Nonlinear eigenvalue problems for higher order model equations

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1 Nonlinear eigenvalue problems for higher order model equations L.A. Peletier Mathematical Institute, Leiden University PB 951, 3 RA Leiden, The Netherlands peletier@math.leidenuniv.nl 1 Introduction In studying comple spatio-temporal pattern formation, new, often higher order, model equations have recently been proposed. When looking for special solutions such as stationary solutions or traveling wave solutions of these equations there are many instances in which one is led to an equation which can be reduced to one of the form d 4 u d 4 + q d u + f(u) =. (1.1) d Here q R is a parameter and f a given, usually nonlinear, function. In this chapter we study the eistence, and qualitative properties of bounded solutions of equation (1.1) in unbounded and bounded domains, Equation (1.1) belongs to a class of Pattern Forming Equations [],[1] in which the parameter q is seen as a measure of what has been called the pattern forming tendency. It determines the relative strength of the second and the fourth order derivative, and as we shall see, it so plays an important role in determining the qualitative properties of solutions of (1.1). Specifically, we shall find that in general, as q increases the compleity of the patterns increases. As to the nonlinearity f, we shall assume throughout that it is a smooth function defined on the whole of R, which vanishes at the origin, i.e. f() =. Therefore, equation (1.1) will always have the trivial solution u =. Typical functions we shall consider in this paper are f(s) = s ± s p 1 s and f(s) = s ± s p 1 s, p > 1. (1.) Eamples of equations which lead to (1.1) are the Swift-Hohenberg (SH) equation: u ( t = κ u 1 + ) u u 3, κ R, (1.3) proposed in 1977 by Swift and Hohenberg [41] in studies of Rayleigh-Bénard convection [16], [7], and the Etended Fisher-Kolmogorov (EFK) equation: u t = γ 4 u 4 + u + u u3, γ >, (1.4) 1

2 proposed in 1988 by Dee and van Saarloos [17] as a higher order model equation for systems which are bi-stable. Stationary solutions of these equations yield equation (1.1) after an appropriate transformation. Traveling wave solutions: u(, t) = v( ct) (c R) of the Beam equation [4]: u t + P 4 u + Q(u) =, P R, (1.5) 4 also yield equation (1.1), as do solutions of the Nonlinear Schrödinger equation []: i u + u t 4 u t 4 + u u =, (1.6) with a sinusoidal spatial profile, i.e. solutions of the form u(, t) = v(t)e ik, where v(t) R and k R. Finally, we mention an important equation in this field of study, which arises in the theory of water waves: d 4 u d 4 + P d u d + u u =, P R, (1.7) and is already in the form of equation (1.1). Equation (1.1) is endowed with two properties, which make it more accessible than most fourth order equations. (i) Equation (1.1) fits into a variational structure and is the Euler-Lagrange equation of the functional { 1 J(u) = (u ) q s (u ) + F (u)} d where F (s) = f(t) dt (1.8) Ω and Ω is an appropriate interval. (ii) Equation (1.1) can be integrated once to yield E(u) def = u u 1 (u ) + q (u ) + F (u) = E, (1.9) where E is a constant, which is often referred to as the Energy. In recent years equation (1.1) has been found to possess a rich structure and, for certain values of q and E, a multitude of qualitatively different solutions. We mention the work of Buffoni, Champneys and Toland et. al., [11], [13], [43], who utilize the Hamiltonian structure of (1.1) and that of Peletier and Troyet. al., [8], [31], [3], [33] who devised a method of topological shooting, combined with continuation arguments. We also mention the work of Kalies and Van der Vorst [3] and [5], that of M.A. Peletier [38], and that of Chaparova and Tersian, [14], [4] et. al., whose work is based on the variational structure of (1.1), and that of Van den Berg who applied the maimum principle to this fourth order equation [4], [5] and [6]. For an etensive bibliography of papers up to 1, we refer to the monograph [33] of that year. As with second order equations, the solution set of equation (1.1) depends crucially on the choice of function f. In the studies mentioned above the functions f(s) = s + s 3 and f(s) = s s, (1.1)

3 have received a great deal of attention. The cubic function (cf. [34], [35], [36]) arises in bi-stable systems and the quadratic equation arises in water wave theory (cf. (1.7)). Here we should also mention the two related functions (one is a smooth version of the other) f(s) = (s + 1) + 1 and f(s) = e s 1, (1.11) where s + = ma{s, }, which were introduced by McKenna and Walter et. al., [8], [9] in connection with the study of traveling waves in suspension bridges (see also [33] and [37]). In this paper we sketch some of the results that have been obtained about the solution structure of equation (1.1) for different functions f, present new results about nonlinearities f(s) which are decreasing for large s, and establish a series of asymptotic results for large values of the parameter q. About decreasing nonlinearities we prove the following theorem: Theorem 1.1 Let f C 1 (R) in (1.1) have the following properties: f (s) < for s R, and f() =, (1.1) and let q be an arbitrary constant in R. Then, for every E, there eists no periodic solution of (1.1) and for every E < there eists precisely one symmetric periodic solution u() on R. This solution is symmetric with respect to each of its critical points. Plainly, the solutions described in Theorem 1.1 have a very simple structure. For more comple structures f(s) will have to be increasing for some values of s R. As a first model nonlinearity we shall discuss the function f(s) = s + s p 1 s where p > 1, (1.13) which arises in the contet of the Swift-Hohenberg equation (1.3) when < α < 1. The following noneistence theorem was proved in [33] : Theorem 1. Let f in (1.1) be given by (1.13). Then, if q, there eist no periodic solutions of (1.1). For q >, the situation becomes more interesting and a great variety of qualitatively different periodic solutions emerge as q increases. Specifically, at the values q n,m = n m + m, n = 1,,..., m = 1,,..., n m, (1.14) n branches of periodic solutions with zero energy (E = ) bifurcate super-critically from the trivial solution and etend al the way to q =. In Figure 1 (a) we show branches bifurcating from q 1,1, q 3,1, q 5,1 and q 7,1. The qualitative properties of solutions on these branches are characterized by the numbers n and m: 3

4 M 4 M q q (a) Branches for f(s) = s + s 3 (b) Branches for f(s) = s s 3 Figure 1: Branches of 1-lap, 3-lap, 5-lap and 7-lap solutions of equation (1.1) in the (q, M)-plane, where M = sup{ u() : R} u() 6 u u() 6 u u u() Figure : Periodic solutions of (1.1) with f(s) = s + s 3 corresponding to n = 1, n = 3 and n = 5 and m = (a) The number of monotone segments, or Laps, between points of symmetry, i.e. points where u = and u =, is given by n in (1.14). (b) The number of zeros between points of symmetry is given by m. In Figure 1 we show solutions on branches which bifurcate from q 1,1, q,1 and q 3,1, which clearly ehibit the properties (a) and (b). As a second model equation we discuss equation (1.1) when f is given by f(s) = s s p 1 s, p > 1. (1.15) which arises in the contet of the Nonlinear Schrödinger equaiton (1.6). When linearizing about the trivial solution, we obtain the same linear equation as with (1.13) and so periodic solutions bifurcate from u = at the same critical values q n,m of q. However, in this case the branches all bifurcate sublinearly and etend back to q =. In Figure 1 (b) we show branches bifurcating from q 1,1, q 3,1, q 5,1 and q 7,1, and we see that they all bend back and converge to one point as q. The solutions on these branches have the same qualitative properties as those shown in Figure. 4

5 As q increases, the two length scales become evident and it is possible to use multi-scale asymptotic methods to obtain accurate epressions for the different periodic solutions. Introducing the scaled variables, we can write equation (1.1) as = q 1/ and u ( ) = u(), (1.16) ε u iv + u + f(u) =, ε = q 1, (1.17) where we have omitted the asterisks again. In this section, we choose f(s) = s + g(s), g(s) = o(s) as s, and we assume that g is an odd function. We shall show that for E = O(1) and large values of q, any periodic solution u() can be approimated by an epression of the form ( ) u() = B () ± ε ρ(ε, α) cos + O(ε ) as ε, (1.18) ε where B () is a solution of the reduced equation u + u + g(u) =, (1.19) and ρ(ε, α) is a known constant which depends on ε and on α = u(). Thus, we see that u() can be viewed as a Baseline solution B with superimposed on it a small amplitude high frequency oscillation. In the presentation of this results we shall make frequent use of recent results of Kuske and Peletier [6]. The asymptotic epression (1.18) for periodic solutions makes it possible to obtain accurate approimations of bifurcation curves in the region in which q is large. The main objective of this chapter is to present methods which have been successful in analyzing fourth order equations such as equation (1.1) and to indicate basic elements of the structure of solution sets of their often comple solution sets. For the sake of transparency we have focussed on equation (1.1) with just a few typical nonlinearities f(u). However, the methods in this chapter are applicable to a wide class of equations. As further eamples we mention the equation u iv + h(u)u 1 h (u)(u ) + f(u) =, (1.) which has been studied in [4] and more recently in [9], [1] when f(u) = u 3 u, and the system w + w(v 1) =, v αv + 1 (1.1) w =. This system arises in the study of coupled nonlinear Schrödinger equations [45] and has been studied in [46], [45] and [44] The plan of the paper is the following. We begin in Section with a detailed analysis of periodic solutions for decreasing nonlinearities f: their eistence, uniqueness, estimates 5

6 for their period, and their asymptotic behaviour for large values of q and E. Then, in Section 3 we turn to an analysis of the solution set when f is given by (1.13), and in Section 4 we discuss equation (1.1) when f is given by (1.15). Finally, in Section 5 we present the multi-analysis of periodic solutions when q is large and E = O(1). The solution graphs in this chapter have been made with XPPAUT [19], the bifurcation graphs with AUTO [18] and the numerical comparison in Section 5 with Matlab. Decreasing nonlinearity f(u) In this section we study periodic solutions of the basic equation when the function f is strictly decreasing: u iv + qu + f(u) = (1.1) f (s) < for s R, and f() =. (.1) We shall show that for any E < and any q R, there eists a unique periodic solution of equation (1.1), and we study its dependence on the eigenvalue q and the energy E, and its asymptotic behavior as q ± and E. We begin with a few preliminary observations. Let u() be a periodic solution of equation (1.1). Then there eist points where u =, and at those critical points the energy identity (1.9) becomes E = 1 (u ) + F (u). (.) From (.1) we conclude that sf(s) < for s R \ {} so that F (s) = s This implies the following noneistence result: f(t) dt < for s R \ {}. Lemma.1 If u is a nontrivial periodic solution of equation (1.1), then its energy must be negative: E(u) = E <. Net, we derive a-priori upper and lower bounds for any periodic solution u() of equation (1.1) in terms of its energy E. Let M + = ma{u() : R} and M = min{u() : R}. (.3) Plainly, there eist points at which u = M + and where u = M. At these points we have u =, and we conclude from the energy identity (1.9) that F (M ± ) E. 6

7 Hence, since M u() M + on R, and F = f < on R, we conclude that Let us denote the roots of the equation F (u()) E for R. (.4) F (s) = E (.5) by c (E) and c + (E), where c < < c +. Then (.4) yields the following a-priori bounds: Lemma. Let u() be a periodic solution of equation (1.1) with energy E <. Then c (E) u() c + (E) for R. (.6) In the net subsection we shall show that for every E < there eists a periodic solution of equation (1.1)..1 Eistence of a periodic solution We construct a periodic solution which is symmetric with respect to all its critical points, i.e., at each critical point u = as well. As in [33] we call such points, points of symmetry. The number of monotone segments between two nearest points of symmetry will be called the number of Laps. Thus, in this section we construct One-Lap periodic solutions. Without loss of generality we shall place the origin at a local minimum, so that u (). (.7) If the first positive critical point a local maimum is located at = ξ, then integration of equation (1.1) over (, ξ) shows that u(ξ) u() f(s) ds =, which implies that u() < and u(ξ) >. (.8) The main result of this subsection is the following eistence theorem. Theorem.1 For each q R and each E < there eists a periodic solution u() of equation (1.1) with energy E, which is symmetric with respect to each of its critical points. We prove Theorem.1 by means of the topological shooting argument that was first developed in [34], [35] and [36] (see also [33]). Since the solution is assumed to be symmetric with respect to the origin we shall be considering the initial value problem { u iv + qu + f(u) = > (u, u, u, u (.9) )() = (α,, β, ), 7

8 where α and β are constants which need to be determined. When we compute the energy E at the origin, where we have put a local minimum, we obtain the following relation between α, β and E: This relation enables us to epress β in terms of α and E: E = 1 β + F (α). (.1) β = ± { E + F (α)}. (.11) When we choose α [c, c + ], then F (α) + E, so that the epression for β is well defined. In view of the assumption (.7), we choose u() = α < and u () = β >. (.1) Plainly, Problem (.9) has a unique local solution; we denote it by u = u(, α). The basic idea of the shooting argument is to find a value of α such that at the first positive critical point ξ, not only u =, but also u =. The solution is then symmetric with respect to ξ. Since by construction it is also symmetric with respect to the origin, the solution can be continued over the whole real line as a periodic solution with period ξ. Thus, we write ξ(α) = sup{ > : u (, α) > on (, )}. Lemma.3 There eists a constant δ > such that if c < α < c + δ, then Proof Note that if α = c, then u (ξ(α), α) = and u (ξ(α), α) <. (.13) u () =, u () =, u iv () = f(c ) <. Hence, in a right-neighborhood of the origin we have u < and u <. If we now increase α slightly, we find that u changes sign in a right-neighborhood of the origin, so that ξ(α) eists, and that ξ(α) and u(ξ(α), α) c as α c. Moreover, integration of the equation shows that u (ξ(α), α) < for α close to c, as asserted.. Net, we seek a value of α for which u (ξ(α), α) >. Note that if u(ξ) < c +, then by the energy identity u (ξ) = { E + F (u(ξ))} <. Therefore, by the Implicit Function Theorem, if u(ξ( α), α) (c, c + ) for some α > c, then ξ(α) depends continuously on α in a neighborhood of α. Define the set A = {α > c : u(ξ(α ), α ) < c + for c < α < α}. Plainly, A is an interval of the form (c, α ). 8

9 Lemma.4 (a) ξ(α) is finite for every α A, and (b) ξ(α ) < and u(ξ(α ), α ) = c + ; (c) u (ξ(α ), α ) = and u (ξ(α ), α ) >. Proof (a) Suppose that ξ = for some α A. Then This means that u () > and u() c + for all >. def lim u() eists = l. Since u = is the only constant solution of the differential equation it follows that l =. However, integration of the equation over R + shows that l >, a contradiction. (b) By definition, u(ξ(α ), α ) c +. However, by the energy identity (1.9), u cannot vanish if u > c +. Hence u(ξ(α ), α ) = c +, as asserted. That ξ(α ) < follows as in the proof of Part (a). (c) Let α = α. Then, since u(ξ) = c +, the energy identity implies that u (ξ) =. Since u > in a left-neighborhood of ξ, it follows that u (ξ). Suppose that u (ξ) =. Then, because u iv (ξ) = f(c + ) >, it follows that u > c + in left-neighborhood of ξ, a contradiction. We are now ready to prove that u (ξ(α), α) has a zero on (c, α ]. For convenience we write ϕ(α) def = u (ξ(α), α). (.14) Since ξ(α) and u(, α) depend continuously on α, so does ϕ(α). In Lemma.3 we have shown that ϕ(α) < for α close to c, and that ϕ(α ) >. Therefore, there there eists a α (c, α ) such that ϕ(α ) =. Plainly, by symmetry arguments, we see that the solution u(, α ) is a 1-lap periodic solution on R with period ξ. Since for the energy E we had chosen an arbitrary negative number, this completes the proof of Theorem.1. Let u() be a 1-lap periodic solution which is symmetric with respect to its critical points, and let its period be de denoted by L. In the following lemma we establish an upper bound for the half period when q >. Lemma.5 Let u() be a periodic solution of equation (1.1) which is symmetric with respect to its critical points. If q, its half-period L satisfies the inequality L < π q. Proof. We shift u() so that its minimum lies at the origin. Then u > on (, L). We differentiate equation (1.1), multiply by sin(π/l) and integrate over (, L). This yields, after repeated integrations by part, L sin ( π ) { π u ( π ) } () L L L q + f (u()) d =. (.15) 9

10 Since f < on R and u > on (, L), this implies that π L > q or L < π q. Remark. Lemma.5 shows that as q, the period shrinks to zero at a rate of O(q 1/ ). More precise estimates will be given in Section.3. Remark. Lemma.5 can be further refined. Let m = min{ f (s) : c s c + }. Then f (s) < m, and we deduce from (.15) that This means that and hence that. Uniqueness L sin ( π ) { π u ( π ) } () L L L q m d >. (.16) π ( π ) L L q m >, π L < q +. (.17) q + 4m In this section we show that the periodic solution obtained in Theorem.1 is unique. Theorem. Suppose that the nonlinearity f(u) has the properties (.1). Then for any q R and any E <, there eists at most one bounded solution of equation (1.1). Proof. We deal with the cases q and q > separately. Case 1: q. Suppose that u 1 and u are periodic solutions of equation (1.1). Without loss of generality we may assume that u 1 and u have local minima at the origin, and that u () u 1 (). By the energy identity (.) evaluated at critical points, this implies that u () u 1 (). Finally, by possibly applying the transformation we can ensure that u () u 1 (). Thus, writing v = u u 1 we obtain v, v =, v, v = at =, (.18) where, by uniqueness, at least one of the inequalities must be strict, and v iv = q v f (θ)v, θ (u 1, u ). (.19) It is evident from (.18) and (.19) that v > and v > in a right-neighborhood of the origin. Let y = sup{ > : v > on (, )}. 1

11 Since v() is bounded it follows that y < and the v () as y. But, integration of (.) over (, ), yields v () = v () + q v () f (θ)v dt > for < < y. Since v () > this implies that v () > on (, y), and in particular v () cannot tend to as y, a contradiction.. Case : q >. Let u 1 be a periodic solution as constructed in Theorem.1 with minimum at the origin and maimum at = L, and let u be a bounded solution of equation (1.1), also with a local minimum at the origin. Without loss of generality we may assume that one of the following two sets of inequalities hold: (A) (B) u u 1 and u u 1 at =, u u 1 and u u 1 at =. By uniqueness, in (A) as well as in (B), one of the two inequalities must be strict. Let ξ be the first zero of u on R+. Then we shall prove the following inequalities: (i) If (A) holds, then (ii) If (B) holds, then u (ξ) u 1 (L) > u () u 1 () and u (ξ) >. (.) u (ξ) u 1 (L) < u () u 1 () and u (ξ) <. (.1) By applying (.) and (.1) alternatively, we conclude that the sequence of maima of u increases and that its sequence of minima decreases. Since u is bounded these etrema tend to limits, so that u tends to a periodic solution u, whose maima and minima lie, respectively, above and below those of u 1. Repeating the argument with u instead of u we are led to a contradiction. Suppose that (A) holds. Let v = u u 1. Then, Also, define w = v + qv, and let Then v, v =, v, v at =. y = sup{ > : v > v() on (, )}. w = v iv + qv = {f(u ) f(u 1 )} > on (, y), and, since w = and w at =, w > and w > on (, y]. (.) We claim that y > π q and v > on (, π/ q). (.3) 11

12 Let a = sup{ > : v > on (, )}. Plainly, a < y. We shall prove that a π/ q. Suppose, to the contrary, that a < π/ q. Then v + qv = w > on (, a] and hence < a sin ( π ) (v + qv ) d = (q π ) a a a sin ( π ) v () d. a Since, by assumption, q < π /a we have a contradiction. Remembering from Lemma.5 that π/ q > L, we may now conclude that which shows that u > u 1 on (, L], ξ > L and u (ξ) > u 1 (L). Thus, since v(l) > v(), or u (L) u 1 (L) > u () u 1 (), it follows that u (ξ) u 1 (L) > u (L) u 1 (L) > u () u 1 (), as we set out to prove. In addition, Since it follows that ξ u (ξ) u () = = L ξ u iv () d = f(u ()) d ξ f(u ()) d L f(u ()) d. ξ > L and u () > u (L) > u 1 (L) > for L < < ξ, and because u > u 1 on (, L) we have Therefore L f(u ()) d > ξ f(u ()) d >, L L f(u 1 ()) d = u 1 (L) u 1 () =. u (ξ) > u () as required in (.). Thus the inequalities (.) have been proved. The inequalities in (.1) can be proved in an identical manner; we shall omit the details. This completes the proof of Theorem.. 1

13 Corollary.1 Let u() be a periodic solution of equation (1.1) in which f( s) = f(s) for s R. Suppose that u() =. Then u( ) = u() for R. Proof. Let u() be a periodic solution of equation (1.1), shifted so that u() =, u () >. Note that u (), because critical values of u() are either positive (maima) or negative (minima). Then the function v() = u( ) satisfies the equation v iv + qv f( v) = Remembering the asymmetry of f, we find that By uniqueness, v = u, so that v iv + qv + f(v) =, v() =, v () >. u( ) = u() for R. Theorems.1 and. together state that for every q R and for every E < there eists a unique nontrivial solution u(; q, E) of equation (1.1). It is symmetric with respect to its critical points. We define the amplitude M(q, E) def = ma{ u(; q, E) : R}. (.4) In Figure 3 we show graphs of M when f(s) = s s 3. In one we have put E = 1 and in the other q =. M 1.5 M q E Figure 3: Branches of periodic solutions of equation (1.1) with f(s) = s s 3 : M versus q (E = 1) and M versus E (q = ) 13

14 .3 Asymptotics In this section we study how the periodic solution evolves as the two parameters q and E become large, and when E becomes small. We first focus on the asymptotic properties of the periodic solution if q and if q +. As in comparable studies for the bi-stable nonlinearity [33], we find that as q, they converge to the periodic solution of the second order equation with the given value of E, and as q + the periodic solutions shrink at a rate of O(q 1 ). We shall do this analysis for the nonlinearity f(s) = s s 3. (.5) By symmetry, c + (E) = c (E), and we shall write c(e) = c ± (E)..3.1 Large q We first prove the following limit for q : Theorem.3 Let u(; q) be the periodic solution for given E < and q R, shifted so that u(; q) = and u (; q) >. Then where V is the solution of the equation u(; q) V (y), y = q as q, (.6) with V () =, V () > and energy E, i.e. V + V + V 3 =, ma{ V (y) : y R} = E 1. Proof. We put y = / q and u(; q) = v(y; q). Then, when we transform (1.1) to the new variables y and v, we obtain, if q <, ε v iv v v v 3 =, ε = 1 q, (.7) and the energy identity becomes ε ( v v 1 (v ) ) + 1 (v ) 1 v 1 4 v4 = E. (.8) Since v(y; q) < c(e) on R by Lemma., it follows from a simple Maimum Principle argument for w = v, that v (y; q) < c(e) + c 3 (E) for y R. Let ϕ C (R). Then, when we multiply (.7) by ϕ, integrate over R, and perform a few integrations by part, we find that ε ϕ v + {ϕ v ϕ(v + v 3 )} =. R R 14

15 Since the sets {v( ; q)} and {v ( ; q)} are both equicontinuous, we can let q along a sequence to find that v(y; q) V (y) and v (y; q) V (y), were V + V + V 3 =, V () =, V () >. Doing the same in the energy identity, we find that 1 (V ) 1 V 1 4 V 4 = E. These two relations together determine V uniquely, so that we may conclude that lim v(y; q) = V (y). q This is the limit we set out to prove. Net, let us consider the limit in the other direction, i.e. for q. Theorem.4 Let u(; q) be the periodic solution for given E < and q R, shifted so that u(; q) = and u (; q) >. Then u(; q) E sin( q) as q +. (.9) q Proof. We follow the proof of a comparable result in [33] (cf. Theorem 4..3) and begin with a preliminary estimate. Lemma.6 Suppose that q >. Then E u(; q) < 1 + q for R. Proof. Write w = u + qu. Then w = u iv + qu = u + u 3 > on (, ξ), where ξ is the first positive zero of u. Since w (ξ) = u (ξ) + qu (ξ) = it follows that w < on (, ξ) and hence, because w() =, that so that At ξ, the energy identity yields w(ξ) = u (ξ) + qu(ξ) <, u (ξ) > qu(ξ). (.3) 1 (u ) = E 1 u 1 4 u4. (.31) Using (.3) to estimate the left hand side of (.31) from below, we arrive at the required upper bound. 15

16 We continue with the proof of Theorem.4 and scale the variables as suggested by Lemma.5: = t, u(; q) = 1 q q v(t; q) and ξ = τ. q For v, we then obtain the equation and the energy identity v iv + v ε v ε 4 v 3 =, ε = 1 q, v v 1 (v ) + 1 (v ) ε v ε4 4 v4 = E. When we formally take the limit as q, or ε, we find that v(t; q) V (t) where V satisfies V iv + V =, V () = and V V 1 (V ) + 1 (V ) = E. This implies that V (t) = E sin(t), as asserted. For the proof of this limit we refer to Section 4. of [33]..3. Small and large negative energy It follows from Lemma. that u() < c(e) = E 1 < E. (.3) This suggests that we scale u() with E 1/ for E small, and with E 1/4 for E large and negative. Starting with small energy, we write u() = E 1/ v() and substitute into (1.1) with f given by (.5). This yields the equation v iv + qv v E v 3 =. (.33) When we let E, and assume that v() V (), we find that V is a solution of the equation V iv + qv V =. (.34) The characteristic equation of (.34) is with roots λ = ±a and λ = ±ib, where λ 4 + qλ 1 =, a = 1 ( q + 4 q) and b = 1 ( q q). (.35) Using arguments employed in establishing the limit for q ± we obtain the following result: 16

17 Theorem.5 Let q R be fied, and let u(; E) be the periodic solution with energye <, shifted so that u(; E) = and u (; E) >. Then u(; E) A(q) E sin(b) as E, (.36) with A(q) = q + q. q Proceeding in an entirely similar manner we find the following limit for E : Theorem.6 Let q R be fied, and let u(; E) be the periodic solution with energy E <, shifted so that u(; E) = and u (; E) >. Then u(; E) E 1/4 V (y), y = E 1/8 as E, (.37) where V is the solution of the equation V iv V 3 =, V () = with first integral V V 1 (V ) 1 4 V 4 = 1. The results of Theorems.3,.4,.5 and.6 are clearly borne out in the bifurcation graphs shown in Figure 3. 3 A super-linear bifurcation problem Whereas the solution structure we found when f(u) is decreasing turned out to be very simple for every negative value of the energy E we found a unique branch of periodic solutions etending from q = to q = + this no longer is the case for functions f(u) which are increasing or non-monotone. In this section we consider the relatively simple case of an increasing nonlinearity: f(s) = s + s p, p > 1, (3.1) where we shall mean s p = s p 1 s. Thus, we consider the equation u iv + qu + u p + u =. (3.) We shall find that results developed for equation (3.) can be used to analyze more complicated nonlinearities. One such nonlinearity will be discussed in Section 4. We begin with some preliminaries, including a noneistence theorem restricting the range of q in which we need look, then discuss an infinite family of branches of periodic solutions which branch off the trivial solution, and finally study their behavioras q. We shall make frequent use of methods and results developed in [33]. 17

18 3.1 Preliminaries We begin with a noneistence theorem. Theorem 3.1 Equation (3.) has no periodic solutions for q. Proof. We use an energy argument. Suppose that u is a non-trivial periodic solution with period (, L). We multiply equation (3.) by u and integrate over (, L). Then, after some integrations by parts, we obtain But, L L (u ) d q (u ) d = L L (u ) d + uu d L L Using this interpolation inequality in (3.3) we obtain (q ) L (u ) d L (u + u p+1 ) d =. (3.3) u d + L (u ) d. u p+1 d. (3.4) Since u is nontrivial, the right hand side in (3.4) is positive and it follows that q >. This implies that there can be no periodic solutions, with whatever energy, if q. We shall particularly focus on periodic solutions which bifurcate from the trivial solution u =. Thus, we need to inspect the linear equation, obtained from (3.) by omitting the nonlinear term: v iv + qv + v =. (3.5) The roots of its characteristic equation are λ ± = ±ia and λ ± = ±ib, (3.6) where a and b are the positive roots of a = 1 ( q + ) q 4 and b = 1 ( q ) q 4. (3.7) Plainly, a and b are well defined if q. We shall find that branches of solutions bifurcate from u = at values of q where resonance occurs, i.e. where the fraction a/b becomes rational, so that for some integers n 1 and m 1, a b = n m q = q n,m = n m + m, m n. (3.8) n The integers n and m will translate into specific geometric properties of the solutions. It will be convenient to write q n = q n,1. The sequence {q n } n=1 is seen to be increasing, starting from q 1 =. We shall be studying single and multi bump periodic solutions of equation (3.) which are either even or odd. These solutions will often have more than two critical points per 18

19 period; to keep track of them we introduce the following notation. Let u() be a solution. We denote its local maima on R + by ξ 1, ξ, ξ 3,... and its local minima by η 1, η, η 3,..., where ξ k η k ξ k+1 η k+1. If u () > for small positive, then the first maimum is ξ 1, and the first minimum is therefore η 1. If u () < for small >, then the first minimum is η 1 and hence the first maimum is ξ. With the nonlinearity f(s) defined by (3.1) it was shown in [33] that these sequences of critical points are infinite. Sometimes, it will be convenient to refer to these critical points collectively. We then denote them by ζ j, where ζ j ζ j+1, and ζ 1 is the first positive critical point. In the analysis of multibump solutions of equation (3.), the location and position of critical points of the zero energy solutions of the linear equation (3.5) play a pivotal role. In particular, we are interested in the solution v() of the problem { v iv + qv + v =, > v =, v = 1, v =, v (3.9) = q/ at =, where the initial data have been chosen so that E(v) =. We shall repeatedly use the following important property of solutions of Problem (3.9): Lemma 3.1 Let n 1 and q > q n 1. Then v(ζ j ) > and v (ζ j ) > for j = 1,,..., n. Proof. The solution of Problem (3.9) is given by v() = 1 ( sin(a) a + sin(b) b ), (3.1) where a and b are given by (3.7). An elementary computation shows that a critical point ζ can be found as a root of the equation and that cos(aζ) + cos(bζ) =, (3.11) v (ζ) = 1 q 4 cos(bζ). (3.1) Using (3.1) and (3.1) to determine the value of v, respectively v, at the roots of equation (3.11), we establish the desired properties.. In order to characterize the shape of solutions, we recall the notions of a point of symmetry and a lap of a function. We say a point R is a point of symmetry of the function φ C(R) if φ( + y) = φ( y) for y R. By a lap of the function φ C 1 (R) we mean an interval ( 1, ), where 1 and are finite, such that φ () for ( 1, ) and φ ( 1 ) =, φ ( ) =. Thus, the interval (, π) is a lap of the function cos(). 19

20 3. Multi bump periodic solutions We begin with the construction of a family of relatively simple, periodic solutions u of equation (3.). Subsequently we turn to more comple periodic solutions. We first construct a family of odd periodic solutions. Theorem 3. Let n 1. For each q > q n 1 there eist an odd periodic solution u n () of equation (3.) which has n 1 laps in each half period, such that u n() >. Its first positive point of symmetry is ζ n and its period is 4ζ n. It has the following properties: u n () > for < < ζ n u j+1() < u j() for j = 1,,..., n 1 (n ), u n (ζ j+1 ) < u n (ζ j ) for j = 1,,..., n 1 (n ). (3.13) Since u n (ζ n ) =, the third inequality implies that u n (ζ j ) > for all j = 1,,..., n 1. In Figure 4 we show the first three solutions u 1, u and u 3 of this theorem. 6 u u() u() 6 u u() 6 u Figure 4: The 1-lap, 3-lap and 5-lap solution of equation (3.) for p = 3 (cf. Theorem 3.) The proof of Theorem 3., much like those to follow, is based on a shooting argument. Let u() be an odd solution. Then it can be viewed as a solution of the the initial value problem { u iv + qu + u p + u =, > u =, u = α, u =, u (3.14) = β at =, where α and β are suitable constants. Conversely, if u() is a solution of Problem (3.14) on R + for some α and β, then by reflection we can etend this solution to one on the whole real line. Thus it suffices to find periodic solutions of Problem (3.14) with the desired geometric properties. Without loss of generality we may choose α in (3.14). Evaluating the energy identity at the origin we find that α, β and E are related through αβ + q α =, (3.15) where we have put E =. We can distinguish two types of solutions: Type I: α >, β = q α and Type II: α =, β >.

21 Thus, in Type I, the parameter α > is arbitrary and in Type II, β is arbitrary. We denote the solution of Problem (3.14) by, respectively, u(; α) and u(; β). It is easily established that these solutions eist in a neighbourhood of the origin and that they depend continuously on α, respectively β, on bounded intervals. Proof. Since, by assumption, α >, we seek a solution of Type I and we have the freedom to choose α. We select α so that u(; α) becomes a solution with the desired geometric properties. This is done by carefully following the way the local maima u(ξ 1 ), u(ξ ), u(ξ 3 ),... and the local minima u(η 1 ), u(η ), u(η 3 ),... vary with α, and tuning α so that they end up in the right positions. This is possible because ξ j C(R + ) and η j C(R + ) for all j 1 [33]. We first prove the theorem for n = 1. For values n the result then follows upon iteration. In tuning α we specifically follow the value of u (ξ 1 (α), α). For convenience, we write ϕ 1 (α) def = u (ξ 1 (α), α). (3.16) Suppose that ϕ 1 (α) = for some α >. Then u = and u = at ξ 1 (α) and so, u() is symmetric with respect to ξ 1. Thus, by defining the function { u() for ξ1, w() = u(ξ 1 ) for ξ 1 ξ 1, we obtain a solution of equation (3.) on (, ξ 1 ): the first half of the periodic solution. The second half is obtained by translating w() over a distance ξ 1 and changing the sign. The function so constructed is one period of a periodic solution of equation (3.) with period 4ξ 1. The essential ingredient in the construction above was the eistence of a zero of ϕ 1. The eistence of such a zero will be established by finding values of α for which ϕ 1 is positive and values for which it is negative. Continuity then ensures the eistence of a zero. We first determine the sign of ϕ 1 (α) when α is large. Lemma 3. Let q R. Then there eists a constant α + > such that ϕ 1 (α) < for α > α +. Proof. We use a scaling argument and introduce the new variables t = α (p 1)/(p+3) and v(t; α) = α 4/(p+3) u(; α). In terms of these variables Problem (3.14) becomes v iv + α (p 1)/(p+3) qv + α 4(p 1)/(p+3) v + v p = v =, v = 1, v =, v = q α (p 1)/(p+3) at t =. Let V be the solution of the limit problem, formally obtained by letting α : { V iv + V p = V =, V = 1, V =, V = at t =. (3.17) (3.18) 1

22 Because Problem (3.17) is a regular perturbation of the limit Problem (3.18), it follows that v (j) (t; α) V (j) (t) as α uniformly on bounded intervals for j =, 1,, 3, 4. We see that V < and V < as long as V >. Hence T = sup{t > : V > on (, t)} is finite and V (T ) < and V (T ) <. Plainly, } ξ 1 (α) α (p 1)/(p+3) T u (ξ 1 (α), α) α (3p+1)/(p+3) V (T ) < so that ϕ 1 (α) < for α large enough. Net, we determine the sign of ϕ 1 (α) when α is small. as α, Lemma 3.3 If q > q 1 =, there eists a constant α > such that ϕ 1 (α) > for < α < α. Proof. In [33] it was shown that for α sufficiently small, the solution u() inherits the qualitative properties of the solution v() of Problem (3.9) on bouned intervals. In particular, because n = 1, we deduce from Lemma 3.1 that u (ξ 1 ) >, as required. We conclude from Lemmas 3. and 3.3, and the continuity of ϕ 1 (α) that there eists a point α1 (α, α + ) such that ϕ 1 (α1 ) =. This implies that the function u 1() = u(; α1 ) is a periodic solution of equation (3.). Plainly, all its critical points are points of symmetry and hence it is a one-lap solution, and u(; α1 ) > for < < ξ 1(α1 ). This concludes the proof of the case n = 1. Net, let n =. We now look for a periodic solution which is symmetric with respect to its second critical point ζ = η 1. Lemma 3.1 states that if q > q 3, then and hence, for α > small Let v(η 1 ) > and v (η 1 ) >, u(η 1 ) > and u (η 1 ) >. α = sup{α > : u(η 1 ) > for < α < α}. Since u 1 (η 1 ) = u 1 (ξ 1 ) < when α = α1, it follows that α (, α1 ). Note that u(η 1 ) = u (η 1 ) = and u (η 1 ) < at α. (3.19) The first equality is obvious and the second one follows from the energy identity. As to u, by uniqueness, u. This means that u must be negative. Hence, the function ϕ (α) = u (η 1 (α), α) changes sign on the interval (, α ) so that there eists a point α (, α ) such that ϕ (α ) =. Thus u () = u(; α ) is a two-lap periodic solution with period 4η 1 (α ) such that u () > for < < η 1. Note that α < α 1, which proves the second property. The third property of u follows from the following lemma.

23 Lemma 3.4 Let u() be a solution of equation (1.1) and a and b are two critical points of u() such that a < b and f(u) > on (a, b). Then Proof Integration of (1.1) over (a, b) yields u (a) > u (b). u (b) u (a) + b Since the integral is positive, the assertion follows. a f(u()) d =. In particular we see that u (ζ 1) > u (ζ ). This completes the proof of Theorem 3. for n =. Remark 3.1 Since by (3.19) it follows that In particular, u(η 1 ) = and u (η 1 ) = at α, u(η 1 + y) = u(η 1 y) for y R. u(η 1 ) = and u (η 1 ) =. Therefore, the solution ũ () = u(; α ) is also periodic, with period η 1, and ũ () > on (, η 1 ) and ũ (η 1 ) =. Plainly, nowhere on [, η 1 ] does this solution have a point of symmetry, i.e., a point where both ũ = and ũ =. Suppose that n = 3. Then we look for a periodic solution which is symmetric with respect to its third critical point ζ 3 = ξ. Lemma 3.1 states that if q > q 5, then u(ξ ) > and u (ξ ) > for α > small. Since u (η 1 ) = at α, it follows from Lemma 3.4 that Thus, we have shown that u (ξ ) < at α. u (ξ ) > for α small and u (ξ ) < for α = α. It follows that there eist an α 3 (, α ) such that u (ξ ) = at α 3, and u 3() = u(; α 3 ) is a periodic solution which is symmetric with respect to ξ. This establishes the eistence of a three-lap periodic solution u 3. By construction, it has the properties u 3 () > for < < ξ, u 3() < u () and u 3 (ξ 1 ) > u 3 (η 1 ) >, 3

24 where the last two inequalities follow from an application of Lemma 3.4. Remark 3. Note that when α = α, then u(η ) <, and when α = α 3, then u(η ) >. Hence, α 3 = sup{α > α 3 : u(η ) > for α 3 < α < α )} < α, and u(η ) =, u (η ) = and u (η ) < at α 3. Hence, ũ 3 () = u(, α 3 ) is a periodic solution with period η, and ũ 3 () > on (, η ). Continuing in this manner we can construct for every n 1 a periodic solution u n if q > q n 1, which is symmetric with respect to = ζ n and has n laps on (, ζ n ), and which is positive on (, ζ n ), and has the required properties. This completes the proof of Theorem 3.. An easy corollary of the proof of Theorem 3., based on the reasoning given in the Remarks 3.1 and 3., is a second family of periodic solutions. Its properties are formulated in the the net theorem Theorem 3.3 Let n 1. For each q > q n+1 there eist an odd periodic solution ũ n of equation (3.) which has 4n 1 laps in each period, such that ũ n() >. It has the properties ũ n () = ũ n (η n ) for < < η n ũ n () > for < < η n In Figure 5 we show graphs of ũ 1, ũ and ũ 3 established in Theorem 3.3. u() u() u() 6 u u Figure 5: The 3-lap, 7-lap and 11-lap solution of equation (3.) for p = 3 (cf. Theorem 3.3) u In an entirely analogous manner we can establish the eistence of a Type II family of periodic solutions of equation (3.). They are characterized by the fact that α = u () =. Specifically, we obtain: 4

25 Theorem 3.4 Let n 1. For each q > q n+1 there eist an odd periodic solution u n of equation (3.) which has n laps in each half period, such that u n() =. It has the properties u n () = ũ n (η n ) for < < η n u n () > for < < η n In Figure 6 we show graphs of u 1, u and u 3. u() u u() u u u() Figure 6: The -lap, 4-lap and 6-lap solution of equation (3.) for p = 3 (cf. Theorem 3.4) Remark 3.3 Of course a solution ũ n as found in Theorem 3.3 can be viewed as one of Type II by shifting it over a distance η n. In Theorems we have established the eistence of odd periodic solutions for q in intervals of the form (q n+1, ), for n = 1,, 3,.... They lie on branches which bifurcate from the trivial solutions at the odd eigenvalues q n+1. Similarly, there eist branches of even periodic solutions, which bifurcate from the even eigenvalues q n and etend al the way to q =. The first three of these are shown in Figure 7. u() u() u() u 6 4 q=1, a= u q=1, a=-.85 u 3 1 q=1, a= Figure 7: The -lap, 4-lap and 6-lap solution of equation (3.) for p = 3 (cf. Theorem 3.5) Their eistence is ensured by the following theorem Theorem 3.5 Let n 1. For each q > q n there eist an even periodic solution u n of equation (3.) which has n laps in each half period. 5

26 So far we have discussed branches of solutions which bifurcate from the eigenvalues q n, n = 1,, 3,.... They all possess the property that that they have precisely two zeros in each period. In [33] it has been shown that there are also branches of periodic solutions which bifurcate from the eigenvalues q n,m (1 m < n). In Figure 1 we show graphs of solutions on branches which bifurcate from q 5,1, q 5,, q 5,3 and q 5,4. Notice that they have, respectively,, 4, 6 and 8 zeros in each period. In fact, on each branch bifurcating from q n,m the solution can be shown to possess m zeros per period. u() u() u() u() u u u u t t t t Figure 8: Periodic solutions of the linear equation in (3.9) for q = q 5,1, q = q 5,, q = q 5,3 and q = q 5,4 3.3 Branches of periodic solutions In Section 3. we have constructed three countable families of odd periodic solutions, each type of solution eisting on a half line of values of q, which etends to infinity. In Figure 11 we show the branches of 1-lap, -lap and 3-lap solutions of the type obtained in Theorem 3. (u () > ) together with branches of -lap, 4-lap and 6-lap solutions of the type obtained in Theorem 3.4 (u () = ). 8 6 M q Figure 9: Branches in the (q, M)-plane of 1-lap, 3-lap, 5-lap and 7-lap (per half-period) solutions of equation (3.) obtained in Theorem 3. and branches of -lap, 4-lap and 6-lap (per half-period) solutions obtained in Theorem 3.4, when p = 3 In the net theorem, which is due to van den Berg [6], we present an upper bound for bounded solutions of equation (3.) on R. 6

27 Theorem 3.6 There eists a positive constant K, which does not depend on q, such that any bounded solution u(, q) of equation (3.) satisfies u(, q) K(1 + q) /(p 1) for q >. Proof We proceed in two steps: we first fi q R + and show that there eists a constant C(q) such that u(, q) C(q), and then we show that C(q) must be O(q /(p 1) ) as q. Thus, let q > and suppose first that there eists a sequence of bounded solutions {u n } of equation (3.) such that u n as n. Write µ n = u n 1. Then µ n as n. We now scale the solutions u n, and define the new variables y = µ (p 1)/4 ( n ) and v n (y) = µ n u n (), (3.) where we have chosen the translations n in such a way, that v n () > 1 for every n 1. (3.1) When we transform equation (3.) to these new variables, we obtain v iv n + qµ (p 1)/ n v n + v p n + µ p 1 n v n =, (3.) By construction, the sequence {v n } is bounded in L (R); in fact Hence, when we write (3.) as v n = 1 for all n 1. v iv n + qµ (p 1)/ n v n = µ p 1 n v n v p n, (3.3) we see that the right hand side is uniformly bounded, so that all the derivatives of v n are uniformly bounded on compact sets in R. Therefore, there eists a subsequence, which we denote again by v n, which converges in C 4 ([ L, L]) to a function V for any L >. Taking the limit in (3.) we find that V satisfies the reduced equation V iv + V p =. (3.4) From (3.1) we conclude that V () 1, so that V must be a bounded nontrivial solution of equation (3.4). But, by [33] (Eercise 3..1), equation (3.4) has no bounded nontrivial solutions. Therefore we have obtained a contradiction. Net, we show that C(q) < K(1 + q) /(p 1) for some K > when q >. Suppose to the contrary, that there eists a sequence {q n } tending to infinity as n with corresponding bounded solutions u n such that q /(p 1) n u n as n. (3.5) We now repeat the argument given in the first part of the proof. Because by (3.5), µ (p 1)/ n q n as n, we obtain the same limit equation (3.4), and thus the same contradiction. This completes the proof of Theorem

28 4 A sub-linear bifurcation problem In this section we investigate families of periodic solutions of the equation u iv + qu + u u p 1 u =, (p > 1) (4.1) i.e., we put f(s) = s s p 1 s in equation (1.1). In contrast to the super-linear equation (3.), this equation has three constant solutions, u = and u = ±1. The spectrum at u = is the same as that of the trivial solution of the equation discussed in Section 3. At u = ±1 the spectrum consists of two real eigenvalues and two imaginary eigenvalues, for any value of q R. Specifically, λ = ±α and λ = ±iβ, (4.) where α = 1 q + 8 q and β = 1 q q. (4.3) As for (3.), zero energy periodic solutions bifurcate from u = at the critical values q n,m. However, we shall see that here they bifurcate sub-critically, whereas in Section 3 they bifurcated super-critically. In Figure 1 we show branches of solutions bifurcating from the values q 1, q 3, q 5 and q 7. As in Figure 7, one branch bifurcates from q 1, whilst from the values q 3, q 5 and q 7 two branches with different solutions bifurcate, one with u () > and one with u () = (see Figure 5). 1.5 M q Figure 1: Branches of 1-lap, 3-lap, 5-lap and 7-lap periodic solutions of equation (4.1) for p = 3 As in Section, we can show that any periodic solution is bounded above by the unique postive root c + of the equation F (s) = 1 s + G(s) = : Theorem 4.1 Let u be a periodic solution of equation (4.1) with zero energy. Then u() < c + def = ( ) p + 1 1/(p 1) for R. 8

29 The proof is an easy consequence of the energy identity (1.9). In the net two theorems establish the eistence of odd periodic solutions with a prescribed number of laps between nearest points of symmetry for different values of q >. We begin with values of q (, ). Theorem 4. Let q (, ). Then for any odd number n there eists an odd periodic solution u n of equation (4.1) with n laps between points of symmetry. Proof Clearly, linearization of equation (4.1) about u = yields the same equation as we obtained for equation (3.). Thus, we are led to the same linear problem as was discussed in Subsection 3.1, with solution v(). We now find that so that Let < q < = v (ξ 1 ) <, (4.4) u (ξ 1 (α), α) < for α small. (4.5) α 1 = sup{α > : u(ξ 1 ) < c + on (, α)} where c + has been defined in Theorem 4.1. As in Section, one can show that α 1 < and that (a) ξ 1 (α) C([, α 1 ]), (b) u(ξ 1 (α 1 ), α 1 ) = c + and (c) u (ξ 1 (α 1 ), α 1 ) >. It follows from (4.5) and properties (b) and (c) that the function ϕ 1 (α) def = u (ξ 1 (α), α) (i) is continuous, and (ii) changes sign, on (, α 1 ). Let α1 be a zero of ϕ 1 on this interval. Then u 1 () = u(; α1 ) can be continued to a 1-lap periodic solution. In order to construct a 3-lap periodic solution, we notice that u 1 (η 1 ) = u 1 (ξ 1 ) <. Let α 1 = sup{α > α 1 : u(η 1 ) < on (α 1, α)}. Then by continuity u(η 1 (α 1 ), α 1 ) = and u (η 1 (α 1 ), α 1 ) <. Since η 1 (α 1 ) = ξ 1 (α 1 ) and u (ξ 1 (α 1 ), α 1 ) >, it follows that the function ϕ 3 (α) def = u (η 1 (α), α) changes sign at some point α3 (α 1, α 1 ), and the function u 3 () = u(; α3 ) is a 3-lap periodic solution. Net, let α 3 = sup{α > α 3 : u(ξ ) < on (α 3, α)}. 9

30 Since u 3 (ξ 1 ) = u 3 (ξ ), this supremum is well defined. Plainly, α 3 < α 1, and as before (a) ξ (α) C([α 1, α 3 ]), (b) u(ξ (α 3 ), α 3 ) = c + and (c) u (ξ 1 (α 3 ), α 3 ) <. Recall that η 1 (α 1 ) = ξ 1 (α 1 ), and hence u (ξ ) < at α 1. Hence, the function ϕ 5 (α) def = u (ξ (α), α) changes sign on the interval (α 1, α 3 ), say at α 5 and the function u 5() = u(; α 5 ) is a 5-lap periodic solution. Continuing in this manner we can successively construct periodic solutions with any odd number of laps between adjacent points of symmetry. Remark 4.1 As in Section 3, in the iteration process we also pick up a sequence of periodic solutions for which both u, u and u vanish in the middle of the period (see Figure 4). Net, we turn to values of q (, ). As we have seen in Lemma 3.1, the inequality (4.4) no longer holds for q > q 1 =. However, we do have so that q 1 < q < q 3 = v(η 1 ) <, (4.6) u(η 1 (α), α) < for α small. (4.7) This allows us to define α 1 again, as in the proof of Theorem 4., and continue to construct a sequence of periodic n-lap solutions where n is odd and n 3. For q > q 3 we have: q 3 < q < q 5 = v(ξ ) > and v (ξ ) <, (4.8) and we can pick up the construction of periodic n-lap solutions at n = 5. Summarizing we can prove the following eistence theorem: Theorem 4.3 Let n be any positive odd integer. Then for < q < q n there eists an odd n-lap periodic solution of equation (4.1) of Type I. Proof The above argument applied successively proves Theorem 4.3 for q q k, where k < n. However, since all the local etrema are non-degenerate, it follows from the continuous dependence of u, u and u on q, that the statement remains true for the isolated values {q k : 1 k < n}. Remark 4. Let n 3. Then for < q < q n there also eist a n-lap periodic solution of equation (4.1) of type II. For q the situation is quite different. In the net theorem we show that there are no periodic solutions which have points of symmetry and one zero between each of them, with respect to which they are odd. Theorem 4.4 Let q and E. Then there are no odd periodic solutions u() of equation (4.1), which are positive on (, L), where = L is its first point of symmetry. 3