Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 291, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
MULTIPLE POSITIVE SOLUTIONS FOR A
KIRCHHOFF TYPE PROBLEM
WENLI ZHANG
Abstract. In this article, we study the existence and multiplicity of positive
solutions of a Kirchhoff type equation on a smooth bounded domain Ω ⊂ R3 ,
and we show that the number of positive solutions of the equation depends on
the topological properties of the domain. The technique is based on LjusternikSchnirelmann category and Morse theory.
1. Introduction
This article concerns the multiplicity of positive solutions to the elliptic problem
Z
− ε2 a + εb
|∇u|2 ∆u + u = |u|p−1 u, x ∈ Ω
(1.1)
Ω
u = 0, x ∈ ∂Ω
where Ω is a smooth bounded domain of R3 , ε > 0, a, b > 0 are constants, 3 < p < 5.
In recent years, some mathematicians considered the problem
Z
− a+b
|∇u|2 ∆u = f (x, u), x ∈ Ω
(1.2)
Ω
u = 0, x ∈ ∂Ω
where Ω ⊂ R3 is a smooth bounded domain. See for example[2, 9, 10, 14, 17, 18,
19, 20].
When a = 1, b = 0, R3 is replaced by RN , and |u|p−1 u is replaced by f (u),
Equation (1.1) reduces to
−ε2 4 u + u = f (u),
x∈Ω
x ∈ ∂Ω
u = 0,
(1.3)
where Ω is a bounded domain of RN . Benci and Cerami [4] used Morse theory to
estimate the number of positive solutions of the problem (1.3). They proved that
for ε sufficiently small the number of positive solutions depends on the topology of
Ω, actually on the Poincaré polynomial of Ω, Pt (Ω), defined below. Candela and
Lazzo [8] considered the same equation with mixed Dirichlet-Neumann boundary
conditions and f (t) = |t|p−2 t. It was proved that the number of positive solutions is
2010 Mathematics Subject Classification. 35J50.
Key words and phrases. Positive solutions; Kirchhoff type equation; Morse theory;
Ljusternik-Schnirelmann category.
c
2015
Texas State University - San Marcos.
Submitted September 2, 2014. Published November 23, 2015.
1
2
WENLI ZHANG
EJDE-2015/291
influenced by the topology of the part Γ1 of the boundary ∂Ω where ε is sufficiently
small. Recently, Benci, Bonanno, Ghimenti and Micheletti [3, 13] proved that the
number of solutions of (1.2) with f (t) = |t|p−2 t on a smooth bounded domain Ω ⊂
R3 depends on the topological properties of the domain. More recently, Ghimenti
and Micheletti [12] extended the result in [3, 13] to the Klein-Gordon-Maxwell and
Schröedinger-Maxwell system and showed that the geometry of the 3-dimensional
Riemannian manifold has effects on the number of solutions of both systems.
Moreover, as far as we known, the existence and multiplicity of nontrivial solutions to the Kirchhoff equation have not ever been studied by using Morse theory.
Motivated by the works described above and the fact, we will try to get the multiplicity of positive solutions to (1.1) by using Ljusternik-Schnirelmann category and
Morse theory. So in this paper we shall fill this gap.
Our main results read as follows.
Theorem 1.1. Let 3 < p < 5. For ε > 0 sufficiently small, the problem (1.1) has
at least cat(Ω) positive solutions.
Theorem 1.2. Let 3 < p < 5. Assume that for ε > 0 sufficiently small all the
solutions of the problem (1.1) are nondegenerate. Then there are at least 2P1 (Ω)−1
positive solutions,
2. Notation and preliminary results
Throughout this article, we use the following norms for u ∈ H01 (Ω):
1/2
1 Z
1/p
1 Z
2
2
ε |∇u| dx
, |u|ε,p = 3
|u|p dx
,
kukε = 3
ε Ω
ε
Z
Z
1/2
1/p
kuk =
|∇u|2 dx
, |u|p =
|u|p dx
R3
R3
H01 (Ω)
and we denote by Hε the Hilbert space
endowed with k · kε norm.
Following the work by He and Zou [15], we let U (x) be the positive ground state
solution of
Z
− a+b
|∇u|2 ∆u + u = |u|p−1 u, x ∈ R3
(2.1)
R3
u ∈ H 1 (R3 ), u(x) > 0, x ∈ R3
and I∞ (U ) = m∞ = inf u∈M∞ I∞ (u), where
1
b
1
a
|u+ |p+1
I∞ (u) = kuk2 + |u|22 + kuk4 −
p+1 .
2
2
4
p+1
0
M∞ = {u ∈ H 1 (R3 )\{0} : G∞ (u) = hI∞
(u), ui = 0}
For ε > 0 we set Uε (x) = U (x/ε). Obviously Uε (x) is the solution of the problem
Z
− ε2 a + εb
|∇u|2 ∆u + u = |u|p−1 u, x ∈ Ω
Ω
u ∈ H01 (Ω),
u > 0,
x∈Ω
Now we shall recall some topological tools.
Definition 2.1 ([16]). Let X a topological space and consider a closed subset
A ⊂ X. We say that A has category k relative to X(catX (A) = k) if A is covered by
k closed sets Ai , i = 1, 2, . . . , k, which are contractible in X, and k is the minimum
integer with this property. We simply denote cat(X) = catX (X).
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MULTIPLE POSITIVE SOLUTIONS
3
Remark 2.2 ([5]). Let X1 and X2 be topological space. If g1 : X1 → X2 and
g2 : X2 → X1 are continuous operators such that g2 ◦ g1 is homotopic to the
identity on X1 , then cat(X1 ) ≤ cat(X2 ).
Definition 2.3. Let X is a topological space and let Hk (X) denotes its k-th homology group with coefficients in Q. The Poincaré polynomial Pt (X) of X is defined
as the following power series in t,
X
Pt (X) =
(dim Hk (X))tk .
k≥0
If X is a compact space, we have that dim Hk (X) < ∞ and this series is finite. In
the case Pt (X) is a polynomial and not a formal series.
Remark 2.4 ([4]). Let X and Y be topological spaces. If f : X → Y and g : Y →
X are continuous operators such that g ◦ f is homotopic to the identity on X, then
Pt (Y ) = Pt (X) + Z(t) where Z(t) is a polynomial with nonnegative coefficients.
3. Proof of main results
To prove our main results, we consider the functional Iε ∈ C 2 (Hε , R), defined
by
1
b
1
a
kuk2ε + |u|2ε,2 + kuk4ε −
|u+ |p+1
ε,p+1
2
2
4
p+1
Obviously, there exists a one to one correspondence between the nontrivial solutions
of problem (1.1) and the nonzero critical points of Iε on Hε .
As the functional Iε is not bounded below on Hε , we introduce the manifold
Iε (u) =
Mε = {u ∈ Hε \{0} : Gε (u) = hIε0 (u), ui = 0}
Next, we present some properties of Iε and Mε .
Lemma 3.1. (1) For any u ∈ Hε \{0}, there is a unique tε > 0 such that utε (x) =
tε u(x) ∈ Mε .
(2) For any ε > 0, Mε is a C 1 submanifold of Hε , and there exists σε > 0 and
Kε > 0 such that for any u ∈ Mε
kukε ≥ σε ,
Iε (u) ≥ Kε .
(3) It holds (P S) condition for the functional Iε on Mε .
Proof. (1) For any u ∈ Hε \{0} and t > 0, set ut (x) = tu(x). Consider
a 2
1
b
1 p+1 + p+1
t kuk2ε + t2 |u|2ε,2 + t4 kuk4ε −
t |u |ε,p+1 .
2
2
4
p+1
By computing, we known that Υε has a unique critical point tε > 0 corresponding
to its maximum. Then Υε (tε ) = maxt>0 Υε (t) and Υ0ε (tε ) = 0. So Gε (utε ) = 0 and
utε ∈ Mε .
(2) By lemma 3.1 (1), Mε 6= ∅. If u ∈ Mε , using that Gε (u) = 0 and 3 < p < 5,
we have
Υε (t) = Iε (ut ) =
hG0ε (u), ui = a(2 − (p + 1))kuk2ε + (2 − (p + 1))|u|2ε,2 + b(4 − (p + 1))kuk4ε < 0.
So Mε is a C 1 manifold. Using that Gε (u) = 0 and the Sobolev embedding, we
have
p+1
akuk2ε + |u|2ε,2 + bkuk4ε = |u+ |p+1
,
ε,p+1 ≤ Ckukε
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WENLI ZHANG
EJDE-2015/291
a ≤ Ckukεp−1 .
So the conclusion kukε ≥ σε follows. For any u ∈ Mε ,
1
1
1
1
1
1
Iε (u) = a( −
)kuk2ε + ( −
)|u|2ε,2 + b( −
)kuk4ε .
2 p+1
2 p+1
4 p+1
Using 3 < p < 5 and kukε ≥ σε , the conclusion Iε (u) ≥ Kε follows.
(3) Let {un } is (P S) sequence for Iε on Mε , that is
Iε (un ) → c, Iε0 |Mε (un ) → 0.
Then it is easy to prove that kun kε is bounded. Going if necessary to a subsequence,
we can assume that kun k2ε → A(> 0), un * u in Hε , un → u in Ls (Ω) (1 ≤ s < 6).
Obviously, we have
ρε (u) = akuk2ε + |u|2ε,2 + bAkuk2ε − |u+ |p+1
ε,p+1 = 0.
Set ωn = un − u. By Brézis-Lieb Lemma, we have kωn k2ε = kun k2ε − kuk2ε + on (1).
Since hIε0 (un ), un i = on (1), we obtain
(a + bA)kωn k2ε + ρε (u) = on (1).
This concludes the proof.
By Lemma 3.1 (2), we obtain Iε |Mε is bounded from below. By using Lagrange
multiplier method, we known that Mε contains every nonzero solution of problem
(1.1), and define the minimax mε as
mε = inf Iε (u)
u∈Mε
Proof of Theorem 1.1. Since the functional Iε ∈ C 2 is bounded below and satisfies the (PS) condition on the complete manifold Mε , we have, by the classical
Ljusternik-Schnirelmann category result [7], that Iε has at least catIεd critical points
in the sublevel
Iεd = {u ∈ Hε : Iε (u) ≤ d}
In the following, we will prove that, for ε and δ sufficiently small, it holds
cat(Ω) ≤ cat(Mε ∩ Iεm∞ +δ )
To prove this, we build two continuous functions
Φε : Ω− → Mε ∩ Iεm∞ +δ ,
β : Mε ∩
Iεm∞ +δ
+
→Ω ,
(3.1)
(3.2)
where
Ω− = {x ∈ Ω : d(x, ∂Ω) < r},
Ω+ = {x ∈ R3 : d(x, ∂Ω) < r}
with r > 0 small enough so that cat(Ω− ) = cat(Ω+ ) = cat(Ω). Following the idea
in [6], we can find two functions Φε and β such that β ◦ Φε : Ω− → Ω+ is homotopic
to the immersion i : Ω− → Ω+ . By Remark 2.2 we obtain the inequality which
completes the proof.
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MULTIPLE POSITIVE SOLUTIONS
5
Proof of Theorem 1.2. By Remark 2.4, (3.1) and (3.2), we have
Pt (Mε ∩ Iεm∞ +δ ) = Pt (Ω) + Z(t)
where Z(t) is a polynomial with nonnegative coefficients. Since inf ε mε = c > 0,
we have
c
Pt (Iεm∞ +δ , Iε2 ) = tPt (Mε ∩ Iεm∞ +δ ),
Pt (Hε , Iεm∞ +δ )
(3.3)
c
=
t(Pt (Iεm∞ +δ , Iε2 )
− t) .
(3.4)
By Morse theory we have
X
c
tµ(u) = Pt (Hε , Iεm∞ +δ ) + Pt (Iεm∞ +δ , Iε2 ) + (1 + t)Qε (t)
(3.5)
u∈Kε
where Kε be the set of critical points of Iε , µ(u) is the Morse index of u, Qε (t) is a
polynomial with nonnegative coefficients. Using this relation with (3.3)–(3.5), we
obtain
X
tµ(u) = tPt (Ω) + t2 (Pt (Ω) − 1) + t(1 + t)Qε (t)
(3.6)
u∈Kε
Theorem 1.2 easily follows by evaluating the power series (3.6) for t = 1.
4. The function Φε
For ξ ∈ Ω− we define the function
ωξ,ε (x) = Uε (x − ξ)χr (|x − ξ|)
where χr is a smooth cut off function χr ≡ 1 for t ∈ [0,
|χ0r (t)| ≤ 2/r.
We define Φε : Ω− → Mε by
(4.1)
r
2 ),
χr ≡ 0 for t > r and
Φε (ξ) = tε (ωξ,ε )ωξ,ε (x)
Remark 4.1. We have that the following limits hold uniformly with respect to
ξ ∈ Ω− ,
kωξ,ε kε → kU k, |ωξ,ε |ε,p → |U |p .
Proposition 4.2. For any ε > 0 the map Φε is continuous. Moreover for any
δ > 0 there exists ε0 > 0 such that if ε < ε0 then Iε (Φε (ξ)) < m∞ + δ.
Proof. Obviously, Φε is continuous. We claim that tε (ωξ,ε ) → 1 uniformly with
respect to ξ ∈ Ω− . In fact, by Lemma 3.1 tε (ωξ,ε ) is the unique solution of
+ p+1
atkωξ,ε k2ε + t|ωξ,ε |2ε,2 + bt3 kωξ,ε k4ε = tp |ωξ,ε
|ε,p+1
By Remark 4.1 we have the claim.
Now, by Remark 4.1 and the above claim we have
Iε (Φε (ξ))
1
1
1
1
1
1
=( −
)at2ε kωξ,ε k2ε + ( −
)t2ε |ωξ,ε |2ε,2 + ( −
)bt4 kωξ,ε k4ε
2 p+1
2 p+1
4 p+1 ε
1
1
1
1
1
1
→( −
)akU k2 + ( −
)|U |2 + ( −
)bkU k4 = m∞
2 p + 1)
2 p+1
4 p+1
this completes the proof.
Remark 4.3. Note that lim supε→0 mε ≤ m∞ .
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WENLI ZHANG
EJDE-2015/291
5. The map β
For u ∈ Mε we can define a point β(u) ∈ R3 by
R
x|u+ |p+1 dx
β(u) = RΩ + p+1
|u | dx
Ω
The function β is well defined in Mε since if u ∈ Mε then u+ 6= 0.
We shall prove that if u ∈ Mε ∩ Iεm∞ +δ then β(u) ∈ Ω+ . First of all we consider
partitions of the compact manifold Ω. Given ε > 0, a finite partition Pε = {Pjε }j∈Λε
is called a “good” partition if: for any j ∈ Λε the set Pjε is closed; Piε ∩ Pjε ⊆
∂Piε ∩ ∂Pjε for i 6= j; there exist r1 (ε), r2 (ε) > 0 such that, for any j, there exists a
point qjε ∈ Pjε such that
B(qjε , ε) ⊂ Pjε ⊂ B(qjε , r2 (ε)) ⊂ B(qjε , r1 (ε)),
with r1 (ε) ≥ r2 (ε) ≥ Cε for some positive constant C; lastly, there exists a finite
number ι ∈ N such that every x ∈ Ω is contained in at most ι balls B(qjε , r1 (ε)),
where ι does not depends on ε.
Lemma 5.1. There exists γ > 0 such that, for any δ > 0 and any ε ∈ (0, ε0 (δ))
where ε0 (δ) is as in Proposition 4.2, given any “good” partition Pε of the domain
Ω and for any u ∈ Mε ∩ Iεm∞ +δ there exists a set Pjε such that
Z
1
|u+ |p+1 dx ≥ γ
ε3 Pjε
Proof. Taking into account that Gε (u) = 0 we have
X 1 Z
akuk2ε ≤ |u+ |p+1
=
|u+ |p+1 dx
ε,p+1
3
ε
ε
P
j
j
X
X
p−1
+ p+1
+ 2
≤
|uj |ε,p+1 =
|u+
j |ε,p+1 |uj |ε,p+1
j
≤
j
p−1
max{|u+
j |ε,p+1 }
j
X
2
|u+
j |ε,p+1
j
+
ε
where u+
j is the restriction of the function u on the set Pj .
Arguing as in [3], we prove that there exists a constant C > 0 such that
X
2
+ 2
|u+
j |ε,p+1 ≤ Cιku kε ,
j
thus
p−1
max{|u+
j |ε,p+1 } ≥
j
that concludes the proof.
a
Cι
Proposition 5.2. For any η ∈ (0, 1) there exists δ0 > 0 such that for any δ ∈ (0, δ0 )
and any ε ∈ (0, ε0 (δ)) as in Proposition 4.2, for any u ∈ Mε ∩ Iεm∞ +δ there exists
a point q = q(u) ∈ Ω such that
√
Z
2(p + 1) 1
2a2 − 2a a2 + 3bm∞ + p+1
|u | dx > (1 − η)
4m∞ +
ε3 B(q, r2 )
p−1
b
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MULTIPLE POSITIVE SOLUTIONS
7
Proof. We prove only the proposition for any u ∈ Mε ∩ Iεmε +2δ . Indeed, by this
result and by Remark 4.3 we get
lim mε = m∞
ε→0
Hence it holds Iεm∞ +δ ⊂ Iεmε +2δ for δ, ε small enough. So the thesis holds.
We argue by contradiction. Suppose that there exists η ∈ (0, 1) such that we can
mε +2δk
find vanishing sequences {δk }, {εk } and a sequence {uk } ⊂ Mεk ∩ Iε k
such
that
mεk ≤ Iεk (uk )
1
1
1
1
1
1
)akuk k2εk + ( −
)|uk |2εk ,2 + ( −
)bkuk k4εk (5.1)
=( −
2 p+1
2 p+1
4 p+1
≤ mεk + 2δk ≤ m∞ + 3δk .
for k large enough, and for any q ∈ Ω,
√
Z
2a2 − 2a a2 + 3bm∞
2(p + 1)
1
+ p+1
(4m∞ +
).
|u | dx ≤ (1 − η)
ε3k B(q, r2 ) k
p−1
b
(5.2)
By Ekeland variational principle and by definition of Mεk we can assume that
Iε0 k (uk ) → 0.
(5.3)
By Lemma 5.1, there exists a set Pkεk ∈ Pεk such that
Z
1
|u+ |p+1 dx ≥ γ.
ε3 Pkεk k
(5.4)
◦
So we can choose a point qk ∈ Pkεk , and define, for z ∈ Ωεk =
1
εk (Ω
− qk ),
ωk (z) = uk (εk z + qk ) = uk (x),
where x ∈ Ω. We obtain that ωk ∈ H01 (Ωεk ). By (5.1), we have
kωk k2H 1 (Ωε
0
k
)
≤ C.
1
So we obtain ωk * ω in Hloc
(R3 ), ωk → ω in Lsloc (R3 )(2 ≤ s < 6) and kωk k2 → A1 .
Thus we prove that ω 6≡ 0 and A1 > 0 by (5.4).
Next we claim
dist(qk , ∂Ω)
lim
= ∞.
(5.5)
k→∞
εk
We argue by contradiction. Suppose that
dist(qk , ∂Ω)
= d < ∞.
εk
It is easy to verify that ω is a solution of
lim
k→∞
−(a + bA1 )∆u + u = |u|p−1 u,
u(x) = 0,
x ∈ ∂R3+
x ∈ R3+
(5.6)
where R3+ is a half space. We know that (5.6) has no nontrivial solution from the
work by [1, 11]. So ω ≡ 0, this contradicts with ω 6≡ 0. This concludes the claim.
By (5.5), Ωεk converges to the whole space R3 as k → ∞. Using (5.1) (5.3) and
computing, we have
0
I∞ (ωk ) → m∞ , I∞
(ωk ) → 0.
(5.7)
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WENLI ZHANG
EJDE-2015/291
This implies {ωk } ⊂ M∞ is a minimizing sequences for m∞ . Arguing as in [15], we
0
have ωk → ω, ω ∈ M∞ , I∞ (ω) = m∞ and I∞
(ω) = 0.
By using (5.7) and Pohozaev identity, we have
a
1
b
1
kωk k2 + |ωk |22 + kωk k4 −
|ωk |p+1
p+1 = m∞ + ok (1),
2
2
4
p+1
akωk k2 + |ωk |22 + bkωk k4 − |ωk |p+1
p+1 = ok (1)
a
3
b
3
kωk k2 + |ωk |22 + kωk k4 −
|ωk |p+1
p+1 = ok (1)
2
2
2
p+1
Thus, we obtain that
|ωk |p+1
p+1
√
2(p + 1) 2a2 − 2a a2 + 3bm∞ →
4m∞ +
p−1
b
So by ωk → ω, for T and k large enough, we have
√
Z
2(p + 1) 2a2 − 2a a2 + 3bm∞ |ωk+ |p+1 dz > (1 − η)
4m∞ +
p−1
b
B(0,T )
On the other hand by (5.2) and the definition of ωk , for any T > 0 we have, for
k large enough,
Z
Z
1
|ωk+ |p+1 dz ≤ 3
|uk |p+1 dx
εk B(qk ,εk T )
B(0,T )
Z
1
≤ 3
|uk |p+1 dx
εk B(qk , r2 )
√
2a2 − 2a a2 + 3bm∞
2(p + 1)
(4m∞ +
).
≤ (1 − η)
p−1
b
This leads to a contradiction.
Proposition 5.3. There exists δ0 > 0 such that for any δ ∈ (0, δ0 ) and any
ε ∈ (0, ε(δ0 )) as in Proposition 5.2, for any u ∈ Mε ∩ Iεm∞ +δ it holds β(u) ∈ Ω+ .
Moreover the composition
β ◦ Φε : Ω− → Ω+
is homotopic to the immersion i : Ω− → Ω+ .
Proof. Arguing by contradiction, we suppose that there exist sequences {δk }, {εk } ⊂
R and {uk } ⊂ Mεk ∩ Iεm∞ +δk such that δk , εk → 0+ , as k → ∞, and β(uk ) 6∈ Ω+
for all k.
By Ekeland variational principle and by definition of Mεk we can assume that
Iε0 k (uk ) → 0. So by Proposition 5.2 we can find qk ∈ Ω such that
√
R
1
2a2 −2a a2 +3bm∞
|uk |p+1 dx
(1 − η) 2(p+1)
4m
+
∞
ε3k B(qk , r2 )
p−1
b
R
√
>
1
2a2 −2a a2 +3b(m∞ +δk ) |uk |p+1 dx
2(p+1)
ε3k Ω
4(m
+
δ
)
+
∞
k
p−1
b
Finally,
|β(uk ) − qk |
R
| ε13 Ω (x − qk )|uk |p+1 dx|
≤ k 1 R
|uk |p+1 dx
ε3 Ω
k
EJDE-2015/291
≤
| ε13
k
MULTIPLE POSITIVE SOLUTIONS
R
(x − qk )|uk |p+1 dx|
B(qk , r2 )
R
1
ε3k Ω
|uk |p+1 dx
r
≤ + 2 diam(Ω) 1 −
2
+
| ε13
k
R
Ω\B(qk , r2 )
R
1
ε3k Ω
(1 − η) 2(p+1)
p−1 (4m∞ +
2(p+1)
p−1 (4(m∞
+ δk ) +
9
(x − qk )|uk |p+1 dx|
|uk |p+1 dx
√
2a2 −2a a2 +3bm∞
)
b
2a2 −2a
√
a2 +3b(m∞ +δk )
)
b
The above expression implies that β(uk ) ∈ Ω+ , which contradicts β(uk ) 6∈ Ω+ . Acknowledgments. This work was supported by the Shanxi University Technology research and development (project 2013158).
References
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half space, Comm. Partial Differential Equations 15 (1990), 1421–1446.
[2] C. O. Alves, F. J. S. A. Corrêa, T. F. Ma; Positive solutions for a quasilinear elliptic equation
of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85-93.
[3] V. Benci, C. Bonanno, A. M. Micheletti; On the multiplicity of a nonlinear elliptic problem
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WENLI ZHANG
Wenli Zhang
Department of Mathematics, Changzhi University, Shanxi 046011, China
E-mail address: ywl6133@126.com
EJDE-2015/291