Existence and multiplicity of bound state solutions to a Kirchhoff type equation with a general nonlinearity
aa r X i v : . [ m a t h . A P ] F e b EXISTENCE AND MULTIPLICITY OF BOUND STATE SOLUTIONS TO AKIRCHHOFF TYPE EQUATION WITH A GENERAL NONLINEARITY
ZHISU LIU, HAIJUN LUO, AND JIANJUN ZHANG
Abstract.
In this paper, we consider the following Kirchhoff type equation − (cid:18) a + b Z R |∇ u | (cid:19) ∆ u + V ( x ) u = f ( u ) , x ∈ R , where a, b > f ∈ C ( R , R ), and the potential V ∈ C ( R , R ) is positive, bounded and satisfiessuitable decay assumptions. By using a new perturbation approach together with a new version ofglobal compactness lemma of Kirchhoff type, we prove the existence and multiplicity of bound statesolutions for the above problem with a general nonlinearity. We especially point out that neitherthe corresponding Ambrosetti-Rabinowitz condition nor any monotonicity assumption is requiredfor f . Moreover, the potential V may not be radially symmetry or coercive. As a prototype, thenonlinear term involves the power-type nonlinearity f ( u ) = | u | p − u for p ∈ (2 , p ∈ (2 ,
3] is left open there.
Contents
1. Introduction 11.1. Overview and motivation 21.2. Our problem 31.3. Our results 32. Preliminary results 53. Existence 53.1. The perturbed problem 53.2. Proof of Theorem 1.3 164. Multiplicity 164.1. Proof of Theorem 1.4 16References 181.
Introduction
In the present paper, we investigate the existence and multiplicity of bound state solutions tothe following Kirchhoff equation − (cid:18) a + b Z R |∇ u | (cid:19) ∆ u + V ( x ) u = f ( u ) , x ∈ R , u ∈ H ( R ) , (K)where V ∈ C ( R , R ) and a, b > V ( x ) = 0 and a domain Ω ⊂ R and replace f ( u ) by f ( x, u ), Mathematics Subject Classification.
Key words and phrases.
Kirchhoff type equation. Perturbation method. Existence. Multiplicity. Variationalmethod.Z. Liu is supported by NSFC (No.11626127) and Hunan Natural Science Excellent Youth Fund (No.2020JJ3029).H. Luo is supported by the Fundamental Research Funds for the Central Universities(No.531118010205) andNSFC(No.11901182). J. Zhang was supported by NSFC(No.11871123). problem (K) becomes as the following Dirichlet problem:(1.1) − (cid:18) a + b Z Ω |∇ u | (cid:19) ∆ u = f ( x, u ) , in Ω ,u = 0 , on ∂ Ω , which is the general form of the stationary counterpart of the hyperbolic Kirchhoff equation(1.2) ρ ∂ u∂t = " p h + E L Z L (cid:18) ∂u∂x (cid:19) dx ∂ u∂x + f ( t, x, u ) . This equation was proposed by Kirchhoff in [19] as an existence of the classical D’Alembert’s waveequations for free vibration of elastic strings, and takes into account the changes in length of thestring produced by transverse vibrations. In (1.2), L denotes the length of the string, E the Youngmodulus of the material, h is the area of the cross section, ρ stands for mass density and p is theinitial tension, f ( t, x, u ) stands for the external force. The function u denotes the displacement, b is the initial tension while a is related to the intrinsic properties. Besides, we also point out thatKirchhoff problems appear in other fields like biological systems, such as population density, where u describes a process which depends on the average of itself. For the further physical background,we refer the readers to [5, 9, 11].1.1. Overview and motivation.
Due to the presence of the integral term, Kirchhoff equationsare no longer a pointwise identity and therefore, are viewed as being nonlocal. This fact bringsmathematical challenges to the analysis, and meanwhile, makes the study of such a problemparticularly interesting. In the past decades, Kirchhoff problems have been receiving extensiveattention. In particular, initiated by Lions [23], the solvability of Kirchhoff type equation (1.1) hasbeen investigated in many studies, see [1, 2, 22, 29, 30, 32, 34, 37, 38, 46] and the references therein.There also have been many interesting works about the existence and multiplicity of boundstate solutions to Kirchhoff type equation (K) via variational methods, see for instance [3, 6, 12, 13,15–17, 20, 21, 26, 27, 31, 40, 42–44] and the references therein. We note that minimax methods areused to study the existence and multiplicity as a typical way. In this process, one has to overcomethe difficulties arising from the effect of non-local property and showing the boundedness andcompactness of Palais-Smale ((PS) for short) sequences. For this aim, one usually assumes thatthe function f satisfies either the 4-superlinear growth condition:lim | t |→ + ∞ F ( u ) /u = + ∞ , (4-superlinear)where F ( u ) = R u f ( s ) ds , or the well-known Ambrosetti-Rabinowitz ((AR) for short) type condition0 < F ( u ) ≤ µ f ( u ) u, µ > , or the monotonicity condition f ( u ) u is strictly increasing in (0 , + ∞ ) . The above conditions are crucial in proving the existence and boundedness of (PS) sequences.Furthermore, nontrivial solutions can be obtained by providing some further conditions on f and V to guarantee the compactness of the (PS) sequence, such as the radial symmetric setting orcoercive condition. It worth of pointing out that, without above conditions, Li and Ye [20] provedthe existence of positive ground state solutions to problem (K) with f ( u ) = | u | p − u , p ∈ (3 , XISTENCE AND MULTIPLICITY OF BOUND STATE SOLUTIONS 3 arguments. Recently, there results of [20] were extended in [26] to the more general case, seealso [14, 39].Compared with the existence results on nontrivial solutions, there is few works published on theinfinitely many solutions of Kirchhoff type problem in R , see [7, 18, 31, 42]. As mentioned above,(AR)-condition or 4-superlinear growth condition and some compactness conditions play importantroles in this literatures. More specifically, Sun et al [36] obtained infinitely many sign-changingsolutions to problem (K) without 4-superlinear growth condition but the coercive condition of V ,by using a combination of invariant sets and the Ljusternik-Schnirelman type minimax method.Under some weak compactness assumptions on V without radial symmetry setting or compactnesshypotheses, Zhang et al. [45] established the existence of infinitely many solutions to problem (K)with f satisfying 4-superlinear growth condition. Very recently, Liu et al. [28] employed a novelperturbation approach and the method of invariant sets of descending flow to prove the existenceof infinitely many sign-changing solutions to problem (K) with a general nonlinearity in the radialsymmetry setting.1.2. Our problem.
These results above left one question:
Does problem (K) admit infinitely many nontrivial solutions without the radial symmetriccondition or coercive condition in the case f ( u ) ∼ | u | p − u, p ∈ (2 , f does not satisfy (AR)-condition (or the 4-superlinear) ormonotonicity assumptions mentioned as before. To the best of our knowledge, so far there hasbeen no results known in this aspect. The main interest of the present paper is to give an affirmativeanswer to this question.1.3. Our results.
Throughout this paper, we assume nonlinearity f satisfies the followinghypotheses( f ) f ∈ C ( R , R ) and lim u → f ( u ) u = 0;( f ) lim sup | u |→∞ | f ( u ) || u | p − < ∞ for some p ∈ (2 , f ) there exists µ > uf ( u ) ≥ µF ( u ) > u = 0, where F ( u ) = R u f ( s ) ds .These are quite natural assumptions when dealing with general subcritical nonlinearities. Inparticular by ( f )-( f ) it follows that for any ε >
0, there exists C ε > | f ( u ) | ≤ ε | u | + C ε | u | p and | F ( u ) | ≤ εu + C ε | u | p +1 . Remark 1.1.
It follows from ( f ) - ( f ) that < µ ≤ p < . As a reference model, f ( u ) = | u | p − u satisfies ( f )-( f ) for p ∈ (2 , . Moreover, the potential V ∈ C ( R , R ) enjoys the following condition:( V ) there exist V , V > V ≤ V ( x ) ≤ V for all x ∈ R ;( V ) for all γ >
0, lim | x |→∞ ∂V∂r ( x ) e γ | x | = + ∞ , where ∂V∂r ( x ) = ( x | x | , ∇ V ( x ));( V ) there exists ¯ c > |∇ V ( x ) | ≤ ¯ c ∂V∂r ( x ) for all x ∈ R and | x | ≥ ¯ c ;( V ) for all almost x ∈ R , ( ∇ V ( x ) , x ) ∈ L ∞ ( R ) ∪ L ( R ) and µ − µ V ( x ) ≥ ( ∇ V ( x ) , x ) ≥ Remark 1.2.
We note that ( V ) and ( V ) were firstly given in Cerami et al [10] to studythe existence of infinitely many bound state solutions for nonlinear scalar field equations. Thisassumptions are key in recovering the compactness of solution sequence when one uses localPohozaev indentity together with decay estimates to study the behavior of solution, see also Liu Z. S. LIU, H. J. LUO, AND J. J. ZHANG and Wang [24]. Of course, ( V ) is also a very natural condition to ensure the boundedness ofsolution sequence, see Li and Ye [20]. It is not difficult to find some concrete function V satisfyingassumptions ( V )-( V ), such as V ( x ) = V −
11 + | x | , V > (3 µ − Cµ − , V ∈ (0 , V − or V ( x ) = V + Ce − | x | , V > µ + 22 µ Ce − , V ∈ ( V + C, + ∞ ) , where C is a positive constant. Our main result is as follows:
Theorem 1.3.
If ( V )-( V ) and ( f )-( f ) hold, then problem (K) admits at least one least energysolution in H ( R ) . Theorem 1.4.
If ( V )-( V ) and ( f )-( f ) hold, then problem (K) has infinitely many bound statesolutions in H ( R ) provided that f ( u ) is odd in u . Now we summarize two main difficulties in finding bound state solutions to problem (K) underthe effect of nonlocal term R R |∇ u | . On one hand, when p ∈ (2 , V . It is mainly motivated by [10,20,24,25]that we make use of a new perturbation approach together with symmetric mountain-pass theoremto study problem (K). More precisely, in order to get boundedness and compactness of (PS)sequences, we modify problem (K) by adding a conceive term and a nonlinear term growing fasterthan 4, see the modified problem (K λ ), and then the corresponding Pohozaev type identity enablesus to get a bounded solution sequence independent of the parameter λ . As a result, by passing tothe limit, a convergence argument allows us to get nontrivial solutions of the original problem (K).In this process, we also need to establish a version of global decomposition of solution sequences(may be containing sign-changing solutions) which seems new for Kirchhoff type equations. Thisdecomposition is crucial in using the local Pohozaev identity and some decay estimates of solutionsto prove compactness of the sequence of solutions. Moreover, we believe that this perturbationapproach should be of independent interest in other problems. Remark 1.5.
The first result is not surprising. Indeed, we can see [26] where they proved theexistence of positive ground states to problem (K) with a general nonlinearity, and even some moregeneral assumptions for f were used in [13, 14, 39] to study the existence of ground state solutions.However, the methods used in this paper are different from ones in [13, 14, 20, 26, 39]. The core ofthis paper is proving the existence of infinite many solutions which seems nontrivial. But it seemsdifficult to obtain infinitely many solutions by using those arguments in [13, 14, 20, 26, 39]. Hereafter, the letter C will be repeatedly used to denote various positive constants whose exactvalues are irrelevant. We omit the symbol dx in the integrals when no confusion can arise. Thispaper is organized as follows. Firstly, some notations are given in Section 2, and Section 3 isdevoted to the existence of positive ground state solution. Then in Section 4, we investigate theexistence of infinitely many bound state solutions. XISTENCE AND MULTIPLICITY OF BOUND STATE SOLUTIONS 5 Preliminary results
To proceed, we first define the Hilbert space H = (cid:26) u ∈ H ( R ) : Z R V ( x ) u < ∞ (cid:27) with the inner product h u, v i = Z R a ∇ u ∇ v + V ( x ) uv and the norm k u k := q h u, u i = (cid:18)Z R a |∇ u | + V ( x ) u (cid:19) . The associated energy functional I : H → R is given by I ( u ) = 12 k u k + b (cid:18)Z R |∇ u | (cid:19) − Z R F ( u ) . It is a well-defined C functional in H and its derivative is given by I ′ ( u ) v = Z R ( a ∇ u ∇ v + V ( x ) uv ) + b Z R |∇ u | Z R ∇ u ∇ v − Z R f ( u ) v, v ∈ H. We introduce the following coercive function which will be of use(2.1) W ( x ) := 1 + | x | α , < α < µ − µ , x ∈ R . Obviously,(2.2) W ( x ) ≥ > , lim | x |→∞ W ( x ) → ∞ , and(2.3) µ − µ W ( x ) ≥ ( ∇ W ( x ) , x ) ≥ x ∈ R . Let E λ := { u ∈ H : R R λW ( x ) u d x < ∞} equipped with the norm k u k E λ = (cid:18) Z R ( |∇ u | + V ( x ) u + λW ( x ) u ) (cid:19) . Note that E = E ⊂ E λ ⊆ H for λ ∈ (0 , Existence
The perturbed problem.
It is known that the boundedness of the Palais-Smale sequenceis not easy to prove for the case p ∈ (2 , λ ∈ (0 , r ∈ (max { p, } , − (cid:18) a + b Z R |∇ u | (cid:19) ∆ u + V ( x ) u + λW ( x ) u = f λ ( u ) , in R ,u ∈ E λ , ( K λ )where f λ ( u ) = f ( u ) + λ | u | r − u. Z. S. LIU, H. J. LUO, AND J. J. ZHANG
An associated functional can be constructed as I λ ( u ) = I ( u ) + λ Z R W ( x ) u − λr Z R | u | r , u ∈ E λ , and for u, v ∈ E λ ,(3.1) I ′ λ ( u ) v = Z R [ a ∇ u ∇ v + V ( x ) uv + λW ( x ) uv ]+ b Z R |∇ u | Z R ∇ u ∇ v − Z R ( f ( u ) v + λ | u | r − uv ) . It is known that I λ belongs to C ( E λ , R ) or C ( E, R ) and a critical point of I λ is a weak solutionof problem ( K λ ). As we know, the original problem can be seen as the limit system of ( K λ ) as λ → + .We will make use of the following Pohozaev type identity, whose proof is standard and can befound in [8]. Lemma 3.1.
Let u be a critical point of I λ in E λ for λ ∈ (0 , , then a Z R |∇ u | + 32 Z R N ( V ( x ) + λW ( x )) u + 12 Z R ( ∇ V ( x ) , x ) u + λ Z R ( ∇ W ( x ) , x ) u + b (cid:18)Z R |∇ u | (cid:19) − Z R ( F ( u ) + λr | u | r ) = 0 . We now verify that the functional I λ has the Mountain Pass geometry uniformly in λ . Lemma 3.2.
Suppose that ( V )-( V ) hold. Then(1) there exist ρ, δ > such that, for any λ ∈ (0 , , I λ ( u ) ≥ δ for every u ∈ S ρ = { u ∈ E λ : k u k E λ = ρ } ;(2) there is v ∈ E \ { } with k v k E λ > ρ such that, for any λ ∈ (0 , , I λ ( v ) < . Proof (1) For any u ∈ E λ , by the definition of I λ , (1.3) and Sobolev’s inequaity, one has I λ ( u ) ≥ k u k E λ − C Z R | u | p − r Z R | u | r ≥ k u k E λ − C k u k pE λ − Cr k u k rE λ . Taking ρ > δ > I λ ( u ) ≥ δ for every u ∈ S ρ .(2) For e ∈ E \ { } , let e t = t / e ( xt ). Observe that Z R F ( e t ) = t Z R F ( t e ) =: t Φ( t ) . By ( f ), a straightforward computation yieldsΦ ′ ( t )Φ( t ) ≥ µ t , ∀ t > , t ], with t >
1, we have Φ( t ) ≥ Φ(1) t µ , implying that(3.2) Z R F ( e t ) ≥ t µ +62 Z R F ( e ) . XISTENCE AND MULTIPLICITY OF BOUND STATE SOLUTIONS 7
Then by the definition of I λ and (V ) and (2.1), one has(3.3) I λ ( e t ) < t k∇ e k + t k∇ e k + t Z R V ( tx ) e + λt Z R W ( tx ) e − t µ +62 Z R F ( e ) ≤ t k∇ e k + t k∇ e k + t V Z R e + t α Z R W ( x ) e − t µ +62 Z R F ( e ) < , which holds for t > α < µ − µ . The proof is complete. (cid:3) By recalling the well-known Mountain-Pass theorem (see [4,41]), there exists a (
P S ) c λ sequence { u n } ⊂ E λ , that is, I λ ( u n ) → c λ and I ′ λ ( u n ) → . We stress that { u n } depends on λ but we omit this dependence in the sequel for convenience. Here c λ is the Mountain Pass level characterized by c λ = inf γ ∈ Γ λ max t ∈ [0 , I λ ( γ ( t ))with Γ λ := n γ ∈ C ([0 , , E λ ) : γ (0) = 0 and I λ ( γ (1)) < o . Remark 3.3.
Observe from Lemma 3.2 that there exist two constants m , m > independentlyon λ such that m < c λ < m . In what follows, we prove the functional I λ satisfies the (PS)-condition. Lemma 3.4.
Assume that there exists { u n } ⊂ E λ such that I λ ( u n ) → c λ and I ′ λ ( u n ) → for anyfixed λ ∈ (0 , as n → ∞ , then there exists a convergence subsequence of { u n } , still denoted by { u n } , such that u n → u in E λ for some u ∈ E λ . Proof
For γ ∈ (4 , r ), by (1.3) we have γI λ ( u n ) − h I ′ λ ( u n ) , u n i = γ − k u n k E λ + b ( γ − (cid:18)Z R |∇ u n | (cid:19) + Z R (cid:18) γf ( u n ) u n − F ( u n ) (cid:19) + λ r − γr Z R | u | r . Then it follows from (1.3) that(3.4) k u n k E λ + b (cid:18)Z R |∇ u n | (cid:19) + λ Z R | u | r ≤ C (1 + k u n k E λ + k u n k pp )for large n . We claim that { u n } is uniformly bounded in E λ . Assume by contradiction that k u n k E λ → ∞ , then by (3.4) we have(3.5) k u n k E λ + b (cid:18)Z R |∇ u n | (cid:19) + λ k u n k rr ≤ C k u n k pp , which implies that k u n k + k u n k rr ≤ C k u n k pp . Let t ∈ (0 ,
1) be such that p = t + − tr . From the interpolation inequality, we deduce that(3.6) k u k + k u k rr ≤ C k u n k pp ≤ C k u n k pt k u n k p (1 − t ) r . Z. S. LIU, H. J. LUO, AND J. J. ZHANG
It follows from (3.6) that there exist C , C > C k u n k r ≤ k u n k r ≤ C k u n k r . In view of (3.6) and (3.7), we have k u n k pp ≤ C k u n k for some C >
0. Therefore, by (3.5), wehave for some C > k u n k E λ + b (cid:18)Z R |∇ u n | (cid:19) + λ k u n k rr ≤ C k u n k . Let v n = u n k u n k Eλ , then(3.8) k v n k ≥ C and b (cid:18)Z R |∇ v n | (cid:19) ≤ C k u n k − E λ , which implies that R R |∇ v n | → n → ∞ . By k v n k E λ = 1, we assume v n ⇀ v in E λ . ByFatou’s lemma we have Z R |∇ v | ≤ lim inf n →∞ Z R |∇ v n | = 0 , which implies v = 0. Then by (3.8) we have k v k ≥ C , a contradiction. Thus, we finish the proofof the claim. Without loss of generality, we assume that there exists u ∈ E λ such that u n ⇀ u weakly in E λ ,u n → u strongly in L q ( R ) for q ∈ [2 , . Note that(3.9) ( I ′ λ ( u n ) − I ′ λ ( u ))( u n − u )= k u n − u k E λ + b Z R |∇ u n | Z R |∇ ( u n − u ) | + b ( Z R |∇ u n | − Z R |∇ u | ) Z R ∇ u ∇ ( u n − u ) − Z R ( f ( u n ) − f ( u ))( u n − u ) − λ Z R ( | u n | r − u n − | u | r − u )( u n − u ) . According to the boundedness of { u n } in E λ , one has b ( Z R |∇ u n | − Z R |∇ u | ) Z R ∇ u ∇ ( u n − u ) → . Similarly, we also have Z R ( f ( u n ) − f ( u ))( u n − u ) → ,λ Z R ( | u n | r − u n − | u | r − u )( u n − u ) → , as n → ∞ . Based on the above facts, from (3.9) we deduce that u n → u in E λ . (cid:3) It follows from Lemma 3.4 that for each λ ∈ (0 , u λ ∈ E λ such that I λ ( u λ ) = c λ and I ′ λ ( u λ ) = 0 . That is to say, u λ is a nontrivial solution of ( K λ ). We now expect that { u λ } converges to anontrivial solution of (K) as λ → { u λ } in a proper way. XISTENCE AND MULTIPLICITY OF BOUND STATE SOLUTIONS 9
Lemma 3.5.
Suppose that λ n → + as n → ∞ , { u n } ⊂ E λ n are nontrivial solutions of ( K λ n ) with | I λ n ( u n ) | ≤ C . Then there exists M > such that k u n k E λn ≤ M for some M > independentlyof n , and, up to subsequence, there is a solution u ∈ H such that u n ⇀ u in H . Proof
By sequence { λ n } ⊂ (0 ,
1] satisfying λ n → + , we can find a subsequence of { u λ n } (stilldenoted by { u n } ) of I λ n with I λ n ( u n ) = c λ n . We claim that { u n } is bounded in H . By theconditions of this lemma, we have(3.10) C ≥ I λ n ( u n ) = a Z R |∇ u n | + 12 Z R ( V ( x ) + λ n W ( x )) u n + b (cid:18)Z R |∇ u n | (cid:19) − Z R F ( u n ) − λ n r Z R | u n | r and(3.11) 0 = a Z R |∇ u n | + Z R ( V ( x ) + λ n W ( x )) u n + b (cid:18)Z R |∇ u n | (cid:19) − Z R f ( u n ) u n − λ n Z R | u n | r . Moreover, from Lemma 3.1, the following identity holds(3.12) a Z R |∇ u n | + 32 Z R ( V ( x ) + λ n W ( x )) u n + 12 Z R ( ∇ V ( x ) + λ n ∇ W ( x ) , x ) u n + b (cid:18)Z R |∇ u n | (cid:19) − Z R ( F ( u n ) + λ n r | u n | r ) = 0 . Multiplying (3.10), (3.11) and (3.12) by 4, − µ and − C ≥ a µ − µ Z R |∇ u n | + µ − µ Z R ( V ( x ) + λ n W ( x )) u n − Z R ( ∇ V ( x ) + λ n ∇ W ( x ) , x ) u n + µ − µ b (cid:18)Z R |∇ u n | (cid:19) + λ n r − µµr Z R | u n | r + Z R ( 1 µ f ( u n ) u n − F ( u n )) . It then follows from ( V ) and (2.3) that4 C ≥ a µ − µ Z R |∇ u n | + µ − µ b (cid:18)Z R |∇ u n | (cid:19) + λ n r − µµr Z R | u n | r , which implies that there exists C > λ n such that(3.13) Z R |∇ u n | < C . Moreover, combining (1.3), (3.10) and hypotheses (V ), we obtain that for small ε >
0, there exists C ε > C > a Z R |∇ u n | + 12 Z R ( V ( x ) + λ n W ( x )) u n − Z R F ( u n ) − λ n r Z R | u n | r > − ε Z R V ( x ) u n − C ε Z R u n + λ n Z R W ( x ) u n > − ε Z R V ( x ) u n − C ε S − (cid:18)Z R |∇ u n | (cid:19) + λ n Z R W ( x ) u n . Combining (3.13) and (3.14), there exists C > λ n such that(3.15) Z R |∇ u n | + Z R ( V ( x ) + λ n W ( x )) u n ≤ C . The conclusions follow immediately. (cid:3)
The following lemma is devoted to the behavior of solution sequence to problem ( K λ ). Lemma 3.6.
Let { u n } ⊂ E λ be a solution sequence of problem ( K λ ) with λ = λ n ≥ and λ n → ,and k u n k E λn ≤ M for M > independent of n . Then there exist a subsequence of { u n } , stilldenoted by { u n } , a number k ∈ N ∪ { } , and finite sequences ( a , ..., a k ) ⊂ R , ( u , w , ..., w k ) ⊂ H, a j ≥ , w j , and A ≥ and k sequences of points { y jn } ⊂ R , ≤ j ≤ k , such that (i) u n ⇀ u , u n ( · + x jn ) ⇀ w j in H as n → ∞ , (ii) | y jn | → + ∞ , | y jn − y in | → + ∞ if i = j, n → + ∞ , (iii) k u n − u − P ki =1 w i ( · − y in ) k → , (iv) A = k∇ u k + P ki =1 k∇ w i k , (v) for any ϕ ∈ C ∞ ( R ) with ϕ ≥ a + bA ) Z R ∇| w j |∇ ϕ + ( V + a j ) Z R | w j | ϕ ≤ Z R | f ( w j ) | ϕ. Proof
Note that { u n } is a bounded sequence in H . There exists u ∈ H and A > u n ⇀ u weakly in H and k∇ u n k → A as n → ∞ after extracting a subsequence. For any ψ ∈ C ∞ ( R ), we have J ′ λ n ( u n ) ψ ≡
0, where J λ ( u ) := 12 k u k + λ Z R W ( x ) u + Ab Z R |∇ u | − Z R F ( u ) − λr Z R | u | r . Moreover, one has for any ψ ∈ C ∞ ( R ) (cid:12)(cid:12)(cid:12)(cid:12) λ n Z R W ( x ) u n ψ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) λ n Z R W ( x ) u n ψ (cid:19) (cid:18) λ n Z R W ( x ) ψ (cid:19) ≤ Cλ n → , which, together with the fact that J ′ λ n ( u n ) = 0, implies thatlim n →∞ J ′ ( u n ) ψ = lim n →∞ (cid:18) J ′ λ n ( u n ) ψ − λ n Z R W ( x ) u n ψ + λ n Z R | u n | r − u n ψ (cid:19) = 0 , where the functional J = J λ with λ = 0. It then follows that J ′ ( u ) = 0, that is,(3.17) Z R ( a ∇ u ∇ ψ + V ( x ) u ψ ) + bA Z R ∇ u ∇ ψ = Z R f ( u ) ψ. We claim that the following differential inequality holds for any ϕ ∈ C ∞ ( R ) with ψ ≥ Z R ( a ∇| u |∇ ϕ + V | u | ϕ ) + bA Z R ∇| u |∇ ϕ ≤ Z R | f ( u ) | ϕ. XISTENCE AND MULTIPLICITY OF BOUND STATE SOLUTIONS 11
Set u ε = p | u | + ε − ε , ε >
0. It is clear that u ε → | u | in H as ε →
0. By (3.17), we have for ϕ ∈ C ∞ ( R ) with ϕ ≥ a + bA ) Z R ∇ u ε ∇ ϕ = ( a + bA ) Z R u ( | u | + ε ) ∇ u ∇ ϕ = ( a + bA ) (cid:18) Z R ∇ u ∇ (cid:18) u ϕ ( | u | + ε ) (cid:19) − Z R |∇ u | ε ϕ ( | u | + ε ) (cid:19) ≤ ( a + bA ) Z R ∇ u ∇ (cid:18) u ϕ ( | u | + ε ) (cid:19) = − Z R V ( x ) u u ϕ ( | u | + ε ) + Z R f ( u ) u ϕ ( | u | + ε ) . So, from ( V ) we deduce that(3.20) a Z R ∇ u ε ∇ ϕ ≤ − Z R V | u | ϕ ( | u | + ε ) − bA Z R ∇ u ε ∇ ϕ + Z R f ( u ) u ϕ ( | u | + ε ) . Let ε → ϕ ∈ H ( R ) with ϕ ≥
0. The claim is true. Wenow apply the concentration compactness principle to the sequence of { v ,n } with v ,n = u n − u .Clearly, v ,n ⇀ H . If vanishing occurs,sup y ∈ R Z B ( y ) | u n − u | dx → , as n → ∞ . Then v ,n → L s ( R ) for s ∈ (2 , J ′ ( u ) = J ′ λ n ( u n ) = 0, we arrive at( a + bA ) Z R |∇ u | + Z R V ( x ) u ≤ lim inf n →∞ (cid:18) ( a + bA ) Z R |∇ u n | + Z R V ( x ) u n (cid:19) ≤ lim sup n →∞ (cid:18) ( a + bA ) Z R |∇ u n | + Z R V ( x ) u n + λ n Z R W ( x ) u n (cid:19) = lim sup n →∞ (cid:18) Z R f ( u n ) u n + λ n Z R | u n | r (cid:19) ≤ Z R f ( u ) u = ( a + bA ) Z R |∇ u | + Z R V ( x ) u , which implies that u n → u strongly in H . So the conclusions of Lemma 3.6 hold for k = 0. Ifnon-vanishing occurs, then there exist m > { y n } ⊂ R such that(3.21) lim inf n →∞ Z B ( y n ) | v ,n ( x ) | ≥ m > . Let us consider the sequence { v ,n ( · + y n ) } . The boundedness of { v ,n } in H implies that thereexists w such that v ,n ( · + y n ) ⇀ w in H . Furthermore, by (3.21) one has Z B (0) | w ( x ) | > m , and, thus, w = 0. Recalling the fact that v ,n ⇀ H , we know that { y n } must be unboundedand, up to a subsequence, we suppose that | y n | → + ∞ .Now we show the following inequality holds:(3.22) ( a + bA ) Z R ∇| w |∇ ψ + Z R ( a + V ) | w | ψ ≤ Z R | f ( w ) | ψ for ψ ∈ C ∞ ( R ) with ψ ≥
0. Recalling (3.15), we have λ n R R W ( x ) u n ≤ C . So, (3.21) impliesthat C ≥ λ n Z R W ( x ) | v ,n ( x ) | ≥ λ n W ( y n ) Z B ( y n ) | v ,n ( x ) | − λ n Z B ( y n ) | W ( x ) − W ( y n ) || v ,n ( x ) | ≥ λ n W ( y n ) m − λ n C, which implies that, up to subsequence, λ n W ( y n ) → a ∈ [0 , + ∞ ). Based on the above facts, wehave for ψ ∈ C ∞ ( R ) with ψ ≥ λ n Z R W ( x + y n ) v ,n ( x + y n ) ψ = λ n W ( y n ) Z R v ,n ( x + y n ) ψ + λ n Z R ( W ( x + y n ) − W ( y n )) v ,n ( x + y n ) ψ = a Z R v ,n ( x + y n ) ψ + o (1)= a Z R w ψ + o (1) . Recalling the fact that v ,n ⇀ H as n → ∞ , we have J ′ λ n ( v ,n ) ψ ( · − y n ) → ψ ∈ C ∞ ( R ), and(3.24) J ′ λ n ( v n ) ψ ( · − y n )= ( a + bA ) Z R ∇ v ,n ( x + y n ) ∇ ψ + Z R V ( x + y n ) v ,n ( x + y n ) ψ + Z R λ n W ( x + y n ) v ,n ( x + y n ) ψ − Z R f ( v ,n ( x + y n )) ψ = o n (1) , which implies by (3.23) that(3.25) ( a + bA ) Z R ∇ w ∇ ψ + Z R V ( x + y n ) v ,n ( x + y n ) ψ + a Z R w ψ − Z R f ( w ) ψ = o n (1) . Set w ε = p | w | + ε − ε , ε >
0. It is clear that w ε → | w | in H as ε →
0. As arguing as theprevious
Claim , we obtain (3.22). Let us set(3.26) v ,n ( x ) = v ,n ( x ) − w ( x − y n ) , then v ,n ( · + y n ) ⇀ H . It follows from the Brezis-Lieb lemma that(3.27) k v ,n k ss = k u n k ss − k u k ss − k w k ss + o (1) , for s ∈ [2 , , Applying the concentration compactness principle to { v ,n } , we have two possibilities: eithervanishing or non-vanishing. If vanishing occurs we havesup y ∈ R Z B ( y ) | v ,n ( x ) | → , then v ,n → L s ( R ) for s ∈ (2 , k = 1. Otherwise, { v ,n } isnon-vanishing, there exist m ′ > { y n } ⊂ R such that(3.28) lim inf n →∞ Z B ( y n ) | v ,n ( x ) | ≥ m ′ > . XISTENCE AND MULTIPLICITY OF BOUND STATE SOLUTIONS 13
We repeat the arguments. By iterating this procedure we obtain sequences of points { y jn } ⊂ R such that | y jn | → + ∞ , | y jn − y in | → + ∞ if i = j as n → + ∞ and v j,n = v j − ,n − w j − ( x − y j − n )(like (3.26)) with j ≥ v jn ⇀ H . Based on the properties of the weak convergence,we have ( a ) k u n k ss − k u k ss − j − X i =1 k w i k ss = k u n − u − j − X i =1 w i ( · − y in ) k ss + o (1) ≥ , ( b ) for any ψ ∈ C ∞ ( R ) with ψ ≥ i = 1 , ..., j − , ( a + bA ) Z R ∇| w i |∇ ψ + ( V + a i ) Z R | w i | ψ ≤ Z R | f ( w i ) | ψ. By the Sobolev embedding theorem and conclusion (b), we have for i = 1 , ..., j − k w i k p ≤ S p Z R (( ∇| w i | ) + | w i | ) ≤ C k w i k pp , where S p is the Sobolev constant of embedding from H ( R ) to L p ( R ). Hence, there exists c > w i such that | w i | p ≥ c . Since { u n } is bounded sequence in H , conclusion (a)implies that the iteration stop at some finite index k . The proof is complete. (cid:3) Remark 3.7.
The proof of Lemma 3.6 is in the spirit of the works Struwe [35] and Li and Ye [20].It is worth of pointing out that this is the first result on decomposition of (PS) sequences (familiesof approximating solutions, may be sign-changing solutions) with general energy level for Kirchhofftype equation. We can find the decomposition of positive solution sequences with mountain passenergy level in [20, 26, 39], which is used to recover the compactness.
Now we investigate the exponential decay property of there approximating solutions { u n } . Fornotations simplicity, in Lemma 3.6, we define y n = 0, a = 0 and u = w . Thus the conclusion inLemma 3.6 can be restated as | y jn − y in | → ∞ , 0 ≤ i < j ≤ k , k u n − k X i =0 w i ( · − y in ) k → , for any ψ ∈ C ∞ ( R ) with ψ ≥ a + bA ) Z R ∇ w i ∇ ψ + ( V + a i ) Z R w i ψ ≤ Z R | f ( w i ) | ψ, i = 0 , , ..., k. Lemma 3.8.
There exists δ > such that (3.30) Z Ω ( n ) R ( |∇ u n | + | u n | ) ≤ Ce − δR , λ n Z Ω ( n ) R W ( x ) | u n | ≤ Ce − δR , where Ω ( n ) R = R \ S ki =0 B R ( y in ) and C > is independent of n, R . Proof
Using Moser’s iteration to the differential inequality (3.29), we can obtain for i = 1 , ..., k Z R \ B R (0) ( |∇ w i | + | w i | ) ≤ Ce − δR , k w i k L ∞ ( R \ B R (0)) ≤ Ce − δR . So by property (iii) of Lemma 3.6, we have for s ∈ [2 , Z Ω ( n ) R | u n | s ≤ k u n − k X i =0 w i ( · − y in ) k sL s (Ω ( n ) R ) + k X i =0 Z R \ B R (0) w si ≤ o n (1) + Ce − δR . So we use Moser’s iteration to prove the L ∞ -estimate | u n ( x ) | ≤ o n (1) + Ce − δR , for all x ∈ Ω ( n ) R , which implies that for any ε >
0, there exist n , R > n ≥ n there holds | u n ( x ) | ≤ ε, ∀ x ∈ Ω ( n ) R . Thus, in view of ( V ) and ( f ), by choosing ε, R such that for R > R , we have(3.31) Z R (cid:18) a ∇ u n ∇ ϕ + λ n W ( x ) u n ϕ + V u n ϕ (cid:19) ≤ , for all x ∈ Ω ( n ) R . For any
R >
0, define ϕ R as ϕ R ( x ) = 0 for x = Ω ( n ) R , ϕ R ( x ) = 1 for x = Ω ( n ) R +1 and |∇ ϕ R | ≤
2. Let ϕ = ϕ R u n , then (3.31) can be estimated as follows:(3.32) Z Ω ( n ) R (cid:18) a ∇ u n ( ϕ R ∇ u n + 2 u n ϕ R ∇ ϕ R ) + ( λ n W ( x ) + V u n ϕ R (cid:19) ≤ , which implies(3.33) Z Ω ( n ) R ( a |∇ u n | + V u n ) ϕ R ≤ C Z Ω ( n ) R | u n ∇ u n ϕ R ∇ ϕ R |≤ C Z Ω ( n ) R \ Ω ( n ) R +1 ( a |∇ u n | + V u n ) , where C > n, R . From (3.33) we infer that Z Ω ( n ) R +1 ( a |∇ u n | + V u n ) ≤ C C Z Ω ( n ) R ( a |∇ u n | + V u n ) . Thus, there exist
C > n, R ) and δ such that Z Ω ( n ) R ( |∇ u n | + | u n | ) ≤ Ce − δR . Returning to (3.32) we also have λ n Z Ω ( n ) R W ( x ) | u n | ≤ Ce − δR . The proof is complete. (cid:3)
Motivated by [10], we derive a local Pohozaev-type identity which is of use in proving theconvergence of solution sequences.
Lemma 3.9. If u ∈ E λ solves equation ( K λ ), then the following identity holds: Z R t · ∇ V ( x ) | u | ψ + λ Z R t · ∇ W ( x ) | u | ψ = − Z R |∇ u | t · ∇ ψ + Z R t · ∇ u ∇ u · ∇ ψ − Z R ( V ( x ) + λW ( x )) | u | t · ∇ ψ + Z R (cid:0) F ( u ) + λr | u | r (cid:1) t · ∇ ψ for t ∈ R and ψ ∈ C ∞ ( R ) . XISTENCE AND MULTIPLICITY OF BOUND STATE SOLUTIONS 15
Proof
Choose ψ ∈ C ∞ ( R ), t ∈ R . Taking t · ∇ uψ as test function in equation ( K λ ) andintegrating by parts, we get the local Pohozaev-type identity. We can see [10] for the details ofproof. (cid:3) Without loss of generality, we assume that | y n | = min {| y in | , i = 1 , ..., k } . Denote y n = y n forsimplicity of notations. Borrowing from the idea in [10], we construct a sequence of cones C n ,having vertex y n and generated by a ball B R n ( x n ) as follows: C n = (cid:26) z ∈ R | z = 12 y n + l ( x − y n ) , x ∈ B R n ( y n ) , l ∈ [0 , ∞ ) (cid:27) , where R n satisfies γk · | y n | r n ≤ R n ≤ kr n = γ · | y n | , γ = 15(¯ c + 1) , where ¯ c is the constant in the definition of the condition ( V ). It is known in [10] that the cone C n has the following property:(3.34) ∂ C n ∩ k [ i =0 B rn ( y in ) = ∅ . Lemma 3.10.
Let { u n } ⊂ E λ be a solution sequence of ( K λ ) with λ = λ n . Assume that k u n k ≤ M for some M > independent of n , then, up to subsequence, there exists u ∈ H such that u n → u in H . Proof
We now apply the local Pohozaev identity. Take u = u n , t = t n = y n | y n | and ψ = ηϕ R ,where η, ϕ R ∈ C ∞ ( R ) such that η ( x ) = 0 for x
6∈ C n , η ( x ) = 1 for x ∈ C n and dist( x, ∂ C n ) ≥ ϕ R ( x ) = 1 for x ∈ B R , and ϕ R ( x ) = 0 for x ∈ R \ B R . By letting R → ∞ , we have(3.35) 12 Z R t n · ∇ V ( x ) | u n | η + λ n Z R t n · ∇ W ( x ) | u n | η = − Z R |∇ u n | t n · ∇ η + Z R t n · ∇ u n ∇ u n · ∇ η − Z R V ( x ) | u n | t n · ∇ η + Z R ( F ( u n ) + λ n r | u n | r ) t n · ∇ η − λ n Z R W ( x ) | u n | t n · ∇ η. From (3.34) and the definition of η , we see that the support of ∇ η is contained in the domainΩ = Ω ( n ) R with R = r n −
1. In view of Lemma 3.8, we know that the right-hand side of (3.35) decaysexponentially, say less than Ce − δ | y n | . Observe that by Lemma 4.2 of [10], we have t n · ∇ V ≥ ∂V∂r for x ∈ C n . Besides, by the definition of W , we see that R R t n · ∇ W ( x ) | u n | η is bounded uniformlyfor λ n . So the left-hand side of can be estimated as(3.36) 12 Z R t n · ∇ V ( x ) | u n | η + λ n Z R t n · ∇ W ( x ) | u n | η = 12 Z R t n · ∇ V ( x ) | u n | η + o (1) ≥
12 inf x ∈ B ( y n ) ∂V ( x ) ∂r Z R | u n | + o (1) ≥ m x ∈ B ( y n ) ∂V ( x ) ∂r , where R B ( y n ) u n dx ≥ m >
0. Thus, together (3.35) and (3.36), we obtain m x ∈ B ( y n ) ∂V ( x ) ∂r ≤ Ce − δ | y n | , which contradicts with (V ). Thus k = 0 and by Lemma 3.6 (iii), we have u n → u in H . (cid:3) In view of Lemma 3.10, u is a nontrivial solution of problem (K). Actually we have proved thefollowing fact. Proposition 3.11.
Assume { u λ } λ ∈ (0 , satisfies I ′ λ ( u λ ) = 0 and c λ = I λ ( u λ ) ∈ [ m , m ] , thenthere exists u ∈ H \ { } such that on a sequence { λ n } tending to zero, it holds u λ n → u in H, c λ n → c , I ( u ) = c and I ′ ( u ) = 0 . Based on Proposition 3.11, we are now able to give the3.2.
Proof of Theorem 1.3.
Define the set of solutions S := { u ∈ H \ { } : I ′ ( u ) = 0 } that, for what we have proved, is nonempty. For u ∈ S , by Sobolev’s inequality, for any ε > C ε > k u k + b k∇ u k ≤ ε Z R u + C ε Z R | u | which implies that S is bounded away from zero. Besides, we can also see from the above inequalitythat k∇ u k ≥ C for all u ∈ S . By recalling (3.10)-(3.12), there exists some C > I ( u ) ≥ C k∇ u k for all u ∈ S . So we infer that c ∗ := inf u ∈S I ( u ) > . Choose finally a minimising sequence { u n } ⊂ S so that I ( u n ) → c ∗ . Similarly to Lemma 3.5 weknow that { u n } is bounded in H . Like the modified functional I λ , we can also prove some factsfor solution sequence { u n } of I corresponding to Lemmas 3.6-3.10. As a consequence, there exists u ∗ ∈ H so that u n → u ∗ in H and I ′ ( u ∗ ) = 0. Then u ∗ is a ground state solution of (K).4. Multiplicity
In this section, we are attempt to use the perturbation approach together with the SymmetricMountain-Pass theorem to prove the existence of infinitely many high energy solutions to problem(K).4.1.
Proof of Theorem 1.4.
We recall that I λ belongs to C ( E, R ). Denote B R by the ball ofradius R > E . Choose a sequence of finite dimensional subspaces E j of E such that dim E j = j and E ⊥ j denotes the orthogonal complement of E j . We define ∂ P by ∂ P := (cid:26) u ∈ E \ { } (cid:12)(cid:12)(cid:12)(cid:12) ( µ + 2) a µ Z R |∇ u | + 2 + 3 µ µ Z R V ( x ) u + 12 Z R ( ∇ V ( x ) , x ) u + (2 + 3 µ ) λ µ Z R W ( x ) u + λ Z R ( ∇ W ( x ) , x ) u + ( µ + 2) b µ (cid:18) Z R |∇ u | (cid:19) = Z R ( 1 µ f ( u ) u + 3 F ( u )) + ( r + 3 µ ) λµr Z R | u | r (cid:27) . XISTENCE AND MULTIPLICITY OF BOUND STATE SOLUTIONS 17
Recalling assumption ( V ) and (2.3), it follows from Sobolev’s inequality that for any ε >
0, thereexists C ε > µ + 2) a µ Z R |∇ u | + 2 + 3 µ µ Z R V ( x ) u ≤ Z R ( 1 µ f ( u ) u + 3 F ( u )) + ( r + 3 µ ) µr Z R | u | r ≤ ε Z R | u | + C ε Z R | u | , ∀ u ∈ ∂ P ∩ E ⊥ j , which implies that there exists m > λ such that k∇ u k ≥ m . For any u ∈ ∂ P ∩ E ⊥ j , using the definition of I λ and (4.1), we arrive at(4.2) I λ ( u ) ≥ a µ − µ Z R |∇ u n | + µ − µ b (cid:18)Z R |∇ u n | (cid:19) + λ r − µ µr Z R | u n | r ≥ a µ − µ m + µ − µ bm =: δ. Moreover, we can choose R j > I λ ( u ) < u ∈ E j ∩ ∂ B R j . Actually, such an R j canbe found by the fact that in the proof of (2) of Lemma 3.2 the element e ∈ C ∞ ( R ) is arbitrary.Note that R j does not depends on λ , that is to say, ∀ λ ∈ (0 ,
1] : I λ ( u ) < u ∈ E j ∩ ∂ B R j . Thus, the functional I λ satisfies all the assumptions of the Symmetric Mountain Pass Theorem,and we define the minimax values c λ ( j ) = inf B ∈ Γ j sup u ∈ B I λ ( u )where Γ j = (cid:26) B = φ ( E j ∩ B R j ) | φ ∈ C ( E j ∩ B R j , E ) , φ is odd , φ = Id on E j ∩ ∂ B R j (cid:27) . It is easy to prove that the following intersection property holds (see [33, Proposition 9.23]): for B ∈ Γ j , B ∩ ∂ P ∩ E ⊥ j = ∅ , which implies by (4.2) that c λ ( j ) > δ >
0. For any fixed j , by the definition of c λ ( j ), we have, inview of (2) of Lemma 3.2, c λ ( j ) ≤ sup u ∈ E j ∩B Rj I λ ( u ) ≤ sup u ∈ E j ∩B Rj (cid:26) C k u k E + C k u k E (cid:27) := C R j , where C R j is indeed independent of λ ∈ (0 ,
1] and k · k E is any norm in E j . Based on the abovearguments, one has c λ ( j ) ∈ [ δ, C R j ]. Using again Lemmas 3.6-3.10 and Proposition 3.11, we knowthat there exists u ( j ) ∈ H \ { } such that on a sequence λ n → + , u λ n ( j ) → u ( j ) in H, c λ n ( j ) → c ( j ) ≥ δ, I ( u ( j )) = c ( j ) and I ′ ( u ( j )) = 0 , that is, u ( j ) is a nontrivial solution of problem (K).Once we show that c ( j ) → + ∞ as j → + ∞ , problem (K) has infinitely many bounded statesolutions and the proof of Theorem 1.4 is finish. Now we give an estimate on I λ as follows I λ ( u ) = I ( u ) + λ Z R W ( x ) u − λr Z R | u | r ≥ Z R ( |∇ u | + V ( x ) u ) − r Z R | u | r := L ( u ) . Define the set ∂ Θ ⊂ H by ∂ Θ := (cid:26) u ∈ H \ { } : Z R ( |∇ u | + V ( x ) u ) = Z R | u | r (cid:27) , which is the Nehari manifold associated to energy functional L , which, by classical arguments,is bounded away from zero and homeomorphic to the unit sphere. Then, for B ∈ Γ j , an easymodification of the proof of [33, Proposition 9.23] shows that an intersection property holds sothat γ ( B ∩ ∂ Θ) ≥ j , for all j ∈ N . Here γ ( · ) denotes the Krasnoselski genus of a symmetric set.Hence, c λ ( j ) = inf B ∈ Γ j sup u ∈ B I λ ( u ) ≥ inf A ⊂ ∂ Θ ,γ ( A ) ≥ j sup u ∈ A L ( u ) := b ( j ) . It is not hard to verify that the functional J is bounded below on ∂ Θ. Moreover, We observethat the boundedness of the Palais-Smale sequence is easy to verify for functional J . As a result,with some suitable modification, the arguments of functional I λ are still valid for J without anyperturbation. So, J satisfies the Palais-Smale condition. Then the Ljusternick-Schnirelmanntheory guarantees that b ( j ) are diverging critical values for J . Therefore, c ( j ) = lim λ → + c λ ( j ) ≥ b ( j ) → + ∞ , as j → + ∞ . That is to say, problem (K) has infinitely many higher energy solutions. The proof is complete. (cid:3)
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Center for Mathematical Sciences,China University of Geosciences,Wuhan, Hubei, 430074, PR China
Email address : [email protected] (H. J. Luo) School of Mathematics,Hunan University,Changsha 410082, Hunan, P.R. China
Email address : [email protected] (J. J. Zhang) College of Mathematics and Statistics,Chongqing Jiaotong University,Chongqing 400074, PR China
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