Volume 10 (2009), Issue 2, Article 59, 8 pp.
ON THE EXISTENCE OF SOLUTIONS TO A CLASS OF p-LAPLACE ELLIPTIC
EQUATIONS
HUI-MEI HE AND JIAN-QING CHEN
F UJIAN N ORMAL U NIVERSITY
F UZHOU , 350007 P.R. C HINA
hehuimei20060670@163.com
jqchen@fjnu.edu.cn
Received 19 September, 2008; accepted 07 December, 2008
Communicated by C. Bandle
A BSTRACT. 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.
Key words and phrases: p-Laplace elliptic equations, Radial solutions, Nonradial solutions.
2000 Mathematics Subject Classification. 35J20.
1. I NTRODUCTION AND M AIN R ESULT
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.
255-08
2
H UI -M EI H E AND J IAN -Q ING C HEN
In this paper, we consider the nonlinear elliptic problem

−∆p u + |x|a |u|p−2 u = |x|b |u|q−2 u


u > 0, u ∈ W 1,p (Ω),
(1.2)


u = 0,
in Ω
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 ) .
1,p
Wr,a
(RN ) is denoted by the space of radially symmetric functions in
Z
N
1,p
N
a
p
1,p
|x| |u| < ∞ .
Wa (R ) = u ∈ W (R ) :
RN
We also denote by Dr1,p (RN ) the space of radially symmetric functions in
n
o
Np
D1,p (RN ) = u ∈ L N −p RN : ∇u ∈ Lp RN .
Our main results are:
Theorem 1.1. If a ≥ 0, b ≥ 0, 1 < p < N and
Np
bp
p < q < q̃ =
+
,
N −p N −p
(p − 1)(q − p)
pb − a p +
< (q − p)(N − 1),
p
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
4b
2N
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.
Theorem 1.2. Suppose that a ≥ 0, b ≥ 0, 1 < p < N and
Np
p<q<
,
N −p
(p − 1)(q − p)
pb − a p +
< (q − p)(N − 1), aq < pb,
p
then for every R, problem (1.2) with Ω = B(0, R), R large enough has a radial and a nonradial
solution.
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
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 59, 8 pp.
http://jipam.vu.edu.au/
R ADIAL S OLUTION AND N ONRADIAL S OLUTIONS
3
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).
2. R ADIAL S OLUTION
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 u ∈
p
1,p
Wr,a
(RN ), u ∈ C(RN \{0}), for a ≥ p−1
(1 − N ), we have that
Z
p−1
Z
12
a(p−1)
p2
p
N −1
+
p
a
p
p2
|x| p
|x| |u|
|∇u|
.
|u(x)| ≤ AN
Proof. Since
p
p −1
d p a· p−1
du p−1
N −1
p
|u| r
r
=
|u|2 2 · 2u · ra· p rN −1
dr
2
dr
p−1
p−1
p
+ |u| a ·
+ N − 1 ra· p −1 rN −1 ,
p
and
p
a≥
(1 − N ),
p−1
we get that
du p−1
d p a· p−1
|u| r p rN −1 ≥ pu|u|p−2 ra· p rN −1
dr
dr
and obtain
Z +∞
du p−1
a· p−1
r p rN −1 |u(r)|p ≤ AN
|u|p−1 S N −1 S a· p dS
dr
r
Z
du p−1
≤ AN |u|p−1 |x|a· p dx
dr
Z
p−1
Z
p1
p
a
p
p
|∇u|
.
≤ AN
|x| |u|
It follows that
p
N −1+a· p−1
|x|
Z
p
|u(x)| ≤ AN
a
p
p−1
Z
p
|x| |u|
p
p1
|∇u|
,
and we have
|x|
a(p−1)
N −1
+
p
p2
Z
|u(x)| ≤ AN
a
p
p−1
Z
2
|x| |u|
p
p
12
|∇u|
p
.
Lemma 2.2. If 1 < p < N, p ≤ r <
Z
r
|u| dx ≤ C
pN
,
N −p
Z
then for any u ∈ W 1,p (RN ), we have that
p
|∇u|
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 59, 8 pp.
N (r−p)
Z
2
p
p
|u|
)
N p+r(p−N
2
p
.
http://jipam.vu.edu.au/
4
H UI -M EI H E AND J IAN -Q ING C HEN
Proof. The proof can be adapted directly from the Gagliardo-Nirenberg inequality.
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
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
Z |x|
=
Z
≤
a(p−1)
N −1
+
p
p2
a
p
p−1
2 ·
p
|x| |u|
Z
·
Z
=
|∇u|p
·
|∇u|p
Z
≤ BN,p,c
c
(N −1)/p+a(p−1)p−2
c
(N −1)/p+a(p−1)p−2
N2 (q−p−
p
|x|a |u|p dx
Z
c
c
q−
(N −1)/p+a(p−1)p−2
(N −1)/p+a(p−1)p−2
|u|
|u|
dx
Z
p
c
p(N −1)+a(p−1)
+ N2
p
|u|p
12 ·
p
|∇u|
c
)
(N −1)/p+a(p−1)p−2
c(p−1)
p(N −1)+a(p−1)
Z
|∇u|p
Z
|u|p
N2p + p−N
2
p
N2p + p−N
2
p
2
cp
q−p− p(N −1)+a(p−1)
c
p(N −1)+a(p−1)
+ N2
p
2
c
(N −1)/p+a(p−1)p−2
p
p
2
q−
c
(N −1)/p+a(p−1)p−2
cp
q− p(N −1)+a(p−1)
cp
q−p− p(N −1)+a(p−1)
.
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
Np
bp
+
,
N −p N −p
(p − 1)(q − p)
pb − a p +
< (q − p)(N − 1),
p
p < q < q̃ =
then m(a, b, p, q) is achieved.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 59, 8 pp.
http://jipam.vu.edu.au/
R ADIAL S OLUTION AND N ONRADIAL S OLUTIONS
5
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.
1,p
By going (if necessary) to a subsequence, we can assume that un * u in Wr,a
(RN ). Hence, by
weak lower semicontinuity, we have
Z
(|∇u|p + |x|a |u|p ) dx ≤ m,
Z
|x|b |u|q dx ≤ 1.
If c is defined by q =
pN
N −p
+
Z
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
.
So we get that, for every t < 1, there exists ε > 0, such that for every n,
Z
|x|b |un |q dx ≥ t.
ε≤|x|≤ 1ε
By the Rellich theorem and Lemma 2.1,
Z
Z
b
q
1 ≥ |x| |un | dx ≥
|x|b |un |q dx ≥ t.
ε≤|x|≤ 1ε
Finally
R
|x|b |u|q 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 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.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 59, 8 pp.
http://jipam.vu.edu.au/
6
H UI -M EI H E AND J IAN -Q ING C HEN
3. N ONRADIAL S OLUTIONS
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# =
p2 b
pN
−
,
N − p a(N − p)
then M (a, b, p, q) is achieved.
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 p
a
p
|∇un | + |x| |un | dx → M.
By going (if necessary) to a subsequence, we can assume that un * u in Wa1,p (RN ). 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
a
r= ,
c
s=
aN p
N −p
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
1s
Z
r1 Z
Np
c−b
a
p
|un | N −p dx
≤ε
|x| |un | dx
≤ Cεc−b .
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
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 59, 8 pp.
http://jipam.vu.edu.au/
R ADIAL S OLUTION AND N ONRADIAL S OLUTIONS
7
By the compactness of the Sobolev theorem in the bounded domain, for 1 < p < N, p < q <
Np
,
N −p
Z
Z
b
q
|x|b |un |q dx ≥ t.
1 ≥ |x| |u| dx ≥
|x|≤ 1ε
Hence
R
|x|b |u|q dx = 1 and M = M (a, b, p, q) is achieved at u.
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
M (a, b, p, q, R) =
inf
|∇u|p + |x|a |u|p dx,
1,p
R u∈Wa (B(0,R))
b
q
B(0,R) |x| |u| dx=1
B(0,R)
Z
m(a, b, p, q, R) =
inf
1,p
R
u∈Wr,a (B(0,R))
|x|b |u|q dx=1
|∇u|p + |x|a |u|p dx.
B(0,R)
B(0,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.
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
Hence u = λ q−p υ is a solution of (1.2), with λ = pq M (a, b, p, q, R). Thus, Problem (1.2) has a
nonradial solution.
4. N ECESSARY C ONDITIONS
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 −p
N +a
N +b
p
a
p
|∇u| dx +
|x| |u| dx −
|x|b |u|q dx = 0.
p
p
q
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).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 59, 8 pp.
http://jipam.vu.edu.au/
8
H UI -M EI H E AND J IAN -Q ING C HEN
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
So, if u is a solution of problem (1.2), we must have
N −p
N +b
N +a
N +b
<
,
>
.
p
q
p
q
Remark 2. The second assumption of Theorem 2.4,
(q − p)(p − 1)
pb − a p +
< (q − p)(N − 1)
p
implies that
N +b
N +a
<
.
q
p
R EFERENCES
[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 systems, Diff. Int.
Eq., 5 (1992), 115–1120.
[3] H. EGNELL, Semilinear elliptic equations involving critical Sobolev exponents, Arch. Rat. Mech.
Anal., 104 (1988), 27–56.
[4] P.H. RABINOWITZ, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43
(1992), 270–291.
[5] W . ROTHER, Some existence results for the equation −4u + k(x)up = 0, Comm. Partial Differential Eq., 15 (1990), 1461–1473.
[6] P. SINTZOFF AND M. WILLEM, A semilinear elliptic equation on RN with unbounded coefficients, in Variational and Topological Methods in the Study of Nonlinear Phenomena, V. Benci et
al. eds., PNLDE Vol. Birkhäuser, Boston, 49 (2002), 105–113.
[7] P. SINTZOFF, Symmetry of solutions of a semilinear elliptic equation with unbound coefficients,
Diff. Int. Eq., 7 (2003), 769–786.
[8] B. SIRAKOV, Existence and multiplicity of solutions of semi-linear elliptic equations in RN , Calc.
Var. Partial Differential Equations, 11 (2002), 119–142.
[9] D. SMETS, J. SU AND M. WILLEM, Non radial ground states for the Hénon equation, Commun.
Contemp. Math., 4 (2002), 467–480.
[10] W.A. STRAUSS, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977),
149–162.
[11] J. SU, Z.-Q. WANG AND M. WILLEM, Weighted Sobolev embedding with unbounded decaying
radial potentials, J. Differential Equations, 238 (2007), 201–219.
[12] J. SU, Z.-Q. WANG AND M. WILLEM, Nonlinear Schrödinger equations with unbounded and
decaying radial potentials, Commun. Contemp. Math., 9(4) (2007), 571–583.
[13] M. WILLEM, Minimax Theorems, Birkhäuser, Boston, 1996.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 59, 8 pp.
http://jipam.vu.edu.au/