Nonresonance for one-dimensional p-laplacian with regular restoring

Transcription

1 J. Math. Anal. Appl ) Nonresonance for one-dimensional p-laplacian with regular restoring Ping Yan Department of Mathematical Sciences, Tsinghua University, Beijing 184, China Received 19 December 21 Submitted by R. Manásevich Abstract In this paper, the existence of periodic solutions of the equation φ p x )) + ft,x)= is proved, where f : [,T] R R has some nonlinear growth near x =±, and the nonresonance condition S P ), which can be explained using the periodic eigenvalues concerning p-laplacian, is satisfied. 23 Elsevier Inc. All rights reserved. 1. Introduction This paper is devoted to the solvability of the following problem φp x ) ) + ft,x)=, 1.1) x) = xt ), x ) = x T ), P ) where 1 <p<+, φ p ) : R R is φ p s) = s p 2 s for any s, and φ p ) =. Here f : [,T] R R, satisfies the Carathéodory condition see Section 2 for the definition) and the following growth condition: there exist a ), b ) L 1,T)such that ft,x) at) lim inf x φ p x) ft,x) lim sup x φ p x) bt) S ) uniformly in a.e. t. We will refer ft,x)in this case as a regular restoring. We introduce the following nonresonance condition: Supported in part by the National 973 Project of China and the National NSF of China. address: pyan@math.tsinghua.edu.cn X/$ see front matter 23 Elsevier Inc. All rights reserved. doi:1.116/s22-247x3)383-4

2 142 P. Yan / J. Math. Anal. Appl ) For any m ) L 1,T) with at) mt) bt) for a.e. t [,T], the equation φp x ) ) + mt)φp x) = S P ) has only the trivial solution verifying the periodic boundary condition P ). Then we have the following nonresonance result: Theorem 1.1. Suppose that f : [,T] R R is a Carathéodory function. If ft,x) satisfies S ), and S P ) holds for those at) and bt) from S ),then1.1) has at least one T -periodic solution. For the case p = 2, the nonresonance condition S P ) was firstly obtained by Fonda and Mawhin [2] using a variational approach. Theorem 1.1, which will be proved using the Leray Schauder degree theory, is a generalization of the results in [2]. Note that when p = 2 the condition S P ) can be expressed using the periodic eigenvalues of x + λ + qt) ) x = qt) = at) or qt) = bt) ). See Remark 3.1 below. The Dirichlet boundary value problems of the one-dimensional p- Laplacian were studied in [1,7]. A similar nonresonance condition like S P ) was obtained in [7] using the weighted Dirichlet eigenvalues of the p-laplacian.the periodic boundary value problems of the one-dimensional p-laplacian case were studied in [4], where the authors showed that many ideas for the case p = 2 in [5] also work in the p-laplacian case. The rest of this paper is organized as follows. Section 2 is devoted to the proof of Theorem 1.1 using the Leray Schauder degree theory. It is worth mentioning here that a priori estimates for all possible periodic solutions will be different from the usual technique in most literature because we find that an abstract result of Zhang [6, Theorem 3.1] on families of positively homogeneous compact operators will reduce such estimates to elementary inequalities. In Section 3, we will give some equivalent statements of our nonresonance condition S P ) using the periodic eigenvalues, which were recently defined by Zhang [8], of φp x ) ) ) + λ + qt) φp x) = qt) = at) or qt) = bt) ). 1.2) Note that when qt) is time-dependent, the complete set of the periodic and the antiperiodic eigenvalues of 1.2) is unknown, see [8], like that for the higher-dimensional Dirichlet problems of the p-laplacian [3]. 2. Proof of Theorem 1.1 Suppose f : [,T] R R satisfies the Carathéodory condition, i.e., For a.e. t, themapx ft,x)is continuous. For every x,themapt ft,x)is measurable. For every ρ>, there exists h ρ L 1,T) such that ft,x) h ρ t) for all a.e. t and all x with x ρ.

3 P. Yan / J. Math. Anal. Appl ) In this setting, by a solution xt) of 1.1) we mean that xt) is C 1 and φ p x t)) is absolutely continuous such that Eq. 1.1) is satisfied for a.e. t. We will reduce problem 1.1) + P ) to a fixed point problem in the space C T ={x C[,T]; x) = xt)}, endowed with the sup norm. Following [4], we first consider the T -periodic solutions of the following auxiliary problem φp x ) ) + ht) =, 2.1) where h ) L 1,T). One has x ) t) = φ q l Ht) for some l R, whereht) = t 1/q = 1. Thus t xt) = x) + Since x ) = x T ), it is necessary that T HT)= hs) ds, andq is the conjugate number of p, 1/p + φ q l Hs) ) ds. 2.2) hs) ds =. 2.3) It follows from x) = xt) that l satisfies T φ q l Hs) ) ds =. 2.4) As the left-hand side of 2.4), viewed as a function of l, is strictly increasing in l and goes to ± as l ±, there exists a unique l = LH ) such that 2.4) is satisfied. In conclusion, by combining 2.2) and the necessary condition 2.3), we know that x ) is a solution of the problem 2.1) + P ) if and only if x ) is a solution of the following fixed point problem in C T : t ) [ ] xt) = x) + φ q HT) + φ q LH ) Hs) ds. 2.5) Remark 2.1. There are many ways to reducing problem 2.1) + P ) to a fixed point problem in C T different from 2.5). For other reductions, see, e.g., [4]. LH ), as a functional of H, is well defined for any H ) from the space C[,T] of continuous functions with the sup norm. Some properties on the functional l = LH ), H C[,T], are collected in the following lemma. Lemma 2.1. Let H,H 1,H 2 C[,T].Then

4 144 P. Yan / J. Math. Anal. Appl ) ) min s [,T ] Hs) LH ) max s [,T ] Hs). In particular, LH ) H. 2) LkH ) = klh), k R. 3) Lk + H)= k + LH ), k R. 4) If H 1 H 2,thenLH 1 ) LH 2 ). 5) LH 1 ) LH 2 ) H 1 H 2. Theorem 1.1 will be proved using the Leray Schauder degree theory. The homotopy can be taken as a linear one between 1.1) and the following simple equation φp x ) ) + m t)φ p x) =, 2.6) where m is any fixed function from L 1,T) such that at) m t) bt) a.e. t [,T]. 2.7) That is to say, we will take the homotopy equation as φp x ) ) + fλ t, x) =, 2.8) where f λ t, x) = λm t)φ p x) + 1 λ)f t, x), λ [, 1]. We will not use the usual technique to prove a priori estimates for all possible T -periodic solutions of 2.8) + P ), because we find that the framework of Zhang [6] enables us to obtain a priori estimates using elementary inequalities. We give some definitions before citing the results from Zhang [6]. Suppose M is a topological space. We say that M is a sequentially compact space if any sequence in M has a convergent subsequence. Let X be a normed linear space. We say that a mapping F : X X is completely continuous if F is continuous and maps bounded sets to relatively compact sets. Let M be a sequentially compact space. We say that a mapping F : M X X is uniformly completely continuous with respect to M) if i) F is continuous on M X ;and ii) for any sequence {p n,x n )} in M X such that {x n } is bounded in X, the sequence {Fp n,x n )} has a convergent subsequence in X. A mapping F : X X is positively homogeneous if Fkx)= kfx) for all k>andallx X. The following result, cf. Theorem 3.1 in [6], is important in the a priori estimates below. Theorem 2.1. Let X be a normed linear space and M a sequentially compact space. Let S : M X X be a uniformly completely continuous mapping. Assume that, for each m M, Sm, ) is positively homogeneous and the equation x = Sm,x) 2.9)

5 P. Yan / J. Math. Anal. Appl ) has only the trivial solution x = in X. Then there exists a constant c > such that x Sm,x) c x for all m M and x X. In order to apply Theorem 2.1, let us take the spaces as X = C T, ) and M = { m L 1,T): at) mt) bt) a.e. t }, 2.1) where at) and bt) are from S ).WhenM is endowed with the topology of weak convergence, we know that M is sequentially compact because all functions from M are equi-integrable. The operator Sm,x) comes from the problem of finding T -periodic solutions of φp x ) ) + mt)φp x) =. 2.11) As in 2.5), x is a T -periodic solution of 2.11) + P ) if and only if x C T = X and satisfies t ) [ ] xt) = x) + φ q XT) + φ q LX) Xs) ds t, 2.12) where Xt) is defined by t Xt) = ms)φ p xs) ) ds. 2.13) Note that Xt) depends upon m and x. Let us define the operator S : M X X as the right-hand side of 2.12), i.e., t ) [ ] Sm, x)t) := x) + φ q XT ) + φ q LX) Xs) ds t, 2.14) where Xt) is defined in 2.13). Following the idea in the proof of Proposition 5.1 in [6], it is not difficult to prove that S : M X X is uniformly completely continuous. For any given m M, by Lemma 2.1, one sees that Sm,x) is homogeneous in x X, i.e., Sm,kx) = ksm, x), k R, x X. Since the fixed point problem 2.9) is equivalent to problem 2.11), we know that 2.9) has only the trivial solution x = inx if at) and bt) satisfy condition S P ). Thus we have the following important estimate. Corollary 2.1. Assume that at) and bt) satisfy S P ). Suppose X = C T, ), and M,S are defined by 2.1) and 2.14), respectively. Then there exists a constant c > such that x Sm,x) c x 2.15) for all m M and x X.

6 146 P. Yan / J. Math. Anal. Appl ) After having obtained 2.15), we find that the approach is elementary to obtain the a priori estimates of T -periodic solutions of 2.8). Before giving the proof of Theorem 1.1, we need some elementary inequalities. Lemma ) If 1 <q 2,then φq u + v) φ q u) 2 2 q v q ) for all u, v R. 2) If q>2,then φ q u + v) φ q u) q 1) u + v ) q 2 v 2.17) for all u, v R. Now we give the proof of Theorem 1.1. We consider the homotopy equation 2.8). In order to apply the Leray Schauder degree theory, we reduce 2.8) + P ) to a fixed point problem in X.Foranyx X, letting ht) in 2.1) be f λ t, xt)), we know that x is a solution of 2.8) + P ) if and only if x is a fixed point of the following problem in the space X xt) = x) + φ q Fλ T ) ) t + φ q [ LFλ ) F λ s) ] ds =: N λ x)t), 2.18) where F λ t) is defined by t ) F λ t) = f λ s,xs) ds. Now we are going to prove that all possible solutions of 2.18) are a priori bounded. From condition S ), we know that for any given ε>, it is always possible to decompose the nonlinear function ft,x)as ft,x)= m ε t, x)φ p x) + γ ε t, x), where both m ε t, x) and γ ε t, x) are Carathéodory functions such that at) m ε t, x) bt), a.e. t, x, 2.19) and there exists )l ε t) L 1,T)with the property γε t, x) ε φp x) + lε t), a.e. t, x. 2.2) In fact, it follows from S ) that for any ε>, there exists M ε > such that for any x >M ε, at) ε 2 < ft,x) φ p x) <bt)+ ε, a.e. t. 2 Then we can find a Carathéodory function m ε t, x) with at) m ε t, x) bt), a.e. t, x,

7 P. Yan / J. Math. Anal. Appl ) such that ft,x) φ p x) m εt, x) ε, a.e. t, x >M ε. Let γ ε t, x) = ft,x) m ε t, x)φ p x), Then γ ε t, x) is a Carathéodory function and γε t, x) ε φp x), a.e. t, x >Mε. a.e. t, x. On the other hand, we can find some ) l ε t) L 1,T)such that γε t, x) lε t), a.e. t, x M ε. Hence 2.19) and 2.2) hold and the decomposition of ft,x)is obtained. Given λ [, 1] and x ) X,wehave ) ) ) f λ t,xt) = λm t)φ p xt) + 1 λ)f t,xt) ) ) = mt)φ p xt) + 1 λ)γε t,xt), where mt) = λm t) + 1 λ)m ε t,xt) ). 2.21) By 2.19), m M. Denote and Then h 1 t) = mt)φ p xt) ), a.e. t, h 2 t) = 1 λ)γ ε t,xt) ), a.e. t. f λ t,xt) ) = h1 t) + h 2 t), a.e. t. Define t H i t) = h i s) ds, t [,T], i= 1, 2. We can rewrite the operator N λ x as N λ x)t) = x) + φ q H1 T ) + H 2 T ) ) t + = Sm, x)t) + Gt), where m ) is given by 2.21) and Gt) = G 1 t) + G 2 t) with φ q [ LH1 + H 2 ) H 1 s) H 2 s) ] ds

8 148 P. Yan / J. Math. Anal. Appl ) G 1 t) = φ q H1 T ) + H 2 T ) ) φ q H1 T ) ), 2.22) G 2 t) = t φq [ LH1 + H 2 ) H 1 s) H 2 s) ] φ q [ LH1 ) H 1 s) ]) ds. 2.23) We give some preliminary estimates. Since h1 s) as) + bs) ) x p 1, we have H 1 C x p 1, 2.24) where C = a L 1,T ) + b L 1,T ). By 2.2), h2 s) ε x p 1 + l ε s) for a.e. s [,T]. Thus H 2 Tε x p 1 + C ε, 2.25) where C ε = l ε L 1,T ). For any s [,T],wehave [ LH 1 + H 2 ) H 1 s) H 2 s) ] [ LH 1 ) H 1 s) ] = [ LH 1 + H 2 ) LH 1 ) ] H 2 s) H 2 + H 2 s) 2 H ) and LH1 ) H 1 s) 2 H ) Next we use Lemma 2.2 to give the estimates on G. In case 1 <q 2, we apply 2.16). From 2.22), we have G q H2 T ) q q H 2 q 1. From 2.23), T G q LH1 + H 2 ) LH 1 ) H 2 s) q 1 ds T 2 2 q 2 H 2 ) q 1 Thus = 2T H 2 q 1. G 2 2 q + 2T) H 2 q ) In case q>2, we apply 2.17). Then G 1 q 1) H1 T ) + H2 T ) ) q 2 H2 T ) q 1) H 1 + H 2 ) q 2 H2.

9 P. Yan / J. Math. Anal. Appl ) For any s [,T], [ φ q LH1 + H 2 ) H 1 s) H 2 s) ] [ φ q LH1 ) H 1 s) ] q 1) LH1 ) H 1 s) + LH1 + H 2 ) LH 1 ) H 2 s) ) q 2 LH 1 + H 2 ) LH 1 ) H 2 s) q 1) 2 H 1 +2 H 2 ) q 2 2 H2 = q 1)2 q 1 H 1 + H 2 ) q 2 H2. Thus, by 2.23), we have G 2 q 1)2 q 1 T H 1 + H 2 ) q 2 H2. Consequently, G q 1)1 + 2 q 1 T) H 1 + H 2 ) q 2 H ) Note that p 1)q 1) = 1. We assume at the moment that <ε<1. If 1 <q 2, we get from 2.25) and 2.28) that G 2 2 q + 2T) Tε x p 1 ) q 1 + C ε 22 2 q + 2T )T ε) q 1 x +C ε 2.3) for some positive constant C ε independent of x X. The last inequality in 2.3) holds because if Tε x p 1 >C ε,wehave Tε x p 1 ) q 1 + C ε 2 q 1 T ε) q 1 x 2T ε) q 1 x, and if Tε x p 1 C ε,wehave Tε x p 1 ) q 1 + C ε 2Cε ) q 1. If q>2, we get from 2.24), 2.25) and 2.29) that G q 1)1 + 2 q 1 T) C x p 1 + Tε x p 1 ) q 2 + C ε Tε x p 1 ) + C ε q 1)1 + 2 q 1 T) C x p 1 + T x p 1 ) q 2 + C ε Tε x p 1 ) + C ε 2 q 1 q 1)1 + 2 q 1 T )C + T) q 2 Tε x +C ε 2.31) for some positive constant C ε independent of x X. The last inequality in 2.31) can be proven by discussing the cases C ε <Tε x p 1 and C ε Tε x p 1. We conclude from 2.3) and 2.31) that there must be some positive constant A independent of λ [, 1], x X ), such that, for any <ε<1, we have G A max{ε, ε q 1 } x +B ε 2.32) for some positive constant B ε independent of λ [, 1] and x X. Now the a priori estimates for all the solutions of 2.18) can be obtained easily from 2.15) and 2.32). Actually, if x X is a solution of 2.18) for some λ [, 1], i.e., x = N λ x = Sm,x) + Gt),

10 15 P. Yan / J. Math. Anal. Appl ) then we get from 2.15) and 2.32) that A max{ε, ε q 1 } x +B ε G = x Sm,x) c x. 2.33) Let us additionally restrict ε, 1) such that A max{ε, ε q 1 } <c. Then we get from 2.33) that all possible solutions x ) of 2.18) must satisfy B ε x c A max{ε, ε q 1 } =: R. This proves that all possible solutions x ) of 2.18) in X are a priori bounded. As the final step, we apply the homotopy invariance of Leray Schauder degree to get deg I N,B, 2R ), ) = deg I N 1,B, 2R ), ), where B, 2R ) ={x X : x < 2R }. Note that the operator N 1 : X X, which corresponds to Eq. 2.6), is odd, and the equation x = N 1 x has only the trivial solution x = inx. Thus the Borsuk theorem implies that degi N 1, B, 2R ), ) is an odd number which is necessarily nonzero. This shows that the equation x = N x, which is equivalent to the problem 1.1) + P ), has necessarily at least one solution x in B, 2R ). Theorem 1.1 is thus proved. 3. Further discussion on nonresonance conditions In this section, we aim at giving some explanation to the nonresonance condition S P ) using the periodic eigenvalues found in [8]. We first mention the p-cosine function C p t) and the p-sine function S p t) which admit many similar properties as the cosine and the sine functions with the period replaced by 2π p,where 2πp 1)1/p π p = p sinπ/p). The function S p and the number π p were first introduced in [1]. Consider the following differential equation: φp x ) ) + φp x) =. 3.1) Let y = φ p x ). Then 3.1) is equivalent to the planar Hamiltonian system x = φ q y), y = φ p x). 3.2) C p t), S p t)) is the unique solution of 3.2) with the initial value x), y)) =, 1).

11 P. Yan / J. Math. Anal. Appl ) Lemma 3.1 [1,8]. C p t) and S p t) are well defined on R and have the following properties: 1) Both C p t) and S p t) are 2π p -periodic. 2) C p t) is even in t and S p t) is odd in t. 3) C p t + π p ) = C p t), St + π p ) = S p t). 4) C p t) = if and only if t = π p /2 +mπ p, m Z; and S p t) = if and only if t = mπ p, m Z. 5) C p t) = φ qs p t)) and S p t) = φ pc p t)). 6) C p t) p /p + S p t) q /q 1/p. Let mt) be a T -periodic function with m L 1,T). Consider the eigenvalue problems φp x ) ) ) + λ + mt) φp x) = 3.3) with respect to the periodic boundary condition P ) or the antiperiodic boundary condition x) + xt) =, x ) + x T ) =. A) Let y = φ p x ) in 3.3). Define the p-polar coordinates in R 2 by x = r 2/p C p θ), y = r 2/q S p θ). 3.3) is equivalent to the following system: r = p/2) λ + mt) 1 ) φ p Cp θ) ) φ q Sp θ) ) r, 3.4) θ = p λ + mt) ) Cp θ) p /p + Sp θ) q /q ). 3.5) For any x,y ) R 2, 3.3) has the unique solution xt) satisfying x), x )) = x,y ). Correspondingly, there exists r,θ ) R 2 with x = r 2/p C p θ ), y = r 2/q S p θ ) such that 3.4) + 3.5) has the unique solution rt), θt)) satisfying the initial conditions r) = 1, θ) = θ. Note that 3.5) is independent of r.) Let θt; θ,λ)be the unique solution of 3.5) satisfying θ; θ,λ)= θ. Define the rotation number of 3.5) or of 3.3)) as ρλ,m) := lim t θt; θ,λ) θ. t Then ρλ,m) exists and is independent of θ. See [8]. The following two sequences were defined in [8]: λ n m) := min { λ R: ρλ,m)= nπ p /T }, λ n m) := max { λ R: ρλ,m)= nπ p /T }. Then they have the following order: < λ m) < λ 1 m) λ 1 m) < <λ n m) λ n m) <, and λ n m) +, λ n m) + as n. It is proved in [8, Theorem 3.3] that if n is even, then λ n m) and λ n m) are eigenvalues of problem 3.3) + P ); andifn is odd, then λ n m) and λ n m) are eigenvalues of problem 3.3) + A). Note that when p 2it remains an open problem whether these λ n m) and λ n m) represent all the eigenvalues of 3.3) + P ) and 3.3) + A).

12 152 P. Yan / J. Math. Anal. Appl ) T For c,d L 1,T), write c d if ct) dt) for a.e. t [,T] and T ct)dt < dt)dt. The following results can be found in Section 4 of [8]. Theorem ) The eigenvalues λ n m) and λ n m) are continuous functions of m ) under the L 1 distance on m s: dm 1,m 2 ) = T m1 t) m 2 t) dt. 2) Comparison results) If c d,then λ n c) > λ n d), λ n c) > λ n d), for all n N. Since the complete set of periodic eigenvalues of 3.3) are not known, we can only use those periodic eigenvalues in [8] to give a sufficient condition to the nonresonance condition S P ). Proposition ) The nonresonance condition S P ) is equivalent to the following condition S P ): For any measurable T -periodic function mt) with at) mt) bt) for a.e. t [,T], all the eigenvalues of 3.3) + P ) are nonzero. 2) If there exists some k Z + such that λ 2k a) <, λ 2k+2 b) >, J 2k ) then condition S P ) holds. Proof. It is easy to check the equivalence between S P ) and S P ). Let us prove 2). If λ is a periodic eigenvalue of 3.3), then ρλ,m) = 2nπ p /T for some n Z +,and λ n N[ λ 2n, λ 2n ], λ ]. It follows from J 2k ) and the comparison results in Theorem 3.1 that and λ 2k m) λ 2k a) < λ 2k+2 m) λ 2k+2 b) >. Thus / n N [λ 2n, λ 2n], λ ] and S P ) is satisfied. Let µ k denote the eigenvalues of φp x ) ) + λφp x) = with respect to the Dirichlet boundary value condition x) = xt ), x ) = x T ).

13 P. Yan / J. Math. Anal. Appl ) It is well known that µ k = kπ p /T) p, k N. Corollary 3.1. Suppose that ft,x) satisfies condition S ). If there exists some k N such that µ 2k at) bt) µ 2k+2, then 1.1) has at least one T -periodic solution. Proof. Note that if ct) c is a constant, then λ k c) = λ k c) = µ k c, k N. It follows from the comparison results in Theorem 3.1 that and λ 2k a) < λ 2k µ 2k ) = λ 2k+2 b) > λ 2k+2 µ 2k ) =. Thus condition S P ) is satisfied and the corollary comes from Theorem 1.1 and Proposition 3.1. Remark 3.1. When p = 2, the sequences {λ 2n m)} and {λ 2n m)} cover all the eigenvalues of 3.1) + P ). See [8]. Thus S P ) is equivalent to the following condition: There exists some k Z + such that either or λ 2k a) <, λ 2k b) >, I 2k ) λ 2k a) <, λ 2k+2 b) >. J 2k ) Such a nonresonance condition is also necessary for the nonautonomous equation 1.1) in some sense. Remark 3.2. For general 1 <p<, if the nonresonance condition S P ) in Theorem 1.1 is J ) in Proposition 3.1, i.e., λ a) <, λ 2 b) >, J ) then the existence of T -periodic solutions of the damped regular equation φp x ) ) + gx)x + ft,x)= can be proved, where gx) is any given continuous function. See [7] for the Dirichlet problem.

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