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Multiple solutions for N -Kirchhoff type problems with critical exponential growth in RN
Nonlinear Analysis 117 (2015) 159–168
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Multiple solutions for N-Kirchhoff type problems with critical exponential growth in RN ✩ Qin Li a , Zuodong Yang a,b,∗ a
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China
b
School of Teacher Education, Nanjing Normal University, Jiangsu Nanjing 210097, China
article
info
Article history: Received 21 October 2014 Accepted 10 January 2015 Communicated by Enzo Mitidieri Keywords: N-Kirchhoff type Trudinger–Moser inequality Critical exponential growth Variational methods
abstract In this paper, we consider a class of nonlocal N-Kirchhoff type problems involving a nonlinearity term having critical exponential growth. By applying variational methods together with Trudinger–Moser inequality in whole RN , we obtain the existence of at least two positive solutions. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction This paper concerns with the existence and multiplicity of positive solutions for the following class of N-Kirchhoff type problem
M
RN 1,N
u∈W
(|∇ u|N + V (x)|u|N )dx (−∆N u + V (x)|u|N −2 u) = λA(x)|u|p−2 u + f (u),
x ∈ RN ,
(1.1)
(RN ),
where ∆N u = div(|∇ u|N −2 ∇ u) is the N-Laplacian operator of u, M (s) = sk for k > 0, s ≥ 0, 1 < p < N , λ > 0 is a real parameter, A(x) is a positive function in Lσ (RN ) with σ = NN−p and functions V , f satisfy the following assumptions:
(V1 ) V (x) ∈ C (RN ) and there exists V0 > 0 such that V (x) ≥ V0 in RN . Moreover, lim|x|→∞ V (x) = +∞. (f1 ) f is a continuous function having critical exponential growth, that is, there exists α0 > 0 such that |f (s)| 0, ∀α > α0 , lim = +∞, ∀α < α0 . |s|→∞ α|s| NN−1 e
(f2 )
f (s) lims→0 N (k+1)−1 |s|
= 0.
✩ Project supported by the National Natural Science Foundation of China (No. 11171092 and No. 11471164); project on Graduate Students Eduction and Innovation of Jiangsu Province (No. KYZZ_0209); A project funded by the Priority Academic Program Development of Jiangsu Highed Education Institutions (PAPD). ∗ Corresponding author at: Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China. E-mail address: [email protected] (Z. Yang).
http://dx.doi.org/10.1016/j.na.2015.01.005 0362-546X/© 2015 Elsevier Ltd. All rights reserved.
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Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
(f3 ) There exists µ > N (k + 1) such that 0 < µF (s) ≤ sf (s),
for all |s| > 0,
s
where F (s) = 0 f (t )dt. (f4 ) There exist constants q > N (k + 1) and γ > 0 such that f ( s ) ≥ γ s q −1 ,
∀s ≥ 0,
where
γ >
8N µ[q − N (k + 1)] q−NN(k(+k+1)1) α (N −1)[q−N N (k+1)] 0
q[µ − N (k + 1)]
αN
Sq,
(1.2)
1
αN = N ωNN−−11 , ωN −1 is the (N − 1)-dimensional measure of the (N − 1)-sphere and S will be defined later. In recent years, much attention has been paid to the Kirchhoff-type problem which is related to the stationary analog of the equation
∂ 2u ρ 2 − ∂t
L 2 2 ∂u dx ∂ u = 0, + h 2L 0 ∂ x ∂ x2
ρ0
E
(1.3)
where ρ, ρ0 , h, E and L are constants. (1.3) was presented by Kirchhoff [21] as an extension of the classical D’Alembert wave equation for free vibrations of elastic string produced by transverse vibrations. Moreover, since the equation contains an integral over [0, L], it is no longer a pointwise identity and therefore it is often called nonlocal problem. After Kirchhoff’s work, various models of Kirchhoff-type have been studied by many authors, see, for example, [6,8,9,11, 12,19,20,24–26,32,33], and the references therein. Specially, Chen et al. [10] studied the following problem
M
RN
(|∇ u|p + V (x)|u|p )dx (−∆p u + V (x)|u|p−2 u) = f (x, u) + g (x),
u(x) → 0,
in RN ,
(1.4)
as |x| → +∞,
where ∆p u = div(|∇ u|p−2 ∇ u) is the p-Laplacian operator with 1 < p < N , M (s) = sk for k > 0, s ≥ 0, V (x) ∈ C (RN ) with V (x) → +∞, as |x| → ∞, and nonlinearities f , g satisfy some conditions. By using the Mountain Pass Theorem, Ekeland’s variational principle and Krasnoselskii’s genus theory, the authors obtain some results of existence of nontrivial solutions for (1.4). For such case of p < N, it is known that in the majority of problems treated with variational methods, the maximal possible growth for the nonlinearity term is polynomial at infinity and by consequence the corresponding energy functional could be defined in some appropriate Sobolev spaces. But things change when dealing with the case of p = N. According to the Trudinger–Moser inequality (see [27,30]), the maximal growth on the nonlinearities which allows us to treat the problem variationally is the critical exponential growth. Hence, there has been considerable attention paid to the case of p = N, see [1,3,5,13–16,23,29] and the references therein. Motivated by works mentioned above, more precisely by results found in [10], a natural question is whether the same phenomenon of existence and multiplicity holds, when we consider the N-Laplacian operator and assume that the nonlinearity has a critical exponential growth in RN . The main purpose of this paper is to study these questions for problem (1.1) involving N-Laplacian operator and critical exponential growth. Moreover, as far as we know, there are no works dealing with such problem. Here, we employ variational methods together with Trudinger–Moser inequality in RN . Our main result is stated as follows. Theorem 1.1. Assume that conditions (V1 ), (f1 )–(f4 ) hold. Then there exists a constant Λ > 0 such that if 0 < λ < Λ, problem (1.1) admits at least two positive solutions. To finish this introduction, we state a version of the Trudinger–Moser inequality for whole RN and a technical lemma which is essential to carry out the proof of our result. Proposition 1.2 ([7, Lemma 1]). If N ≥ 2, α > 0 and u ∈ W 1,N (RN ), then
eα|u|
N N −1
RN
− SN −2 (α, u) dx < ∞, Nk
α N −1 . Moreover, if α < α , |∇ u|N ≤ 1 and |u| where SN −2 (α, u) = N N ≤ K for some positive constant K , then there N k=0 k! |u| exists a positive constant C = C (N , K , α) such that
N −2
eα|u|
RN
N N −1
k
− SN −2 (α, u) dx ≤ C .
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
161
Proposition 1.3 ([2, Lemma 2.4]). Let {un } ⊂ W 1,N (RN ) be a sequence satisfying lim sup ∥un ∥N <
α N −1 N
α0
n→∞
.
Then there exist constants α > α0 , ς > 1 and C > 0 (independent of n) such that
eα|un |
N N −1
ς − SN −2 (α, un ) dx ≤ C ,
RN
∀ n ≥ n0 ,
for some n0 sufficiently large. 2. Notations and preliminaries For convenience, we assume that V0 = 1. Moreover, since we intend to find positive weak solutions, we suppose that f (s) = 0 in (−∞, 0). Consider the Sobolev space X = W 1,N (RN ) endowed with the norm
∥ u∥ X =
RN
(|∇ u|N + |u|N )dx
N1
,
∀u ∈ X .
Then by the continuous embedding X ↩→ Lθ (RN ), for all u ∈ X , there exists a constant lθ > 0 such that
|u|θ ≤ lθ ∥u∥X ,
θ ∈ [N , +∞).
(2.1)
Now, we introduce the subspace
E = u ∈ X
V (x)|u| dx < ∞ . N
RN
Obviously, E is a Banach space endowed with the norm
∥ u∥ =
RN
(|∇ u|N + V (x)|u|N )dx
N1
,
∀u ∈ E .
Then by (2.1), for all u ∈ E, we have
|u|θ ≤ lθ ∥u∥X ≤ lθ ∥u∥,
θ ∈ [N , +∞).
(2.2)
Moreover, for q > N (k + 1),
S=
inf
u∈E \{0}
RN
N1 (|∇ u|N + V (x)|u|N )dx 1q q dx | u | N R
is well defined. In the present paper, we will show the multiplicity of weak solutions for (1.1) by looking for critical points of the associated functional I λ ( u) =
1 N (k + 1)
∥u∥N (k+1) −
λ p
RN
A(x)|u|p dx −
RN
F (u)dx.
Clearly, Iλ ∈ C 1 (E , R) and the critical points of the functional Iλ are in fact weak solutions of (1.1). By a weak solution of (1.1), we mean u ∈ E satisfying
∥u∥Nk
RN
(|∇ u|N −2 ∇ u∇ψ + V (x)|u|N −2 uψ)dx − λ
RN
A(x)|u|p−2 uψ dx −
for all ψ ∈ E. Now, we give a compact embedding lemma, which is a key property of E. Lemma 2.1. Let (V1 ) hold true. Then the embedding E ↩→ LN (RN ) is compact.
RN
f (u)ψ dx = 0,
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Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
Proof. Let Y = LN (RN ), BR = x ∈ RN |x| < R and BcR = RN \BR . Denote E (Ω ), Y (Ω ) the spaces of functions u ∈ E , u ∈ Y restricted onto Ω ⊂ RN respectively. By (V1 ), we let αR = infBc V (x) and obviously, αR → +∞, as R → +∞. Then, we get R 1 1 |u|N dx ≤ V (x)|u|N dx ≤ ∥ u∥ N αR BcR αR BcR
and so lim
sup
|u|Y (BcR ) ∥ u∥
R→+∞ u∈E \{0}
= 0.
On the other hand, it is well known that E (BR ) ↩→ Y (BR ) is compact. Then, Theorem 7.9 in [22] shows that the embedding E ↩→ Y is compact, which completes the proof. In the sequel, we will prove the existence of a weak solution of (1.1) having a positive energy by using the Mountain Pass Theorem in [4] (or see [28]). Firstly, we check that the functional Iλ verifies the Mountain Pass geometry. Lemma 2.2. Assume that (f1 )–(f3 ) hold. Then there exists Λ1 > 0 such that if 0 < λ < Λ1 , the functional Iλ satisfies: (i) There exist η > 0 and ρ > 0 such that Iλ (u) ≥ η,
for ∥u∥ = ρ.
(ii) There exists e ∈ E with ∥e∥ > ρ such that Iλ (e) < 0. Proof. First, we show that Iλ satisfies (i). By (f1 ) and (f2 ), given ϵ > 0 and ϱ ≥ N (k + 1), there exists Cϵ > 0 such that, for every α > α0 ,
|F (s)| ≤
N ϵ N −1 |s|N (k+1) + Cϵ |s|ϱ eα|s| − SN −2 (α, s) , N (k + 1)
for all s ∈ R.
(2.3)
It is noted in [17,18] that
RN
N N −1 |s|ϱ eα|s| − SN −2 (α, s) dx ≤ C (α, N )∥s∥ϱ ,
(2.4)
provided that ∥s∥ ≤ δ , where positive constant δ is sufficiently small. Then by (2.2)–(2.4) and Hölder inequality, we have
N ϵ N −1 N (k+1) ∥ u∥ − A(x)|u| dx − |u|N (k+1) − Cϵ |u|ϱ eα|u| − SN −2 (α, u) dx Iλ (u) ≥ N N (k + 1) p RN N (k + 1) R N N (k+1) λ 1 p N (k+1) ϱ α|u| N −1 ≥ 1 − ϵ lN (k+1) ∥u∥ − |A|σ |u|N − Cϵ |u| e − SN −2 (α, u) dx N (k + 1) p RN N (k+1) λ 1 N (k+1) 1 − ϵ lN (k+1) ∥u∥ − |A|σ ∥u∥p − C1 ∥u∥ϱ . ≥ N (k + 1) p 1
N (k+1)
λ
Choosing ϵ sufficiently small and Λ1 = Iλ (u) ≥
≥
1 2N (k + 1) 1 4N (k + 1)
ρ N (k+1) −
λ p
pρ N (k+1)−p 4N (k+1)|A|σ
p
, for 0 < λ < Λ1 and ∥u∥ = ρ , we deduce
|A|σ ρ p − C1 ρ ϱ
ρ N (k+1) − C1 ρ ϱ .
Thus, if we choose ρ > 0 small enough, there exists η > 0 such that Iλ (u) ≥ η for ∥u∥ = ρ , which verifies (i). In order to prove (ii), for s ∈ R, we set
Φ (t ) = t −µ F (ts) − F (s),
t ≥ 1.
Then it follows from (f3 ) that
Φ ′ (t ) = t −µ−1 [−µF (ts) + tsf (ts)] ≥ 0. Hence, Φ (t ) ≥ Φ (1) = 0 for all t ≥ 1 and then F (ts) ≥ t µ F (s) for all s ∈ R, t ≥ 1.
(2.5)
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
163
Fixing u ∈ E \{0}, we have t N (k+1)
Iλ (tu) =
N (k + 1) t N (k+1)
≤
N (k + 1)
∥u∥N (k+1) − ∥u∥N (k+1) −
λt p
p
λt p
A(x)|u|p dx −
RN
p
RN
A(x)|u|p dx − t µ
RN
F (tu)dx
RN
F (u)dx.
Since µ > N (k + 1), Iλ (tu) → −∞ as t → +∞. Taking e = tu with t > 1 sufficiently large, we conclude (ii). By a version of the Mountain Pass Theorem found in [4], there exists a Palais Smale sequence {un } ⊂ E such that I λ ( un ) → c
and
Iλ′ (un ) → 0,
as n → +∞,
where c = inf max Iλ (γ (t )) γ ∈Γ t ∈[0,1]
with
Γ = {γ ∈ C ([0, 1], E )|γ (0) = 0, γ (1) = e}. Then we have the following results. Lemma 2.3. Assume that (f1 )–(f4 ) hold. Then the mountain level c satisfies c<
1
1 N (k + 1)
8N
−
1 αN (N −1)(k+1)
µ
α0
.
Proof. Let w ∈ E \{0} be a function verifying the equation
S=
N1 N N (|∇w| + V ( x )|w| ) dx RN . 1q q |w| dx RN
Then by the definition of c and (f4 ), we get
t N (k+1) λt p ∥w∥N (k+1) − A(x)|w|p dx − F (t w)dx t ≥0 N (k + 1) p RN RN ∥w∥N (k+1) − F (t w)dx
c ≤ max Iλ (t w) = max t ≥0
t N (k+1) ≤ max t ≥0 N (k + 1) RN t N (k+1) γ tq ≤ max ∥w∥N (k+1) − |w|qq . t ≥0 N (k + 1) q N (k+1)
Dividing by |w|q c
|w|qN (k+1)
and by a direct computation, one has
t N (k+1) γ tq S N (k+1) − |w|qq−N (k+1) t ≥0 N (k + 1) q qN (k+1) −N (k+1) 1 1 N (k+1) = − S q−N (k+1) γ q−N (k+1) |w|− , q N (k + 1) q ≤ max
which implies c≤
1
N (k + 1)
−
1 qN (k+1) S q−N (k+1) γ q
−N (k+1) q−N (k+1)
.
By (1.2), we can easily get c<
1 8N
1 N (k + 1)
−
which completes the proof.
1 αN (N −1)(k+1)
µ
α0
,
(2.6)
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Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
Lemma 2.4. Let {un } ⊂ E be a (PS )c -sequence associated with Iλ . Then there exists Λ2 > 0 such that if 0 < λ < Λ2 , there holds
α N −1 N
lim sup ∥un ∥N <
α0
n→∞
.
Proof. First, we claim that {un } is bounded in E. In fact, (2.6) implies that 1 ′ Iλ (un )un ≤ c + on (1) + on (1)∥un ∥.
Iλ (un ) −
µ
On the other hand, it follows from (f3 ) and Hölder inequality that
1 1 1 ∥un ∥N (k+1) − λ − A(x)|un |p dx + f (un )un − F (un ) dx N (k + 1) µ p µ RN RN µ 1 1 1 1 ≥ ∥un ∥N (k+1) − λ − − A(x)|un |p dx N (k + 1) µ p µ RN 1 1 1 1 ∥un ∥N (k+1) − λ |A|σ ∥un ∥p . − − ≥ N (k + 1) µ p µ
1 ′ I λ ( un ) un =
Iλ (un ) −
µ
1
−
1
Then 1
−
N (k + 1)
1
µ
∥un ∥N (k+1) ≤ λ
1 p
−
1
µ
|A|σ ∥un ∥p + c + on (1) + on (1)∥un ∥,
(2.7)
which implies that {un } is bounded in E. Applying Young inequality in (2.7), we get 1
−
N (k + 1)
1
µ
∥un ∥N (k+1) ≤ ϵ
1 p
−
1
µ
∥un ∥N (k+1) + Cϵ
1 p
−
1
µ
N (k+1)
N (k+1)
λ N (k+1)−p |A|σN (k+1)−p + c + on (1)
and then 1
1
−
N (k + 1)
−ϵ
µ
1 p
1
−
µ
∥un ∥N (k+1) ≤ Cϵ
1 p
1
−
µ
N (k+1)
N (k+1)
N (k+1)
N (k+1)
λ N (k+1)−p |A|σN (k+1)−p + c + on (1).
Choosing ϵ small enough, we have 1
1
2 N (k + 1)
−
1
µ
lim sup ∥un ∥N (k+1) ≤ c + Cϵ
1
n→∞
p
−
1
µ
λ N (k+1)−p |A|σN (k+1)−p + on (1),
which leads to
c
N
lim sup ∥un ∥ ≤ n→∞
where C2 =
Cϵ
check that
1 2
1 N (k+1)
N (k+1) N (k+1)−p
1 1 p − µ |A|σ 1 1 1 ( −µ ) 2 N (k+1)
lim sup ∥un ∥N ≤ n→∞
N
1
+ C2 λ
k+1 1 ,
µ
. By Lemma 2.3 and let Λ2 = ( 2C12 )
α N −1 α0
−
N (k+1) N (k+1)−p
N (k+1)−p N (k+1)
( ααN0 )
(N −1)[N (k+1)−p] N
, for 0 < λ < Λ2 , we can easily
.
This proves the lemma. Lemma 2.5. Assume that (V1 ), (f1 )–(f4 ) hold. Then the functional Iλ (u) satisfies the (PS )c condition for 0 < c <
)( ααN0 )(N −1)(k+1) . µ 1
1 8N
( N (k1+1) −
Proof. Let {un } be a (PS )c -sequence. Then by Lemma 2.4, passing to a subsequence, if necessary, we have ∥un ∥ → ν0 ≥ 0. If ν0 = 0, then the proof is finished. In the following, we assume ν0 > 0. Then for n large, ∥un ∥ ≥ 12 ν0 > 0. We now show that {un } has a convergent subsequence in E. By Lemmas 2.1 and 2.4, going if necessary to a subsequence, we can assume that un ⇀ u1
in E ,
(2.8)
un → u
1
in L (R ),
(2.9)
un → u
1
a.e. in R .
(2.10)
N
N
N
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
165
Denote An = ⟨Iλ′ (un ), un − u1 ⟩ = ∥un ∥Nk
−λ
|∇ un |N −2 ∇ un ∇(un − u1 ) + V (x)|un |N −2 un (un − u1 ) dx RN p−2 A(x)|un | un (un − u1 )dx − f (un )(un − u1 )dx
RN
RN
and Bn = ∥un ∥
Nk
RN
|∇ u1 |N −2 ∇ u1 ∇(un − u1 ) + V (x)|u1 |N −2 u1 (un − u1 ) dx.
Then the fact Iλ′ (un ) → 0 implies that An → 0, as n → ∞. Similarly, the fact {un } is bounded and un ⇀ u1 in E implies that Bn → 0, as n → ∞. Now, we claim
lim
n→∞
RN
f (un )(un − u1 )dx = 0
(2.11)
A(x)|un |p−2 un (un − u1 )dx = 0.
(2.12)
and
lim
n→∞
RN
It follows from (f1 ) and (f2 ) that
RN
f (un )(un − u1 )dx ≤
|un |N (k+1)−1 |un − u1 |dx + C3
RN
RN
N N −1 |un − u1 | eα|un | − SN −2 (α, un ) dx.
By (2.2), (2.9) and Hölder inequality, we have
| un |
N (k+1)−1
1
|un − u |dx ≤
β
|un | dx
RN N (k+1)−1
RN
≤ lβ
NN−1
N1 |un − u1 |N dx RN
∥ un ∥
N (k+1)−1
|un − u1 |N → 0,
as n → ∞,
N [N (k+1)−1]
. where β = N −1 On the other hand, by (2.9), Hölder inequality, Proposition 1.3 and Lemma 2.4, we deduce RN
N N −1 − SN −2 (α, un ) dx ≤ |un − u1 | eα|un |
RN
N1 NN−1 N N N −1 (eα|un | |un − u1 |N dx − SN −2 (α, un )) N −1 dx RN
≤ C4 |un − u |N → 0, 1
as n → ∞.
Then (2.11) holds. Applying (2.9) and Hölder inequality, we get
RN
A(x)|un |p−2 un (un − u1 )dx ≤
NN−p N |A(x)| N −p dx
RN
p−1
≤ ∥un ∥
|un |N dx RN
N1 |un − u1 |N dx
RN
|A|σ |un − u |N → 0, 1
p−N 1
as n → ∞,
which proves (2.12). Note that An − Bn = ∥un ∥ Dn − λ Nk
A(x)|un |
p−2
RN
un (un − u )dx − 1
RN
f (un )(un − u1 )dx
(2.13)
with
Dn = RN
|∇ un |N −2 ∇ un − |∇ u1 |N −2 ∇ u1 , ∇ un − ∇ u1 + V (x) |un |N −2 un − |u1 |N −2 u1 , un − u1 dx.
Then by (2.11)–(2.13) and the fact An , Bn → 0, ∥un ∥ ≥ By using the standard inequality
⟨|x|N −2 x − |y|N −2 y, x − y⟩ ≥ CN |x − y|N ,
ν > 0, we have Dn → 0, as n → ∞.
1 2 0
N ≥ 2,
we can easily deduce that ∥un − u1 ∥ → 0, as n → ∞. Thus, Iλ (u) satisfies the (PS )c condition for 0 < c < 1
µ
)( ααN0 )(N −1)(k+1) and the proof is finished.
1 8N
( N (k1+1) −
166
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
3. Proof of Theorem 1.1 In this section, we apply the Mountain Pass Theorem and Ekeland’s variational principle to prove the existence of two positive weak solutions for (1.1). Lemma 3.1. Assume that (V1 ), (f1 )–(f4 ) hold. Then there exists Λ3 > 0 such that if 0 < λ < Λ3 , problem (1.1) admits a weak solution u1 satisfying Iλ (u1 ) > 0. Proof. Let Λ3 = min{Λ1 , Λ2 }. Then the proof follows directly from Lemmas 2.2 and 2.5 and the Mountain Pass Theorem in [4] (or see [28]). In the following, we prove the existence of the second solution u2 different from u1 . Lemma 3.2. Assume that (f1 )–(f4 ) hold. Then there exists Λ4 > 0 such that if 0 < λ < Λ4 , the functional Iλ (u) satisfies the (PS )c0 condition with c0 ≤ 0. Proof. Fix c0 ≤ 0 and assume that {un } ⊂ E satisfies Iλ (un ) → c0 ,
Iλ′ (un ) → 0,
as n → ∞.
(3.1)
Proceeding as in (2.7), we derive 1
N (k + 1)
−
1
µ
∥un ∥N (k+1) ≤ λ
1 p
−
1
µ
|A|σ ∥un ∥p + c0 + on (1)
and then, for a subsequence, we have 1
N (k + 1)
−
1
1
lim sup ∥un ∥N (k+1)−p − λ
µ
n→∞
p
−
1
µ
|A|σ lim sup ∥un ∥p ≤ 0. n→∞
Hence, lim sup ∥un ∥N ≤
N 1 N (k+1)−p λ p − µ1 1 N (k+1)
n→∞
Choosing Λ4 =
p[µ−N (k+1)] N (k+1)(µ−p)
lim sup ∥un ∥N <
( ααN0 )
−
(N −1)[N (k+1)−p]
α N −1
n→∞
N
α0
1
.
µ N
, for 0 < λ < Λ4 , we have
.
Then, by using the same arguments as Lemma 2.5, we prove that Iλ satisfies the (PS )c0 condition for c0 ≤ 0. Lemma 3.3. Assume that (V1 ), (f1 )–(f4 ) hold. Then there exists Λ5 > 0 such that if 0 < λ < Λ5 , problem (1.1) has a weak solution u2 satisfying Iλ (u2 ) < 0. Proof. Choosing a function ω ∈ E \{0} and by (f3 ), we have Iλ (t ω) ≤
t N (k+1) N (k + 1)
λt p
∥ω∥N (k+1) −
p
RN
A(x)|ω|p dx.
(3.2)
Since N (k + 1) > p, it follows from (3.2) that Iλ (t ω) < 0 for t > 0 small. Thus c0 = inf Iλ (u) < 0
and
u∈Bρ
inf Iλ (u) > 0,
u∈∂ Bρ
(3.3)
where ρ > 0 is given by Lemma 2.2(i) and Bρ is an open ball in E centered at the origin with radius ρ . Let εn → 0 be such that 0 < εn < inf Iλ (u) − inf Iλ (u). u∈∂ Bρ
u∈Bρ
(3.4)
By Ekeland’s variational principle, there exists {un } ⊂ Bρ such that c0 ≤ Iλ (un ) ≤ c0 + εn
(3.5)
and Iλ (un ) < Iλ (u) + εn ∥un − u∥,
∀u ∈ Bρ , u ̸= un .
(3.6)
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
167
Then by (3.3)–(3.5), we get Iλ (un ) ≤ c0 + εn ≤ inf Iλ (u) + εn < inf Iλ (u) u∈Bρ
u∈∂ Bρ
and so {un } ⊂ Bρ . Consider the sequence vn = un + t ϕ ⊂ Bρ for ϕ ∈ B1 and t > 0 small enough. Then it follows from (3.6) that t −1 Iλ (un + t ϕ) − Iλ (un ) ≥ −εn ∥ϕ∥.
(3.7)
Passing to the limit as t → 0+ , (3.7) implies that Iλ′ (un )ϕ ≥ −εn ∥ϕ∥,
∀ϕ ∈ B1 .
Replacing ϕ in (3.7) by −ϕ , we have Iλ′ (un )(−ϕ) ≥ −εn ∥ϕ∥,
∀ϕ ∈ B1 .
Then
|Iλ′ (un )ϕ| ≤ εn ,
∀ϕ ∈ B1
and so
∥Iλ′ (un )∥ → 0,
as n → ∞.
(3.8) Iλ′ (un )
Therefore, there exists a sequence {un } ⊂ Bρ such that Iλ (un ) → c0 < 0 and → 0, as n → ∞. By Lemma 3.2, {un } has a convergent subsequence (still denoted by {un }) such that un → u2 in E. Thus, u2 is a nontrivial solution for (1.1) with Iλ (u2 ) < 0, which completes the proof. Proof of Theorem 1.1. It follows from Lemmas 3.1 and 3.3 that problem (1.1) has at least two weak solutions. By using a simple argument as [3], we can assume that u1 ≥ 0 and u2 ≥ 0 in RN . Furthermore, as a consequence of Harnack’s inequality [31], we have u1 > 0 and u2 > 0 in RN . Thus, we finish the proof. References [1] S. Adachi, T. Watanabe, Uniqueness and non-degeneracy of positive radial solutions for quasilinear elliptic equations with exponential nonlinearity, Nonlinear Anal. 108 (2014) 275–290. [2] C.O. Alves, L.R. de Freitas, Multiplicity of nonradial solutions for a class of quasilinear equations on annulus with exponential critical growth, Topol. Methods Nonlinear Anal. 39 (2012) 243–262. [3] C.O. Alves, G.M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN , J. Differential Equations 246 (2009) 1288–1311. [4] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [5] S. Aouaoui, On some semilinear elliptic equation involving exponential growth, Appl. Math. Lett. 33 (2014) 23–28. [6] G. Autuori, F. Colasuonno, P. Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math. 16 (5) (2014) 1450002. 43 pages. [7] J.M. Bezerra do Ó, N-Laplacian equations in RN with critical growth, Abstr. Appl. Anal. 2 (1997) 301–315. [8] J. Chen, Multiple positive solutions to a class of Kirchhoff equation on R3 with indefinite nonlinearity, Nonlinear Anal. 96 (2014) 134–145. [9] C. Chen, J. Huang, L. Liu, Multiple solutions to the homogeneous p-Kirchhoff elliptic equation with concave-convex nonlinearities, Appl. Math. Lett. 26 (2013) 754–759. [10] C. Chen, H. Song, Z. Xiu, Multiple solutions for p-Kirchhoff equations in RN , Nonlinear Anal. 86 (2013) 146–156. [11] C. Chen, Q. Zhu, Existence of positive solutions to p-Kirchhoff-type problem without compactness conditions, Appl. Math. Lett. 28 (2014) 82–87. [12] F. Colasuonno, P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011) 5962–5974. [13] L.R. de Freitas, Multiplicity of solutions for a class of quasilinear equations with exponential critical growth, Nonlinear Anal. 95 (2014) 607–624. [14] M. de Souza, On a singular class of elliptic systems involving critical growth in R2 , Nonlinear Anal. RWA 12 (2011) 1072–1088. [15] M. de Souza, E. de Medeiros, U. Severo, On a class of quasilinear elliptic problems involving Trudinger–Moser nonlinearities, J. Math. Anal. Appl. 403 (2013) 357–364. [16] M. de Souza, João Marcos do Ó, On a singular and nonhomogeneous N-Laplacian equation involving critical growth, J. Math. Anal. Appl. 380 (2011) 241–263. [17] J.M. do Ó, Semilinear Dirichlet problems for the N-Laplacian in RN with nonlinearities in critical growth range, Differential Integral Equations 9 (1996) 967–979. [18] J.M. do Ó, E.S. Medeiros, Remarks on least energy solutions for quasilinear elliptic problems in RN , Electron J. Differential Equations 56 (2008) 1–16. [19] G.M. Figueiredo, R.G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation, Math. Nachr. (2014) 1–13. http://dx.doi.org/10.1002/mana.201300195. [20] X. He, W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 , J. Differential Equations 252 (2012) 1813–1834. [21] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [22] I. Kuzin, S. Pohozaev, Entire Solutions of Semilinear Elliptic Equations, in: Progress in Nonlinear Differential Equations and Their Applications, vol. 33, Birkhäuser, 1997. [23] N. Lam, G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in RN , J. Funct. Anal. 262 (2012) 1132–1165. [24] Y. Li, F. Li, J. Shi, Existence of positive solutions to Kirchhoff type problems with zero mass, J. Math. Anal. Appl. 410 (2014) 361–374. [25] Q. Li, Z. Yang, Existence of multiple solutions for a p-Kirchhoff equation with singular and critical nonlinearities, submitted for publication. [26] D. Liu, P. Zhao, Multiple nontrivial solutions to a p-Kirchhoff equation, Nonlinear Anal. 75 (2012) 5032–5038. [27] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/1971) 1077–1092. [28] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, third ed., Springer-Verlag, New York, 2000.
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Z. Tan, F. Fang, Nontrivial solutions for N-Laplacian equations with sub-exponential growth, Anal. Appl. 11 (03) (2013). N.S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473–483. N. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure. Appl. Math. 20 (1967) 721–747. J. Wang, L. Tian, J. Xu, F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012) 2314–2351. [33] Q. Zhang, H. Sun, J.J. Nieto, Positive solutions for a superlinear Kirchhoff type problem with a parameter, Nonlinear Anal. 95 (2014) 333–338.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Multiple solutions for N-Kirchhoff type problems with critical exponential growth in RN ✩ Qin Li a , Zuodong Yang a,b,∗ a
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China
b
School of Teacher Education, Nanjing Normal University, Jiangsu Nanjing 210097, China
article
info
Article history: Received 21 October 2014 Accepted 10 January 2015 Communicated by Enzo Mitidieri Keywords: N-Kirchhoff type Trudinger–Moser inequality Critical exponential growth Variational methods
abstract In this paper, we consider a class of nonlocal N-Kirchhoff type problems involving a nonlinearity term having critical exponential growth. By applying variational methods together with Trudinger–Moser inequality in whole RN , we obtain the existence of at least two positive solutions. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction This paper concerns with the existence and multiplicity of positive solutions for the following class of N-Kirchhoff type problem
M
RN 1,N
u∈W
(|∇ u|N + V (x)|u|N )dx (−∆N u + V (x)|u|N −2 u) = λA(x)|u|p−2 u + f (u),
x ∈ RN ,
(1.1)
(RN ),
where ∆N u = div(|∇ u|N −2 ∇ u) is the N-Laplacian operator of u, M (s) = sk for k > 0, s ≥ 0, 1 < p < N , λ > 0 is a real parameter, A(x) is a positive function in Lσ (RN ) with σ = NN−p and functions V , f satisfy the following assumptions:
(V1 ) V (x) ∈ C (RN ) and there exists V0 > 0 such that V (x) ≥ V0 in RN . Moreover, lim|x|→∞ V (x) = +∞. (f1 ) f is a continuous function having critical exponential growth, that is, there exists α0 > 0 such that |f (s)| 0, ∀α > α0 , lim = +∞, ∀α < α0 . |s|→∞ α|s| NN−1 e
(f2 )
f (s) lims→0 N (k+1)−1 |s|
= 0.
✩ Project supported by the National Natural Science Foundation of China (No. 11171092 and No. 11471164); project on Graduate Students Eduction and Innovation of Jiangsu Province (No. KYZZ_0209); A project funded by the Priority Academic Program Development of Jiangsu Highed Education Institutions (PAPD). ∗ Corresponding author at: Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China. E-mail address: [email protected] (Z. Yang).
http://dx.doi.org/10.1016/j.na.2015.01.005 0362-546X/© 2015 Elsevier Ltd. All rights reserved.
160
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
(f3 ) There exists µ > N (k + 1) such that 0 < µF (s) ≤ sf (s),
for all |s| > 0,
s
where F (s) = 0 f (t )dt. (f4 ) There exist constants q > N (k + 1) and γ > 0 such that f ( s ) ≥ γ s q −1 ,
∀s ≥ 0,
where
γ >
8N µ[q − N (k + 1)] q−NN(k(+k+1)1) α (N −1)[q−N N (k+1)] 0
q[µ − N (k + 1)]
αN
Sq,
(1.2)
1
αN = N ωNN−−11 , ωN −1 is the (N − 1)-dimensional measure of the (N − 1)-sphere and S will be defined later. In recent years, much attention has been paid to the Kirchhoff-type problem which is related to the stationary analog of the equation
∂ 2u ρ 2 − ∂t
L 2 2 ∂u dx ∂ u = 0, + h 2L 0 ∂ x ∂ x2
ρ0
E
(1.3)
where ρ, ρ0 , h, E and L are constants. (1.3) was presented by Kirchhoff [21] as an extension of the classical D’Alembert wave equation for free vibrations of elastic string produced by transverse vibrations. Moreover, since the equation contains an integral over [0, L], it is no longer a pointwise identity and therefore it is often called nonlocal problem. After Kirchhoff’s work, various models of Kirchhoff-type have been studied by many authors, see, for example, [6,8,9,11, 12,19,20,24–26,32,33], and the references therein. Specially, Chen et al. [10] studied the following problem
M
RN
(|∇ u|p + V (x)|u|p )dx (−∆p u + V (x)|u|p−2 u) = f (x, u) + g (x),
u(x) → 0,
in RN ,
(1.4)
as |x| → +∞,
where ∆p u = div(|∇ u|p−2 ∇ u) is the p-Laplacian operator with 1 < p < N , M (s) = sk for k > 0, s ≥ 0, V (x) ∈ C (RN ) with V (x) → +∞, as |x| → ∞, and nonlinearities f , g satisfy some conditions. By using the Mountain Pass Theorem, Ekeland’s variational principle and Krasnoselskii’s genus theory, the authors obtain some results of existence of nontrivial solutions for (1.4). For such case of p < N, it is known that in the majority of problems treated with variational methods, the maximal possible growth for the nonlinearity term is polynomial at infinity and by consequence the corresponding energy functional could be defined in some appropriate Sobolev spaces. But things change when dealing with the case of p = N. According to the Trudinger–Moser inequality (see [27,30]), the maximal growth on the nonlinearities which allows us to treat the problem variationally is the critical exponential growth. Hence, there has been considerable attention paid to the case of p = N, see [1,3,5,13–16,23,29] and the references therein. Motivated by works mentioned above, more precisely by results found in [10], a natural question is whether the same phenomenon of existence and multiplicity holds, when we consider the N-Laplacian operator and assume that the nonlinearity has a critical exponential growth in RN . The main purpose of this paper is to study these questions for problem (1.1) involving N-Laplacian operator and critical exponential growth. Moreover, as far as we know, there are no works dealing with such problem. Here, we employ variational methods together with Trudinger–Moser inequality in RN . Our main result is stated as follows. Theorem 1.1. Assume that conditions (V1 ), (f1 )–(f4 ) hold. Then there exists a constant Λ > 0 such that if 0 < λ < Λ, problem (1.1) admits at least two positive solutions. To finish this introduction, we state a version of the Trudinger–Moser inequality for whole RN and a technical lemma which is essential to carry out the proof of our result. Proposition 1.2 ([7, Lemma 1]). If N ≥ 2, α > 0 and u ∈ W 1,N (RN ), then
eα|u|
N N −1
RN
− SN −2 (α, u) dx < ∞, Nk
α N −1 . Moreover, if α < α , |∇ u|N ≤ 1 and |u| where SN −2 (α, u) = N N ≤ K for some positive constant K , then there N k=0 k! |u| exists a positive constant C = C (N , K , α) such that
N −2
eα|u|
RN
N N −1
k
− SN −2 (α, u) dx ≤ C .
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
161
Proposition 1.3 ([2, Lemma 2.4]). Let {un } ⊂ W 1,N (RN ) be a sequence satisfying lim sup ∥un ∥N <
α N −1 N
α0
n→∞
.
Then there exist constants α > α0 , ς > 1 and C > 0 (independent of n) such that
eα|un |
N N −1
ς − SN −2 (α, un ) dx ≤ C ,
RN
∀ n ≥ n0 ,
for some n0 sufficiently large. 2. Notations and preliminaries For convenience, we assume that V0 = 1. Moreover, since we intend to find positive weak solutions, we suppose that f (s) = 0 in (−∞, 0). Consider the Sobolev space X = W 1,N (RN ) endowed with the norm
∥ u∥ X =
RN
(|∇ u|N + |u|N )dx
N1
,
∀u ∈ X .
Then by the continuous embedding X ↩→ Lθ (RN ), for all u ∈ X , there exists a constant lθ > 0 such that
|u|θ ≤ lθ ∥u∥X ,
θ ∈ [N , +∞).
(2.1)
Now, we introduce the subspace
E = u ∈ X
V (x)|u| dx < ∞ . N
RN
Obviously, E is a Banach space endowed with the norm
∥ u∥ =
RN
(|∇ u|N + V (x)|u|N )dx
N1
,
∀u ∈ E .
Then by (2.1), for all u ∈ E, we have
|u|θ ≤ lθ ∥u∥X ≤ lθ ∥u∥,
θ ∈ [N , +∞).
(2.2)
Moreover, for q > N (k + 1),
S=
inf
u∈E \{0}
RN
N1 (|∇ u|N + V (x)|u|N )dx 1q q dx | u | N R
is well defined. In the present paper, we will show the multiplicity of weak solutions for (1.1) by looking for critical points of the associated functional I λ ( u) =
1 N (k + 1)
∥u∥N (k+1) −
λ p
RN
A(x)|u|p dx −
RN
F (u)dx.
Clearly, Iλ ∈ C 1 (E , R) and the critical points of the functional Iλ are in fact weak solutions of (1.1). By a weak solution of (1.1), we mean u ∈ E satisfying
∥u∥Nk
RN
(|∇ u|N −2 ∇ u∇ψ + V (x)|u|N −2 uψ)dx − λ
RN
A(x)|u|p−2 uψ dx −
for all ψ ∈ E. Now, we give a compact embedding lemma, which is a key property of E. Lemma 2.1. Let (V1 ) hold true. Then the embedding E ↩→ LN (RN ) is compact.
RN
f (u)ψ dx = 0,
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Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
Proof. Let Y = LN (RN ), BR = x ∈ RN |x| < R and BcR = RN \BR . Denote E (Ω ), Y (Ω ) the spaces of functions u ∈ E , u ∈ Y restricted onto Ω ⊂ RN respectively. By (V1 ), we let αR = infBc V (x) and obviously, αR → +∞, as R → +∞. Then, we get R 1 1 |u|N dx ≤ V (x)|u|N dx ≤ ∥ u∥ N αR BcR αR BcR
and so lim
sup
|u|Y (BcR ) ∥ u∥
R→+∞ u∈E \{0}
= 0.
On the other hand, it is well known that E (BR ) ↩→ Y (BR ) is compact. Then, Theorem 7.9 in [22] shows that the embedding E ↩→ Y is compact, which completes the proof. In the sequel, we will prove the existence of a weak solution of (1.1) having a positive energy by using the Mountain Pass Theorem in [4] (or see [28]). Firstly, we check that the functional Iλ verifies the Mountain Pass geometry. Lemma 2.2. Assume that (f1 )–(f3 ) hold. Then there exists Λ1 > 0 such that if 0 < λ < Λ1 , the functional Iλ satisfies: (i) There exist η > 0 and ρ > 0 such that Iλ (u) ≥ η,
for ∥u∥ = ρ.
(ii) There exists e ∈ E with ∥e∥ > ρ such that Iλ (e) < 0. Proof. First, we show that Iλ satisfies (i). By (f1 ) and (f2 ), given ϵ > 0 and ϱ ≥ N (k + 1), there exists Cϵ > 0 such that, for every α > α0 ,
|F (s)| ≤
N ϵ N −1 |s|N (k+1) + Cϵ |s|ϱ eα|s| − SN −2 (α, s) , N (k + 1)
for all s ∈ R.
(2.3)
It is noted in [17,18] that
RN
N N −1 |s|ϱ eα|s| − SN −2 (α, s) dx ≤ C (α, N )∥s∥ϱ ,
(2.4)
provided that ∥s∥ ≤ δ , where positive constant δ is sufficiently small. Then by (2.2)–(2.4) and Hölder inequality, we have
N ϵ N −1 N (k+1) ∥ u∥ − A(x)|u| dx − |u|N (k+1) − Cϵ |u|ϱ eα|u| − SN −2 (α, u) dx Iλ (u) ≥ N N (k + 1) p RN N (k + 1) R N N (k+1) λ 1 p N (k+1) ϱ α|u| N −1 ≥ 1 − ϵ lN (k+1) ∥u∥ − |A|σ |u|N − Cϵ |u| e − SN −2 (α, u) dx N (k + 1) p RN N (k+1) λ 1 N (k+1) 1 − ϵ lN (k+1) ∥u∥ − |A|σ ∥u∥p − C1 ∥u∥ϱ . ≥ N (k + 1) p 1
N (k+1)
λ
Choosing ϵ sufficiently small and Λ1 = Iλ (u) ≥
≥
1 2N (k + 1) 1 4N (k + 1)
ρ N (k+1) −
λ p
pρ N (k+1)−p 4N (k+1)|A|σ
p
, for 0 < λ < Λ1 and ∥u∥ = ρ , we deduce
|A|σ ρ p − C1 ρ ϱ
ρ N (k+1) − C1 ρ ϱ .
Thus, if we choose ρ > 0 small enough, there exists η > 0 such that Iλ (u) ≥ η for ∥u∥ = ρ , which verifies (i). In order to prove (ii), for s ∈ R, we set
Φ (t ) = t −µ F (ts) − F (s),
t ≥ 1.
Then it follows from (f3 ) that
Φ ′ (t ) = t −µ−1 [−µF (ts) + tsf (ts)] ≥ 0. Hence, Φ (t ) ≥ Φ (1) = 0 for all t ≥ 1 and then F (ts) ≥ t µ F (s) for all s ∈ R, t ≥ 1.
(2.5)
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
163
Fixing u ∈ E \{0}, we have t N (k+1)
Iλ (tu) =
N (k + 1) t N (k+1)
≤
N (k + 1)
∥u∥N (k+1) − ∥u∥N (k+1) −
λt p
p
λt p
A(x)|u|p dx −
RN
p
RN
A(x)|u|p dx − t µ
RN
F (tu)dx
RN
F (u)dx.
Since µ > N (k + 1), Iλ (tu) → −∞ as t → +∞. Taking e = tu with t > 1 sufficiently large, we conclude (ii). By a version of the Mountain Pass Theorem found in [4], there exists a Palais Smale sequence {un } ⊂ E such that I λ ( un ) → c
and
Iλ′ (un ) → 0,
as n → +∞,
where c = inf max Iλ (γ (t )) γ ∈Γ t ∈[0,1]
with
Γ = {γ ∈ C ([0, 1], E )|γ (0) = 0, γ (1) = e}. Then we have the following results. Lemma 2.3. Assume that (f1 )–(f4 ) hold. Then the mountain level c satisfies c<
1
1 N (k + 1)
8N
−
1 αN (N −1)(k+1)
µ
α0
.
Proof. Let w ∈ E \{0} be a function verifying the equation
S=
N1 N N (|∇w| + V ( x )|w| ) dx RN . 1q q |w| dx RN
Then by the definition of c and (f4 ), we get
t N (k+1) λt p ∥w∥N (k+1) − A(x)|w|p dx − F (t w)dx t ≥0 N (k + 1) p RN RN ∥w∥N (k+1) − F (t w)dx
c ≤ max Iλ (t w) = max t ≥0
t N (k+1) ≤ max t ≥0 N (k + 1) RN t N (k+1) γ tq ≤ max ∥w∥N (k+1) − |w|qq . t ≥0 N (k + 1) q N (k+1)
Dividing by |w|q c
|w|qN (k+1)
and by a direct computation, one has
t N (k+1) γ tq S N (k+1) − |w|qq−N (k+1) t ≥0 N (k + 1) q qN (k+1) −N (k+1) 1 1 N (k+1) = − S q−N (k+1) γ q−N (k+1) |w|− , q N (k + 1) q ≤ max
which implies c≤
1
N (k + 1)
−
1 qN (k+1) S q−N (k+1) γ q
−N (k+1) q−N (k+1)
.
By (1.2), we can easily get c<
1 8N
1 N (k + 1)
−
which completes the proof.
1 αN (N −1)(k+1)
µ
α0
,
(2.6)
164
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
Lemma 2.4. Let {un } ⊂ E be a (PS )c -sequence associated with Iλ . Then there exists Λ2 > 0 such that if 0 < λ < Λ2 , there holds
α N −1 N
lim sup ∥un ∥N <
α0
n→∞
.
Proof. First, we claim that {un } is bounded in E. In fact, (2.6) implies that 1 ′ Iλ (un )un ≤ c + on (1) + on (1)∥un ∥.
Iλ (un ) −
µ
On the other hand, it follows from (f3 ) and Hölder inequality that
1 1 1 ∥un ∥N (k+1) − λ − A(x)|un |p dx + f (un )un − F (un ) dx N (k + 1) µ p µ RN RN µ 1 1 1 1 ≥ ∥un ∥N (k+1) − λ − − A(x)|un |p dx N (k + 1) µ p µ RN 1 1 1 1 ∥un ∥N (k+1) − λ |A|σ ∥un ∥p . − − ≥ N (k + 1) µ p µ
1 ′ I λ ( un ) un =
Iλ (un ) −
µ
1
−
1
Then 1
−
N (k + 1)
1
µ
∥un ∥N (k+1) ≤ λ
1 p
−
1
µ
|A|σ ∥un ∥p + c + on (1) + on (1)∥un ∥,
(2.7)
which implies that {un } is bounded in E. Applying Young inequality in (2.7), we get 1
−
N (k + 1)
1
µ
∥un ∥N (k+1) ≤ ϵ
1 p
−
1
µ
∥un ∥N (k+1) + Cϵ
1 p
−
1
µ
N (k+1)
N (k+1)
λ N (k+1)−p |A|σN (k+1)−p + c + on (1)
and then 1
1
−
N (k + 1)
−ϵ
µ
1 p
1
−
µ
∥un ∥N (k+1) ≤ Cϵ
1 p
1
−
µ
N (k+1)
N (k+1)
N (k+1)
N (k+1)
λ N (k+1)−p |A|σN (k+1)−p + c + on (1).
Choosing ϵ small enough, we have 1
1
2 N (k + 1)
−
1
µ
lim sup ∥un ∥N (k+1) ≤ c + Cϵ
1
n→∞
p
−
1
µ
λ N (k+1)−p |A|σN (k+1)−p + on (1),
which leads to
c
N
lim sup ∥un ∥ ≤ n→∞
where C2 =
Cϵ
check that
1 2
1 N (k+1)
N (k+1) N (k+1)−p
1 1 p − µ |A|σ 1 1 1 ( −µ ) 2 N (k+1)
lim sup ∥un ∥N ≤ n→∞
N
1
+ C2 λ
k+1 1 ,
µ
. By Lemma 2.3 and let Λ2 = ( 2C12 )
α N −1 α0
−
N (k+1) N (k+1)−p
N (k+1)−p N (k+1)
( ααN0 )
(N −1)[N (k+1)−p] N
, for 0 < λ < Λ2 , we can easily
.
This proves the lemma. Lemma 2.5. Assume that (V1 ), (f1 )–(f4 ) hold. Then the functional Iλ (u) satisfies the (PS )c condition for 0 < c <
)( ααN0 )(N −1)(k+1) . µ 1
1 8N
( N (k1+1) −
Proof. Let {un } be a (PS )c -sequence. Then by Lemma 2.4, passing to a subsequence, if necessary, we have ∥un ∥ → ν0 ≥ 0. If ν0 = 0, then the proof is finished. In the following, we assume ν0 > 0. Then for n large, ∥un ∥ ≥ 12 ν0 > 0. We now show that {un } has a convergent subsequence in E. By Lemmas 2.1 and 2.4, going if necessary to a subsequence, we can assume that un ⇀ u1
in E ,
(2.8)
un → u
1
in L (R ),
(2.9)
un → u
1
a.e. in R .
(2.10)
N
N
N
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
165
Denote An = ⟨Iλ′ (un ), un − u1 ⟩ = ∥un ∥Nk
−λ
|∇ un |N −2 ∇ un ∇(un − u1 ) + V (x)|un |N −2 un (un − u1 ) dx RN p−2 A(x)|un | un (un − u1 )dx − f (un )(un − u1 )dx
RN
RN
and Bn = ∥un ∥
Nk
RN
|∇ u1 |N −2 ∇ u1 ∇(un − u1 ) + V (x)|u1 |N −2 u1 (un − u1 ) dx.
Then the fact Iλ′ (un ) → 0 implies that An → 0, as n → ∞. Similarly, the fact {un } is bounded and un ⇀ u1 in E implies that Bn → 0, as n → ∞. Now, we claim
lim
n→∞
RN
f (un )(un − u1 )dx = 0
(2.11)
A(x)|un |p−2 un (un − u1 )dx = 0.
(2.12)
and
lim
n→∞
RN
It follows from (f1 ) and (f2 ) that
RN
f (un )(un − u1 )dx ≤
|un |N (k+1)−1 |un − u1 |dx + C3
RN
RN
N N −1 |un − u1 | eα|un | − SN −2 (α, un ) dx.
By (2.2), (2.9) and Hölder inequality, we have
| un |
N (k+1)−1
1
|un − u |dx ≤
β
|un | dx
RN N (k+1)−1
RN
≤ lβ
NN−1
N1 |un − u1 |N dx RN
∥ un ∥
N (k+1)−1
|un − u1 |N → 0,
as n → ∞,
N [N (k+1)−1]
. where β = N −1 On the other hand, by (2.9), Hölder inequality, Proposition 1.3 and Lemma 2.4, we deduce RN
N N −1 − SN −2 (α, un ) dx ≤ |un − u1 | eα|un |
RN
N1 NN−1 N N N −1 (eα|un | |un − u1 |N dx − SN −2 (α, un )) N −1 dx RN
≤ C4 |un − u |N → 0, 1
as n → ∞.
Then (2.11) holds. Applying (2.9) and Hölder inequality, we get
RN
A(x)|un |p−2 un (un − u1 )dx ≤
NN−p N |A(x)| N −p dx
RN
p−1
≤ ∥un ∥
|un |N dx RN
N1 |un − u1 |N dx
RN
|A|σ |un − u |N → 0, 1
p−N 1
as n → ∞,
which proves (2.12). Note that An − Bn = ∥un ∥ Dn − λ Nk
A(x)|un |
p−2
RN
un (un − u )dx − 1
RN
f (un )(un − u1 )dx
(2.13)
with
Dn = RN
|∇ un |N −2 ∇ un − |∇ u1 |N −2 ∇ u1 , ∇ un − ∇ u1 + V (x) |un |N −2 un − |u1 |N −2 u1 , un − u1 dx.
Then by (2.11)–(2.13) and the fact An , Bn → 0, ∥un ∥ ≥ By using the standard inequality
⟨|x|N −2 x − |y|N −2 y, x − y⟩ ≥ CN |x − y|N ,
ν > 0, we have Dn → 0, as n → ∞.
1 2 0
N ≥ 2,
we can easily deduce that ∥un − u1 ∥ → 0, as n → ∞. Thus, Iλ (u) satisfies the (PS )c condition for 0 < c < 1
µ
)( ααN0 )(N −1)(k+1) and the proof is finished.
1 8N
( N (k1+1) −
166
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
3. Proof of Theorem 1.1 In this section, we apply the Mountain Pass Theorem and Ekeland’s variational principle to prove the existence of two positive weak solutions for (1.1). Lemma 3.1. Assume that (V1 ), (f1 )–(f4 ) hold. Then there exists Λ3 > 0 such that if 0 < λ < Λ3 , problem (1.1) admits a weak solution u1 satisfying Iλ (u1 ) > 0. Proof. Let Λ3 = min{Λ1 , Λ2 }. Then the proof follows directly from Lemmas 2.2 and 2.5 and the Mountain Pass Theorem in [4] (or see [28]). In the following, we prove the existence of the second solution u2 different from u1 . Lemma 3.2. Assume that (f1 )–(f4 ) hold. Then there exists Λ4 > 0 such that if 0 < λ < Λ4 , the functional Iλ (u) satisfies the (PS )c0 condition with c0 ≤ 0. Proof. Fix c0 ≤ 0 and assume that {un } ⊂ E satisfies Iλ (un ) → c0 ,
Iλ′ (un ) → 0,
as n → ∞.
(3.1)
Proceeding as in (2.7), we derive 1
N (k + 1)
−
1
µ
∥un ∥N (k+1) ≤ λ
1 p
−
1
µ
|A|σ ∥un ∥p + c0 + on (1)
and then, for a subsequence, we have 1
N (k + 1)
−
1
1
lim sup ∥un ∥N (k+1)−p − λ
µ
n→∞
p
−
1
µ
|A|σ lim sup ∥un ∥p ≤ 0. n→∞
Hence, lim sup ∥un ∥N ≤
N 1 N (k+1)−p λ p − µ1 1 N (k+1)
n→∞
Choosing Λ4 =
p[µ−N (k+1)] N (k+1)(µ−p)
lim sup ∥un ∥N <
( ααN0 )
−
(N −1)[N (k+1)−p]
α N −1
n→∞
N
α0
1
.
µ N
, for 0 < λ < Λ4 , we have
.
Then, by using the same arguments as Lemma 2.5, we prove that Iλ satisfies the (PS )c0 condition for c0 ≤ 0. Lemma 3.3. Assume that (V1 ), (f1 )–(f4 ) hold. Then there exists Λ5 > 0 such that if 0 < λ < Λ5 , problem (1.1) has a weak solution u2 satisfying Iλ (u2 ) < 0. Proof. Choosing a function ω ∈ E \{0} and by (f3 ), we have Iλ (t ω) ≤
t N (k+1) N (k + 1)
λt p
∥ω∥N (k+1) −
p
RN
A(x)|ω|p dx.
(3.2)
Since N (k + 1) > p, it follows from (3.2) that Iλ (t ω) < 0 for t > 0 small. Thus c0 = inf Iλ (u) < 0
and
u∈Bρ
inf Iλ (u) > 0,
u∈∂ Bρ
(3.3)
where ρ > 0 is given by Lemma 2.2(i) and Bρ is an open ball in E centered at the origin with radius ρ . Let εn → 0 be such that 0 < εn < inf Iλ (u) − inf Iλ (u). u∈∂ Bρ
u∈Bρ
(3.4)
By Ekeland’s variational principle, there exists {un } ⊂ Bρ such that c0 ≤ Iλ (un ) ≤ c0 + εn
(3.5)
and Iλ (un ) < Iλ (u) + εn ∥un − u∥,
∀u ∈ Bρ , u ̸= un .
(3.6)
Q. Li, Z. Yang / Nonlinear Analysis 117 (2015) 159–168
167
Then by (3.3)–(3.5), we get Iλ (un ) ≤ c0 + εn ≤ inf Iλ (u) + εn < inf Iλ (u) u∈Bρ
u∈∂ Bρ
and so {un } ⊂ Bρ . Consider the sequence vn = un + t ϕ ⊂ Bρ for ϕ ∈ B1 and t > 0 small enough. Then it follows from (3.6) that t −1 Iλ (un + t ϕ) − Iλ (un ) ≥ −εn ∥ϕ∥.
(3.7)
Passing to the limit as t → 0+ , (3.7) implies that Iλ′ (un )ϕ ≥ −εn ∥ϕ∥,
∀ϕ ∈ B1 .
Replacing ϕ in (3.7) by −ϕ , we have Iλ′ (un )(−ϕ) ≥ −εn ∥ϕ∥,
∀ϕ ∈ B1 .
Then
|Iλ′ (un )ϕ| ≤ εn ,
∀ϕ ∈ B1
and so
∥Iλ′ (un )∥ → 0,
as n → ∞.
(3.8) Iλ′ (un )
Therefore, there exists a sequence {un } ⊂ Bρ such that Iλ (un ) → c0 < 0 and → 0, as n → ∞. By Lemma 3.2, {un } has a convergent subsequence (still denoted by {un }) such that un → u2 in E. Thus, u2 is a nontrivial solution for (1.1) with Iλ (u2 ) < 0, which completes the proof. Proof of Theorem 1.1. It follows from Lemmas 3.1 and 3.3 that problem (1.1) has at least two weak solutions. By using a simple argument as [3], we can assume that u1 ≥ 0 and u2 ≥ 0 in RN . Furthermore, as a consequence of Harnack’s inequality [31], we have u1 > 0 and u2 > 0 in RN . Thus, we finish the proof. References [1] S. Adachi, T. Watanabe, Uniqueness and non-degeneracy of positive radial solutions for quasilinear elliptic equations with exponential nonlinearity, Nonlinear Anal. 108 (2014) 275–290. [2] C.O. Alves, L.R. de Freitas, Multiplicity of nonradial solutions for a class of quasilinear equations on annulus with exponential critical growth, Topol. Methods Nonlinear Anal. 39 (2012) 243–262. [3] C.O. Alves, G.M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN , J. Differential Equations 246 (2009) 1288–1311. [4] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [5] S. Aouaoui, On some semilinear elliptic equation involving exponential growth, Appl. Math. Lett. 33 (2014) 23–28. [6] G. Autuori, F. Colasuonno, P. Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math. 16 (5) (2014) 1450002. 43 pages. [7] J.M. Bezerra do Ó, N-Laplacian equations in RN with critical growth, Abstr. Appl. Anal. 2 (1997) 301–315. [8] J. Chen, Multiple positive solutions to a class of Kirchhoff equation on R3 with indefinite nonlinearity, Nonlinear Anal. 96 (2014) 134–145. [9] C. Chen, J. Huang, L. Liu, Multiple solutions to the homogeneous p-Kirchhoff elliptic equation with concave-convex nonlinearities, Appl. Math. Lett. 26 (2013) 754–759. [10] C. Chen, H. Song, Z. Xiu, Multiple solutions for p-Kirchhoff equations in RN , Nonlinear Anal. 86 (2013) 146–156. [11] C. Chen, Q. Zhu, Existence of positive solutions to p-Kirchhoff-type problem without compactness conditions, Appl. Math. Lett. 28 (2014) 82–87. [12] F. Colasuonno, P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011) 5962–5974. [13] L.R. de Freitas, Multiplicity of solutions for a class of quasilinear equations with exponential critical growth, Nonlinear Anal. 95 (2014) 607–624. [14] M. de Souza, On a singular class of elliptic systems involving critical growth in R2 , Nonlinear Anal. RWA 12 (2011) 1072–1088. [15] M. de Souza, E. de Medeiros, U. Severo, On a class of quasilinear elliptic problems involving Trudinger–Moser nonlinearities, J. Math. Anal. Appl. 403 (2013) 357–364. [16] M. de Souza, João Marcos do Ó, On a singular and nonhomogeneous N-Laplacian equation involving critical growth, J. Math. Anal. Appl. 380 (2011) 241–263. [17] J.M. do Ó, Semilinear Dirichlet problems for the N-Laplacian in RN with nonlinearities in critical growth range, Differential Integral Equations 9 (1996) 967–979. [18] J.M. do Ó, E.S. Medeiros, Remarks on least energy solutions for quasilinear elliptic problems in RN , Electron J. Differential Equations 56 (2008) 1–16. [19] G.M. Figueiredo, R.G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation, Math. Nachr. (2014) 1–13. http://dx.doi.org/10.1002/mana.201300195. [20] X. He, W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 , J. Differential Equations 252 (2012) 1813–1834. [21] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [22] I. Kuzin, S. Pohozaev, Entire Solutions of Semilinear Elliptic Equations, in: Progress in Nonlinear Differential Equations and Their Applications, vol. 33, Birkhäuser, 1997. [23] N. Lam, G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in RN , J. Funct. Anal. 262 (2012) 1132–1165. [24] Y. Li, F. Li, J. Shi, Existence of positive solutions to Kirchhoff type problems with zero mass, J. Math. Anal. Appl. 410 (2014) 361–374. [25] Q. Li, Z. Yang, Existence of multiple solutions for a p-Kirchhoff equation with singular and critical nonlinearities, submitted for publication. [26] D. Liu, P. Zhao, Multiple nontrivial solutions to a p-Kirchhoff equation, Nonlinear Anal. 75 (2012) 5032–5038. [27] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/1971) 1077–1092. [28] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, third ed., Springer-Verlag, New York, 2000.
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