MULTIPLICITY RESULTS FOR ASYMMETRIC BOUNDARY VALUE PROBLEMS WITH INDEFINITE WEIGHTS

We prove existence and multiplicity of solutions, with prescribed nodal properties, to a boundary value problem of the form u′′ + f (t,u)= 0, u(0)= u(T)= 0. The nonlinearity is supposed to satisfy asymmetric, asymptotically linear assumptions involving indefinite weights. We first study some auxiliary half-linear, two-weighted problems for which an eigenvalue theory holds. Multiplicity is ensured by assumptions expressed in terms of weighted eigenvalues. The proof is developed in the framework of topological methods and is based on some relations between rotation numbers and weighted eigenvalues.

Our main result will be achieved by using a topological degree approach on the lines of [16].A relation between the notions of rotation number and of weighted eigenvalues will be crucial.Multiplicity will be achieved through some information about the number of zeros of the solutions to (1.1).
In the literature, many authors have studied existence and multiplicity of solutions with prescribed nodal properties of problem (1.1) in an asymptotically linear setting.
The first references we wish to quote rely on asymptotically linear problems characterized by the presence of constant bounds in assumptions analogous to (H ± ) and (K ± ).The particular case when a + ∞ ≡ a − ∞ and b + ∞ ≡ b − ∞ in (H ± ) is usually called "symmetric."In this context, we quote the work [49] by Ubilla where the p-Laplacian operator is considered and the work [26] by García-Huidobro and Ubilla where extensions for a wider class of nonhomogeneous differential operators are treated.
In the more general asymmetric context, we refer, among others, to the paper [20] by Dinca-Sanchez.A detailed comparison between our result and the one attained in [20] is developed in the last part of this work (cf.Remark 3.7).We point out that all the articles we have quoted above are written in the spirit of the shooting method and are in the framework of continuous functions.We wish also to mention the paper by Castro and Lazer [11] and the recent contribution attained by Li and Zhou in the work [37], devoted to study with variational methods the existence of multiple nontrivial solutions of p-Laplacian-type PDEs.An interesting result is also provided by Esteban in [22].
In the framework of topological methods, we refer to [9], dealing with asymptotically linear asymmetric systems.It is worth noticing that if we restrict ourselves to the scalar symmetric case with constant weights, then [9,Corollary 4.4] and our main result coincide (for a detailed comparison, see [15]).
Multiplicity of solutions of asymptotically, asymmetric semilinear elliptic problems has been widely investigated in the literature.A classical result in this setting is provided by Fučík in [24].In this direction, we quote, among others, [14,17,28,36] and (for a brief survey) [41].

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The list of results available in the literature as far as multiplicity of solutions for asymptotically linear problems with weights is concerned is definitely shorter.
Before quoting some recent contributions achieved in this context, we wish to remark that a wider number of authors have studied asymmetric, asymptotically linear problems characterized by assumptions involving indefinite weights in order to get existence of solutions.This kind of problems is usually solved by comparing the given equation in (1.1) with prescribed half-linear problems (which reduce to linear ones in the symmetric case).
This procedure has been adopted in the search for positive solutions to elliptic asymptotically linear or semilinear BVPs (cf., among others, [3,8,13,18,30,34,43]).In this framework, results have been given on the existence of at least one solution (not necessarily positive) for semilinear boundary value problems both in the symmetric case (see, e.g., [19] by de Figueiredo and Miyagaki as well as [51,52] by Zhang) and in the asymmetric case (see, e.g., [5] by Arias et al., [21] by Dong,[44] by Reichel and Walter,and [46] by Rynne).We remark that the theory of weighted eigenvalues has been employed in most of the papers quoted above.It is also worth noticing that both in [19] and in [51] additional assumptions are formulated in order to achieve the nontriviality of solutions.
As far as multiplicity of solutions for asymptotically linear problems with weights is concerned, we first mention the works by Sadyrbaev [48] and by Klokov and Sadyrbaev [32] (both generalizing the results in [33]).In these papers the study of an asymmetric boundary value problem is developed through the study of some variation equation.More regularity is required on the nonlinearities.In the asymmetric setting, we wish also to quote the interesting contribution [47] by Rynne.Remarks 3.8 and 3.9 provide more details about these results.Finally, we refer to [16] which handles an asymptotically linear problem of the form (1.1) under symmetric assumptions.For the periodic problems, an analogous result has been attained by Zanini in [50].
It is important to note that the approach used to achieve our result follows the same lines of the one employed in [16].The present paper extends the results in [16] in two directions: an asymmetric situation is treated and indefinite weights are considered.
In the literature the equation in (1.4) is called half-linear, being positively homogeneous and linear in the cones u > 0 and u < 0. A generalized version of the classical Sturm comparison theorems to the half-linear setting will be proved in the following section (see Lemma 2.2).According to this generalization, the existence of a sequence of eigenvalues for problems (1.4) will be ensured.More precisely, the following result (which might be considered of independent interest) will be achieved.
In [15,Chapter 4] we treated the hypotheses (H ± ) and (K ± ) in the particular case where, instead of L 1 -functions, C 1 -functions are considered.In this framework, we used the eigenvalue theory developed in [1,2].Now, we are in position to state our main theorem.
Clearly, if (1.6) or (1.7) is satisfied for every ν ∈ {<, >}, then, for every h ∈ N with m ≤ h ≤ n, Theorem 1.2 guarantees the existence of at least two solutions u h and v h of problem (1.1) with u h (0) > 0 and v h (0) < 0 having exactly h − 1 zeros in (0,T).
This work is organized as follows.Section 2 is devoted to present some preliminary results.More precisely, we first state the abstract topological theorem (Theorem 2.1) providing a first multiplicity result.In the central part of the section, we prove the generalizations of the Sturm theory to the half-linear case.As a consequence, we obtain Theorem 1.1 and a relation between rotation numbers and two-weighted half-eigenvalues (see Lemma 2.6).We conclude Section 2 by recalling the statements of the so-called Elastic lemma.Finally, in Section 3 we obtain the estimates on the rotation numbers which will lead to proving our main theorem.Some comparisons with multiplicity results in the literature are developed in the last part of the paper.

Preliminary results
In this section, we introduce the notion of "rotation number" and explain its role when dealing with multiplicity problems.
We consider polar coordinates (ϑ,ρ), where the angles are counted in the clockwise sense starting from the positive y-axis, so that x = ρ sinϑ, y = ρ cos ϑ.For any continuous curve γ : [0,T] → R 2 \ {0}, we can consider a lifting γ : [0,T] → R × R + 0 to the (ϑ,ρ)-plane.Setting γ(t) = (ϑ γ (t),ρ γ (t)), we have that ϑ γ and ρ γ are continuous functions (ρ(t) = |γ(t)|) and, moreover, ϑ γ (t) − ϑ γ (0) is independent of the lifting of γ which has been considered.Hence, for each t ∈ [0,T], we can define the rotation number which counts the number of half-turns of the vector defined on [0,T] and such that u(t) 2 + u (t) 2 > 0 for every t ∈ [0,T], then we can define the corresponding rotation number.To denote it, we will use the notation Rot (2.2) (t;u).Observe that (2.3) We will now show how some estimates on the rotation number lead to a multiplicity result.The proof of our main result is based on the following theorem which comes from an application of the modified version of the Leray-Schauder continuation theorem stated in [42].
For the proof, we refer to [16].
In order to prove our main result, we are first interested in providing a generalization of the Sturm comparison theorems for half-linear asymmetric two-weighted equations of the form where ϕ,ψ ∈ L 1 ([0,T],R) and λ ∈ R + 0 .We introduce now a simpler notation for rotation numbers of solutions to (2.7).Considering v ∈ S 1 and k ∈ R, it is well known that the Cauchy problem admits a unique solution, denoted by u(•;kv).Furthermore, the rotation number of u(•;kv) is well defined whenever k = 0, since in this case it is easy to show that u(t;kv) 2 + u (t;kv) 2 > 0 for every t ∈ [0,T].As u(t;kv) = ku(t;v), one can also see that the rotation number is independent on k for k = 0. Hence, it is not restrictive to work with k = 1.
For this reason, we will set In order to discuss the dependence of the rotation numbers upon the coefficients of (2.7), we rewrite problem (2.7) in the equivalent form (2.10) We introduce now the polar coordinates (θ,r In particular, the angular coordinate θ := θ λ;ϕ,ψ satisfies As a first step, we generalize the classical Sturm comparison theorem to the asymmetric, indefinite case.This is of fundamental importance for the proof of the existence of a sequence of weighted eigenvalues for (1.4).We refer to Theorem 2.5 for more details. (2.12) Slight variants of the first part of this lemma have been proved in [21,44].In what follows, we develop a complete proof, using different arguments.
Proof.Following a classical procedure (see [12]), we consider θ := θ 2 − θ 1 .It is easy to show that where Notice that the right-hand side of (2.14) is defined almost everywhere, since from (2.11) the function sinθ i has only isolated zeros in t.

Proof.
In what follows, we will provide a proof only of the first part of the lemma.The second part can be proved in an analogous way (just reverse the inequalities).
Let ε > 0 be fixed and suppose that ϕ, ψ, ϕ * , ψ * satisfy (2.22) for some δ > 0. By Lemma 2.3, we know that Rot ϕ * ,ψ * (t;v) ≥ Rot ϕ−δ,ψ−δ (t;v) for every t ∈ [0,T].Hence, to prove the result, it will be sufficient to check that for a suitable choice of δ > 0 (sufficiently small), it follows that (2.26) Assume, by contradiction, that for each n there is Without loss of generality, we can also suppose that t n → τ ∈ [0,T], so that Let u n (•), u(•) be, respectively, the solutions of and of (2.7), with By the continuous dependence of the solutions from the equations data, which is valid also in case of Carathéodory assumptions [27], we know that u n → u in the C 1 -norm, uniformly on [0,T].Hence, which is a contradiction with (2.28).Then, (2.26) is achieved and the proof is complete.
Our aim consists now in generalizing the comparison results contained in Lemma 2.4 to asymptotically linear equations (see Lemma 3.1).To this purpose we need to exhibit a relation between weighted eigenvalues and rotation numbers of solutions to the following 966 Multiplicity results for asymmetric BVPs with weights half-linear Cauchy problems: where ν ∈ {<, >} and ϕ,ψ ∈ L 1 ([0,T],R) with (ϕ + ,ψ + ) (0,0) on (0,T).According to the notation previously introduced, we recall that v < := (0,−1) and v > := (0,1).We first prove that there exists a positive sequence of weighted eigenvalues for halflinear problems of the form (1.4).More precisely, we have the following.

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To complete the proof, we observe that the validity of (2.35) easily follows from the definition of the angular functions and, in particular, from the equalities where (u(t),u (t)) represents a solution of (2.32) and λ,ρ ν λ (•),r λ;ν (•) ∈ R + 0 .Taking into account Lemma 2.2, the angular relations (2.35), and the characterization of the eigenvalues established by Theorem 2.5, we immediately deduce the validity of the following relation between rotation numbers and weighted eigenvalues.
The last part of this section is devoted to recalling the well-known result called Elastic lemma (cf.[25]).We consider the problem This lemma can be seen as a consequence of Lemma 2.7.

The main result
As a first step, we establish a relation between the rotation number of solutions to equation (2.2) and the rotation number of solutions to suitable half-eigenvalue problems, in the spirit of extending the comparison Lemma 2.4 to the framework of nonlinear equations.In Lemma 3.1, we will implicitly assume that the solutions of (2.2) we consider are defined in [0,T] and are such that (u(t),u (t)) = 0, for all t ∈ [0,T].In the sequel, we will confine our applications to nonlinear equations which are asymptotically linear at zero and at infinity and, therefore, all the solutions (but the trivial one) of (2.2) will satisfy such assumptions.
Proof.Assume (3.1) and let ε > 0 be fixed.By Lemma 2.4, there is δ > 0 such that For such a δ, using (3.1) and the Carathéodory assumptions, we can find a function where x + := max{x,0} and x − := max{−x,0}.Now, assume by contradiction that the claimed consequence of (3.1) is not true.This means that for each n there is a solution u n of (2.2) defined on [0,T] with ρ n (t) = |(u n (t),u n (t))| ≥ n for all t ∈ [0,T] and such that, for some t n ∈ [0,T], where Without loss of generality, it is also possible to assume that t n → τ ∈ [0,T] and v n → w = (sin(α),cos(α)) ∈ S 1 with α n → α.Then, passing to the upper limit on the above equation and using the continuity of Rot Next, we use the polar coordinates (ϑ n (t),ρ n (t)) to represent (u n (t),u n (t)).We obtain that holds for almost every t ∈ [0,T].By a result on differential inequalities [35] we know that where θ n is the solution of (3.16) By continuous dependence (with respect to L 1 -perturbations of the vector field), we know that, as n → ∞, θ n → θ, uniformly on [0,T], θ(•) being the solution of In particular, by uniform convergence, we have that θ n (t n ) → θ(τ) for n → ∞.Thus, we can conclude that Finally, recalling (3.13) and the definition of Rot ϕ−δ,ψ−δ (τ;w), we have This, clearly, contradicts (3.9).We have thus proved the first claim in Lemma 3.1.The proof of the other three parts of the statement follows precisely the same steps (even with some simplification for the case of rotations near zero) and therefore it is omitted.

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In order to apply Theorem 2.1, we will now state four propositions (analogous to the ones described in [16]) which provide the needed estimates on the rotation number of a solution to the initial value problem with z 0 ∈ R suitably chosen.
The first proposition provides a bound from below on the rotation number for all the solutions of (3.20) having z 0 "large enough."Proposition 3.2.Suppose that f satisfies the hypothesis (H ± ) and the Carathéodory conditions.Moreover, assume that there exists Then, there exists > 0 such that for every z 0 ∈ R with z 0 ν0, |z 0 | > , every (possible) solution u of (3.20) is such that Rot (2.2) (T;u) > j.
Proof.We take z 0 ∈ R, z 0 ν0, and a solution u of (3.20).Moreover, we define We note that from hypothesis (H ± ) the continuous functions a + ∞ and a − ∞ satisfy, respectively, the assumptions (3.1) in Lemma 3.1.Hence, by applying such a lemma, we deduce the existence of a constant and, in particular, Rot (2.2) (T;u) > j.
In order to determine a lower bound on |z 0 | which leads to |(u(t), u (t))| ≥ R for every t ∈ [0,T], we will apply Lemma 2.7, since its hypotheses are verified.Indeed, by combining the assumption (H ± ) with the Carathéodory conditions, for σ > 0, we can find a function Thus, setting R 1 = R in Lemma 2.7, we infer that there exists > R such that if This completes the proof.
The following proposition guarantees a bound from above on the rotation number of all the (possible) solutions to (3.20) for which z 0 is taken sufficiently large.Since the corresponding proof follows the same steps of Proposition 3.2, we will omit the details.Proposition 3.3.Suppose that f satisfies the hypothesis (H ± ) and the Carathéodory conditions.Moreover, assume that there exists j ∈ N and ν ∈ {<, >} such that Then, there exists * > 0 such that for every z 0 ∈ R with z 0 ν0 and |z 0 | > * , every (possible) solution u of (3.20) is such that Rot (2.2) (T;u) < j.
Finally, the last two propositions provide lower and upper estimates, respectively, for the rotation number of all the solutions of (3.20) characterized by a sufficiently small z 0 .Both their proofs are similar to the one of Proposition 3.2.The only difference consists in the fact that Lemma 2.8 (instead of Lemma 2.7) is used in order to achieve the result.For this reason, we will prove only the following.Proposition 3.4.Suppose that f satisfies the hypothesis (K ± ) and the Carathéodory conditions.Moreover, assume that there exists i ∈ N and ν ∈ {<, >} such that λ ν i (a + 0 ,a − 0 ) < 1.Then, there exists δ > 0 such that for every z 0 ∈ R with z 0 ν0 and |z 0 | < δ, every (possible) solution u of (3.20) is such that Rot (2.2) (T;u) > i.

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Now we prove our main result, Theorem 1.2.Theorem 1.2 can be easily proved combining the previous propositions with Theorem 2.1.
Proof.We take ν ∈ {<, >} and h ∈ N with m ≤ h ≤ n.First of all, assume that λ ν n (a + 0 ,a In particular, by Proposition 3.3, we can find a positive constant * such that for every y 0 ∈ R with y 0 ν0 and |y 0 | > * , every (possible) solution u of , we use Propositions 3.2 and 3.5 in order to obtain the estimates on the rotation number.Hence, following exactly the previous steps, from an application of Theorem 2.1 we easily achieve the thesis.Now, we provide two comparison remarks with the multiplicity results obtained in an asymmetric, asymptotically linear setting by Dinca and Sanchez in [20] and by Sadyrbaev in [48], respectively.Remark 3.7.The paper [20] deals with the existence and multiplicity of solutions for an asymmetric problem with constant weights of the form (1.1) satisfying the following asymptotically linear assumptions.
There exist ε 0 > 0, R > 0, and positive constants a (3.28) A Lipschitz condition on f is required as well.In order to achieve the multiplicity result, in [20] a shooting argument is developed based on the notion of "variation index" (see also [31]) and the implicit function theorem is applied.We point out that The gap condition (analogous to our assumptions (1.6) or (1.7)) between the behavior of the nonlinearity in zero and near infinity is written in terms of the number of zeros k ν (λ,µ) in (0,π) of the solution of where λ, µ are positive constants and ν ∈ {<, >}.Statements (a) and (c) in [20,Theorem 3] ensure the existence of k We recall that Theorem 1.2 guarantees the existence of n − m + 1 solutions whenever (3.32) holds.
On the other hand, the number of zeros k ν can be equivalently defined by the following expression: which, combined with (3.31), leads to  (1.1).Hence, we can conclude that [20, Theorem 3(a) and (c)] follow from our main result.
On the other hand, [20, Theorem 3(b) and (d)], dealing with multiplicity of solutions to problem (1.1) whenever a gap condition involving the constants a ± ∞ and b ± 0 holds, are more general than the ones we have analyzed above, since a condition symmetric to (3.31) is not assumed.

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Remark 3.8.In [48], Sadyrbaev has given a multiplicity result for a scalar equation of the form u + f (t,u) = g(t,u,u ), (3.36)where g is sublinear and f is asymptotically linear at infinity.More precisely, it is supposed that lim x→+∞ ( f (t,x)/x) = c + ∞ (t) and lim x→−∞ ( f (t,x)/x) = c − ∞ (t) uniformly in t.The argument in [48] is developed through the study of some variation equation (cf.[48, (1.5)]) associated to (3.36) and assumptions are given relatively to the number of zeros of the solutions of this auxiliary equation and of the two-weighted, half-linear equation For this approach, it is required that the functions f and g be of class C 1 .No explicit sign condition is assumed on the continuous functions c ± ∞ .It is possible to compare our main result with the one of Sadyrbaev provided that we assume g ≡ 0, (c + ∞ ) + • (c − ∞ ) + ≡ 0, and lim x→0 ( f (t,x)/x) = c 0 (t) uniformly in t, where c + 0 ≡ 0. In this particular situation, the variation equation studied in [48] becomes Arguing as in Remark 3.7, one can transform the hypotheses in [48] involving the number of zeros of the solutions to the Cauchy problems (3.37) and (3.38) into hypotheses involving the weighted eigenvalues.More precisely, the assumptions of [48,Corollary 4.1] can be rewritten as follows: for each ν ∈ {<, >} there exist h ν ,k ∈ N (which represent the number of zeros of the solutions to (3.37) and (3.38), resp.)such that Remark 3.9.Rynne in [47] has studied multiplicity of solutions for asymmetric problems involving superlinear nonlinearities.Multiplicity is achieved by using an alternative approach.Instead of an eigenvalue problem of the form (1.4), [47] considers the halfeigenvalues problem u + µu(t) + ϕ(t)u + (t) − ψ(t)u − (t) = 0, u(0) = 0 = u(T), u (0)ν0, (3.40)where ϕ,ψ ∈ L ∞ ([0,T],R) and ν ∈ {<, >}.The corresponding eigenvalue theory is developed, among others, in [7, Theorem 2], where ϕ, ψ are continuous functions on [0, T], and, more recently, in [45,Theorem 5.1] for ϕ,ψ ∈ L ∞ ([0,T],R).In particular, [45, Theorem 5.1] is the analog (in this alternative setting) of Theorem 1.1 and provides a monotone increasing sequence of half-eigenvalues for problem (3.40).The half-eigenfunction corresponding to µ ν j (ϕ,ψ) has exactly j − 1 zeros on (0, T).
According to (3.42), we infer that Theorem 1.2 holds true even if we replace assumptions (1.6) and (1.7) with the equivalent conditions and respectively.
Rynne in [47] proves the existence of solutions with specified nodal properties to some asymmetric problem by studying the sign of the eigenvalues µ ν j (ϕ,ψ) associated to the two-weighted, half-linear problem (3.40).The result in [47] is not directly comparable with ours, since in [47] a superlinear assumption near +∞ is considered.
We point out that sign conditions like µ ν j (ϕ,ψ) < 0 implicitly contain some hypothesis on the sign of the weights ϕ, ψ.

. 39 )
In this setting,[48, Corollary 4.1] provides the existence of at least |h > − k| + |k − h < | nontrivial solutions to problem (1.1).It is easy to show that such a corollary follows from Theorem 1.2.
27)is such that Rot (2.2) (T;u) < m ≤ h.Moreover, from Proposition 3.4 it follows that there exists δ > 0 such that for every y 0 ∈ R with y 0 ν0 and |y 0 | < δ, every (possible) solution u of (3.27) is such that Rot (2.2) (T;u) > n ≥ h.It is not restrictive to suppose that δ < * .Taking into account Lemmas 2.7 and 2.8 and fixing r ∈ (0,δ) and R ∈ ( * ,+∞), we observe that all the hypotheses of Theorem 2.1 are satisfied.Thus, Theorem 1.2 is proved in this first case.If λ ν n .31) We now compare the assumption (1.6) with the assumptions of [20, Theorem 3(a) and (c)].To this aim, we rewrite them in terms of the rotation numbers Rot b +