Dec 30, 2015 - Kirchhoff Problems, Critical Potential, Concave term, Nehari ...... [1] Kirchhoff, G.R. (1883) Vorlesungen über mathematische PhysikâMechanik.
Applied Mathematics, 2015, 6, 2248-2256 Published Online December 2015 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2015.614198
Four Nontrivial Solutions for Kirchhoff Problems with Critical Potential, Critical Exponent and a Concave Term Mohammed El Mokhtar Ould El Mokhtar Departement of Mathematics, College of Science, Qassim University, Buraidah, Kingdom of Saudi Arabia
Received 13 November 2015; accepted 27 December 2015; published 30 December 2015 Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract In this paper, we consider the existence of multiple solutions to the Kirchhoff problems with critical potential, critical exponent and a concave term. Our main tools are the Nehari manifold and mountain pass theorem.
Keywords Kirchhoff Problems, Critical Potential, Concave term, Nehari Manifold, Mountain Pass Theorem
1. Introduction In this paper, we consider the multiplicity results of nontrivial solutions of the following Kirchhoff problem q −2 4 Lµ ,a ,bu = h u u + λ f u u in Ω, = on ∂Ω u 0
( (∫
where Lµ ,a ,b v :=− a + b
Ω
∇v + µ x 2
−2
) )(
v 2 dx ∆v + µ x
−2
(1.1)
)
v , Ω is a smooth bounded domain of 3 , a > 0 ,
b > 0 , λ ≠ 0 , 1 < q < 2 , µ < 1 4 , λ is a real parameter, f ∈ −1 C ( Ω ) with −1 is the topological dual of 01 ( Ω ) satisfying suitable conditions, h is a bounded positive function on Ω. The original one-dimensional Kirchhoff equation was introduced by Kirchhoff [1] in 1883. His model takes into account the changes in length of the strings produced by transverse vibrations. In recent years, the existence and multiplicity of solutions to the nonlocal problem How to cite this paper: El Mokhtar, E.O. (2015) Four Nontrivial Solutions for Kirchhoff Problems with Critical Potential, Critical Exponent and a Concave Term. Applied Mathematics, 6, 2248-2256. http://dx.doi.org/10.4236/am.2015.614198
(
E. O. El Mokhtar
)
2 − a + b ∇u= dx ∆u g ( x; u ) in Ω, ∫ Ω u 0 on ∂Ω =
(1.2)
has been studied by various researchers and many interesting and important results can be found. For instance, positive solutions could be obtained in [2]-[4]. Especially, Chen et al. [5] discussed a Kirchhoff type problem q −2 = 2∗ 2 N ( N − 2 ) if N ≥ 3 , 2∗ = ∞ if when= g ( x; u ) f ( x ) u p − 2u + λ g ( x ) u u , where 1 < q < 2 < p < N = 1, 2, f ( x ) and g ( x ) with some proper conditions are sign-changing weight functions. And they have obtained the existence of two positive solutions if p > 4 , 0 < λ < λ0 ( a ) . Researchers, such as Mao and Zhang [6], Mao and Luan [7], found sign-changing solutions. As for in nitely many solutions, we refer readers to [8] [9]. He and Zou [10] considered the class of Kirchhoff type problem when g ( x; u ) = λ f ( x; u ) with some conditions and proved a sequence of a.e. positive weak solutions tending to zero in L∞ ( Ω ) . In the case of a bounded domain of N with N ≥ 3 , Tarantello [8] proved, under a suitable condition on f, the existence of at least two solutions to (1.2) for a = 0 , b = 1 and g= ( x; u ) u
4 N −2
u+ f .
Before formulating our results, we give some definitions and notation. The space 01 ( Ω ) is equiped with the norm
(∫
= u
∇u dx 2
Ω
)
12
wich equivalent to the norm
( ∫ ( ∇u
u µ=
2
Ω
−µ x
−2
u
2
) dx )
,
)
u ,
12
with µ < 1 4 . More explicitly, we have
(1 − 4µ ) +
12
u ≤ u
µ
(
≤ 1 − 4µ −
12
for all u ∈ 01 ( Ω ) , with µ + = max ( µ , 0 ) and µ − = min ( µ , 0 ) . Let S µ be the best Sobolev constant, then
S µ12
(∫ ≤
2
u dx Ω u
)
2
(2.1)
12
µ
Since our approach is variational, we define the functional J λ on 01 ( Ω ) by
J λ ( u ) = (1 2 ) a u
2
µ
+ (1 4 ) b u
4
µ
− (1 6 ) ∫Ω h u dx − ( λ q ) ∫Ω f u dx, 6
q
(2.2)
A point u ∈ 01 ( Ω ) is a weak solution of the Equation (1.1) if it is the critical point of the functional J λ . Generally speaking, a function u is called a solution of (1.1) if u ∈ 01 ( Ω ) and for all v ∈ 01 ( Ω ) it holds
( a + b u ) ∫ ( ∇ u∇ v − µ x 2
µ
−2
Ω
)
uv dx − ∫Ω h u uvdx − λ ∫Ω f u 5
q −1
uvdx =0.
Throughout this work, we consider the following assumptions: (F) There exist ν 0 > 0 and δ 0 > 0 such that f ( x ) ≥ ν 0 , for all x in B ( 0, 2δ 0 ) . 1
1 2 (H) h ∈ C ( Ω ) , h ( 0 ) = max h ( x ) = h0 + o x , x ∈ B ( 0, 2δ 0 ) , β > 6 − µ . x∈Ω 4 Here, B ( a, r ) denotes the ball centered at a with radius r. In our work, we research the critical points as the minimizers of the energy functional associated to the problem (1.1) on the constraint defined by the Nehari manifold, which are solutions of our system. Let λ0 be positive number such that
( ) β
λ0 = E ( A + BD ) ( A′ + B ′D ) , k
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E. O. El Mokhtar
where
2−q 4−q 4 2 = A = a, B = b, A′ = a, B ′ b, 6 − q 6 − q 6 − q 6−q 2q ( q − 2 ) a = , E ( 4 − q )( q + 2 ) b
= D
( h0 )
q−2 4
3( 2 − q )
= f H −1 ( S µ ) 2 and k −1
q−2 . 4
Now we can state our main results. Theorem 1. Assume that 1 < q < 2 , −∞ < µ
3(6 − q )
(4 − q)
then there exists
λ1 > 0 such that for all λ verifying 0 < λ < min ( λ0 , λ1 ) the problem (1.1) has at least two positive solutions. Theorem 3. In addition to the assumptions of the Theorem 2, assuming λ < 0 then the problem (1.1) has at least two positive solutions and two opposite solutions. This paper is organized as follows. In Section 2, we give some preliminaries. Section 3 and 4 are devoted to the proofs of Theorems 1 and 2. In the last Section, we prove the Theorem 3.
2. Preliminaries Definition 1. Let c ∈ , E a Banach space and I ∈ C1 ( E , ) . i) ( un )n is a Palais-Smale sequence at level c (in short ( PS )c ) in E for I if
I ( un ) = c + on (1) and I ′ ( un ) = on (1) ,
where on (1) tends to 0 as n goes at infinity. ii) We say that I satisfies the ( PS )c condition if any ( PS )c sequence in E for I has a convergent subsequence. Lemma 1. Let X Banach space, and J ∈ C1 ( X , ) verifying the Palais-Smale condition. Suppose that J ( 0 ) = 0 and that: i) there exist R > 0 , r > 0 such that if u = R, then J ( u ) ≥ r ; ii) there exist ( u0 ) ∈ X such that u0 > R and J ( u0 ) ≤ 0;
(
let c = inf max J ( γ ( t ) ) γ ∈Γ t∈[ 0,1]
)
where
Γ=
{γ ∈ C ([0,1]; X ) such that γ ( 0 )=
}
0 et γ (1)= u0 ,
then c is critical value of J such that c ≥ r .
Nehari Manifold It is well known that the functional J λ is of class C1 in 01 ( Ω ) and the solutions of (1.1) are the critical points of J λ which is not bounded below on 01 ( Ω ) . Consider the lowing Nehari manifold
{
}
1 ′ λ = u ∈ 0 , 0 ( Ω ) \ {0} : J λ ( u ) , u =
Thus, u ∈ λ if and only if
Define
(a + b u ) u 2
2
µ
µ
0 − ∫Ωh u dx − λ ∫Ω f u dx = 6
φλ ( u ) = J λ′ ( u ) , u .
Then, for u ∈ λ
2250
q
(2.3)
(
φλ′ ( u ) , u = 2a + 4b u
2
µ
E. O. El Mokhtar
)u
− 6 ∫Ω h u dx − λ q ∫Ω f u dx
2
6
µ
2 = ( 2 − q ) a + ( 4 − q ) b u µ u
(
q
− ( 6 − q ) ∫Ω h u dx
2
6
µ
=( 6 − q ) λ ∫Ω f u dx − 4a + 2b u q
Now, we split λ in three parts:
{ = {u ∈ : φ ′ ( u ) , u = {u ∈ : φ ′ ( u ) , u
2
µ
)u
2
µ
(2.4)
.
} = 0} < 0} .
λ+ = u ∈ λ : φλ′ ( u ) , u > 0 λ0
λ−
λ
λ
λ
λ
Note that λ contains every nontrivial solution of the problem (1.1). Moreover, we have the following results. Lemma 2. J λ is coercive and bounded from below on λ . Proof. If u ∈ λ , then by (2.3) and the Hölder inequality, we deduce that
J λ ( u ) = (1 2 ) a u
2
≥ (1 3) a u
2
µ
µ
+ (1 4 ) b u
− (1 6 ) ∫ h u dx − ( λ q ) ∫ f u dx Ω Ω
4
+ (1 12 ) b u
q
6
µ 4
µ
1 1 −λ − f q 6
q
−1
u µ.
Thus, J λ is coercive and bounded from below on λ . We have the following results. Lemma 3. Suppose that u0 is a local minimizer for J λ on λ . Then, if u0 ∉ λ0 , u0 is a critical point of J λ . Proof. If u0 is a local minimizer for J λ on λ , then u0 is a solution of the optimization problem
min
{u /φλ ( u ) = 0}
Jλ (u ).
Hence, there exists a Lagrange multipliers θ ∈ such that
J λ′ ( u0 ) = θφλ′ ( u0 ) in −1 Thus, J λ′ ( u0 ) , u0 = θ φλ′ ( u0 ) , u0 .
But φλ′ ( u0 ) , u0 ≠ 0 , since u0 ∉ λ0 . Hence θ = 0 . This completes the proof. Lemma 4. There exists a positive number λ0 such that, for all λ ∈ ( 0, λ0 ) we have λ0 = ∅ . Proof. Let us reason by contradiction. Suppose λ0 ≠ ∅ such that 0 < λ < λ0 . Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain
u
4
µ
and
u
4
µ
(
≥ A′ + B ′ u
(
≤ A+ B u
2
µ
2
µ
)
)
4 q−2
−1 ( 2 − q )
f
−4 q−2 −1
λ
−4 q−2
h0 ( S µ ) , −6
with
2−q 4−q 4 2 = A = a, B = b, A′ = a, B ′ b. 6−q 6−q 6−q 6−q From (2.5) and (2.6), we obtain λ ≥ λ0 , which contradicts an hypothesis.
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E. O. El Mokhtar
Thus λ = λ+ λ− . Define − = c : inf = J λ ( u ) , c + : inf + J= inf − J λ ( u ) . λ ( u ) and c : u∈λ
u∈λ
u∈λ
For the sequel, we need the following Lemma. Lemma 5. i) For all λ such that 0 < λ < λ0 , one has c ≤ c + < 0 . ii) There exists λ1 > 0 such that for all 0 < λ < λ1 , one has
(
c − > C0 = C0 a, b, q, f Proof. i) Let u ∈ λ+ . By (2.4), we have
(
(2 − q) a + (4 − q)b u
2
µ
−1
) ( 6 − q ) u
).
2
> ∫Ω h u dx 6
µ ,a
and so
1 1 1 1 1 1 4 2 6 J λ ( u ) = − b u µ + − a u µ + − ∫Ω h u dx 4 2 6 q q q (4 − q) (2 − q) a u 2 . 4 b uµ+ < − µ 12 3q q We conclude that c ≤ c + < 0 . ii) Let u ∈ λ− . By (2.4) and the Hölder inequality we get
1 1 q − λ − f −1 u µ . q 6 1 1 4 q ≥ (1 3) a + (1 12 ) b min u µ , u µ − λ − f q 6
J λ ( u ) ≥ (1 3) a u
2
µ
+ (1 12 ) b u
4
µ
(
)
−1
(
4
max u µ , u
q
µ
).
( 4a + b ) q Thus, for all λ such that 0 < λ < λ1 = , we have J λ ( u ) ≥ C0 . ( 6 − q ) f −1 with P For each u ∈ 01 ( Ω ) =
∫Ω h u
6
dx > a u µ , we write 2
3 P − a u = = tm : tmax (u )
(
2
µ
)
2 1 + 1 + 20aP u µ 2 9 P−a u µ 3P
(
12
)
> 0.
Lemma 6. Let λ real parameters such that 0 < λ < λ0 . For each u ∈ 01 ( Ω ) with there exist unique t + and t − such that 0 < t + < tm < t − , t + u ∈ λ+ , t − u ∈ λ− ,
( ) = J ( t u ) inf = J ( tu ) and J ( t u ) λ
+
0 ≤t ≤tm
λ
λ
−
( )
∫Ω h u
6
dx > a u µ , 2
sup J λ ( tu ) . t ≥0
Proof. With minor modifications, we refer to [11]. Proposition 1. (see [11]) i) For all λ such that 0 < λ < λ0 , there exists a ( PS )c+ sequence in λ+ . ii) For all λ such that 0 < λ < λ1 , there exists a a
( PS )c
−
sequence in λ− .
3. Proof of Theorem 1 Now, taking as a starting point the work of Tarantello [8], we establish the existence of a local minimum for J λ
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E. O. El Mokhtar
on λ+ . Proposition 2. For all λ such that 0 < λ < λ0 , the functional J λ has a minimizer u0+ ∈ λ+ and it satisfies: i) J λ u0+ = c= c + ,
( ) ( u ) is a nontrivial solution of (1.1).
+ ii) 0 Proof. If 0 < λ < λ0 , then by Proposition 1. i) there exists a ( un )n ( PS )c+ sequence in λ+ , thus it bounded by Lemma 2. Then, there exists u0+ ∈ 01 ( Ω ) and we can extract a subsequence which will denoted by ( un )n such that 1 un u0+ weakly in 0 (Ω)
un u0+ weakly in L6 ( Ω )
(3.1)
un → u0+ strongly in Lq ( Ω ) un → u0+ a.e in Ω
Thus, by (3.1), u0+ is a weak nontrivial solution of (1.1). Now, we show that un converges to u0+ strongly < lim inf un µ , a in 01 ( Ω ) . Suppose otherwise. By the lower semi-continuity of the norm, then either u0+ µ ,a
n →∞
and we obtain
( ) ( a 3) u
c ≤ Jλ = u0+
+ 2 0 µ
+ ( b 12 ) u0+
q 1 1 − λ − ∫Ω f u0+ dx < lim inf J λ = ( un ) c. n →∞ q 6
4
µ
We get a contradiction. Therefore, un converge to u0+ strongly in 01 ( Ω ) . Moreover, we have u0+ ∈ λ+ . If not, then by Lemma 6, there are two numbers t0+ and t0− , uniquely defined so that t0+ u0+ ∈ λ+ and t − u0+ ∈ λ− . In particular, we have t0+ < t0− = 1 . Since
(
(
)
d d2 0 and J λ tu0+ J λ tu0+ = = t t0+= t dt dt 2
( ) ( ) > 0, there exists t < t ≤ t such that J ( t u ) < J ( t u ) . By Lemma 6, we get J (t u ) < J (t u ) < J (t u ) = J (u ) , which contradicts the fact that J ( u ) = c . Since J ( u ) = J ( u ) and u + 0
−
− 0
λ
+ + 0 0
+ + 0 0
λ
−
λ
λ
−
)
t0+
+ 0
+ 0
λ
+ 0
− + 0 0
λ
+ 0
+ 0
∈ λ+ , then by Lemma 3, we may assume that u is a nontrivial nonnegative solution of (1.1). By the Harnack inequality, we conclude that u0+ > 0 and v0+ > 0 , see for exanmple [12]. + 0
λ
+ 0
+
λ
λ
+ 0
4. Proof of Theorem 2 Next, we establish the existence of a local minimum for J λ on λ− . For this, we require the following Lemma. 3(6 − q ) then for all λ such that 0 < λ < λ1 , the functional J λ has a miniLemma 7. Assume that b > (4 − q) mizer u0− in λ− and it satisfies: i) J λ u0−= c − > 0, ii) u0− is a nontrivial solution of (1.1) in 01 ( Ω ) . Proof. If 0 < λ < λ1 , then by Proposition 1. ii) there exists a ( un )n , ( PS )c− sequence in λ− , thus it bounded by Lemma 2. Then, there exists u0− ∈ 01 ( Ω ) and we can extract a subsequence which will denoted by ( un )n such that 1 un u0− weakly in 0 (Ω)
( )
un u0− weakly in L6 ( Ω ) un → u0− strongly in Lq ( Ω ) un → u0− a.e in Ω
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E. O. El Mokhtar
This implies that
∫Ω h un
6
6
dx → ∫Ω h u0− dx, as n goes to ∞.
Moreover, by (H) and (2.4) we obtain
∫Ω h un
6
(4 − q) b un dx > (6 − q) (4 − q) > b un (6 − q)
(2 − q) a (6 − q) (2 − q) a 4 + µ (6 − q) 4
+
µ
2 un µ 2 un µ − un
2
µ
(2 − q) (4 − q) 2(4 − q) = > C1 a b − 3 b − 2 (6 − q) (6 − q) (6 − q) 2
if b >
3(6 − q )
(4 − q)
−2
we get
∫Ω h un
6
dx > C1 > 0.
(4.1)
This implies that − 6
∫Ω h u0
dx ≥ C1.
Now, we prove that ( un )n converges to u0− strongly in 01 ( Ω ) . Suppose otherwise. Then, either u0− < lim inf un µ . By Lemma 6 there is a unique t0− such that t0− u0− ∈ λ− . Since µ
(
n →∞
)
un ∈ λ− , J λ ( un ) ≥ J λ ( tun ) , for all t ≥ 0, we have
)
(
(
)
J λ t0− u0− < lim J λ t0− un ≤ lim J λ ( un ) = c− , n →∞
n →∞
and this is a contradiction. Hence,
( un )n → u0− strongly in 01 ( Ω ) . Thus,
( )
to J λ u0− J λ ( un ) converges = c − as n tends to + ∞.
( ) − 0
( ) − 0
− 0
−
Since J λ u = J λ u and u ∈ λ , then by (4.1) and Lemma 3, we may assume that u0− is a nontrivial nonnegative solution of (1.1). By the maximum principle, we conclude that u0− > 0 . Now, we complete the proof of Theorem 2. By Propositions 2 and Lemma 7, we obtain that (1.1) has two positive solutions u0+ ∈ λ+ and u0− ∈ λ− . Since λ+ λ− = ∅ , this implies that u0+ and u0− are distinct.
5. Proof of Theorem 3 In this section, we consider the following Nehari submanifold of λ
{
λ , = u ∈
1 0
( Ω ) \ {0} :
J λ′ ( u ) , u = 0 and u
µ
}
≥ >0 .
Thus, u ∈ λ , if and only if
(a + b u ) u 2
2
µ
µ
− ∫Ω h u dx − λ ∫Ω = f u dx 0 and u 6
q
µ
≥ > 0.
Firsly, we need the following Lemmas. Lemma 8. Under the hypothesis of theorem 3, there exist 0 such that λ , is nonempty for any λ < 0 and ∈ ( 0, 0 ) .
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E. O. El Mokhtar
Proof. Fix u0 ∈ 01 ( Ω ) \ {0} and let
g ( t ) = J λ′ ( tu0 ) , tu0 = at 2 u0
2
+ bt 4 u0
µ
4
µ
− t 6 ∫Ω h u0 dx − λt q ∫Ω f u0 dx. 6
q
Clearly g ( 0 ) = 0 and g ( t ) → −∞ as n → +∞ . Moreover, we have
g (1) = a u0
2
µ
≥ a u0 If u0
2
µ
≥ > 0 for 0 < < 0 = ( ah0−1 )
14
µ
− ∫Ω h u0 dx − λ ∫Ω f u0 dx
4
+ b u0
6
µ
q
−3 6 − ( S µ ) h0 u0 µ .
( Sµ )
34
, then there exists t0 > 0 such that g ( t0 ) = 0 . Thus,
( t0u0 ) ∈ λ ,
and λ , is nonempty for any λ < 0 . Lemma 9. There exist M positive real such that
φλ′ ( u ) , u < − M < 0, for u ∈ λ , and any λ < 0. Proof. Let u ∈ λ , , then by (2.3), (2.4) and the Holder inequality, allows us to write
φλ′ ( u ) , u ≤ u µ ( 6 − q ) λ f 2
−1
− ( 4a + 2b ) .
Thus, if λ < 0 then we obtain that
φλ′ ( u ) , u < 0, for any u ∈ λ , .
(5.1)
Lemma 10. There exist r and η positive constants such that i) we have
J λ ( u ) ≥ η > 0 for u
µ ,a
= r.
ii) there exists σ ∈ λ , when σ µ > r , with r = u µ , such that J λ (σ ) ≤ 0 . Proof. We can suppose that the minima of J λ are realized by u0+ and u0− . The geometric conditions of the mountain pass theorem are satisfied. Indeed, we have −3 6 6 i) By (2.4), (5.1), the Holder inequality and the fact that ∫ h u dx ≤ S µ h0 u µ , we get
( ) ( )
Ω
a Jλ (u ) ≥ u 2
2
µ
b + u 4
h − Sµ 0 u 2 −3
4
µ
6
µ
−
λ q
f
q
−1
u µ.
Thus, for λ < 0 there exist η , r > 0 such that
J λ ( u ) ≥ η > 0 when r = u
µ ,a
small.
ii) Let t > 0 , then we have for all θ ∈ λ , J λ ( tθ ) =
a 2 t θ 2
2
µ
b + t4 θ 4
4
µ
1 λ 6 q − t 6 ∫Ω h θ dx − t q ∫Ω f θ dx. 6 q
Letting σ = tθ for t large enough, we obtain J λ (σ ) ≤ 0. For t large enough we can ensure σ Let Γ and c defined by
= Γ:
:γ ( 0 ) {γ : [0,1] → = λ ,
u0− and = γ (1) u0+
µ ,a
>r.
}
and
(
)
c := inf max J λ ( γ ( t ) ) . γ ∈Π t∈[ 0,1]
Proof of Theorem 3. If λ < 0 then, by the Lemmas 2 and Proposition 1. ii), J λ verifying the Palais-Smale condition in λ , . Moreover, from the Lemmas 3, 9 and 10, there exists uc such that
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E. O. El Mokhtar
= J λ ( uc ) c and uc ∈ λ , .
Thus uc is the third solution of our system such that uc ≠ u0+ and uc ≠ u0− . Since (1.1) is odd with respect u, we obtain that −uc is also a solution of (1.1).
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