 3, and 1 <q<p<N, fx, u/u
p−1
tends to a positive constant l as
u → ∞. The authors prove in this paper that the problem possesses a nontrivial solution
even if the nonlinearity fx, t does not satisfy the Ambrosetti-Rabinowitz condition.
In 20, Li and Liang used the mountain pass theorem to study the following elliptic
problem:
− div
|
∇u
|
p−2
∇u
− div
|
∇u
|
q−2
∇u
|
u
|
p−2
u
|
u
|
q−2
u f
x, u
in R
N
,
u ∈ W
1,p
∩ W
1,q
R
N
,
1.8
where 1 <q<p<N. They generalized a similar result for p-Laplacian type equation in 15.
It is our purpose in this paper to study the existence of ground state to the problem
1.1 in R
N
. We call any minimizer a ground state for 1.1. We inspired by 9, 16, 21 try to
use constrained minimization method to study problem 1.1. Let us point out that although
the idea was used before for other problems, the adaptation to the procedure to our problem
is not trivial at all. But since both p-andq-Laplacian operators are involved, careful analysis is
needed. A typical difficulty for problem 1.1 in R
N
is the lack of compactness of the Sobolev
imbedding due to t he invariance of R
N
under the translations and rotations. However, our
method has essential difference with the methods used in 19, 20. In order to obtain the
4 Boundary Value Problems
results, we have to overcome two main difficulties; one is that R
N
results in the loss of
compactness; the other is that W
1,p
R
N
is not a Hilbert space for 1 <p<Nand it does
not satisfy the Lieb lemma, except for p 2.
The 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
Let
F
x, t
t
0
f
x, s
ds, F
t
t
0
f
s
ds
2.1
and we define variational functionals I : W
1,p
∩W
1,q
R
N
→ R and I
∞
: W
1,p
∩W
1,q
R
N
→ R
by
I
u
1
p
R
N
|
∇u
|
p
dx
1
q
R
N
|
∇u
|
q
dx −
R
N
F
x, u
dx,
I
∞
u
1
p
R
N
|
∇u
|
p
dx
1
q
R
N
|
∇u
|
q
dx −
R
N
F
u
dx.
2.2
Solutions to problem 1.1 will be found as minimizers of the variational problem
I
λ
inf
I
u
; u ∈ W
1,p
R
N
,
R
N
a
x
|
u
|
p
b
x
|
u
|
q
dx λ
,λ>0.
I
λ
To find a solution of problem I
λ
we introduce the limit variational problem
I
∞
λ
inf
I
∞
u
; u ∈ W
1,p
R
N
,
R
N
a
x
|
u
|
p
b
x
|
u
|
q
dx λ
,λ>0.
I
∞
λ
Lemma 2.1. Let u
n
⊆ W
1,p
0
Ω a bounded sequence and p ≥ 2. Going if necessary to a subsequence,
one may assume that u
n
uin W
1,p
0
Ω, u
n
→ u almost everywhere, where Ω ⊆ R
N
is an open
subset.
Then,
lim
n →∞
Ω
|
∇u
n
|
p
dx ≥ lim
n →∞
Ω
|
∇u
n
−∇u
|
p
dx lim
n →∞
Ω
|
∇u
|
p
dx.
2.3
Boundary Value Problems 5
Proof. When p 2 from Brezis-Lieb lemma see 21, Lemma 1.32 we have
lim
n →∞
Ω
|
∇u
n
|
2
dx lim
n →∞
Ω
|
∇u
n
−∇u
|
2
dx lim
n →∞
Ω
|
∇u
|
2
dx,
2.4
when 3 ≥ p>2, using the lower semicontinuity of the L
p
-norm with respect to the weak
convergence and u
n
uin W
1,p
Ω, we deduce
|
∇u
n
|
p−2
∇u
n
, ∇u
n
≥
|
∇u
|
p−2
∇u, ∇u
o
1
,
lim
n →∞
|
∇u
n
−∇u
|
p−2
∇u
n
, ∇u
n
≥ lim
n →∞
|
∇u
n
−∇u
|
p−2
∇u
n
, ∇u
lim
n →∞
|
∇u
n
−∇u
|
p−2
∇u, ∇u
n
lim
n →∞
|
∇u
n
−∇u
|
p−2
∇u, ∇u
.
2.5
Then,
lim
n →∞
Ω
|
∇u
n
|
p
−
|
∇u
|
p
dx
lim
n →∞
Ω
|
∇u
n
|
p−2
|
∇u
n
|
2
−
|
∇u
|
2
dx lim
n →∞
Ω
|
∇u
n
|
p−2
−
|
∇u
|
p−2
|
∇u
|
2
dx
lim
n →∞
Ω
|
∇u
n
|
p−2
|
∇u
|
p−2
|
∇u
n
|
2
−
|
∇u
|
2
dx
lim
n →∞
Ω
|
∇u
n
|
p−2
|
∇u
|
2
−
|
∇u
|
p−2
|
∇u
n
|
2
dx.
2.6
From u
n
uin W
1,p
Ω,
lim
n →∞
Ω
|
∇u
n
|
p−2
|
∇u
|
2
−
|
∇u
|
p−2
|
∇u
n
|
2
dx 0
. 2.7
So
lim
n →∞
Ω
|
∇u
n
|
p
−
|
∇u
|
p
dx lim
n →∞
Ω
|
∇u
n
|
p−2
|
∇u
|
p−2
|
∇u
n
|
2
−
|
∇u
|
2
dx
≥ lim
n →∞
Ω
|
∇u
n
−∇u
|
p−2
|
∇u
n
|
2
−
|
∇u
|
2
.
2.8
6 Boundary Value Problems
So we have
|
∇u
n
|
p−2
∇u
n
, ∇u
n
|
∇u
n
−∇u
|
p−2
∇u, ∇u
n
|
∇u
n
−∇u
|
p−2
∇u
n
, ∇u
≥
|
∇u
n
−∇u
|
p−2
∇u
n
, ∇u
n
|
∇u
n
−∇u
|
p−2
∇u, ∇u
|
∇u
|
p−2
∇u, ∇u
o
1
.
2.9
Then,
|
∇u
n
|
p−2
∇u
n
, ∇u
n
≥
|
∇u
n
−∇u
|
p−2
∇u
n
−∇u, ∇u
n
−∇u
|
∇u
|
p−2
∇u, ∇u
o
1
lim
n →∞
Ω
|
∇u
n
|
p
dx ≥ lim
n →∞
Ω
|
∇u
n
−∇u
|
p
dx lim
n →∞
Ω
|
∇u
|
p
dx,
2.10
when p>3, there exists a k ∈ N that 0 <p− k ≤ 1. Then, we only need to prove the following
inequality:
lim
n →∞
Ω
|
∇u
n
|
p
−
|
∇u
|
p
dx ≥ lim
n →∞
Ω
|
∇u
n
−∇u
|
p−k
|
∇u
n
|
k
−
|
∇u
|
k
.
2.11
The proof of it is similar to the above, so we omit it here. So, the lemma is proved.
Lemma 2.2. Let {u
n
} be a bounded sequence in W
1,p
R
N
such that
lim
n →∞
sup
y∈R
N
B
y,R
u
q
n
dx 0,p≤ q<p
∗
2.12
for some R>0.Thenu
n
→ 0 in L
s
R
N
for p<s<p
∗
,wherep
∗
Np/N − p.
Proof. We consider the case N ≥ 3. Let q<s<p
∗
and u ∈ W
1,p
R
N
. Holder and Sobolev
inequalities imply that
|
u
|
L
s
By,R
≤
|
u
|
1−λ
L
q
By,R
|
u
|
λ
L
p
∗
By,R
≤ C
|
u
|
1−λ
L
q
By,R
By,R
|
u
|
p
|
∇u
|
p
λ/p
,
2.13
where λ s − q/p
∗
− qp
∗
/s. Choosing λ p/s,weobtain
By,R
|
u
|
s
≤ C
s
|
u
|
1−λ
s
L
q
By,R
By,R
|
u
|
p
|
∇u
|
p
. 2.14
Boundary Value Problems 7
Now, covering R
N
by balls of radius r, in such a way that each point of R
N
is contained
in at most N 1 balls, we find
R
N
|
u
|
s
≤
N 1
C
s
sup
y∈R
N
By,R
|
u
|
q
1−λ
s/q
By,R
|
u
|
p
|
∇u
|
p
. 2.15
Under the assumption of the lemma, u
n
→ 0inL
s
R
N
, p<s<p
∗
. The proof is
complete.
Corollary 2.3. Let {u
m
} be a sequence in W
1,p
R
N
satisfying 0 <ρ
R
N
|u
m
|
p
dx and such that
u
m
0 in W
1,p
R
N
. Then there exist a sequence {y
m
}⊂R
N
and a function 0
/
u ∈ W
1,p
R
N
such that up to a subsequence u
m
· y
m
uin W
1,p
R
N
.
Lemma 2.4. Let f ∈ CR
N
× R and suppose that
lim
|
s
|
→∞
f
x, s
|
s
|
p
∗
−1
0
2.16
uniformly in x ∈ R
N
and
f
x, s
≤ C
|
s
|
p−1
|
s
|
p
∗
−1
2.17
for all x ∈ R
N
and t ∈ R.Ifu
m
u
0
in W
1,p
R
N
and u
m
→ u
0
a.e. on R
N
,then
lim
m →∞
R
N
F
x, u
m
dx −
R
N
F
x, u
0
dx −
R
N
F
x, u
m
− u
0
dx
0,
2.18
where Fx, u
u
0
fx, tdt.
Proof. Let R>0. Applying the mean value theorem we have
R
N
F
x, u
m
dx
|x|≤R
F
x, u
m
dx
|x|≥R
F
x, u
0
u
m
− u
0
dx
|x|≤R
F
x, u
m
dx
|x|≥R
F
x, u
m
− u
0
f
x, θu
0
u
m
− u
0
u
0
dx,
2.19
8 Boundary Value Problems
where θ depends on x and R and satisfies 0 <θ<1. We now write
R
N
F
x, u
m
dx −
R
N
F
x, u
0
dx −
R
N
F
x, u
m
− u
0
dx
≤
|
x
|
≤R
F
x, u
m
− F
x, u
0
dx
|
x
|
≥R
F
x, u
0
dx
|
x
|
≤R
F
x, u
m
− u
0
dx
|
x
|
≥R
f
x, θu
0
u
m
− u
0
u
0
dx
.
2.20
For each fixed R>0
lim
m →∞
|x|≤R
F
x, u
m
− F
x, u
0
dx 0,
lim
m →∞
|x|≤R
F
x, u
m
− u
0
dx 0.
2.21
Applying 2.20 and the Holder inequality we get that
|
x
|
≥R
f
x, θu
0
u
m
− u
0
u
0
dx
≤ C
|
x
|
≥R
|
θu
0
u
m
− u
0
|
p−1
|
u
0
|
|
θu
0
u
m
− u
0
|
p
∗
−1
|
u
0
|
dx
≤ C
|
x
|
≥R
|
u
0
|
p
1/p
|x|≥R
|
θu
0
u
m
− u
0
|
p
p−1/p
C
|
x
|
≥R
|
u
0
|
p
∗
1/p
∗
|
x
|
≥R
|
θu
0
u
m
− u
0
|
p
∗
p
∗
−1/p
∗
.
2.22
Since {u
m
} is bounded in W
1,p
R
N
we see that
lim
R →∞
|
x
|
≥R
f
x, θu
0
u
m
− u
0
u
0
dx
0. 2.23
The result follows from 2.21 and 2.23.
Lemma 2.5. Functions I
λ
and I
∞
λ
are continuous on 0, ∞ and minimizing sequences for problems
I
λ
and I
∞
λ
are bounded in W
1,p
R
N
.
Boundary Value Problems 9
Proof. From condition A, we observe that for each ε>0 there exists C
ε
> 0 such that
F
u
,
|
F
x, u
|
≤ ε
R
N
|
u
|
p
dx ε
R
N
|
u
|
pp
2
/N
dx C
ε
R
N
|
u
|
α
dx,
2.24
where p<α<p p
2
/N and ε>0.
By the Holder and Sobolev inequalities we have
R
N
|
u
|
pp
2
/N
dx
R
N
|
u
|
pp
∗
−p−p
2
/N/p
∗
−pp
∗
p
2
/N/p
∗
−p
dx
≤
R
N
|
u
|
p
p
∗
−p−p
2
/N/p
∗
−p
R
N
|
u
|
p
∗
p
2
/N/p
∗
−p
≤ S
−1
R
N
|
u
|
p
p/N
R
N
|
∇u
|
p
dx,
2.25
where |u|
p
p
∗
≤ S
−1
|∇u|
p
p
.
Similarly we have
R
N
|
u
|
α
dx
R
N
|
u
|
pp
∗
−α/p
∗
−pp
∗
α−p/p
∗
−p
dx
≤
R
N
|
u
|
p
dx
p
∗
−α/p
∗
−p
R
N
|
u
|
p
∗
dx
α−p/p
∗
−p
≤ S
−p
∗
α−p/pp
∗
−p
R
N
|
u
|
p
dx
p
∗
−α/p
∗
−p
R
N
|
∇u
|
p
dx
p
∗
α−p
/p
p
∗
−p
.
2.26
Consequently by the Young inequality we have
R
N
|
u
|
α
dx ≤ η
R
N
|
∇u
|
p
dx K
η
R
N
|
u
|
α
dx
pp
∗
−α/p
2
p
∗
−p
2
−p
∗
α
2.27
for η>0, where Kη > 0 is a constant.
Because u ∈ W
1,p
∩ W
1,q
R
N
so we can by Sobolev embedding and λ
R
N
ax|u|
p
bx|u|
q
dx letting
λ
R
N
|u|
p
dx < ∞, we derive the following estimates for Iu and I
∞
u:
I
u
,I
∞
u
≥
1
p
− εS
−1
λ
p/N
− C
ε
η
R
N
|
∇u
|
p
dx
1
q
R
N
|
∇u
|
q
dx − ε
λ − K
η
C
ε
λ
pp
∗
−α/p
2
p
∗
−p
2
−p
∗
α
.
2.28
10 Boundary Value Problems
Choosing ε>0andη>0sothat
1
p
− εS
−1
λ
p/N
− C
ε
η>0,
2.29
we see that I
λ
and I
∞
λ
are finite and moreover minimizing sequences for problems I
λ
and
I
∞
λ
are bounded. It is easy to check that I
λ
and I
∞
λ
are continuous on 0, ∞.
We observe that I
∞
μ
≤ 0 for all μ>0. Indeed, let u ∈ C
∞
0
R
N
and
R
N
a
x
u
x/σ
σ
N/q
p
dx
R
N
b
x
u
x/σ
σ
N/q
q
dx μ,
2.30
then for each σ>0 we have
I
∞
μ
≤
1
pσ
pp/q−1N
R
N
|
∇u
|
p
dx
1
qσ
q
R
N
|
∇u
|
q
dx − σ
N
R
N
F
σ
−N/q
u
dx −→ 0
2.31
as σ →∞.
Lemma 2.6. Suppose that I
∞
λ
< 0 for some λ>0,thenI
∞
μ
/μ is nonincreasing on 0, ∞ and
lim
μ → 0
I
∞
μ
/μ0. Moreover there exists λ
∗
≤ λ such that
I
∞
μ
μ
>
I
∞
λ
λ
for μ ∈
0,λ
∗
. 2.32
Proof. We observe that
inf
I
∞
u
R
N
a
x
|
u
|
p
b
x
|
u
|
q
dx
inf
I
∞
u
x/σ
1/N
R
N
a
x/σ
1/N
u
x/σ
1/N
p
dx b
x/σ
1/N
u
x/σ
1/N
q
dx
.
2.33
So if
R
N
ax|u|
p
bx|u|
q
dx k and
R
N
ax/σ
1/N
|ux/σ
1/N
|
p
dx bx/σ
1/N
|ux/
σ
1/N
|
q
dx k then I
∞
ux I
∞
ux/σ
1/N
I
∞
k
.
We have that if σ>0andα>0with
R
N
ax|u|
p
bx|u|
q
dx α, then
R
N
a
x
σ
1/N
u
x
σ
1/N
p
dx b
x
σ
1/N
u
x
σ
1/N
q
dx σα, I
∞
u
x
σ
1/N
I
∞
σα
.
2.34
Boundary Value Problems 11
Consequently, for all α
1
> 0andα
2
> 0 we have
I
∞
α
1
inf
1
p
α
1
α
2
N−p/N
R
N
|
∇u
|
p
dx
1
q
α
1
α
2
N−q/N
R
N
|
∇u
|
q
dx −
α
1
α
2
R
N
F
u
dx;
R
N
a
x
|
u
|
p
b
x
|
u
|
q
dx α
2
.
2.35
If 0 <α
1
<α
2
, then for each ε>0 there exists u ∈ W
1,p
∩ W
1,q
R
N
with
R
N
ax|u|
p
bx|u|
q
dx α
2
such that
I
∞
α
1
ε>
1
p
α
1
α
2
N−p/N
R
N
|
∇u
|
p
dx
1
q
α
1
α
2
N−q/N
R
N
|
∇u
|
q
dx −
α
1
α
2
R
N
F
u
dx
≥
α
1
α
2
1
p
R
N
|
∇u
|
p
dx
1
q
R
N
|
∇u
|
q
dx −
R
N
F
u
dx
≥
α
1
α
2
I
∞
α
2
.
2.36
This inequality yields
I
∞
α
1
α
1
>
I
∞
α
2
α
2
.
2.37
Since I
∞
μ
≤ 0 for all μ>0, we see that
lim
μ → 0
I
∞
μ
μ
c ≤ 0.
2.38
We claim that c 0. Indeed, it follows from 2.36 and from the estimate obtained in the
Lemma 2.1 that for every 0 <μ<λthere exists an u
μ
∈ W
1,p
∩ W
1,q
R
N
,with
R
N
ax|u
μ
|
p
bx|u
μ
|
q
dx λ such that
I
∞
μ
μ
2
>
1
p
μ
λ
N−p/N
R
N
∇u
μ
p
dx
1
q
μ
λ
N−q/N
R
N
∇u
μ
q
dx −
μ
λ
R
N
F
u
μ
dx
≥
μ
λ
1
p
R
N
∇u
μ
p
dx
1
q
R
N
∇u
μ
q
dx −
R
N
F
u
μ
dx
≥
μ
λ
C
1
λ
R
N
∇u
μ
p
dx C
2
λ
R
N
∇u
μ
q
dx − C
3
λ
,
2.39
where C
1
λ > 0, C
2
λ > 0, and C
3
λ > 0 are constants. Hence
μ
2
≥
μ
λ
C
1
λ
R
N
∇u
μ
p
dx C
2
λ
R
N
∇u
μ
q
dx − C
3
λ
, 2.40
12 Boundary Value Problems
that is,
R
N
|∇u
μ
|
p
dx ≤ C
4
λ,
R
N
|∇u
μ
|
p
dx ≤ C
5
λ for some constant C
4
λ,C
5
λ > 0
independent of μ. We see that there exists ε
0
> 0 and a sequence μ
n
→ 0 such that
R
N
∇u
μn
p
dx ≥ ε
0
,
R
N
∇u
μn
q
dx ≥ ε
0
.
2.41
If
R
N
|∇u
μ
n
|
p
dx ≥ ε
0
then
R
N
|∇u
μ
n
|
q
dx ≥ η ≥ 0.
Then, using the fact that
R
N
Fu
μ
n
dx ≤ C for some constant C>0, we get
I
∞
μ
n
μ
n
μ
n
≥
1
p
λ
−N−p/N
μ
n
−p/N
ε
0
1
q
λ
−N−q/N
μ
n
−q/N
η −
C
λ
−→ ∞
2.42
as μ
n
→ 0 and this contradicts the fact that lim
μ → 0
I
∞
μ
/μc ≤ 0. Therefore
lim
μ → 0
R
N
∇u
μ
p
dx 0, lim
μ → 0
I
∞
μ
0,
2.43
when
R
N
|∇u
μ
n
|
p
dx ≥ ε
0
> 0 we can use the same method to obtain that lim
μ → 0
R
N
|∇u
μ
|
q
dx
0.
So
lim
μ → 0
R
N
∇u
μ
p
dx lim
μ → 0
R
N
∇u
μ
q
dx 0, lim
μ → 0
I
∞
μ
0
, 2.44
this implies that lim
μ → 0
R
N
Fu
μ
dx 0 and consequently
I
∞
μ
μ
μ ≥−
1
λ
R
N
F
u
μ
dx −→ 0.
2.45
This shows that lim
μ → 0
I
∞
μ
/μ0. Finally, we observe that lim
μ → 0
I
∞
μ
/μ0 >I
∞
λ
/λ
which obtain 2.32.
3. Proof of Main Theorems
Theorem 3.1. Suppose that I
∞
λ
< 0 for some λ>0, then there exists 0 <α
0
≤ λ such that problem
I
∞
α
0
has a minimizer. Moreover each minimizing sequence for I
∞
α
0
up to a translation is relatively
compact in W
1,p
∩ W
1,q
R
N
.
Proof. According to Lemma 2.6 the set
α
1
;
I
∞
α
α
>
I
∞
λ
λ
for each α ∈
0,α
1
3.1
Boundary Value Problems 13
is nonempty. We define
α
0
sup
α
1
;
I
∞
α
α
>
I
∞
λ
λ
for each α ∈
0,α
1
. 3.2
It follows from the continuity of I
∞
λ
that
0 <α
0
≤ λ,
I
∞
α
0
α
0
λ
I
∞
λ
< 0,
I
∞
α
>
α
λ
I
∞
λ
,
3.3
for all 0 <α<α
0
. This yields
I
∞
α
0
α
0
λ
I
∞
λ
α
0
− α
λ
I
∞
λ
α
λ
I
∞
λ
<I
∞
α
0
−α
I
∞
α
3.4
for each α ∈ 0,α
0
.
Let {u
m
}⊂W
1,p
∩ W
1,q
R
N
be a minimizing sequence for I
∞
α
0
. Since {u
m
} is bounded
we may assume that u
m
uin W
1,p
∩ W
1,q
R
N
, u
m
→ u a.e. on R
N
. First we consider the
case u ≡ 0. In this case by Lemma 2.2 u
m
→ 0forq<α<p
∗
or Corollary there exists a
sequence {u
m
}⊂R
N
such that u
m
· y
m
v
/
0inW
1,p
∩ W
1,q
R
N
.
In the first case lim
m →∞
R
N
Fu
m
dx 0 and consequently
I
∞
α
0
lim
m →∞
I
∞
u
m
lim
m →∞
1
p
R
N
|
∇u
m
|
p
dx
1
q
R
N
|
∇u
m
|
q
dx −
R
N
F
u
m
dx
≥ 0,
3.5
which is impossible. Hence u
m
· y
m
v
/
0inW
1,p
∩ W
1,q
R
N
holds and letting v
m
x
u
m
x y
m
from Brezis-Lieb lemma see 21, Lemma 1.32 we have
R
N
a
x
|
u
m
|
p
b
x
|
u
m
|
q
dx
R
N
a
x y
m
|
v
m
|
p
b
x y
m
|
v
m
|
q
dx
R
N
a
x y
m
|
v
|
p
b
x y
m
|
v
|
q
dx
R
N
a
x y
m
|
v
m
− v
|
p
b
x y
m
|
v
m
− v
|
q
dx o
1
.
3.6
We now show that
R
N
a
x y
m
|
v
|
p
b
x y
m
|
v
|
q
dx α
0
.
3.7
14 Boundary Value Problems
In the contrary case from Lemma 2.1 we have
0 <
R
N
a
x y
m
|
v
|
p
b
x y
m
|
v
|
q
dx < α
0
.
3.8
By 3.21 we have
lim
m →∞
R
N
a
x y
m
|
v
m
− v
|
p
b
x y
m
|
v
m
− v
|
q
dx −→ α
0
− λ,
λ
R
N
a
x y
m
|
v
|
p
b
x y
m
|
v
|
q
dx.
3.9
On the other hand, by Lemmas 2.1 and 2.4 we have
R
N
F
v
m
dx
R
N
F
v
dx
R
N
F
v
m
− v
dx o
1
,
R
N
|
∇v
m
|
p
|
∇v
m
|
q
dx ≥
R
N
|
∇v
|
p
|
∇v
|
q
dx
R
N
|
∇
v
m
− v
|
p
|
∇
v
m
− v
|
q
dx o
1
,
3.10
and this implies that
I
∞
α
0
≥ I
∞
v
I
∞
v
m
− v
o
1
≥ I
∞
λ
I
∞
α
0
−λ
0
o
1
.
3.11
Letting m →∞we get I
∞
α
0
≥ I
∞
λ
I
∞
α
0
−λ
0
which contradicts 3.4. Therefore
R
N
ax y
m
|v|
p
bx y
m
|v|
q
dx α
0
. It then follows from 3.6 that v
m
→ v in L
p
∩L
q
R
N
. By the Gagliardo-
Nirenberg inequality v
m
→ v in L
s
R
N
, q ≤ s<∞. Obviously this implies that I
∞
α
0
I
∞
v
I
∞
v·−y
m
and
R
N
ax|v·−y
m
|
p
bx|v·−y
m
|
q
dx α
0
. To complete the proof we show
that v
m
→ v in W
1,p
∩ W
1,q
R
N
. Indeed, we have
I
∞
α
0
1
p
R
N
|
∇v
m
|
p
dx
1
q
R
N
|
∇v
m
|
q
dx −
R
N
F
v
m
dx o
1
≥
1
p
R
N
|
∇v
|
p
dx
1
q
R
N
|
∇v
|
q
dx
1
p
R
N
|
∇v
m
− v
|
p
dx
1
q
R
N
|
∇v
m
− v
|
q
dx
−
R
N
F
v
dx
R
N
F
v
−
F
v
m
dx o
1
.
3.12
Since lim
m →∞
R
N
Fv−Fv
m
dx 0, we deduce from 3.12 that ∇v
m
→∇v in L
p
∩L
q
R
N
and hence v
m
→ v in W
1,p
∩ W
1,q
R
N
.
If u
/
0, we repeat the previous argument to show that I
∞
α
0
is attained.
Theorem 3.2. Suppose that Fx, t ≥ Ft on R
N
× R and that I
λ
< 0 for some λ>0, then the
infimum I
λ
0
is attained for some 0 <λ
0
≤ λ.
Boundary Value Problems 15
Proof. Since Fx, t ≥
Ft on R
N
× R we have I
μ
≤ I
∞
μ
for μ ≥ 0. We distinguish two cases: i
I
λ
I
∞
λ
< 0, ii I
λ
<I
∞
λ
.
Case i.ByTheorem 3.1 there exists λ
0
∈ 0,λ such that
I
∞
λ
0
I
∞
u
,
R
N
a
x
|
u
|
p
b
x
|
u
|
q
dx λ
0
for some u ∈ W
1,p
∩ W
1,q
R
N
.
3.13
Thus
I
λ
0
≤ I
u
1
p
R
N
|
∇
u
|
p
dx
1
q
R
N
|
∇
u
|
q
dx −
R
N
F
x, u
dx
≤
1
p
R
N
|
∇
u
|
p
dx
1
q
R
N
|
∇
u
|
q
dx −
R
N
F
u
dx I
∞
u
I
∞
λ
0
.
3.14
If I
λ
0
I
∞
λ
0
, then I also attains its infimum I
λ
0
at u. Therefore it remains to consider the case
I
λ
0
<I
∞
λ
0
. Consequently we need to prove the following claim.
If I
λ
<I
∞
λ
for some λ>0, then there exists α
0
∈ 0,λ such that problem I
α
0
has a
solution. This obviously completes the proof of case i and also provides the proof of case
ii.
By virtue of Lemma 2.5, I
β
I
∞
λ−β
is continuous for β ∈ 0,λ and also I
0
I
∞
0
0. If
I
λ
<I
∞
λ
for some λ>0, then there exists γ>0 such that
I
λ
<I
β
I
∞
λ−β
3.15
for all β ∈ 0,γ.Let
α
0
sup
γ; I
λ
<I
β
I
∞
λ−β
, for 0 ≤ β<γ
. 3.16
Then we have
I
λ
I
α
0
I
∞
λ−α
0
,
I
λ
<I
α
I
∞
λ−α
3.17
for 0 ≤ α<α
0
. This implies that
I
α
0
I
∞
λ−α
0
I
λ
<I
∞
λ
≤ 0,
3.18
and hence
I
α
0
<I
∞
λ
− I
∞
λ−α
0
≤ I
∞
α
0
≤ 0
, 3.19
we show that I
α
0
is attained by a u ∈ W
1,p
∩ W
1,q
R
N
and every minimizing sequence for
I
α
0
is relatively compact in W
1,p
∩ W
1,q
R
N
.Let{u
m
} be a minimizing sequence for I
α
0
. Since
16 Boundary Value Problems
{u
m
} is bounded, we may assume that u
m
u in W
1,p
∩ W
1,q
R
N
, u
m
u a.e. on R
N
.
Arguing indirectly we assume that
u ≡ 0onR
N
.Weseethat
lim
m →∞
B
0,R
|
F
x, u
m
|
dx lim
m →∞
B
0,R
F
u
m
dx 0
3.20
for each R>0. We now write
I
u
m
1
p
R
N
|
∇
u
m
|
p
dx
1
q
R
N
|
∇
u
m
|
q
dx −
R
N
F
x, u
m
dx
I
∞
u
m
R
N
F
u
m
− F
x, u
m
dx.
3.21
We show that
lim
m →∞
R
N
F
u
m
− F
x, u
m
dx 0.
3.22
Towards this end we write
R
N
F
u
m
− F
x, u
m
dx
≤
B
0,R
|
F
x, u
m
|
dx
B
0,R
F
u
m
dx
|x|≥R,|u
m
|≤δ
|x|≥R,δ≤|u
m
|≤1/δ
|x|≥R,|u
m
|>1/δ
F
u
m
− F
x, u
m
.
3.23
We now define the following quantities:
1
δ
sup
0<|t|<δ,x∈R
N
F
t
− F
x, t
|
t
|
p
R
sup
δ≤|t|≤1/δ,|x|≥R
F
t
− F
x, t
,
2
δ
sup
|t|≥1/δ
F
t
− F
x, t
|
t
|
pN/N−p
.
3.24
Boundary Value Problems 17
It follows from assumption A that lim
δ → 0
1
δlim
δ → 0
2
δ0andby
B lim
R →∞
R0 for each fixed δ>0. Inserting these quantities into 3.23 we derive
the following estimate:
R
N
F
u
m
− F
x, u
m
dx
≤
1
δ
R
N
|
u
m
|
p
dx
R
δ
p
R
N
|
u
m
|
p
dx
2
δ
R
N
|
u
m
|
pN/N−p
dx
B
0,R
|
F
x, u
m
|
dx
B
0,R
F
u
m
dx.
3.25
First letting m →∞, R →∞, and then δ → 0, relation 3.22 readily follows. Combining
3.21 and 3.22 we obtain
I
u
m
I
∞
u
m
o
1
, 3.26
which implies Iu
m
≥ I
∞
α
0
o1 and consequently I
α
0
≥ I
∞
α
0
and this contradicts 3.19.
Therefore 0 <
R
N
ax|u|
p
bx|u|
q
dx ≤ α
0
. Suppose that λ
R
N
ax|u|
p
bx|u|
q
dx < α
0
.
Writing
R
N
|
∇u
m
|
p
|
∇u
m
|
q
dx ≥
R
N
|
∇
u
|
p
|
∇u
|
q
dx
R
N
|
∇
u
m
− u
|
p
|
∇
u
m
− u
|
q
dx o
1
R
N
a
x
|
u
m
|
p
b
x
|
u
m
|
q
dx
R
N
a
x
|
u
|
p
b
x
|
u
|
q
dx
R
N
a
x
|
u
m
− u
|
p
b
x
|
u
m
− u
|
q
dx o
1
,
3.27
R
N
F
x, u
m
dx
R
N
F
x, u
dx
R
N
F
x, u
m
− u
dx o
1
, 3.28
we deduce that
I
α
0
≥ I
u
I
u
m
− u
o
1
. 3.29
By a similar method used to obtain 3.22 we also have
I
u
m
− u
I
∞
u
m
− u
o
1
. 3.30
Hence the last two relations yield
I
α
0
≥ I
u
I
∞
u
m
− u
o
1
≥ I
λ
I
∞
α
0
−λ
,
3.31
18 Boundary Value Problems
and this contradicts 3.19. Consequently
R
N
ax|u|
p
bx|u|
q
dx α
0
and 3.27 yields
u
m
→ u in L
p
∩ L
q
R
N
. By the Gagliardo-Nirenberg inequality we have u
m
→ u in L
s
R
N
,
q ≤ s<p
∗
. This obviously show that I
α
0
Iu and
R
N
ax|u|
p
bx|u|
q
dx α
0
;thatis,u is
a solution of problem I
α
0
. Finally, writing
I
α
0
≥
1
p
R
N
|
∇
u
|
p
dx
1
q
R
N
|
∇
u
|
q
dx
1
p
R
N
|
∇
u
n
− u
|
p
dx
1
q
R
N
|
∇
u
n
− u
|
q
dx
−
R
N
F
u
dx −
R
N
F
x, u
m
−
F
u
m
dx
R
N
F
u
− F
u
m
dx o
1
,
3.32
and using 3.22 we deduce from this that ∇u
m
→∇u in L
p
∩ L
q
R
N
and hence u
m
→ u in
W
1,p
∩ W
1,q
R
N
.
Theorem 3.3. Suppose that Fx, t ≥
Ft on R
N
× R and that Fζ > 0 for some ζ ∈ R,then
problem I
λ
has a minimizer for some λ>0.
Proof. The condition
Fζ > 0 for some ζ>0 implies that
R
N
Fuxdx > 0 for some u ∈
W
1,p
∩ W
1,q
R
N
. Letting vxux/σ, σ>0, we have
I
∞
v
σ
N
1
pσ
p
R
N
|
∇u
|
p
dx
1
qσ
q
R
N
|
∇u
|
q
dx −
R
N
F
u
x
dx
< 0
3.33
for σ>0sufficiently large. Hence there exists λ>0 such that I
λ
≤ I
∞
λ
< 0 and the result
follows from Theorem 3.2.
Remark 3.4. It is a standard argument that minimizers of I
μ
correspond to weak solutions of
problem 1.1 with λ appearing as a Lagrange multiplier. Such a λ is then called the principal
eigenvalue for problem 1.1.
Remark 3.5. If a ∈ L
N/p
R
N
, b ∈ L
N/q
R
N
, a, b < 0, we can use the similar method to study
it, where I
λ
inf{Iu; u ∈ W
1,p
∩ W
1,q
R
N
,
R
N
a
−
x|u|
p
b
−
x|u|
q
dx λ},λ > 0,a
−
−a, b
−
−b.
Acknowledgments
This project was supported by the National Natural Science Foundation of China no.
10871060 and the Natural Science Foundation of Educational Department of Jiangsu
Province no. 08KJB110005.
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