Second order, multi-point problems with variable coefficients
aa r X i v : . [ m a t h . C A ] J un SECOND ORDER, MULTI-POINT PROBLEMS WITH VARIABLECOEFFICIENTS
FRANC¸ OIS GENOUD AND BRYAN P. RYNNE
Abstract.
In this paper we consider the eigenvalue problem consisting of theequation − u ′′ = λru, on ( − , , where r ∈ C [ − , , r > λ ∈ R , together with the multi-point boundaryconditions u ( ±
1) = m ± X i =1 α ± i u ( η ± i ) , where m ± > i = 1 , . . . , m ± , α ± i ∈ R , η ± i ∈ [ − , η + i = 1, η − i = −
1. We show that if the coefficients α ± i ∈ R are sufficientlysmall (depending on r ) then the spectral properties of this problem are similarto those of the usual separated problem, but if the coefficients α ± i are notsufficiently small then these standard spectral properties need not hold. Thespectral properties of such multi-point problems have been obtained before forthe constant coefficient case ( r ≡ Introduction
In this paper we consider the eigenvalue problem consisting of the equation − u ′′ = λru, on ( − , , (1.1)where r ∈ C [ − , , r > λ ∈ R , together with the multi-point boundaryconditions u ( ±
1) = m ± X i =1 α ± i u ( η ± i ) , (1.2)where m ± > i = 1 , . . . , m ± , α ± i ∈ R , η ± i ∈ [ − , η + i = 1, η − i = −
1. An eigenvalue is a number λ for which (1.1), (1.2) has a non-trivial solution u (an eigenfunction ). The spectrum , σ , is the set of eigenvalues. Aneigenvalue is termed simple if its algebraic multiplicity (defined in Section 4.3) isequal to 1. Date : 6 June 2011This work was supported by the Engineering and Physical Sciences Research Council[EP/H030514/1].
For any integer m > α = ( α , . . . , α m ) ∈ R m , the notation α = 0, α >
0, will mean α i = 0, α i > i = 1 , . . . , m , respectively, and we define the norm | α | := m X i =1 | α i | , α ∈ R m . (1.3)For the coefficients in (1.2) we will use the notation α ± := ( α ± , . . . , α ± m ± ) ∈ R m ± , α := ( α − , α + ) ∈ R m − × R m + , and similarly for η ± , η ; we also let := (0 , ∈ R m − × R m + . For any γ > A γ := { α : | α ± | < γ } . When α = the boundary conditions (1.2) reduce to the standard, separatedDirichlet boundary conditions at x = ±
1, so when α = we call the conditions(1.2) Dirichlet-type boundary conditions. For the separated conditions the spectralproperties of the problem are well known, and this case will play a central role inour analysis, with the results for general α ∈ A γ being obtained by continuationfrom α = .We will obtain various properties of the spectrum of the problem (1.1), (1.2),including the existence of eigenvalues, their algebraic multiplicity, continuity prop-erties, and the positivity of the principal eigenfunction. As in the classical Sturm-Liouville theory for separated boundary conditions, the eigenfunctions will be shownto have certain ‘oscillation’ properties. However, in the multi-point case these can-not be characterised simply by counting nodal zeros of the eigenfunctions, andinstead will be described in terms of certain sets of functions T ± k , k = 1 , , . . . (firstintroduced in [11]), which will be defined in Section 2.2. In particular, we will provethe following theorem in Section 4. Theorem 1.1.
For any r ∈ C [ − , , r > , there exists γ = γ ( r ) ∈ (0 , suchthat if α ∈ A γ then the spectrum σ of (1.1) , (1.2) consists of a strictly increasingsequence of simple eigenvalues λ k = λ k ( r ) > , k = 1 , , . . . . Each eigenvalue λ k has an eigenfunction u k ∈ T + k . In addition, lim k →∞ λ k = ∞ . In the constant coefficient case, with r ≡
1, it was shown in [12, Theorem5.1] that Theorem 1.1 is valid with γ = 1, but can fail if γ >
1. Here, in thevariable coefficient case, it will be shown that Theorem 1.1 is valid for ‘sufficientlysmall’ γ ( r ), but may fail for values of γ <
1. The spectral properties described inTheorem 1.1 have not previously been obtained for the variable coefficient problem.Principal eigenvalues (with positive eigenfunctions) have been discussed for thevariable coefficient case in [14] and [16].We also consider nonlinear problems of the form − u ′′ = f ( · , u ) , on ( − , , (1.4)together with the boundary conditions (1.2), where f ∈ C ([ − , × R , R ). Undera suitable ‘nonresonance’ condition on f we show that (1.2), (1.4) has a solution.In addition, we consider the special case where equation (1.4) has the form − u ′′ = g ( · , u ) u, (1.5)with g ∈ C ([ − , × R , R ) (the differentiability condition on g is technical andcould be removed under suitable hypotheses, see Remark 6.4 below). Since u ≡ trivial solution of (1.2), (1.5), and the nonresonance result only yields existenceof at least one solution, additional arguments are required to obtain non-trivial ULTI-POINT PROBLEMS WITH VARIABLE COEFFICIENTS 3 solutions of this problem. In fact, we will prove the existence of nodal solutions, thatis, solutions belonging to specific sets T ± k . We do this by studying the bifurcationproblem − u ′′ = λg ( · , u ) u. (1.6)Under certain hypotheses on g we prove a (partial) Rabinowitz-type global bifurca-tion theorem for the problem (1.2), (1.6), showing that global (unbounded) continuaof non-trivial solutions bifurcate from the eigenvalues of the linearisation of the bi-furcation problem at u = 0. Nodal solutions for (1.2), (1.5) will then be obtainedas solutions of (1.2), (1.6) with λ = 1. This programme will be carried out usingthe spectral properties of the linear problems corresponding to the asymptotes of g as u → | u | → ∞ .In the special case where g has the form g ( x, u ) = r ( x ) e g ( u ) , (1.7)with r ∈ C [ − , , r >
0, and e g ∈ C ( R ), we obtain nodal solutions u ∈ T ± k when e g ‘crosses’ an eigenvalue λ k = λ k ( r ). Such results are well known in other settings,and similar results for autonomous multi-point problems (with g independent of x ) were obtained in [2, 12]. Nonautonomous multi-point problems similar to (1.2),(1.5) (with a separated boundary condition at one end-point) have been discussedrecently and nodal solutions were obtained in, for example, [1, 8] (see also thereferences therein for other results in the same spirit). These papers do not havethe eigenvalues of the full, variable coefficient, multi-point problem available, andthey obtain nodal solutions when e g crosses intervals between consecutive eigenvaluesof a related problem with separated boundary conditions at both end-points. Wewill describe these results further in Section 6 below, and compare them with ourresults.Multi-point problems have received much attention recently. For instance, inthe constant coefficient case some partial results regarding spectral properties wereobtained in [9] for a problem with a separated boundary condition at one end-pointand a multi-point condition at the other end. Improved results for more generalproblems were then obtained in [2, 11, 12, 13]. Once the spectral properties areknown they can be used to obtain nonresonance conditions and nodal solutionsin a standard manner. The variable coefficient case has not been considered tothe same extent. Principal eigenvalues, with positive principal eigenfunctions, havebeen obtained in [14, 15, 16], and these have been used to obtain positive solutionsof nonlinear multi-point problems. As mentioned above, the papers [1, 8] obtainnodal solutions for multi-point problems, but they do so using the eigenvalues ofa related, separated problem, instead of those of the multi-point problem. Forbrevity, we will not discuss the background material any further here, but simplyrefer the reader to the review paper [3] for more discussion and references.1.1. Neumann-type boundary conditions.
If the values of u are replaced withthe values of the derivative u ′ in the conditions (1.2) we obtain so called Neumann-type boundary conditions. Most of our results can be extended to deal with suchconditions (or a mixture of Dirichlet and Neumann-type conditions). The onlydifficulty is that the multi-point operator introduced in Section 2.1 below is notinvertible, since constant functions lie in its null space. This can be dealt with usingthe methods in [13], which deals with such boundary conditions in the constantcoefficient case, so we will say no more about this here. The paper [7] deals with a
FRANC¸ OIS GENOUD AND BRYAN P. RYNNE variable coefficient problem with a separated boundary condition at one end and aNeumann-type multi-point condition at the other end, using a similar approach tothat in the papers [1, 8] (which deal with a Dirichlet-type condition).1.2.
More general, nonlocal boundary conditions.
Non-local boundary con-ditions more general than the above multi-point conditions have also been consid-ered recently by several authors in various contexts, see for example [1, 15] and thereferences therein. For instance, the Dirichlet-type conditions (1.2) can be replacedby integral conditions of the form u ( ±
1) = Z − u ( y ) dµ A ± ( y ) , (1.8)where A ± are functions of bounded variations and the corresponding measures µ A ± satisfy suitable restrictions of the form Z − d | µ A ± | < γ. (1.9)Here, the right-hand side in (1.8) is a Lebesgue-Stieltjes integral with respect tothe signed measure µ A ± generated by A ± , and in (1.9) the term | µ A ± | denotes thetotal variation of µ A ± (we refer the reader to [5, Section 19] and [6, Section 36] forthe required measure and integration theory).By choosing A ± to be suitable step functions we see that the Dirichlet-typeboundary conditions (1.2) can be regarded as a special case of the condition (1.8).Also, it is clear that (1.9) generalizes the condition α ∈ A γ of Theorem 1.1.It is explained in [3, Section 6] (for equations involving the p -Laplacian, withconstant coefficients) how to generalize the multi-point setting to boundary condi-tions of the form (1.8). The results of the present paper can be readily extendedto such boundary conditions in a similar manner. The restriction on the coeffi-cient α in Theorem 1.1 must be replaced by the condition (1.9), with a suitable γ = γ ( r ) ∈ (0 , Preliminary results
In this section we will describe various preliminary results that will be used inthe following sections.2.1.
Function spaces.
For any integer n >
0, let C n [ − ,
1] denote the usualBanach space of n -times continuously differentiable functions on [ − , | · | n . A suitable space in which to search forsolutions of (1.1), and which incorporates the boundary conditions (1.2), is thespace X := { u ∈ C [ − ,
1] : u satisfies (1.2) } , k u k X := | u | , u ∈ X. We also let Y := C [ − , k · k Y := | · | .We define ∆ : X → Y by ∆ u := u ′′ , u ∈ X. By the definition of the spaces X , Y , the operator ∆ is well-defined and continuous.The following result is proved in [12, Theorem 3.1]. ULTI-POINT PROBLEMS WITH VARIABLE COEFFICIENTS 5
Theorem 2.1. If | α ± | < then the operator ∆ : X → Y is bijective, and theinverse operator ∆ − : Y → X is continuous. In addition, ∆ − : Y → C [ − , iscompact. For h ∈ C ([ − , × R , R ), we also define the Nemitskii operator h : Y → Y by h ( u )( x ) := f ( x, u ( x )) , u ∈ Y (we will use the same notation for a function andits associated Nemitskii operator — this should cause no confusion). The operator h : Y → Y is bounded and continuous.2.2. Nodal properties.
For any C function u , if u ( x ) = 0 then x is a simple zero of u if u ′ ( x ) = 0. Now, for any integer k > ν ∈ {±} , we define T νk ⊂ X to be the set of functions u ∈ X satisfying the following conditions:(a) u ′ ( ± = 0 and νu ′ ( − > u ′ has only simple zeros in ( − , k such zeros;(c) u has a zero strictly between each consecutive zero of u ′ .We also define T k := T + k ∪ T − k . Remark . The sets T ± k were first introduced in [11] to characterise the oscillationproperties of the eigenfunctions of multi-point Dirichlet-type problems with con-stant coefficients. It was also shown in [11] that the usual method of characterisingthe oscillations by counting the nodes of the eigenfunctions may fail for multi-pointproblems. There is a longer discussion of various methods of characterising theoscillation properties of multi-point problems in [13, Section 9.4].2.3. Solution estimates.
We will now obtain various estimates on solutions of(1.1). Letting r ′± > r ′ , we define r min / max := (min / max) r, ( r ′± ) min / max := (min / max) r ′± and c min := min n exp (cid:16) − r ′± ) max r min (cid:17)o , c max := 1 c min . For any λ > θ ∈ R we let w ( λ, θ ) ∈ C ( R ) denote the solution of (1.1)satisfying the initial conditions w ( λ, θ )(0) = sin θ, w ( λ, θ ) ′ (0) = ( λr (0)) / cos θ. (2.1)Clearly, any solution of (1.1) has the form u = Cw ( λ, θ ), for suitable C, θ ∈ R .Also, for any solution u of (1.1) we define the Lyapunov function E ( λ, u )( x ) := u ′ ( x ) + λr ( x ) u ( x ) , x ∈ [ − , , (2.2)and when no confusion is possible we will simply write E ( x ). By (2.1), E ( λ, w ( λ, θ ))(0) = λr (0) , θ ∈ [0 , π ] . (2.3) Lemma 2.3.
For λ > , if u is a non-trivial solution of (1.1) then c min E ( x ) E (0) c max for all x ∈ [ − , . (2.4) Hence, for θ ∈ [0 , π ] and x ∈ [ − , , λr min c min w ( λ, θ ) ′ ( x ) + λr ( x ) w ( λ, θ )( x ) λr max c max . (2.5) FRANC¸ OIS GENOUD AND BRYAN P. RYNNE
Proof.
From (1.1) we obtain − r ′− ( x ) r ( x ) E ( x ) E ′ ( x ) = λr ′ ( x ) u ( x ) r ′ + ( x ) r ( x ) E ( x ) , (2.6)and the result follows by integration. (cid:3) Corollary 2.4.
For λ > and θ ∈ [0 , π ] , | w ( λ, θ ) | ( r max c max /r min ) / , | w ( λ, θ ) ′ | ( r max c max ) / λ / . (2.7) Lemma 2.5.
For λ > Λ := (4 r max c max ) /r c and θ ∈ [0 , π ] , ± Z ± w ( λ, θ ) r > c := 14 r min c min > . (2.8) Proof.
We write w := w ( λ, θ ) and prove the ‘+’ case, the other one being similar.Multiplying (1.1) by w and integrating by parts yields − w (1) w ′ (1) + w (0) w ′ (0) + Z ( w ′ ) = λ Z w r, and by (2.5) we have Z ( w ′ ) > λr min c min − λ Z w r. Combining these inequalities and using (2.7) yield2 λ Z w r > − w (1) w ′ (1) + w (0) w ′ (0) + λr min c min > − | w | | w ′ | + λr min c min > − r max c max r − / λ / + λr min c min , and the claim follows by a straightforward calculation. (cid:3) For reference, we now state a simple Lagrange identity that will be used severaltimes below.
Lemma 2.6.
Suppose that u, v, z ∈ C [ − , , u satisfies (1.1) and z satisfies − z ′′ = λrz + rv, on ( − , . (2.9) Then Z x uvr = (cid:2) u ′ z − uz ′ (cid:3) x , x ∈ [ − , . (2.10) Proof.
Multiply (1.1) by z , (2.9) by u , subtract and integrate by parts. (cid:3) We will also need some information regarding the derivatives w λ = w λ ( λ, θ ) and w θ = w θ ( λ, θ ). By (1.1) and (2.1), these derivatives satisfy the following initialvalue problems: − w ′′ λ = λrw λ + rw, w λ (0) = 0 , w ′ λ (0) = λ − / r (0) / cos θ, (2.11) − w ′′ θ = λrw θ , w θ (0) = cos θ, w ′ θ (0) = − ( λr (0)) / sin θ. (2.12) ULTI-POINT PROBLEMS WITH VARIABLE COEFFICIENTS 7
Lemma 2.7.
For λ > Λ := r / (4 r c ) and θ ∈ [0 , π ] , λ / | w λ ( λ, θ ) | c := 3 r ( c max /r min ) / | w λ ( λ, θ ) ′ | c := 3 r c / /r min . (2.13) Proof.
Let e ψ := w ( λ, / ( λr (0)) / , ψ := w ( λ, π/ e ψ and ψ shows that | e ψ | ( c max /r min ) / λ − / , | e ψ ′ | c / , | ψ | ( c max r max /r min ) / , | ψ ′ | ( c max r max ) / λ / . (2.14)The variation of constants formula for the differential equation in (2.11) yields w λ ( x ) = w ′ λ (0) e ψ ( x ) + Z x (cid:2) ψ ( x ) e ψ ( y ) − ψ ( y ) e ψ ( x ) (cid:3) r ( y ) w ( y ) dy,w ′ λ ( x ) = w ′ λ (0) e ψ ′ ( x ) + Z x (cid:2) ψ ′ ( x ) e ψ ( y ) − ψ ( y ) e ψ ′ ( x ) (cid:3) r ( y ) w ( y ) dy, so by (2.11), | w λ | r / λ − / | e ψ | + 2 r max | w | | ψ | | e ψ | , | w ′ λ | r / λ − / | e ψ ′ | + r max | w | (cid:0) | ψ ′ | | e ψ | + | ψ | | e ψ ′ | (cid:1) , and (2.13) now follows from (2.7) and (2.14). (cid:3) Lemma 2.8. If λ > Λ := max { c − r max , Λ } and w ( ±
1) = 0 then :( a ) w ′ ( ± w θ ( ± > b ) ± w ′ ( ± w λ ( ± > .Proof. Combining (2.11) and (2.12) with Lemma 2.6 yields w ′ ( ± w θ ( ±
1) = ( λr (0)) / (2.15)and w ′ ( ± w λ ( ±
1) = Z ± rw − λ − / r (0) / sin θ cos θ. (2.16)Hence (a) follows immediately (for all λ > (cid:3) The constant coefficient case.
To illustrate the above estimates, let usbriefly consider the case r ≡
1. For w = w ( λ, w ( x ) = sin( λ / x ) , w ′ ( x ) = λ / cos( λ / x ) ,w λ ( x ) = 12 λ − / x cos( λ / x )and w ′ λ ( x ) = 12 (cid:2) λ − / cos( λ / x ) − x sin( λ / x ) (cid:3) . In this case, the Lyapunov function E ≡
1, and all the above estimates hold with c min / max = 1. We particularly want to emphasize here that a lower bound of theform (2.8) cannot hold uniformly for all λ > ± Z ± w = 12 (cid:18) − sin(2 λ / )2 λ / (cid:19) . FRANC¸ OIS GENOUD AND BRYAN P. RYNNE
Notation.
In principle, the coefficients α ± are η ± are fixed, but many of theproofs will be by continuation with respect to the coefficients α ± so we will oftenexplicitly display the dependence of various functions on these coefficients. Ofcourse, most of the functions we introduce also depend on the coefficients η ± , butwe will usually regard these coefficients as fixed and omit them from the notation.In particular, it will be important to observe that all estimates we obtain will beuniform with respect to α , as α varies over the set { α : | α ± | < } . Also, even though the results of Theorem 1.1 depend on r , we will generallyconsider r to be fixed in Sections 3 and 4, so, except where necessary, we will notexplicitly display the dependence on r .3. Solutions with a single boundary condition
In this section we consider the problem − u ′′ = λru, on ( − , , (3.1) u ( η ) = m X i =1 α i u ( η i ) , (3.2)with a single, multi-point boundary condition, and fixed λ > m > α ∈ R m , η ∈ [ − ,
1] and η ∈ [ − , m . We will show that the set of solutions of (3.1),(3.2), is 1-dimensional. This result will show that the eigenspace correspondingto an eigenvalue is 1-dimensional. For separated boundary value problems this 1-dimensionality is a consequence of the uniqueness of the solutions of the initial valueproblem with a single point initial condition. Theorem 3.2 below can be regardedas an analogue of this result for the multi-point ‘initial condition’ (3.2).For the following results, it will be convenient to regard λ as fixed, and α asvariable, so we will omit λ from the notation and include α . Also, we recall thatany solution u of (3.1) must have the form u = Cw ( θ ), for suitable C, θ ∈ R .We first prove a preliminary lemma which will also be useful later. Lemma 3.1. If | α | < a := ( r min c min /r max c max ) / and u is a non-trivial solutionof (3.1) , (3.2) , then u ′ ( η ) = 0 . Proof.
Suppose that u ′ ( η ) = 0 . Since u = Cw ( θ ), for some C, θ ∈ R , w ( θ ) ′ ( η ) = 0and it follows by (2.3)-(2.4) that | w ( θ )( η ) | = (cid:18) E ( w ( θ ))( η ) λr ( η ) (cid:19) / > c / (cid:18) r (0) r ( η ) (cid:19) / , | w ( θ )( η i ) | (cid:18) E ( w ( θ ))( η i ) λr ( η i ) (cid:19) / c / (cid:18) r (0) r ( η i ) (cid:19) / , i = 1 , . . . , m. Hence, by (3.2), c / (cid:18) r (0) r ( η ) (cid:19) / | w ( θ )( η ) | m X i =1 | α i || w ( θ )( η i ) | c / m X i =1 | α i | (cid:18) r (0) r ( η i ) (cid:19) / , which yields m X i =1 | α i | (cid:18) r ( η ) r ( η i ) (cid:19) / > (cid:18) c min c max (cid:19) / . But this contradicts the assumption that | α | < a , and so completes the proof. (cid:3) ULTI-POINT PROBLEMS WITH VARIABLE COEFFICIENTS 9
We can now prove the main result of this section.
Theorem 3.2. If | α | < a then the set of solutions of (3.1) , (3.2) , is 1-dimensional.Proof. Since any solution of (3.1) has the form u = Cw ( θ ) for suitable C, θ ∈ R ,we see that u satisfies (3.2) if and only ifΓ( θ, α ) := w ( θ )( η ) − m X i =1 α i w ( θ )( η i ) = 0 , θ ∈ R . (3.3)The function Γ : R × R m → R is C , and we will denote the partial derivative of Γwith respect to θ by Γ θ . Lemma 3.3. If | α | < a then Γ( θ, α ) = 0 = ⇒ Γ θ ( θ, α ) = 0 , θ ∈ R . Proof.
Suppose that Γ( θ, α ) = Γ θ ( θ, α ) = 0, for some ( θ, α ) ∈ R × R m . Defining alinear functional B α : C [ − , → R by B α z := z ( η ) − m X i =1 α i z ( η i ) , z ∈ C [ − , , we see that (cid:18) (cid:19) = (cid:18) B α ( w ( θ )) B α ( w θ ( θ )) (cid:19) = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) B α ( w (0)) B α ( w ( π/ (cid:19) , so B α ( w (0)) = B α ( w ( π/ θ such that w ( θ ) ′ ( η ) = 0 and B α ( w ( θ )) = 0. However, this contradicts Lemma 3.1, which proves Lemma 3.3. (cid:3) Now, by definition, Γ( · ,
0) = w ( · )( η ) has exactly 1 zero in [0 , π ), and so it followsfrom Lemma 3.3 and continuity that Γ( · , α ) has exactly 1 zero in [0 , π ), for all α with | α | < a (of course, by periodicity and linearity, there are other zeros outside [0 , π ),but these do not yield distinct solutions of (3.3)). This proves Theorem 3.2. (cid:3) Eigenvalues
We now consider the eigenvalue problem (1.1), (1.2), which can be rewritten as − ∆( u ) = λru, u ∈ X. (4.1)In the following subsections we will prove various properties of the eigenvalues andeigenfunctions, including Theorem 1.1.4.1. Proof of Theorem 1.1.
We first establish some useful properties of solutions.Let ( λ, u ) be a nontrivial solution of (4.1), with | α ± | < Lemma 4.1. | u ( ± | | α ± || u | < | u | .Proof. By (1.2), | u ( ± | m ± X i =1 | α ± i || u ( η ± i ) | | α ± || u | . (cid:3) Lemma 4.2. If λ < then u cannot have a strictly positive local max ( respectively,a strictly negative local min ) in ( − , . Proof.
Suppose that u ′ ( x ) = 0, x ∈ ( − , u ′ ( x ) = − λ Z xx ru, x ∈ [ − , , and the result follows by inspecting the sign of u ′ in a neighbourhood of x . (cid:3) Combining Theorem 2.1 with Lemmas 4.1 and 4.2 shows that λ > | u | is attained in ( − , u must have a global max or min in ( − , | α ± | are sufficiently small then λ is uniformlybounded away from zero. Lemma 4.3.
There exists Λ > such that if | α ± | then λ > Λ .Proof. Suppose, on the contrary, that for each n = 1 , , . . . , there exist α ± n , with | α ± n | , and a corresponding solution ( λ n , u n ) of (1.1)-(1.2) with | u n | = 1 and λ n >
0, such that λ n → n → ∞ . Then, by (1.1), | u ′′ n | →
0, so thereexist constants c , m , and a subsequence such that u n ( x ) → c + mx , uniformly for x ∈ [ − , | α ± | . (cid:3) The next lemma shows that if | α ± | are sufficiently small then any eigenfunctionmust lie in one of the nodal sets T k . Lemma 4.4. If | α ± | < a then u ′ ( − u ′ (1) = 0 and u ∈ T k , for some k > .Proof. The first part of the lemma follows from Lemma 3.1. The second part thenfollows from this, together with the definition of the sets T k and the properties ofsolutions of the differential equation (1.1). (cid:3) We now define C functions Γ ± : (0 , ∞ ) × R × R m ± → R byΓ ± ( λ, θ, α ± ) := w ( λ, θ )( ± − m ± X i =1 α ± i w ( λ, θ )( η ± i ) (4.2)(where w ( λ, θ ) is as in Section 2). Substituting w ( λ, θ ) into (1.2) shows that, forgiven coefficients α ± , a number λ is an eigenvalue if and only ifΓ ± ( λ, θ, α ± ) = 0 , (4.3)for some θ ∈ R . Hence it suffices to consider the set of solutions of (4.3).We will prove Theorem 1.1 by continuation with respect to α , away from α = ,where the required information on the solutions of (4.3) follows from the standardtheory of the separated Dirichlet problem, with boundary conditions u ( ±
1) = 0.For reference, we state this in the following lemma.
Lemma 4.5.
Suppose that α = . For each k = 1 , , . . . , there exists an eigenvalue λ k > and a unique θ k ∈ [0 , π ) such that u k = w ( λ k , θ k ) ∈ T k is a correspondingeigenfunction. Also, λ k < λ k +1 , and λ k → ∞ as k → ∞ . By construction, each ( λ k , θ k ) satisfies (4.3) (with α = ). Of course, by theperiodicity properties of Γ ± with respect to θ , there are other solutions of (4.3)than those in Lemma 4.5, but these do not yield distinct solutions of the eigenvalueproblem (4.1). In fact, to remove these extra solutions and to reduce the domainof θ to a compact set, from now on we will regard θ as lying in the circle obtainedfrom the interval [0 , π ] by identifying the points 0 and 2 π , which we denote by S ,and we regard the domain of the functions Γ ± as (0 , ∞ ) × S × R m ± . ULTI-POINT PROBLEMS WITH VARIABLE COEFFICIENTS 11
We now wish to solve (4.3) for small α by applying the implicit function theoremat the points ( λ k , θ k , ), k >
1. We would like to obtain solutions ( λ k ( α ) , θ k ( α ) , α ),for all k >
1, for all α lying in a fixed set A γ (with γ independent of k ). Simplyapplying the implicit function theorem at each of the points ( λ k , θ k , ) would not,of course, yield a fixed set A γ , so we proceed in two steps: we first obtain a fixedset A γ for all ‘large’ eigenvalues using uniform estimates and continuation; we thendeal with the remaining (finite) set of ‘small’ eigenvalues.The following result will allow us to apply the implicit function theorem at thepoints ( λ k , θ k , ), for sufficiently large k > Proposition 4.6. If λ k > Λ then ± Γ ± λ ( λ k , θ k , ± θ ( λ k , θ k , > . (4.4) Proof.
Since λ k is a Dirichlet eigenvalue w ( λ k , θ k )( ±
1) = 0, so by Lemma 2.8, if λ k > Λ then ± w λ ( λ k , θ k )( ± w θ ( λ k , θ k )( ± > , which, by the definition of Γ ± , is (4.4). (cid:3) We now wish to extend the result of Proposition 4.6 to small α = . To do thiswe first prove a partial result. We will suppose from now on that | α ± | min { a , } so that Theorem 3.2 and Lemmas 3.3, 4.3 and 4.4 hold. Lemma 4.7.
There exists a s ∈ (cid:0) , min { a , } (cid:1) and Λ s > such that, if ν ∈ {±} , λ > Λ s , | α ν | < a s , and θ ∈ S , then Γ ν ( λ, θ, α ν ) = 0 = ⇒ Γ νλ ( λ, θ, α ν ) Γ νθ ( λ, θ, α ν ) = 0 . (4.5) Proof.
We prove the result for Γ + , and we omit the superscript throughout theproof; the other case is similar. We also omit the argument ( λ, θ, α ) throughout.By Lemma 3.3, Γ = 0 ⇒ Γ θ = 0, so we need only prove that Γ = 0 ⇒ Γ λ = 0 . Suppose, on the contrary, that there exists some ( λ, θ, α ) such thatΓ = w (1) − m X i =1 α i w ( η i ) = 0 , Γ λ = w λ (1) − m X i =1 α i w λ ( η i ) = 0 . (4.6)Then | w (1) | | α || w | , | w λ (1) | | α || w λ | . (4.7)It follows from (2.10) with u = v = w and z = w λ that Z w r = [ w ′ w λ − ww ′ λ ] . Inserting the estimates (2.7), (2.8), (2.11), (2.13) and (4.7) into this equation showsthat, for λ > max { Λ , Λ } , c Z w r c | α | + 12 r (0) / λ − / , (4.8)where c := ( r max c max ) / ( c + c r − / ) >
0. This proves that if λ is sufficientlylarge and | α ± | is sufficiently small then (4.6) cannot hold, which completes theproof of Lemma 4.7. (cid:3) We now extend the result of Lemma 4.7 to yield similar sign conditions to thosein (4.4) in Proposition 4.6.
Proposition 4.8. If ν ∈ {±} , λ > Λ s , | α ν | < a s and θ ∈ S , then Γ ν ( λ, θ, α ν ) = 0 = ⇒ ν Γ νλ ( λ, θ, α ν ) Γ νθ ( λ, θ, α ν ) > . (4.9) Proof.
Suppose that Γ ν ( λ, θ, α ν ) = 0. We regard ( λ, θ, α ν ) as fixed, and considerthe equation G ( e θ, t ) := Γ ν ( λ, e θ, tα ν ) = 0 , ( e θ, t ) ∈ S × [0 , . (4.10)Clearly, if t ∈ [0 ,
1] then | tα ν | < a s , so by (4.5), G ( θ,
1) = 0 and G ( e θ, t ) = 0 = ⇒ G θ ( e θ, t ) = 0 . (4.11)Hence, by (4.11), the implicit function theorem, and the compactness of S , thereexists a C solution function t → e θ ( t ) : [0 , → S , for (4.10) such that e θ (1) = θ and Γ ν ( λ, e θ ( t ) , tα ν ) = 0 , t ∈ [0 , t = 1, is trivial; standard argu-ments show that its domain can be extended to include the interval [0 , λ, θ, α ν ) = ( λ, e θ (0) , λ, e θ ( t ) , tα ν ) for all t ∈ [0 , t = 1 shows that (4.9) holds at ( λ, θ, α ν ), which completes theproof of Proposition 4.8. (cid:3) We now return to the pair of equations (4.3). To solve these using the implicitfunction theorem we define the Jacobian determinant J ( s, θ, α ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ − λ ( λ, θ, α − ) Γ − θ ( λ, θ, α − )Γ + λ ( λ, θ, α + ) Γ + θ ( λ, θ, α + ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , for ( λ, θ, α ) ∈ (0 , ∞ ) × S × A a s . The sign properties of Γ ± λ , Γ ± θ proved in Propo-sitions 4.6 and 4.8 now yield the following results. Corollary 4.9. ( a ) If λ k > Λ s then J ( λ k , θ k , α ) = 0 . ( b ) If λ > Λ s and | α ± | < a s then Γ + ( λ, θ, α + ) = Γ − ( λ, θ, α − ) = 0 = ⇒ J ( λ, θ, α ) = 0 . The following lemma will be useful to separate ‘small’ and ‘large’ eigenvalues.
Lemma 4.10.
There exists an integer k s and a constant κ > such that if λ is aneigenvalue with eigenfunction u ∈ T k with k > k s then Λ s + 1 λ κk .Proof. This follows immediately by applying the Sturm comparison theorem to thedifferential equation (1.1). (cid:3)
Now suppose that ( λ, θ, α ) ∈ (Λ s , ∞ ) × S × A a s is an arbitrary solution of (4.3),with w ( λ, θ ) ∈ T k for some k > k s . By Corollary 4.9 and the implicit functiontheorem, there exists a maximal open interval e N containing 1 and a C solutionfunction t → ( e λ ( t ) , e θ ( t )) : e N → (0 , ∞ ) × S , such that ( e λ (1) , e θ (1)) = ( λ, θ ) , Γ ± ( e λ ( t ) , e θ ( t ) , tα ± ) = 0 , t ∈ e N .
Also, it follows immediately from Lemma 4.4 and continuity that w ( e λ ( t ) , e θ ( t )) ∈ T k , t ∈ e N. (4.12) ULTI-POINT PROBLEMS WITH VARIABLE COEFFICIENTS 13
Furthermore, Lemma 4.10 and (4.12) show thatΛ s + 1 e λ ( t ) κk , t ∈ e N , and combining this estimate with Corollary 4.9 and a standard continuation argu-ment (similar to the argument in the proof of [3, Lemma 4.8(b)]) shows that, since e N is a maximal interval, in fact [0 , ⊂ e N .Hence, the above arguments have shown that, for any α ∈ A a s and k > k s : C -( a ) if ( λ, θ, α ) ∈ (Λ s , ∞ ) × S × A a s is a solution of (4.3) with w ( λ, θ ) ∈ T k ,then ( λ, θ, α ) can be continuously connected to the solution ( λ k , θ k , ).Similar arguments show that: C -( b ) any solution ( λ k , θ k , ), k > k s , can be continuously connected to exactlyone solution of (4.3) in (Λ s , ∞ ) × S × A a s , say ( λ k ( α ) , θ k ( α ) , α ).We now extend these properties to the full set of solutions, for sufficiently small | α ν | . Proposition 4.11.
There exists γ ∈ (0 , a s ] such that if α ∈ A γ then the properties C - ( a ) and C - ( b ) hold for all k > .Proof. For any λ > θ ∈ R we now let e w ( λ, θ ) ∈ C ( R ) denote the solution of(1.1) satisfying the initial conditions e w ( λ, θ )(0) = sin θ, e w ( λ, θ ) ′ (0) = cos θ, (4.13)and define functions e Γ ± as in (4.2), but using e w instead of w . Analogously to equa-tion (4.3), eigenvalues of (4.1) also correspond to solutions of the pair of equations e Γ ± = 0. However, the fact that the initial conditions for e w do not depend on λ now allows us to establish the analogue of Lemma 2.8, and hence Proposition 4.6,for all λ > e Γ ± = 0 at each of the points ( λ k , θ k , ), 1 k < k s . Combining this withthe above results for k > k s then yields a common ‘continuation domain’ A γ ⊂ A a s for all the eigenvalues. (cid:3) Remark . We used the function w in the continuation argument to obtainthe ‘large’ eigenvalues in the interval (Λ s , ∞ ) since this function yielded uniformestimates in Section 2.3 for all large λ , based on the equation (2.3). Unfortunately,although the initial conditions for e w (and, more particularly, for e w λ ) are simplerthan those for w , the analogue of (2.3) for e w is E ( λ, e w ( λ, θ ))(0) = cos θ + λr (0) sin θ, θ ∈ [0 , π ] , which would not yield uniform estimates for large λ .We conclude from the above results that, for all k > α ∈ A γ , there is aneigenvalue λ k ( α ) with corresponding eigenfunction u k ( α ) := w ( λ k ( α ) , θ k ( α )) ∈ T k (or u k ( α ) := e w ( λ k ( α ) , θ k ( α )) when 1 k < k s ), and there is no eigenvalue ˆ λ = λ k ( α ) having an eigenfunction ˆ u ∈ T k . Also, by Lemma 4.5, λ k < λ k +1 , and byTheorem 3.2, λ k ( α ) = λ k +1 ( α ) for any α ∈ A γ , so it follows from the continuationconstruction that λ k ( α ) < λ k +1 ( α ) for all α ∈ A γ . The fact that λ k ( α ) hasalgebraic multiplicity equal to 1 (and so is simple) will be proved in Lemma 4.14below. Finally, for fixed α , the fact that u k ( α ) ∈ T k , for k >
1, shows that as k → ∞ theoscillation count tends to ∞ , so by standard properties of the differential equation(1.1), we must have lim k →∞ λ k = ∞ . This concludes the proof of Theorem 1.1. (cid:3) Continuity of the eigenvalues and eigenfunctions.
The implicit func-tion theorem construction of λ k and u k in the proof of Theorem 1.1 also impliescontinuity properties which will be useful below, so we state these in the followingcorollary (continuity of u k will be in the space C [ − , Corollary 4.13.
For each k > , λ k ∈ R and u k ∈ C [ − , depend continuouslyon ( α , η ) ∈ A γ × ( − , m − × [ − , m + . Algebraic multiplicity and topological degree.
By Theorem 2.1, for | α ± | < K r : Y → Y by K r u := − ∆ − ( ru ), u ∈ Y , and then the eigenvalue problem (4.1) is equivalent to the prob-lem u = λK r u, u ∈ Y. (4.14)In particular, for α ∈ A γ , the eigenvalues λ k ( α ) obtained in Theorem 1.1 are thecharacteristic values of K r , so we may define the algebraic multiplicity of λ k ( α ) tobe the algebraic multiplicity of λ k ( α ) as a characteristic value of K r , that is m ( λ k ( α )) := dim ∞ [ l =1 N (cid:0) ( I Y − λ k ( α ) K r ) l (cid:1) , where N denotes null-space and I Y is the identity on Y . Lemma 4.14.
Suppose that α ∈ A γ . Then m ( λ k ( α )) = 1 , k > .Proof. The proof follows by the same continuation argument as in [12, Lemma 5.13],starting from α = , where the result is easily verified. (cid:3) Now, if λ is not a characteristic value of K r then the Leray-Schauder degreedeg( I Y − λK r , B R ,
0) is well defined for any
R >
0, where B R denotes the open ballin Y centred at 0 with radius R , see [17, Ch. 13]. Proposition 4.15.
Suppose that α ∈ A γ . Then, for any R > , deg( I Y − λK r , B R ,
0) = ( , if λ < λ , ( − k , if λ ∈ ( λ k , λ k +1 ) , k > .Proof. The result follows from Lemma 4.14 and the Leray-Schauder index theorem(see, for example, [17, Proposition 14.5]). (cid:3)
Positivity of the principal eigenfunction.
A slight extension of the aboveproof of Theorem 1.1 also proves the following result.
Corollary 4.16. ( a ) If each α ± > , then u > on [ − , . ( b ) If either α − < or α + < , then u changes sign on ( − , .Proof. If α ± > u cannot have a strictly positive global max or strictly negative global min at theend-points x = ±
1. Hence, if min u < < max u , then u ′ must have at leasttwo zeros in ( − , u ∈ T +1 . Therefore, u > − , u ( ± >
0, which proves case (a). Case(b) follows immediately from the boundary conditions (1.2). (cid:3)
ULTI-POINT PROBLEMS WITH VARIABLE COEFFICIENTS 15
Counterexamples.
As mentioned in the introduction, in the constant coef-ficient case with r ≡
1, Theorem 1.1 holds for all α ∈ A . In this section we givetwo examples showing that this is not true in the variable coefficient case. The firstexample shows that, for certain α ∈ A and r ∈ C [ − , Example 1.
For δ ∈ (0 , / r δ ( x ) := ( , x ∈ [ − δ, δ ] , , x ∈ [ − , − δ ) ∪ ( δ, , together with the symmetric boundary conditions u ( ±
1) = √ u (0) . (4.15)We will show that there are no eigenvalues in the interval I δ := [( π δ ) , ( π δ ) ].Since, for small δ , the interval I δ can be arbitrarily long and an eigenvalue λ cor-responds to an oscillation count of ‘roughly’ 4 λ / /π (rigorous estimates can easilybe given using the form of r and the solutions of (1.1)), this result shows that theremay be arbitrarily many oscillation counts k for which there are no correspondingeigenvalues.We first observe that if u is an eigenfunction corresponding to an eigenvalue λ ,then the function u e ( x ) := ( u ( x ) + u ( − x )) is an even eigenfunction correspondingto λ , so it suffices to show that there are no even eigenfunctions corresponding toeigenvalues λ ∈ I δ .We now search for an even solution u of (1.1), (4.15), which we suppose to benormalized to u (0) = 1. Thus, u must have the form u ( x ) = ( cos λ / x, x ∈ [0 , δ ] ,C δ cos 2 λ / ( x − ǫ δ ) , x ∈ ( δ, , (4.16)where C δ = 0 and ǫ δ are chosen so that u ∈ C [ − , E ( λ, u ) defined in (2.2) is piecewise constant here, with u + 1 λ ( u ′ ) ≡ , in [0 , δ ], u + 14 λ ( u ′ ) ≡ u ( δ ) + 14 λ u ′ ( δ ) = u ( δ ) + 14 (1 − u ( δ ) )= 14 + 34 u ( δ ) , in [ δ, . Now suppose that λ ∈ I δ . Then u ( δ ) /
2, and hence u (1)
14 + 34 u ( δ ) < u (0) , so u cannot satisfy (4.15), and hence there are no eigenvalues in I δ .Finally, a similar example with a C coefficient function r can be constructed bychoosing a suitable C approximation of the function r δ . (cid:3) Our second example shows that, for some α ∈ A and r ∈ C [ − , Example 2.
Consider equation (1.1) with r ( x ) := 2 − cos( π x ) , x ∈ [ − , . Let µ D > − u ′′ = µru on (0 , , u (0) = u (1) = 0 , with corresponding eigenfunction u D . Clearly, u D can be extended by antisymme-try to a solution u of (1.1) satisfying u ( ±
1) = αu (0) , (4.17)for any α ∈ ( − , v be the (even) solution of the initialvalue problem − v ′′ = µ D rv on ( − , , v (0) = 1 , v ′ (0) = 0 . Setting α := v (1) in (4.17), we see that u and v are two linearly independenteigenfunctions of the problem (1.1), (4.17), corresponding to the eigenvalue λ = µ D .We only need to check that | α | = | v (1) | <
1. The Lyapunov function associatedwith v (defined in (2.2)) satisfies0 E ′ ( x ) E ( x ) r ′ ( x ) r ( x ) < r ′ ( x ) , x ∈ (0 , , so by integration, v ′ (1) + 2 µ D v (1) = E (1) < E (0) Z r ′ = µ D , which yields | v (1) | < − / . (cid:3) A non-resonance condition
In this section we consider the problem (1.2), (1.4), which we can rewrite as − ∆ u = f ( u ) , u ∈ X, (5.1)and we establish the existence of solutions under a ‘nonresonance’ condition. Wesuppose that f ∈ C ([ − , × R , R ) satisfies the following condition.(F ∞ ) There exists r ∞ ∈ C [ − ,
1] such thatlim | ξ |→∞ f ( x, ξ ) ξ = r ∞ ( x ) > , uniformly for x ∈ [ − , . We also assume that | α ± | < γ ( r ∞ ), and we define the eigenvalues λ ∞ k := λ k ( r ∞ ), k = 1 , , . . . . Theorem 5.1.
Suppose that (F ∞ ) holds, and λ ∞ k = 1 , for all k > . Then (5.1) has a solution.Proof. Equation (5.1) is equivalent to u = T ( u ) , u ∈ Y, where we define T : Y → Y by T ( u ) := − ∆ − ( f ( u )). We now define a homotopy H : [0 , × Y → Y by H ( t, u ) = (1 − t ) T ( u ) + tK r ∞ ( u ) , ( t, u ) ∈ [0 , × Y. It follows from Theorem 2.1 that H is completely continuous. Furthermore, ar-guments similar to the proof of [2, Theorem 5.3] show that there exists R > t, u ) ∈ [0 , × Y of u = H ( t, u ) satisfy | u | < R . Hence, ULTI-POINT PROBLEMS WITH VARIABLE COEFFICIENTS 17 it follows from the invariance property of the degree (see [17, Section 13.6]) andProposition 4.15 thatdeg( I Y − T , B R ,
0) = deg( I Y − K r ∞ , B R , = 0 , which proves the theorem. (cid:3) Nodal solutions
In this section we consider the following special case of (5.1), − ∆ u = g ( u ) u, u ∈ X, (6.1)and we establish the existence of nodal solutions of (6.1) (that is, solutions u lyingin specific sets T ± k ). We suppose throughout this section that g ∈ C ([ − , × R )satisfies the following condition.(G) There exists r , r ∞ ∈ C [ − ,
1] such thatlim ξ → g ( x, ξ ) = r ( x ) > , lim | ξ |→∞ g ( x, ξ ) = r ∞ ( x ) > , uniformly for x ∈ [ − , | g x | + sup | ξg ξ | < ∞ . It follows from (G) that the Nemitskii mapping g : Y → Y satisfieslim | u | → | g ( u ) − r | = 0 . (6.2)We also assume throughout that | α ± | < min { γ ( r ) , γ ( r ∞ ) } , and we define theeigenvalues λ k := λ k ( r ), λ ∞ k := λ k ( r ∞ ), k = 1 , , . . . . We will establish the existence of nodal solutions of (6.1) by studying the solutionset of the bifurcation-type problem − ∆ u = λg ( u ) u, ( λ, u ) ∈ (0 , ∞ ) × X (6.3)(solutions of (6.3) with λ = 1 yield solutions of (6.1)). Clearly, ( λ, u ) = ( λ,
0) is a(trivial) solution of (6.3) for any λ ∈ R , and we let S denote the set of non-trivialsolutions of (6.3). We will use global bifurcation arguments to obtain unbounded,closed, connected sets of solutions of (6.3), and then use these sets to obtain solu-tions of (6.1). We first prove two preliminary lemmas. Lemma 6.1.
For any Λ > there exists γ = γ g, Λ ∈ (0 , such that, if α ∈ A γ and ( λ, u ) ∈ S with λ Λ , then u ∈ T k for some k > .Proof. The proof is similar to that of Lemma 4.4, and we need only show that u ′ ( − u ′ (1) = 0. For ( λ, u ) ∈ S , we define the Lyapunov function E ( x ) := u ′ ( x ) + λg ( x, u ( x )) u ( x ) , x ∈ [ − , . Then, by (6.3), E ′ ( x ) = λ [ g x ( x, u ( x )) + g ξ ( x, u ( x )) u ′ ( x )] u ( x ) and so, by (G), there exist C , C > | E ′ | C λu + C λ / | u ′ | λ / | u | C λu + C λ / [( u ′ ) + λu ] ( C + C λ / )[( u ′ ) + λu ] = ( C + C λ / )[( u ′ ) + g − λg min u ] max { , g − } ( C + C λ / )[( u ′ ) + λg min u ] max { , g − } ( C + C λ / ) E =: R ( λ ) E. It follows by integration that E (0)e − R ( λ ) E ( x ) E (0)e R ( λ ) . A similar argument to the proof of Lemma 3.1 now shows that if | α ± | < γ g, Λ := (cid:18) g min g max (cid:19) / e − R (Λ) then u ′ ( − u ′ (1) = 0. (cid:3) Lemma 6.2. If ( λ, u ) ∈ S with u ∈ T k , for some k > , then λ < Λ( k ) := g − (( k + 2) π/ .Proof. The function sin (cid:0) ( k +2) π ( x + 1) (cid:1) has k + 3 zeros in [ − ,
1] so, by the Sturmcomparison theorem, if λ > Λ( k ) then u must have at least k + 2 zeros in [ − , u ′ must have at least k + 1 zeros in [ − , u ∈ T k . (cid:3) We can now prove the following Rabinowitz-type global bifurcation theorem, forindividual nodal counts k > Theorem 6.3.
For a given k > , let γ k := γ g, Λ( k ) ( recall Lemmas 6.1 and 6.2 ) ,and suppose that α ∈ A γ k . Then there exists closed connected sets C ± k ⊂ S with theproperties :(a) ( λ k , ∈ C ± k ;(b) C ± k \{ ( λ k , } ⊂ (0 , Λ( k )] × T ± k ;(c) C ± k is unbounded in (0 , Λ( k )] × Y .Proof. By Theorem 2.1, equation (6.3) is equivalent to the equation u = G ( λ, u ) , ( λ, u ) ∈ (0 , ∞ ) × Y, (6.4)where G ( λ, u ) := − λ ∆ − ( g ( u ) u ) , ( λ, u ) ∈ (0 , ∞ ) × Y, and by (6.2), | G ( λ, u ) − λK r u | = o( | u | ) , as | u | → λ intervals. This shows that equation (6.4) has the sameform as the bifurcation problems considered in [10]. Hence, by the methods in [10](in particular, the proof of [10, Theorem 2.3], which deals with separated Sturm-Liouville problems), we can show that: • there exist closed, connected sets C ± k ⊂ S satisfying (a) and (b) in aneighbourhood of ( λ k , ν ∈ {±} :(i) ( λ k ′ , ∈ C νk , with k ′ = k , or (ii) C νk is unbounded in (0 , ∞ ) × Y ; • by Lemmas 6.1 and 6.2, the sets C ± k cannot intersect either (0 , Λ( k )] × ∂T ± k or { Λ( k ) } × T ± k , so (b) must hold globally; • since (b) holds, it now follows, as in the proof of [10, Theorem 2.3], thatalternative (i) above cannot hold, which proves (c).The proof is complete. (cid:3) Remark . For fixed α = , Theorem 6.3 does not yield unbounded continua C ± k for arbitrarily large k , since lim k →∞ γ k = 0. This is due to the dependenceof γ g, Λ on Λ in Lemma 6.1. Also, the proof of this lemma is where we used theassumptions in condition (G) on the derivatives of g . By a more detailed argumentwe could remove these assumptions and obtain Lemma 6.1 for (suitably bounded) ULTI-POINT PROBLEMS WITH VARIABLE COEFFICIENTS 19 C functions g . However, it is not clear that the dependence on Λ can be removed, orwhat would be the most general result, so for simplicity we have retained condition(G) and the above method of proof of Lemma 6.1, which fits in with our previousarguments using the Lyapunov functions.We now prove the existence of nodal solutions for (6.1). Theorem 6.5.
Suppose that ( λ k − λ ∞ k − < and α ∈ A γ k , for some k > .Then (6.1) has solutions u ± k ∈ T ± k .Proof. By Theorem 6.3 (c) and standard arguments (see, for example, the proof of[2, Theorem 5.3]), we may choose sequences { ( µ ± n , u ± n ) } ⊂ C ± k such that µ ± n → λ ∞ k and | u ± n | → ∞ , as n → ∞ . Hence, by connectedness, the sets C ± k intersect thehyperplane { } × Y , which proves Theorem 6.5. (cid:3) Finally, we briefly consider the special case where g has the form (1.7) with r ∈ C [ − , , r >
0, and e g ∈ C ( R ). The hypothesis (G) is satisfied if | e g ′ ( ξ ) ξ | isbounded and the limits e g := lim ξ → e g ( ξ ) > , e g ∞ := lim | ξ |→∞ e g ( ξ ) > k >
1, if we write λ k := λ k ( r ) then λ k = λ k / e g , λ ∞ k = λ k / e g ∞ , andTheorem 6.5 now takes the following form. Theorem 6.6.
Suppose that ( λ k − e g )( λ k − e g ∞ ) < and α ∈ A γ k , for some k > .Then (6.1) has solutions u ± k ∈ T ± k . The condition ( λ k − e g )( λ k − e g ∞ ) < e g ‘crosses’ the eigenvalue λ k . The use of such crossing conditions to obtain solutionsis well known in many settings.Similar results to Theorem 6.6 have been obtained recently in, for example, [1, 8]and references cited therein. Roughly speaking, these papers deal with problems ofthe form (6.1) with boundary conditions of the form c u ( −
1) + c u ′ ( −
1) = 0 , u (1) = m + X i =1 α + i u ( η + i ) , (6.5)with constants c , c satisfying | c | + | c | >
0, that is, a single, separated boundarycondition at one end-point, and a multi-point condition at the other end-point.They also assume that g has the form (1.7) (or a sum of such terms — to simplifythe discussion we merely consider the form (1.7) here). Although our results, asstated, do not cover the multi-point boundary conditions (6.5), it is trivial to extendthem to do so. In particular, the analogue of Theorem 1.1 holds here. The proofis by a similar (but simpler) continuation argument in which the function w cannow be specified to satisfy the boundary condition at x = − λ and α (sothere is now no analogue of the variable θ ), and then the boundary condition at x = 1 leads to a single equation to solve for λ ( α ) (instead of a pair of equations).This considerably simplifies the construction, and yields a set of eigenvalues λ k ( r ), k >
1, of (1.1), (6.5), for all α in a suitable set A γ ( r ) . Once the eigenvalues havebeen obtained the discussion of the nonlinear problem (6.1), (6.5) proceeds as aboveand, in particular, yields the analogue of Theorem 6.6 for this problem.The main results and hypotheses in [1, 8] for the problem (6.1), (6.5) can besummarised as follows (for brevity we omit various detailed conditions; in particular, it is hard to describe the conditions on the size of | α | in [1, 8], or to compare themwith our conditions). Letting µ k , k >
1, denote the eigenvalues of (1.1) togetherwith the separated boundary conditions c u ( −
1) + c u ′ ( −
1) = 0 , u ′ (1) = 0 , (6.6)they obtain nodal solutions u ± k ∈ T ± k of (6.1), (6.5) if e g < µ k < µ k +1 < e g ∞ or e g ∞ < µ k < µ k +1 < e g , that is, if e g ‘crosses’ the interval [ µ k , µ k +1 ]. In other words,the range of e g is compared with the eigenvalues of the linear problem obtainedby replacing the original multi-point boundary conditions (6.5) with the separatedboundary conditions (6.6). Heuristically, one could say that using the eigenvalues µ k corresponding to the changed boundary conditions leads to the necessity for thefunction e g to cross the interval [ µ k , µ k +1 ], rather than crossing the single eigenvalue λ k obtained from the original boundary conditions. The following lemma shows thatif e g crosses [ µ k , µ k +1 ] then it crosses λ k , but obviously the converse need not hold. Lemma 6.7. If α ∈ A γ ( r ) then, for each k > , µ k < λ k ( α ) < µ k +1 .Proof. We denote by u k ( α ) the eigenfunctions of problem (1.1), (6.5) obtained byadapting our arguments in Section 4, as outlined above. First, at α = , we applySturm’s comparison theorem to the problems (1.1), (6.5) and (1.1), (6.6). It followsthat µ k < λ k ( ) < µ k +1 , for all k >
1. Now suppose that λ k ( α ) = µ k , for some k > α ∈ A γ ( r ) , and let v k be an eigenfunction of (1.1), (6.6), correspondingto the eigenvalue µ k . Since u k ( α ) and v k each satisfy the boundary condition in(6.6) at x = −
1, by a rescaling we may suppose that u k ( α ) = v k . But the analogueof Lemma 4.4 for this setting shows that v ′ k (1) = 0, which contradicts v k being aneigenfunction of (1.1), (6.6). Therefore, for all α ∈ A γ ( r ) and k > λ k ( α ) = µ k and a similar argument shows that λ k ( α ) = µ k +1 . Hence, by continuation, µ k <λ k ( α ) < µ k +1 for all α ∈ A γ ( r ) , k > (cid:3) We conclude that the crossing condition in [1, 8] is more restrictive than thatin Theorem 6.6 above. In addition, we have obtained results for the boundaryconditions (1.2), having multi-point conditions at both end-points. It is mucheasier to deal with the conditions (6.5) than with (1.2) since shooting methods canbe used, as in [1, 8] (shooting from x = −
1, using the separated boundary conditionto provide an ‘initial condition’; this is not possible with multi-point conditions atboth ends). Finally, as explained in the introduction, it is straightforward to extendour methods to the case of integral boundary conditions at both end-points.
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Department of Mathematics and the Maxwell Institute for Mathematical Sciences,Heriot-Watt University, Edinburgh EH14 4AS, Scotland.
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