Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 214289, 19 pages
doi:10.1155/2011/214289
Research Article
Multiple Solutions of p-Laplacian with
Neumann and Robin Boundary Conditions for
Both Resonance and Oscillation Problem
Jing Zhang and Xiaoping Xue
Department of Mathematics, Harbin Institute of Technology, Harbin 150025, China
Correspondence should be addressed to Jing Zhang, [email protected]
Received 29 June 2010; Revised 7 November 2010; Accepted 18 January 2011
Academic Editor: Sandro Salsa
Copyright q 2011 J. Zhang and X. Xue. This is an open access article distributed u nder the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We discuss Neumann and Robin problems driven by the p-Laplacian with jumping nonlinearities.
Using sub-sup solution method, Fuc
´
ık spectrum, mountain pass theorem, degree theorem together
with suitable truncation techniques, we show that the Neumann problem has infinitely many
nonconstant solutions and the Robin problem has at least four nontrivial solutions. Furthermore,
we study oscillating equations with Robin boundary and obtain infinitely many nontrivial
solutions.
1. Introduction
Let Ω be a bounded domain of R
n
with smooth boundary ∂Ω, we consider the following
problems:
i Neumann problem:
−Δ
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
|
p−2
u 0, on ∂Ω,
p
2
2 Boundary Value Problems
where Δ
p
u div|∇u|
p−2
∇u is the p-Laplacian operator of u with 1 <p<∞, α>0,
bx ∈ L
∞
∂Ω, bx ≥ 0, and bx
/
0on∂Ω, fx, 00fora.e.x ∈ Ω,and∂u/∂ν
denotes the outer normal derivative of u with respect to ∂Ω.Ourpurposeistoshow
the multiplicity of solutions to p
|
p
dx −
Ω
F
x, u
dx,
J
2
u
1
p
Ω
|
∇u
|
p
dx
α
p
Ω
|
fx, sds. The critical
points of functionals correspond to the weak solutions of problems. In Li 1 and Zhang et al.
2, the authors study the existence and multiple solutions of p
1
and p
2
using the critical
points theory for the semilinear case p 2. There also have been some papers dealing with the
quasilinear case p
/
2 using the critical point theory, and some existence results of solutions
have been generalized to this case in the work of Perera 3, Zhang et al. 4,andZhang-Li5.
Most of these papers use the minimax arguments, and nontrivial solutions are obtained with
the assumption that the nonlinearity is superlinear at 0. In this paper, we give the nontrivial
solutions of p
1
and p
2
with a jumping nonlinearity when the asymptotic limits of the
nonlinearity fall in the regions formed by the curves of the Fuc
´
ık spectrum. Our technique is
based on mountain pass theorem, computing the critical groups and Fuc
´
ıkspectrum.
Our general assumptions are the following.
f
1
There is constant C>0suchthatfx, t satisfies the following subcritical condi-
tions:
},wherea
i
,b
i
∈ R, i 1, 2, , which satisfy a
i
> 0, b
i
< 0
and a
i
∞, b
i
−∞as i →∞. And at the same time {a
i
}, {b
i
} satisfy
f
x, a
i
αa
p−1
i
,f
x, b
i
i
,a
i1
,wherei is an even number, i ≥ 0; fx, t < −α|t|
p−1
if t ∈ b
i1
,b
i
,
where i is an even number, i ≥ 0; fx, t > −α|t|
p−1
if t ∈ b
i1
,b
i
,wherei is an odd number,
i ≥ 1, for every x ∈ Ω.
f
3
For all t
/
a
i
,b
i
, f is C
1
; f
respectively.
Boundary Value Problems 3
f
4
Let a, bf
x, a
i
− α, f
−
x, a
i
− α for i is an even number, i ≥ 2. For a, b ∈ R
2
,
the problem
−Δ
p
u a
u − c
p−1
− b
i
x, t
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
0,t<0,
f
x, t
, 0 ≤ t ≤ a
i
,
f
x, a
i
,t>a
i
,
i
}, {b
i
}, i 0, 1, 2, In this paper, we mainly discuss whether it has many nonconstant
solutions and what their locations are.
Then we have the main results of this paper.
Theorem 1.1. Assume that (f
1
)–(f
5
)hold.Thenp
1
has infinitely many nonconstant solutions.
Moreover, if one chooses some order intervals which have two pairs of strict constant sub-sup solutions,
then p
1
has at least two nonconstant solutions in some order intervals.
Furthermore, if we assume that f
−
x, 0
/
f
x, 0 under the same c onditions as in
Theorem 1.1, we can have at least one sign-changing solution which is of mountain pass type
from the mountain pass theorem in order interval. When we discuss multiple solutions of
p
1
mountain pass type but with positive local degree.
For the Robin problem, if ∃M
1
> 0, M
2
> 0suchthatfx, M
1
0, fx, −M
2
0for
a.e. x ∈ Ω, then we give the following assumptions:
4 Boundary Value Problems
g
1
f ∈ C
1
Ω × R
1
\{0}, f
−
x, 0
/
f
x, 0,andmin{f
x, 0,f
u a
u
p−1
− b
u
−
p−1
, in Ω,
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
|
p−2
u 0, on ∂Ω,
1.6
only has trivial solution 0, where u
±
2
,whereE
2
{u ∈ W
1,p
Ω : u kϕ
1
tϕ
2
},
C C
2
/2bx
L
∞
∂Ω
.HereC is the imbedding
constant of Sobolev Trace Theorem see 6, M is large enough, ε
0
is small enough,
λ
2
is the second of the eigenvalue problems with Robin boundary value condition,
and ϕ
1
, ϕ
2
are the first and the second eigenfunction, respectively.
p
1
dx, |t|≥M,whereϕ
1
is the first
eigenvalue of the Robin problem and
Ω
ϕ
p
1
dx 1.
Then we have the following.
Theorem 1.5. Assume that f is satisfied as in Theorem 1.3 and (f
2
), (F
), one can get infinitely
many nontrivial solutions of problem p
2
. Some of them are minimum points; others are mountain
pass points.
2. Preliminaries
Now we recall the notion of critical groups of an isolated critical point u of a C
1
functional J
briefly. Let U ⊂ M be an isolated neighborhood of u such that there are no critical points of J
in U \{u}; M is a Banach space. The critical groups of u are defined as
C
q
by H
q
X. Assume that J ∈ C
2
M, R, and a critical point u is called nondegenerate if the
Hessian J
u at this point has a bounded inverse. Let u be a nondegenerate critical point of
J; we call the dimension of the negative space corresponding to the spectral decomposition
of J
u, that i s, the dimension of the subspace of negative eigenvectors of J
u,theMorse
index of u, and denote it by indJ
u.IfC
1
J, u
/
0, then we call an isolated critical point u
of J as a mountain pass point. For the details, we refer to 7.
We have the following basic facts on the critical groups for an isolated critical point
of J.
a Let u be is an isolated minimum point of J,thenC
q
J, uδ
q0
G.
b If J ∈ C
, in Ω,
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
|
p−2
u 0, on ∂Ω,
2.2
with u
≤ u a.e. in Ω,wheregx, s is a Carath
´
eodory function on Ω × R with the property that, for
any s
0
> 0, there exists a constant A such that |gx, s|≤A for a.e. x ∈ Ω and all s ∈ −s
0
,s
0
.
Consider the associated functional
Φ
Ω which is is uniformly convex 1 <p<∞ and
equipped with the norm u
Ω
|∇u|
p
dxmα
Ω
|u|
p
dx
1/p
.LetE be a Hilbert space and
P
E
⊂ E a closed convex cone such that X is densely embedded in E. Assume that P X ∩ P
E
,
P has nonempty interior
˙
P and any order interval is bounded. It is well known that PS
condition implies the compactness of the critical set at each level c ∈ R,onthecaseofthe
above condition. Then we assume the following:
J
1
J ∈ C
2
E, R and satisfies PS condition in E and deformation property in X;
6 Boundary Value Problems
is a pair of strict supersolution of ∇J 0. v
0
<v
1
is
a subsolution of ∇J 0. Suppose that v
0
,v
1
and v
0
,v
2
are admissible invariant sets for J.IfJ
has a local strict minimizer w in v
0
,v
2
\ v
0
,v
1
.ThenJ has mountain pass points u
0
in v
0
,v
2
\
v
∅.ThenJ has a mountain pass point u
0
,u
0
∈ v
1
,ω
2
\ v
1
,v
2
∪ ω
1
,ω
2
.
More precisely, let v
0
be the maximal minimizer of J in v
1
,v
2
and ω
0
the minimal m inimizer of J in
ω
1
,ω
2
p
u is strictly increasing in u. The assumption is not
essential but is assumed for simplicity. If such m does not exist then we can approximate f
by a sequence of functions so that m as above exists, and obtain the solutions by passing to
limits. For m>α, we need the operator
A
m
: X −→ X, A
m
u
−Δ
p
mg
p
·
−1
x, u
mg
p
|
u
|
p−2
u mg
p
u
2.5
is strongly orderpreserving. Fr om the above discussion, we have the mountain pass theorem
in order intervals of J
1
and J
2
.
Next, let us recall some notions and known results on Fuc
´
ıkspectrum.
The Fuc
´
ıkspectrumofp-Laplacian on W
1,p
Ω is defined as the set Σ
p
of those points
a, b ∈ R
2
for which the problem
l
where {λ
l
}
l∈N
are the distinct eigenvalues of −Δsee, e.g.,
12. It was shown in Schechter 13 that Σ
2
contains continuous and strictly decreasing
curves C
l
1
, C
l
2
through λ
l
,λ
l
such that the points in the square Q
l
λ
l−1
,λ
l1
2
that are
either below the lower curve C
l
ık 12 in connection with jumping nonlinearities. A first nontrivial curve C
2
in Σ
p
through λ
2
,λ
2
that is continuous, strictly decreasing, and asymptotic to λ
1
× R and R × λ
1
at infinity was constructed and variationally characterized by a mountain-pass procedure in
Cuesta et al. 15.
Consider the problem
−Δ
p
u a
u − c
p−1
− b
u − c
x
|
u
|
p−2
u 0, on ∂Ω,
2.8
from the variational point of view; solutions of 2.7 and 2.8 are the critical points of the
functional
I
1
u
I
1
u, a, b
Ω
|
∇u
|
p
− a
|
∇u
|
p
− a
u
p
− b
u
−
p
dx
1
p
∂Ω
b
x
|
u
|
p
ds,
and
C
12
λ
1
× λ
1
, ∞ ∪ λ
1
, ∞ × λ
1
.
8 Boundary Value Problems
Lemma 2.6. i If a, b lies below C
11
,thenC
q
I, cδ
q0
Z.
ii If a, b lies between C
11
and C
12
,thenC
q
I, c0 for all q.
iii If a, b lies between C
12
and C
u − c
p
,u∈ X,
2.10
and
I
s
is the restriction of I
s
to the C
1
manifold
S
u ∈ X :
Ω
|
u − c
|
p
1
.
2.11
u ∈ S :
I
s
≤ d
2.13
are related by
I
d
∩ S
I
db
a−b
.
2.14
Lemma 2.7. If a, b does not belong to Σ
p
,then
C
q
I, c
∼
⎧
replaced by
b
{u ∈ S :
I
a−b
u >b}.
Boundary Value Problems 9
3. The Proof of the Main Results
Let
f
i
x, t
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
0,t<0,
f
x, t
1
p
Ω
|
∇u
|
p
dx
α
p
Ω
|
u
|
p
dx −
Ω
F
i
x, u
dx.
3.1
It is well known that critical points of J
E, R. We can discuss similar case for b
i
.
Next, we give the relation of the solutions of p
and the solutions of p
1
,thatis,
Lemma 3.2 below. In order to prove Lemma 3.2, we firstly give the comparison principle.
Let
L
p
: −Δ
p
a
x
|
u
|
p−2
u,
λ
1,p
a
inf
3.2
Lemma 3.1 comparison principle see 17. Assume a ∈ L
∞
Ω, λ
1,p
a > 0.TheL
p
u ∈
L
∞
Ω with u|
∂Ω
∈ C
1α
∂Ω,andL
p
u ≤ 0 with u ∈ W
1,p
Ω ∩ L
∞
Ω,thenu ≤ 0.
Lemma 3.2. If u
i
x is a solution of p
,thenu
i
x is also a solution of p
1
and satisfies 0 ≤
, on ∂U
i
,
3.3
where −Δ
p
u f
i
x, u − α|u|
p−2
u fx, a
i
− α|u|
p−2
u ≤ fx, a
i
− αa
p−1
i
0 ,x ∈ U
i
by the
definition of f
i
x, u. By the comparison principle, we can conclude that u
i
x ≤ 0inU
i
.Itis
a contradiction, so we have that U
completes the proof of the lemma.
Remark 3.3. From the above discussion, by applying Lemma 3.2, we know that solutions of
p
are also the solutions of p
1
if we want to pr ove Theorem 1.1, we only need to prove that
p
has infinitely many nonconstant solutions under the assumptions as in Theorem 1.1 and
p
has two nonconstant solutions in every order interval.
Theorem 3.4. There ar e infinitely many nonconstant solutions of p
. Moreover, if there exists some
order intervals which have two pairs of strict constant sub-sup solutions, then there are at least two
nonconstant solutions in these order intervals.
Proof. We treat the case of a
i
; the other case of b
i
is proved by a similar argument.
If f
2
holds, then
−Δ
p
a
i
for each k, k 1, 2, ,i/2.
Now, we study the order interval u
1
, u
3
in X which includes two suborder intervals
u
1
, u
1
and u
3
, u
3
, a
2
∈ u
1
, u
3
.
We infer that J
1i
u satisfies deformation properties and is bounded from below on
u
1
, u
3
and so we get a mountain pass point u
1
u f
i
x, u
− α
|
u
|
p−2
u, in Ω,
∂u
∂ν
0, on ∂Ω,
3.5
where f
i
∈ CΩ × R and as u → a
2
we have
f
i
x, u
− α
|
u
|
p−2
u − a
2
−
p−1
◦
|
u − a
2
|
p−1
.
3.6
We take a f
i
x, a
2
− α,b f
i−
x, a
2
− α, then from assumption f
4
and the definition of
Σ
2
C
q
J
1i
,u
1
,andwe
have u
1
/
a
2
.
2 Denote
J
a−b
u
Ω
|
∇u
|
p
−
a − b
|
p
1
,
3.9
where a f
i
x, a
2
− α,b f
i−
x, a
2
− α as shown above.
From f
4
, we know that a, b does not belong to Σ
p
,andif
J
a−b
u >b,a.e.
J
b
a−b
1
, and we have u
1
/
a
2
.
Similarly, applying the mountain pass theorem in order interval to u
3
, u
5
which
contain two sub-order intervals u
3
, u
3
and u
5
, u
5
, we get a mountain pass point u
2
and
prove that C
q
J
1i
,a
4
C
i
, we have the computing formular
deg
id − K
i
,B
u
1
,r
, 0
−1, 3.11
where r>0 is small enough, K
i
−Δ
p
m αg
p
·
−1
f
∗
i
|
X
: X → X is of class C
0
l
.
3.12
Furthermore, for minimum points a
1
, a
3
,
C
q
J
1i
,a
1
∼
δ
q0
G, C
q
J
1i
,a
3
∼
deg
id − K
i
,
u
3
, u
3
, 0
deg
id − K
i
,B
a
2
,r
, 0
deg
id − K
i
∗
1
/
u
1
and is nonconstant.
Similarly, we can discuss the order interval u
3
, u
5
, and we get another critical point
u
∗
2
/
u
2
. We let the procedure go on.
This completes the proof of Theorem 3.4.
Thus, we prove that the conclusion of Theorem 1.1 holds.
The Proof of Corollaries 1.2 and 1.4.
Proof. See Theorem 3.5 of Li 1.
Proof of Theorem 1.3. From the variational point of view, solutions of p
2
are the critical points
of the functional
J
2
u
p
ds −
Ω
F
x, u
dx,
3.15
defined on X : W
1,p
Ω,whereFx, u
u
0
fx, sds.
We show that J
2
belongs to C
1
X, R.Infact,weset
J
21
u
1
p
∂Ω
b
x
|
u
|
p
ds.
3.16
Under the condition f
1
,itiswellknownthatJ
21
is a C
1
-functional. Next, we consider J
22
.If
we let u, v ∈ X,0< |t| < 1,
J
22
u tv
− J
22
p−q
|
v
|
q
ds
−→
∂Ω
b
x
|
u
|
p−2
uv ds,
t −→ 0
.
3.17
So we have that J
22
has a Gateaux derivative and J
22
u,v
∂Ω
∂Ω
b
x
|
u
n
|
p−2
u
n
−
|
u
|
p−2
u
vds
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
cb
L
∞
∂Ω
∂Ω
|
u
n
|
|
u
|
p−2
|
u
n
− u
||
v
u
n
− u
L
p
∂Ω
Tv
L
p
∂Ω
if p ≥ 2,
cb
L
∞
∂Ω
T
u
n
− u
p−1
L
p
∂Ω
Tv
,v
≤
⎧
⎪
⎨
⎪
⎩
cb
L
∞
∂Ω
T
u
n
− u
L
p
∂Ω
Tv
L
p
∂Ω
if p ≥ 2,
cb
L
− u
W
1,p
Ω
v
W
1,p
Ω
if p ≥ 2,
cb
L
∞
∂Ω
u
n
− u
p−1
W
1,p
Ω
v
W
1,p
Ω
if p<2,
3.19
where 1/p 1/p
1, T : W
1,p
≤
⎧
⎨
⎩
c
|
u
|
|
v
|
p−2
|
u − v
|
if p ≥ 2,
c
|
u − v
|
p−1
if p<2,
3.20
which hold for a convenient c>0, u, v ∈ R
n
.So
J
L
∞
∂Ω
u
n
− u
p−1
W
1,p
Ω
if p<2.
−→ 0,
n −→ ∞
. 3.21
So J
22
u is continuous and J
2
∈ C
1
X, R.
14 Boundary Value Problems
Consider the truncated functions
f
x, t
u
1
p
Ω
|
∇u
|
p
dx
α
p
Ω
|
u
|
p
dx
1
p
∂Ω
b
x
|
u
Now, we construct the sub-sup solutions of p
2
. It is easy to see that M
1
is a constant
sup-solution of p
2
and −M
2
is a constant subsolution. Moreover, we consider εϕ
1
for all
ε>0 small enough. From 14 we know that ϕ
1
x > 0, x ∈ Ω. In fact, with u : εϕ
1
,byg
1
we have
−Δ
p
u α
u
p−2
u − f
⎦
≤ 0. 3.24
Furthermore, ϕ
1
∈ W
1,p
Ω ∩ L
∞
Ω satisfies −Δ
p
ϕ
1
λϕ
p−1
1
in the weak sense, then the
regularity theory for the p-Laplacian e.g., 19 implies ϕ
1
∈ C
1,α
Ω for some α αn, p ∈
0, 1.Moreoverϕ
1
≥ 0. In addition, by the strong maximum principle of 20 and ϕ
1
/
0, then
ϕ
1
> 0inΩ and ∂ϕ
is a pair of strict sub-sup solutions.
Now we study the order interval −M
2
,M
1
in X which includes two suborder
intervals −M
2
, −εϕ
1
and εϕ
1
,M
1
.ByLemma 2.2, there exists weak solutions of p
2
relative minimum points u
2
, u
3
in −M
2
, −εϕ
1
and εϕ
1
,M
1
, respectively. We can infer
2
, we know that the left and the right derivatives of
f at 0 are
different, we consider the problem
−Δ
p
u
f
x, u
− α
|
u
|
p−2
u, in Ω,
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
u
p−1
−
f
−
x, 0
− α
u
−
p−1
◦
|
u
|
p−1
. 3.26
We take a
by Lemma 2.6iv.Inthiscase,C
1
J,00, so C
q
J,0 C
q
J,u
1
,wehaveu
1
/
0.
2 Denote
J
a−b
u
Ω
|
∇u
|
S
u ∈ X :
Ω
|
u
|
p
1
,
3.29
where a
f
x, 0 − α, b
f
−
x, 0 − α as shown above.
From g
2
, we know that a, b does not belong to Σ
p
,if
1
, and we have u
1
/
0.
Now, we discuss the solutions in −M
2
,M
1
more deeply. We already have four
solutions 0, u
1
, u
2
, u
3
,whereu
1
is the mountain pass point and u
2
, u
3
are the local minimum
points of J
2
. For the minimum points u
2
, u
3
,wehave
id −
K, B
u
1
,r
, 0
−1, 3.32
where r>0 is small enough,
K −Δ
p
m αg
p
·
−1
f
∗
|
X
: X → X is of class C
0
and
16 Boundary Value Problems
strongly order preserving, f
∗
−M, M
, 0
deg
id −
K,
−M, −εϕ
1
, 0
deg
id −
K,
εϕ
1
,M
, 0
deg
.
3.34
It is impossible. From the above discussion, we conclude that there must exist another critical
point u
∗
∈ −M
2
,M
1
, which satisfies u
∗
/
u
1
and is nontrivial.
This completes the proof of Theorem 1.3.
Proof of Theorem 1.5. Consider the truncated function
f
i
x, t
⎧
⎪
⎪
⎪
⎨
⎪
⎪
|
∇u
|
p
dx
α
p
Ω
|
u
|
p
dx
1
p
∂Ω
b
x
|
u
|
p
ds −
Ω
F
i
, in Ω,
|
∇u
|
p−2
∂u
∂ν
b
x
|
u
|
p−2
u 0, on ∂Ω.
3.37
Boundary Value Problems 17
By the standard argument w e know that J
i
satisfies J
1
–J
3
and the order intervals consisted
by sub-super-solutions are admissible invariant set of J
i
. Taking v
0
−M
2
p
Ω
∇ϕ
1
p
dx
α
p
t
p
1
Ω
ϕ
1
p
dx
t
p
1
p
∂Ω
b
p
dx −
λ
1
α ε
0
C
t
p
1
p
Ω
ϕ
1
p
dx < J
2
u
1
.
3.38
If we take v
2
1
3.39
which implies that J
i
u has a minimizer u
2
∈ v
0
,v
2
\ v
0
,v
1
such that J
i
u
2
<J
i
u
1
.By
Lemma 2.3 we get a mountain pass point u
3
.Moreover,v
0
<u
i
2
t
2
ϕ
1
<J
2
u
2
. 3.40
If we take v
2
a
n
2
>t
2
ϕ
1
,wheren
2
<i,then
J
i
t
1
such that J
i
u
4
<J
i
u
2
.By
Lemma 2.3 we get a mountain pass point u
5
.Moreover,v
0
<u
i
<v
2
, i 1, 2, 3, 4, 5, and u
i
are
all positive. Continue making the procedure we obtain the result.
The proof is complete.
Corollary 3.5. Moreover, p
1
has infinitely many nonconstant negative energy solutions {u
k
},
which are mountain pass types, if the conditions as in Theorem 1.1 hold and J
1
2k1
,
W
1
u
2k−1
, u
2k−1
, W
2
u
2k1
, u
2k1
, c
∗
Ja
2k
, k 1, 2, Wediscusstheproblem
in W which have two minimum points a
2k−1
and a
2k1
.Wehavethata
2k−1
and a
2k1
are in
the same radial direction A {ke
1
2k
,whereγ
∗
is a special path between
a
2k−1
and a
2k1
, which is a path of radial direction A {ke
1
| k ∈ R}.Soa
2k
is a mountain
pass point. But according to assumptions f
3
and f
4
, we know that C
1
J
1
,a
2k
0, l
/
2,
that is, a
2k
is not a mountain pass type. This is a contradiction. We draw the conclusion.
18 Boundary Value Problems
J
1i
,u
1
,
C
q
J,0
∼
⎧
⎨
⎩
Z, q 1,
0,q
/
1,
∼
C
q
J,u
1
2
{u ∈ E |
u k
1
e
1
k
2
e
2
}, e
1
,e
2
are the first and the second eigenfunctions of −Δ
p
α} with
Neumann boundary, respectively, for all k
1
,k
2
∈ R, e
1
e
2
1, ε
0
> 0, and M
is large enough,
then under f
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Boundary Value Problems 19
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