A TOPOLOGICAL DEGREE APPROACH TO SUBLINEAR SYSTEMS OF SECOND ORDER DIFFERENTIAL EQUATIONS

. In this paper we study the existence of radial solutions to sublinear systems of elliptic equations. We ﬁrst give a multiplicity result on solutions with prescribed nodal properties; then, we show the existence of positive solutions. The proofs are based on topological degree arguments.


1.
Introduction. In this paper we are concerned with the study of radial solutions to Dirichlet problems for systems of elliptic equations; more precisely, we consider where Ω = {ξ ∈ R N : 0 < a < |ξ| < b} is an annulus in R N , N ≥ 1, and γ > 0; for 0 > 0, we set I 0 := [− 0 , 0 ] 2 and we assume that G : I 0 −→ R and q : [a, b] −→ R are C 1 -functions. For simplicity, all the results are stated and proved in the case of two equations, but they are valid, by means of suitable changes, in the general case of K equations (K ≥ 1). It is well-known that the search of radial solutions to (1.1) leads to the study of a system of ordinary differential equations of the form    x + m(t, x) = 0 for a suitable function m ∈ C 1 ([0, π] × R 2 , R 2 ), with x = (x 1 , x 2 ) (see [27]). We remark that, when one attacks systems, many classical techniques used in the search of solutions to a single equation (as, for instance, the shooting method and phaseplane analysis) cannot be applied. For results on second order differential systems we refer to the books [21,22]: in many cases, the approach is of variational type, due to the natural fitness of this method for the study of problems involving more than one equation; however, it is quite difficult to find, by means of these techniques and in absence of restrictive conditions, multiplicity results. As far as a topological degree method is concerned, we recall the works [9,12,18], where the so-called 862 ANNA CAPIETTO AND WALTER DAMBROSIO "weakly-coupled" systems are considered, and the recent paper by R. Manásevich and J. Mawhin [19], where it is given an important outburst towards the study of systems with nonlinear differential operators. Existence of radial solutions in annular domains in the case of one equation has been proved by many authors (see e.g. [11,13,27]), under various conditions both at infinity and near the origin. The aim of this paper is to treat nonlinearities G having only a subquadratic behaviour near the origin; on these lines, in [4] we gave a multiplicity theorem in the case of one equation. Though some techniques contained in that paper are useful, when one tries to prove an analoguous result for systems some new ideas are needed. We assume: (H1) the following relations hold: Moreover, setting It is well-known that assumption (H1) represents a subquadratic condition near the origin for G. As far as hypothesis (H2) is concerned, it guarantees the uniqueness of solutions to some Cauchy problems related to (1.1) (cf. also the papers [11], [25,] and [17]). Finally, assumption (H3) is a technical condition which arises when dealing with systems (it is trivially fulfilled in the case of one equation). It is satisfied e.g. by "subquadratic" polynomials and it enables us to develop some crucial energy estimates (cf. Example 2.4 and the proof of Lemma 2.7). We point out that (H3) could be replaced by a formally more complicated hypothesis which enables to study the case when a more general function of the form F (|ξ|, u, v) is considered in (1.1).
For the proof we use a topological degree approach; more precisely, we combine the application of a continuation theorem (see e.g. [7]) with some estimates on the number of zeros of each component of a radial solution to (1.1). Moreover, we use a time-map technique for autonomous equations. We are also able to make an homotopy which carries our original problem into an uncoupled problem, for which we can use phase-plane analysis (on the lines of [4,8]). To the best of our knowledge, there are very few results in the literature concerning subquadratic nonlinearities; beside the papers [4,6], we recall the early work by G. J. Butler [3], where a periodic problem is considered. Recent contributions on this subject have been given, among others, by A. Ambrosetti, J. Garcia-Azorero and I. Peral [1,2], L. H. Erbe, S. Hu and H. Wang [15], V. Moroz [23] and H. Wang [27]. We stress the fact that, apart from [2,Sect. 2], in all these papers, which deal with a single equation, a combination of assumptions in zero and at infinity is required, while (H1)-(H3) are merely conditions of local nature.
We remark that Theorem A guarantees only the existence of radial solutions whose components have a sufficiently large number of zeros. Indeed, the subquadratic behaviour of the nonlinearity G is not sufficient to ensure, for all γ > 0, the existence of radial solutions with an arbitrary number of zeros (cf. Remark 3.6). However, provided that γ is sufficiently small, it is possible to show that such solutions do exist (see Proposition 3.5).
The proof of the above theorem follows from the estimates on the number of zeros proved in Section 2 and a detailed phase-plane analysis on the solutions of autonomous equations (see Lemma 3.4). The study of positive solutions, in the case of a single equation, has been faced by means of different methods and, in general, existence results have been given depending on the values of the parameter γ > 0. We recall the papers by H . Dang, R. Manásevich and K. Schmitt [13] and H. Wang [27], where it has been proved, under conditions on the behaviour of the nonlinearity both in zero and at infinity, that positive solutions exist for small values of γ. We also quote the paper by A. Ambrosetti, J. Garcia-Azorero and I. Peral [2], which deals (using bifurcation techniques) with sublinear functions which, in a neighbourhood of zero, are asymptotic to a power. For other related results, see the work of R. Manásevich, F. I. Njoku and F. Zanolin [20]. Among the few results on systems, we also refer to [10], where the existence of positive solutions is proved by using the method of sub-super solutions. As a final remark, we observe that when studying Dirichlet problems associated to one equation the sublinear and the superlinear conditions, near zero and at infinity, respectively, represent dual situations, in the sense that in the former case we are able to find the existence of infinitely many solutions with small norm, while in the latter we have infinitely many solutions with large norm. The generalization of this fact to systems is shown by a comparison between Theorem 2.2 and the results in the forthcoming paper [5].
In what follows, we will denote by X = C 1 0 ([0, π], R 2 ) the space of the functions x ∈ C 1 ([0, π], R 2 ) such that x(0) = 0 = x(π) and we will indicate with ||x|| 1 the corresponding C 1 -norm. Moreover, ∂ i will denote the partial derivative with respect to the variable x i , for i = 1, 2, and ∂ u , ∂ v the partial derivatives with respect to the variables u and v. Finally, by deg we mean the Leray-Schauder degree.

2.
A multiplicity result. Let 0 < a < b and let Ω = {ξ ∈ R N : a < |ξ| < b} be an annulus in R N . Let us consider a system of the form We suppose: (H1) the following relations hold: (H2) there exists a positive constant q 0 > 0 such that q(r) ≥ q 0 for every r ∈ [a, b].
Remark 2.1. We observe that assumption (H1) implies some information on the sign of the function G; more precisely, since we have and from (H1) we also deduce that G is always positive inĨ 0 . In what follows, for simplicityĨ 0 will be identified with I 0 .
from Remark 2.1 we know that G u and G v are positive functions. We then assume that: (H3) there exists a positive constant Υ such that and It is well-known that assumption (H1) represents a sublinear condition near zero for ∇G. As far as hypothesis (H2) is concerned, it guarantees the uniqueness of solutions to some Cauchy problems related to (2.4) (cf. Proposition 2.4 in [4]). It is immediate to see that (H1) and (H2) are precisely the conditions we assumed in [4] when dealing with one equation. Assumption (H3) is a technical condition which arises when dealing with systems (it is trivially fulfilled in the case of one equation). It is satisfied by "sublinear" polynomials and it enables us to develop some crucial energy estimates (cf. Example 2.4 and the proof of Lemma 2.7).
Under conditions (H1)-(H2)-(H3) we are able to prove the following result, which generalizes to the case of systems the results contained in [4] (recall the definition of τ given in (1.3) in the Introduction): Remark 2.3. It is easy to prove, on the lines of [4,6], that the C 1 -norm of the solutions u k , v k , which depends on (n 1 , n 2 ), tends to zero as n 1 , n 2 tend to infinity.
As it is well-known, radial solutions to (2.4) correspond, for r = |ξ|, to functions Now, by means of a change of variable (see [27]), it is easy to see that there exists (2.6) Therefore, we shall concentrate on the nodal properties of the solutions to (2.6).

ANNA CAPIETTO AND WALTER DAMBROSIO
Before giving the proof of Theorem 2.2, we show a class of systems of the form (2.6) satisfying (H1)-(H2)-(H3). (2.7) According to the change of variables in [27], this means that we are treating the case when, in the system of PDEs (2.4), we have q(s) = s −2(K−1) . Then, it is easy to see that Theorem 2.2 applies to (2.7) when we assume: (1) the functions G 1 and G 2 are sublinear near zero, i.e.
(2) the following relations hold: Note that in the particular case L 1 (s) ≡ l 1 > 0, L 2 (s) ≡ l 2 > 0 we are in presence of a so-called "weakly-coupled" system.
The proof of Theorem 2.2 follows from the application of a continuation theorem given in [6] for an abstract equation of the form and such that In particular there is at least one solutionũ ∈ (B \Ā) 1 of the operator equation In view of the application of Theorem 2.5, we make an homotopy by introducing the function G 0 : R 2 −→ R given by In what follows, we will use the notation G λ (x) = G(x, λ) and p λ (t) = p(t, λ) for every t ∈ [0, π], x ∈ I 0 and λ ∈ [0, 1]. We then consider the Dirichlet problem which can be written (see [21]) in the form (2.8) with respect to the Banach space X = C 1 0 ([0, π], R 2 ). Moreover, we denote by Σ ⊂ X × [0, 1] the set of the solutions to (2.12). Now, we observe that assumptions (H1)-(H2)-(H3) are preserved along the homotopy; more precisely, we have lim uniformly in x 1 ∈ [− 0 , 0 ] and λ ∈ [0, 1]. Moreover, it is easy to check that there exists a positive constant p 0 > 0 such that Finally, we have and where Υ is given in assumption (H3) and G 1 λ (x 1 , We are now ready to give some preliminary lemmas. Lemma 2.6. Assume (H1)-(H2)-(H3); then, for every ∈ (0, 0 ) there exists µ > 0 such that each solution of and satisfies The proof of this lemma is based on some estimates on the energy function E λ (t) = x 2 (t)); the details, being similar to the case of a single equation treated in [4], are omitted for brevity.
We now give a lemma which describes some qualitative properties of the solutions to Cauchy problems associated to the system in (2.12) and which is crucial for the validity of Proposition 2.8. For its proof, it has been necessary to introduce a new "energy-like" function and to use (H3). We denote by µ 0 the number µ 0 given in Lemma 2.6. Lemma 2.7. Assume (H1)-(H2)-(H3); then, the following statements hold:

Let x be a solution of the initial value problem
2. For every µ ∈ R 2 with µ i = 0 for i = 1, 2 and |µ| ≤ µ 0 , there exists η µ > 0 such that every solution x of is defined in [0, π] and satisfies Proof. The results are easy consequences of some estimates on suitable energy functions associated to the system x + p λ (t)∇G λ (x) = 0. Indeed, as far as Statement 1 is concerned, let x be a solution of (2.20) and let us suppose, for simplicity, that (x 0 ) 1 = (x 0 ) 1 = 0 (the other case is completely analogous). From Lemma 2.6, we already know that x is defined on [0, π]; moreover, from the choice of µ 0 , we also have |x 2 (t)| ≤ 0 for every t ∈ [0, π]. Let us consider with G 1 λ as in (2.17); by differentiating E 1,λ , we obtain . Using (2.15) and (2.17) we deduce that there exists a constant Z > 0 such that (2.23) and so Since E 1,λ (t 0 ) = 0, we have that E 1,λ (t) ≡ 0 in [0, π] and, by simple arguments, Finally, as for Statement 2, from (2.23) we deduce that for any solution x of (2.21) we have uniformly in t, λ and x 2 , from (2.24) we see that there exists η 1 > 0 such that In a similar way, using analogous estimates on the function E 2,λ defined by we obtain the existence of η 2 > 0 such that x 2 (t) 2 + x 2 (t) 2 ≥ η 2 for every t ∈ [0, π]; taking η µ = min{η 1 , η 2 }, we get the result. Now, for every µ ∈ R 2 , with µ i = 0 for i = 1, 2 and |µ| ≤ µ 0 , let us set From Lemma 2.7, it is easy to see that, for any (x, λ) ∈ Σ µ , every component of x has only simple zeros in [0, π). Hence, for i = 1, 2, the number n i (x, λ) of zeros of x i in [0, π) is well defined. Recalling [16, Lemma 3.1], we deduce that the map λ)) is continuous. Moreover, arguing as in [4,6], it is possible to prove the following: Proposition 2.8. There exists n * ∈ N 2 such that for every solution (x, λ) of (2.12) and for every i = 1, 2 we have: Proposition 2.9. For every m = (m 1 , m 2 ) ∈ N 2 there exists µ m ∈ (0, µ 0 ) such that for every solution (x, λ) of (2.12) we have: We are now in position to prove Theorem 2.2: Proof of Theorem 2.2. We apply four times Theorem 2.5. Let n = (n 1 , n 2 ) ∈ N 2 be such that n i > n * i for i = 1, 2 and, consistently with the notation of Theorem 2.5, let us choose If (x, λ) ∈ Σ ∩ (∂C), then there exists i ∈ {1, 2} such that x i (0) = µ 0 or x i (0) = µ n ; therefore, by Proposition 2.8 and Proposition 2.9, we have n i (x, λ) < n * i or n i (x, λ) > n i . In any case, this is absurd and so (2.9) holds. As far as (2.10) is concerned, we observe that if (x, λ) ∈ Σ n then, by Lemma 2.7, ||x|| 1 ≤ 0 . In this way, also (2.10) is fulfilled. Finally, we refer to the paper [4] (see the Proof of Theorem 3.2 therein) to show that there exists an open subset U n 0 such that (2.11) is satisfied. Henceforth, all the assumptions of Theorem 2.5 are fulfilled and we deduce the existence of a solution x of (2.6) with x i (0) > 0 for i = 1, 2 and such that x i has exactly n i zeros in [0, π). By the previous discussion, this leads to the existence of a radial nodal solution of (2.4). In order to prove the existence of the other solutions to (2.6), it is sufficient to modify the choice of the set A defined above: let us fix (s 1 , s 2 ) ∈ τ and suppose for instance that s 1 = −1 and s 2 = +1 (the other cases are similar). It is now easy to see that, defining then we obtain the existence of a solution x with sgn (x ) i (0) = s i for i = 1, 2.
3. Positive solutions. In this section, we shall be concerned with the following system (3.25) With the same notation of Section 2, we take Ω = {ξ ∈ R N : 0 < a < |ξ| < b}, Moreover, we assume that γ is a positive real number. Under the same assumptions on q and G we made in Section 2, it is immediate to see that (3.25) has infinitely many radial solutions for every γ > 0. However, Theorem 2.2 guarantees only the existence of solutions with a "sufficiently large" number of zeros (see also Proposition 3.5). It is our goal in this section to show the existence of positive radial solutions to (3.25), in the case when the parameter γ is sufficiently small. The validity of this fact is supported, in the case of an autonomous equation, by some elementary phase-plane analysis (see also Remark 3.6). As above, for simplicity we will only consider the case of two equations. We will prove the following result: The proof of Theorem 3.1 will be performed by means of the abstract continuation theorem 2.5. For this reason, and since we shall be dealing with sublinear nonlinearities (as in Section 2), in what follows we argument on the lines of Section 2: we skip all the proofs that do not significantly differ from those we already performed, and we give the details of the ones that have required new ideas.
As usual, positive radial solutions to (3.25) correspond to positive solutions of where p is a positive C 1 -function on [0, π]. Now, we still denote by G the extension . Then, we are led to study the homotopized problem 2 )), for every t ∈ [0, π], (x 1 , x 2 ) ∈ I 0 and λ ∈ [0, 1]. From (H1)-(H2)-(H3), arguing as in Section 2, it is easy to check that there is p 0 > 0 such that (3.28) Moreover, with the same notation as in (2.16)-(2.17), we have and where Υ is given in assumption (H3). Now, according to Section 2, we need to check the validity of Lemma 2.6 and 2.7 when we consider, instead of G, the function γG; indeed, this can be done since (3.28), (3.29) and (3.30) hold. It is important to observe that, due to the fact that the constant p 0 in (3.28) does not depend on γ > 0, the constant µ 0 = µ 0 given in Lemma 2.6 is also independent from γ > 0. As a consequence, like in Section 2, the functional "number of zeros" is continuous. The final (and crucial) estimates of Section 2 (cf. Proposition 2.9 and Proposition 2.8) contain an lower (respectively, upper) estimate on the number of zeros of the components of the solutions to (3.27); since we are now dealing with positive solutions (i.e. solutions with only one zero in [0, π)), those statements need to be reformulated as follows.
For the proof it is sufficient to take, in Proposition 2.9, (m 1 , m 2 ) = (1, 1). As for the upper estimate on the number of zeros, a new argument has to be developed; indeed, this is precisely the point where a restriction on the constant γ is needed. We have the following: There exists γ 1 > 0 such that for every γ ∈ (0, γ 1 ) problem (3.27) has no positive solutions (x, λ) with |x 1 (0)| = µ 0 or |x 2 (0)| = µ 0 .
Proof. We show that there exists γ 1 > 0 such that if problem (3.27) has a positive solution (x, λ) with |x 1 (0)| = µ 0 or |x 2 (0)| = µ 0 , then γ ≥ γ 1 . Let us suppose that (x, λ) is a solution of (3.27) with |x 1 (0)| = µ 0 , the other case being similar. By integrating the first equation in (3.27) we obtain Since x 1 (0) = 0 = x 1 (π), there exists t 0 ∈ (0, π] such that x 1 (t 0 ) = 0; hence Now, we denote by S the maximum of the function ∂ 1 G on I 0 , by S 0 the maximum of the function ∂ 1 G 0 on I 0 and by P the maximum of the function p on [0, π]. Then we have This implies that We have now developed all the estimates relative to the homotopized system (3.27). In order to apply Theorem 2.5 (cf. assumption (2.11)), we need to study problem (3.27) with λ = 0; we are then led to discuss the existence of positive solutions to the Dirichlet problem u + γ 3 √ u = 0 where u : [0, π] −→ R and γ is sufficiently small. To this aim, we develope some time-map estimates and a detailed phase-plane analysis for solutions to (3.31). Indeed, for µ γ given in Proposition 3.2, we can prove the following result: Lemma 3.4. There exists γ 2 > 0 such that for every γ ∈ (0, γ 2 ) problem (3.31) has a positive solution u with u (0) ∈ (µ γ , µ 0 ).
As already observed, the proof of Lemma 3.4 consists on the study of the range of the so-called time-map T γ : (0, +∞) −→ R + ; more precisely, we use the fact that (3.31) has a positive solution u with u (0) = µ > 0 if and only if T γ (µ) = π/2. A more complete discussion on this approach can be found e.g. in [4,8].
We are now in position to conclude the proof of Theorem 3.1.
Finally, due to the fact that we obtained positive solutions as solutions having one zero in [0, π), it is not difficult to prove also the following: Proposition 3.5. For every k = (k 1 , k 2 ) ∈ N 2 , there exists γ k > 0 such that for every γ ∈ (0, γ k ) problem (3.25) has a radial solution (u, v) with u and v having k 1 , k 2 zeros in [0, π), respectively. Remark 3.6. It is possible to check, with time-map arguments, that a subquadratic behaviour of the nonlinearity G (without any assumption at infinity) is not sufficient to guarantee the existence of positive solutions to (3.25) for all γ.
Remark 3.7. 1. The results contained in Theorems 2.2 and 3.1 can be obtained also in the case when we replace the Laplace operator by the p-laplacian; indeed, it is sufficient to observe that, also for the p-laplacian, it is possible to compute the number of zeros of the radial solutions by means of a suitable integral formula (see [12]). 2. We stress the fact that we could generalize Theorem 2.2 to a system containing general boundary conditions of Sturm-Liouville type.