ON THE EXISTENCE OF SOLUTIONS TO A CLASS
OF p-LAPLACE ELLIPTIC EQUATIONS
Radial Solution and
Nonradial Solutions
HUI-MEI HE AND JIAN-QING CHEN
vol. 10, iss. 2, art. 59, 2009
Fujian Normal University
Fuzhou, 350007 P.R. China
EMail: hehuimei20060670@163.com
Hui-mei He and Jian-qing Chen
jqchen@fjnu.edu.cn
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Contents
Received:
19 September, 2008
Accepted:
07 December, 2008
Communicated by:
JJ
II
C. Bandle
J
I
2000 AMS Sub. Class.:
35J20.
Page 1 of 19
Key words:
p-Laplace elliptic equations, Radial solutions, Nonradial solutions.
Abstract:
We study the equation −∆p u + |x|a |u|p−2 u = |x|b |u|q−2 u with Dirichlet
boundary condition on B(0, R) or on RN . We prove the existence of the radial solution and nonradial solutions of this equation.
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1
Introduction and Main Result
3
2
Radial Solution
6
3
Nonradial Solutions
12
4
Necessary Conditions
16
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
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1.
Introduction and Main Result
Equations of the form
(
−∆t u + g(x)|u|s−2 u = f (x, u)
in Ω
u = 0,
on ∂Ω
(1.1)
have attracted much attention. Many papers deal with the problem (1.1) in the case of
t = 2, Ω = RN , s = 2, g large at infinity and f superlinear, subcritical and bounded
in x, see e.g. [1], [2] and [4]. The problem (1.1) with t = 2, Ω = B(0, 1), g(x) = 0
and f (x, u) = |x|b ul−1 was studied in [9]; in particular, it was proved that under
some conditions the ground states are not radial symmetric. The case t = 2, Ω =
B(0, 1) or on RN , g(x) = 1 and f (x, u) = |x|b |u|l−2 u was studied in [7]. The
problem (1.1) with t = s, Ω = RN , g(x) = V (|x|) and f (x, u) = Q(|x|)|u|l−2 u was
studied by J. Su., Z.-Q. Wang and M. Willem ([11], [12]). They proved embedding
results for functions in the weighted W 1,p (RN ) space of radial symmetry. The results
were then used to obtain ground state and bound state solutions of equations with
unbounded or decaying radial potentials.
In this paper, we consider the nonlinear elliptic problem

−∆p u + |x|a |u|p−2 u = |x|b |u|q−2 u in Ω


u > 0, u ∈ W 1,p (Ω),
(1.2)


u = 0,
on ∂Ω
and prove the existence of the radial and the nonradial solutions of the problem (1.2).
Here, ∆p u = div(|∇u|p−2 ∇u) is the p-Laplacian operator, and 1 < p < N, a ≥
0, b ≥ 0.
We denote by Wr1,p (RN ) the space of radially symmetric functions in
W 1,p (RN ) = u ∈ Lp (RN ) : ∇u ∈ Lp (RN ) .
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
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1,p
Wr,a
(RN ) is denoted by the space of radially symmetric functions in
Z
1,p
N
1,p
N
a
p
Wa (R ) = u ∈ W (R ) :
|x| |u| < ∞ .
RN
We also denote by Dr1,p (RN ) the space of radially symmetric functions in
n
o
Np
1,p
N
N
p
N
N
−p
D (R ) = u ∈ L
R : ∇u ∈ L R
.
Our main results are:
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
Theorem 1.1. If a ≥ 0, b ≥ 0, 1 < p < N and
Np
bp
+
,
N −p N −p
(p − 1)(q − p)
pb − a p +
< (q − p)(N − 1),
p
p < q < q̃ =
then the problem (1.2) has a radial solution.
Remark 1. In [8], Sirakov proves that the problem (1.2) with p = 2 has a solution
for
2N
4b
2 < q < q# =
−
.
N − 2 a(N − 2)
In [6], P. Sintzoff and M. Willem proved the existence of a solution of the problem (1.2) with
q
p = 2, q ≤ 2∗ , 2b − a 1 +
< (N − 1)(q − 2).
2
Theorem 1.1 extends the results of [6] to the general equation with a p−Laplacian
operator.
vol. 10, iss. 2, art. 59, 2009
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Theorem 1.2. Suppose that a ≥ 0, b ≥ 0, 1 < p < N and
p<q<
(p − 1)(q − p)
pb − a p +
p
Np
,
N −p
< (q − p)(N − 1),
aq < pb,
then for every R, problem (1.2) with Ω = B(0, R), R large enough has a radial and
a nonradial solution.
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
This paper is organized as follows: In Section 2, we study (1.2) in the case of
Ω = RN . We prove the existence of a radial least energy solution of (1.2) when
(p − 1)(q − p)
1 < p < N, p < q < q̃, pb − a p +
< (q − p)(N − 1).
p
In Section 3, we consider the existence of nonradial solutions of (1.2) with Ω =
B(0, R), R large enough. Finally, in Section 4, we consider necessary conditions
for the existence of solutions of (1.2).
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2.
Radial Solution
In this paper, unless stated otherwise, all integrals are understood to be taken over
all of RN . Also, throughout the paper, we will often denote various constants by the
same letter.
Lemma 2.1. Suppose that 1 < p < N. There exist AN > 0, such that, for every
p
1,p
u ∈ Wr,a
(RN ), u ∈ C(RN \{0}), for a ≥ p−1
(1 − N ), we have that
|x|
a(p−1)
N −1
+
p
p2
Z
a
p
p−1
Z
2
|x| |u|
|u(x)| ≤ AN
p
p
|∇u|
12
Hui-mei He and Jian-qing Chen
p
.
Proof. Since
p
p −1
d p a· p−1
du p−1
|u|2 2 · 2u · ra· p rN −1
|u| r p rN −1 =
2
dr
dr
p−1
p−1
p
+ |u| a ·
+ N − 1 ra· p −1 rN −1 ,
p
and
a≥
p
(1 − N ),
p−1
we get that
d p a· p−1
du p−1
|u| r p rN −1 ≥ pu|u|p−2 ra· p rN −1
dr
dr
and obtain
r
a· p−1
N −1
p
r
+∞
du p−1
|u(r)| ≤ AN
|u| S N −1 S a· p dS
dr
Zr
p−1
du
≤ AN |u|p−1 |x|a· p dx
dr
p
Z
p−1
Radial Solution and
Nonradial Solutions
vol. 10, iss. 2, art. 59, 2009
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Z
≤ AN
a
p−1
Z
p
p
|x| |u|
p1
p
|∇u|
.
It follows that
p
N −1+a· p−1
|x|
p
Z
|u(x)| ≤ AN
a
p
p−1
Z
p
p
|x| |u|
p1
|∇u|
,
Radial Solution and
Nonradial Solutions
and we have
|x|
Hui-mei He and Jian-qing Chen
a(p−1)
N −1
+
p
p2
Z
|u(x)| ≤ AN
|x|a |u|p
p−1
Z
2
p
|∇u|p
12
vol. 10, iss. 2, art. 59, 2009
p
.
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Lemma 2.2. If 1 < p < N, p ≤ r < NpN
, then for any u ∈ W 1,p (RN ), we have
−p
that
)
Z
N (r−p)
Z
N p+r(p−N
Z
p2
p2
r
p
p
|u| dx ≤ C
|∇u|
|u|
.
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Proof. The proof can be adapted directly from the Gagliardo-Nirenberg inequality.
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The following inequality extends the results of [5] to the general equation with
the p−Laplacian operator.
Lemma 2.3. For
1 < p < N,
p<q<
pN
+
N −p
c
N −1
p
+
a(p−1)
p2
,
a≥
p
(1 − N ),
p−1
Close
there exist BN,p,c such that for every u ∈ Dr1,p (RN ), we have
Z
|x|c |u|q dx ≤ BN,p,c
Z
|∇u|p
c
p(N −1)+a(p−1)
+ N2
p
2
cp
q−p− p(N −1)+a(p−1)
.
Proof. Using Lemma 2.1 and Lemma 2.2, we have
Z
|x|c |u|q dx
c
c
c
q−
Z a(p−1)
(N −1)/p+a(p−1)p−2
(N −1)/p+a(p−1)p−2
(N −1)/p+a(p−1)p−2
N −1
+
2
p
p
=
|x|
|u|
|u|
dx
Z
≤
a
p
p−1
2 ·
p
|x| |u|
Z
·
Z
=
p
c
(N −1)/p+a(p−1)p−2
N2 (q−p−
p
|∇u|
|x|a |u|p dx
Z
p
p
|∇u|
c
)
(N −1)/p+a(p−1)p−2
c(p−1)
p(N −1)+a(p−1)
Z
12 ·
|u|p
c
(N −1)/p+a(p−1)p−2
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
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Contents
Z
p
N2p + p−N
q−
2
p
|u|
N2p + p−N
2
p
Radial Solution and
Nonradial Solutions
p
p
c
(N −1)/p+a(p−1)p−2
cp2
q− p(N −1)+a(p−1)
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Z
·
|∇u|p
Z
≤ BN,p,c
c
p(N −1)+a(p−1)
+ N2
|∇u|p
p
2
cp
q−p− p(N −1)+a(p−1)
c
p(N −1)+a(p−1)
+ N2
p
2
cp
q−p− p(N −1)+a(p−1)
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.
Next, to prove Theorem 1.1, we consider the following minimization problem
Z p
a
p
m = m(a, b, p, q) =
inf
|∇u|
+
|x|
|u|
dx.
1,p
u∈Wr,a (RN )
R
|x|b |u|q dx=1
Theorem 2.4. If a ≥ 0, b ≥ 0, 1 < p < N and
Radial Solution and
Nonradial Solutions
Np
bp
p < q < q̃ =
+
,
N −p N −p
(p − 1)(q − p)
pb − a p +
< (q − p)(N − 1),
p
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
Title Page
then m(a, b, p, q) is achieved.
1,p
Proof. Let (un ) ⊂ Wr,a
(RN ) be a minimizing sequence for m = m(a, b, p, q) :
Z
|x|b |un |q dx = 1,
Z
(|∇un |p + |x|a |un |p ) dx → m.
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By going (if necessary) to a subsequence, we can assume that un * u in
Hence, by weak lower semicontinuity, we have
Z
(|∇u|p + |x|a |u|p ) dx ≤ m,
Z
|x|b |u|q dx ≤ 1.
1,p
Wr,a
(RN ).
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If c is defined by q =
Z
pN
N −p
+
pc
,
N −p
then c < b and it follows from Lemma 2.3 that
Z
b
q
b−c
|x| |un | dx ≤ ε
|x|c |un |q dx ≤ Cεb−c .
|x|≤ε
1,p
Since (un ) is bounded in Wr,a
(RN ). We deduce from Lemma 2.1 that
Z
Z
b
q
|x| |un | dx =
|x|b−a |un |q−p |x|a |un |p dx
|x|≥ 1ε
|x|≥ 1ε
b−a−(q−p)( Np−1 + a(p−1)
Z
)
p2
1
≤
C |x|a |u|p dx
ε
≤ Cε
a( q+1
−
p
(q−p)(N −1)
q
)−b+
p
p2
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By the Rellich theorem and Lemma 2.1,
Z
Z
b
q
1 ≥ |x| |un | dx ≥
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|x|b |un |q dx ≥ t.
ε≤|x|≤ 1ε
Finally
vol. 10, iss. 2, art. 59, 2009
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ε≤|x|≤ 1ε
b
Hui-mei He and Jian-qing Chen
.
So we get that, for every t < 1, there exists ε > 0, such that for every n,
Z
|x|b |un |q dx ≥ t.
R
Radial Solution and
Nonradial Solutions
q
|x| |u| dx = 1 and m = m(a, b, p, q) is achieved at u.
Now we will prove Theorem 1.1.
Proof. By Theorem 2.4, m is achieved. Then by the Lagrange multiplier rule, the
symmetric criticality principle (see e.g. [13]) and the maximum principle, we obtain
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a solution of

 −∆p υ + |x|a |υ|p−2 υ = λ|x|b |υ|q−2 υ,

υ > 0, υ ∈ W 1,p (RN ).
1
Hence u = λ q−p υ is a radial solution of (1.2), with λ = pq m.
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
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3.
Nonradial Solutions
In this section, we will prove Theorem 1.2. We use the preceding results to construct
nonradial solutions of problem (1.2) in the case Ω = B(0, R).
Consider
Z p
a
p
M = M (a, b, p, q) =
inf
|∇u|
+
|x|
|u|
dx.
1,p
N
a (R )
Ru∈W
|x|b |u|q dx=1
It is clear that M ≤ m, and using our previous results, we prove that M is achieved
under some conditions.
Theorem 3.1. If a ≥ 0, b ≥ 0, 1 < p < N and
p < q < q# =
pN
p2 b
−
,
N − p a(N − p)
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
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JJ
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then M (a, b, p, q) is achieved.
J
I
Proof. Let (un ) ⊂ Wa1,p (RN ) be a minimizing sequence for M = M (a, b, p, q) :
Z
|x|b |un |q dx = 1,
Z |∇un |p + |x|a |un |p dx → M.
Page 12 of 19
By going (if necessary) to a subsequence, we can assume that un * u in Wa1,p (RN ).
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Hence, by weak lower semicontinuity, we have
Z p
a
p
|∇u| + |x| |u| dx ≤ M,
Z
|x|b |u|q dx ≤ 1.
If c is defined by q =
pN
N −p
−
p2 c
,
a(N −p)
then c > b and
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
a
r= ,
c
s=
aN p
N −p
vol. 10, iss. 2, art. 59, 2009
aq − pc
are conjugate. It follows from the Hölder and Sobolev inequalities that
b−c Z
Z
1
b
q
|x| |un | dx ≤
|x|c |un |q dx
1
ε
|x|≥ ε
b−c Z
pc
pc
1
=
|x|c |un | a |un |q− a dx
ε
Z
r1 Z
1s
Np
c−b
a
p
≤ε
|x| |un | dx
|un | N −p dx
≤ Cεc−b .
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As in Theorem 2.4, for every t < 1, there exists ε > 0 such that, for every n,
Z
|x|b |un |q dx ≥ t.
|x|≤ 1ε
By the compactness of the Sobolev theorem in the bounded domain, for 1 < p <
N, p < q <
Np
,
N −p
Z
1≥
b
q
Z
|x|b |un |q dx ≥ t.
|x| |u| dx ≥
|x|≤ 1ε
Hence
R
|x|b |u|q dx = 1 and M = M (a, b, p, q) is achieved at u.
Radial Solution and
Nonradial Solutions
Now we will prove Theorem 1.2.
Proof. By Theorem 2.4, m(a, b, p, q) is positive. Since pb > aq, it is easy to verify
that M (a, b, p, q) = 0. Let us define
Z
p
a
p
M (a, b, p, q, R) =
inf
|∇u|
+
|x|
|u|
dx,
1,p
R u∈Wa (B(0,R))
b
q
B(0,R) |x| |u| dx=1
inf
1,p
u∈Wr,a (B(0,R))
b
q
B(0,R) |x| |u| dx=1
vol. 10, iss. 2, art. 59, 2009
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B(0,R)
Contents
Z
m(a, b, p, q, R) =
Hui-mei He and Jian-qing Chen
|∇u|p + |x|a |u|p dx.
B(0,R)
R
It is clear that, for every R > 0, M (a, b, p, q, R) and m(a, b, p, q, R) are achieved
and
lim M (a, b, p, q, R) = M (a, b, p, q) = 0,
R→∞
lim m(a, b, p, q, R) = m(a, b, p, q) > 0.
R→∞
Then from Theorem 1.1, we know that problem (1.2) with B(0, R) has a radial
solution.
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On the other hand, by the Lagrange multiplier rule, the symmetric criticality principle (see e.g.[13]) and the maximum principle, we obtain a solution of

a
p−2
b
q−2

 −∆p υ + |x| |υ| υ = λ|x| |υ| υ in B(0, R)
υ > 0, u ∈ W 1,p (B(0, R)),


υ = 0,
on ∂B(0, R).
1
q−p
Hence u = λ υ is a solution of (1.2), with λ = pq M (a, b, p, q, R). Thus, Problem (1.2) has a nonradial solution.
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
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4.
Necessary Conditions
In this section we obtain a nonexistence result for the solution of problem (1.2) using
a Pohozaev-type identity. The Pohozaev identity has been derived for very general
problems by H. Egnell [3].
Lemma 4.1. Let u ∈ W 1,p (RN ) be a solution of (1.2), then u satisfies
Z
Z
Z
N +a
N +b
N −p
p
a
p
|∇u| dx +
|x| |u| dx −
|x|b |u|q dx = 0.
p
p
q
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
Theorem 4.2. Suppose that
q̃ =
Np
pb
+
≤q
N −p N −p
or
N +a
N +b
≤
.
p
q
Then there is no solution for problem (1.2).
Proof. Multiplying (1.2) by u and integrating, we see that
Z
Z
b
q
|x| |u| dx =
|∇u|p + |x|a |u|p dx.
On the other hand, using Lemma 4.1, we obtain
Z
Z
N +a N +b
N −p N +b
p
−
|∇u| dx +
−
|x|a |u|p dx = 0.
p
q
p
q
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So, if u is a solution of problem (1.2), we must have
N −p
N +b
<
,
p
q
N +a
N +b
>
.
p
q
Remark 2. The second assumption of Theorem 2.4,
(q − p)(p − 1)
pb − a p +
< (q − p)(N − 1)
p
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
implies that
N +a
N +b
<
.
q
p
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References
[1] T. BARTSCH AND Z.-Q. WANG, Existence and multiplicity results for some
superlinear elliptic problem on RN , Comm. Partial Diff. Eq., 20 (1995), 1725–
1741.
[2] W. OMANA AND M. WILLEM, Homoclinic orbits for a class of Hamiltonian
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[4] P.H. RABINOWITZ, On a class of nonlinear Schrödinger equations, Z. Angew.
Math. Phys., 43 (1992), 270–291.
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
Title Page
p
[5] W . ROTHER, Some existence results for the equation −4u + k(x)u = 0,
Comm. Partial Differential Eq., 15 (1990), 1461–1473.
[6] P. SINTZOFF AND M. WILLEM, A semilinear elliptic equation on RN with
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49 (2002), 105–113.
[7] P. SINTZOFF, Symmetry of solutions of a semilinear elliptic equation with
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[9] D. SMETS, J. SU AND M. WILLEM, Non radial ground states for the Hénon
equation, Commun. Contemp. Math., 4 (2002), 467–480.
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Math. Phys., 55 (1977), 149–162.
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[11] J. SU, Z.-Q. WANG AND M. WILLEM, Weighted Sobolev embedding with
unbounded decaying radial potentials, J. Differential Equations, 238 (2007),
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[12] J. SU, Z.-Q. WANG AND M. WILLEM, Nonlinear Schrödinger equations with
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[13] M. WILLEM, Minimax Theorems, Birkhäuser, Boston, 1996.
Radial Solution and
Nonradial Solutions
Hui-mei He and Jian-qing Chen
vol. 10, iss. 2, art. 59, 2009
Title Page
Contents
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