A Lower/Upper Solutions Result for Generalised Radial p-Laplacian Boundary Value Problems

We provide existence results to some planar nonlinear boundary value problems, in the presence of lower and upper solutions. Our results apply to a class of systems generalising radial elliptic equations driven by the p-Laplace operator, and to some problems involving the Laplace–Beltrami operator on the sphere. After extending the definition of lower and upper solutions to the planar system, we prove our results by a shooting method involving a careful analysis of the solutions in the phase plane.

Denoting by q > 1 the conjugate exponent of p, satisfying 1 p + 1 q = 1, Eq. (5) is equivalent to the system u = |y| q−2 y, (a(r)y) = g(r, u), which is a special form of (3). Note that the function a(r) vanishes at r = 0, creating a singularity for our problem, and this fact generates a main difficulty in our study. The problem of the presence of the singularity, concerning existence, uniqueness and continuous dependence on initial data, for the Cauchy problems associated with the second-order differential equation (5), was already faced in [9] (in the Appendix), see also [2][3][4]. In the appendix of this paper, we present the corresponding discussion for system (3). Moreover, we can also allow the possibility of having a second singularity at r = 1. We consider the mixed boundary conditions problem with θ ∈] − π 2 , π 2 ]. Having in mind the radial problem, it is natural to assume the Neumann condition y(0) = 0 at the left endpoint of our interval. Concerning the right endpoint, notice that, in case θ = 0, we have a Neumann-type boundary condition, while in case θ = π 2 , we are dealing with a Dirichlet-type condition.
Concerning the proof of the above theorem, we present an alternative approach to the standard application of degree theory, based on a shooting method, after a careful phase plane analysis of the solutions.
Remark 3. We underline that if we replace assumptions (A2) and (A3) with the hypothesis a(t) > 0, for all t ∈ [0, 1], then the conclusion of Theorem 2 can be proved with simpler computations.
If we are only interested in the Neumann problem (6), with θ = 0, we can weaken the assumptions on the function a by allowing a(1) = 0. We shall assume that a : [0, 1] → R satisfies (A1) and (A2) a(t) > 0, for all t ∈]0, 1[; (A3) a(0) = 0, a(1) = 0, and there exist ρ 0 ≤ ρ 1 in ]0, 1[ such that An example of a function a satisfying these assumptions is It arises, e.g. when dealing with the Laplace-Beltrami operator on the sphere S N −1 ⊆ R N , if we are looking for solutions depending only on the latitude ϕ = πt (asking for symmetry with respect to all the other angle variables). In this case, the problem we need to solve is the following which is a special form of (6), with θ = 0, the function a defined by (8) (6), with θ = 0, satisfying α ≤ β. Then, problem (6) with θ = 0 has a solution (x, y) such that The proof of this second theorem will be provided through a double shooting method.
Remark 5. If the functions f and g are only continuous, similar results can still be proved, adapting the approximation technique used in [8]. However, in this case, we need to assume the existence of strict lower and upper solutions α and β satisfying α(t) < β(t) for all t ∈]0, 1[.

Lower and Upper Solutions
We now provide the definitions of lower and upper solutions for problem (6), thus extending the ones given in [5][6][7].
For an intuitive meaning of the previous definitions, see Fig. 1.

Phase Plane Estimates
In this section, we provide some results which will be later used to prove the main theorems.
We set whereĝ was defined in (7), and take a constant K satisfying where C is the constant introduced in Proposition 8. We first modify the functions f (t, y) and g(t, x) as follows. Defineg : Let us consider the correspondingly modified problem We shall prove the existence of a solution of ( P ) and then verify that such a solution is indeed a solution of problem (6). To this aim, we define some regions in the space [0, 1] × R × R and prove some invariance properties of them with respect to the solutions of the planar system We set (15) We have: Proof. We only prove (i), as the other assertions follow in a similar way. Assume ) and hence, from (12), we get Suppose next that K < y(t 0 ) < K + 1. Then, using (12) again and the fact that β ∞ < K, we obtain

Lemma 10.
Let (x, y) be a solution of ( S) defined at a point t 0 ∈ [0, 1], and suppose that both x(t 0 ) > β(t 0 ) and y(t 0 ) = y β (t 0 ) hold. We have: Proof. We first consider case (i). We recall that, from (13), we have hence we obtain, using (16), Since We consider now case (ii). Let us set Case (iii) can be proved in a similar way.
The following symmetric result can be proved similarly for the lower solution α. Lemma 11. Let (x, y) be a solution of ( S) defined at a point t 0 ∈ [0, 1], and suppose that both x(t 0 ) < α(t 0 ) and y(t 0 ) = y α (t 0 ). We have: The previous results allow us to prove some invariance properties of the regions A NE , A SE , A NW , A SW introduced in (15). To this aim, in the following statement, we consider a solution (x, y) of ( S) defined on a maximal interval of existence I. Notice that, due to the linear growth of the functionsf andg, if (A1), (A2) and (A3) hold, we can have the following two alternatives: On the other hand, if (A1), (A2) and (A3) hold, we can have the following four alternatives: Proof. Let us prove the first assertion, the others follow similarly. Let (t 0 , x(t 0 ), y(t 0 )) ∈ A NE for some t 0 ∈ [0, 1]. By contradiction, assume that there exists t 1 ∈]t 0 , 1] ∩ I such that (t, x(t), y(t)) ∈ A NE , for every t ∈ [t 0 , t 1 [, and (t 1 , x(t 1 ), y(t 1 )) / ∈ A NE . In particular we have either x(t 1 ) = β(t 1 ), or y(t 1 ) = y β (t 1 ).
Finally, using Lemma 10 (i), we get a contradiction also in the case y(t 1 ) = y β (t 1 ).
Proof. We assume, by contradiction, that there exists Suppose now that y(t 0 ) = y β (t 0 ). From Lemma 10, we see that y(t) > y β (t) in a right neighbourhood of t 0 , and we conclude as before.
Hence, we conclude that In a similar way, we can prove that x(t) ≥ α(t) for every t ∈ [t 1 , t 2 ], thus concluding the proof.

Lemma 14.
Let (x, y) be a solution of ( S), defined on a nontrivial interval [0, t 2 ] ⊆ [0, 1], satisfying the following properties: Then Proof. By Lemma 13, we have that Recalling the definition (14) of M and Proposition 8, we deduce that for every t ∈]0, t 2 ]. This inequality is trivially satisfied also in case t = 0, hence the lemma is completely proved.
Arguing similarly, we can prove the following result. Then, So far, we have proved the following a priori bounds. Summing up, to prove Theorem 2, we need to find a solution of ( P ) satisfying the assumptions of Proposition 16. Similarly, to prove Theorem 4, we need to find a solution of ( P ), with θ = 0, satisfying the assumptions of Proposition 17.

Proof of Theorem 2
To prove our result, we shall apply a shooting argument, with the aim of finding σ ∈ R such that the solution (x, y) of the Cauchy problem x =f (t, y), a(t)y =g(t, x), also satisfies x(1) sin θ + y(1) cos θ = 0. We start by defining the flow associated with system ( S). Let X be the set of initial data and consider the solution (x(·), y(·)) = Φ(·; t 0 , σ, τ) = Φ 1 (·; t 0 , σ, τ), Φ 2 (·; t 0 , σ, τ) of ( S) satisfying x(t 0 ) = σ and y(t 0 ) = τ . The proof concerning the existence of such a solution, which is not completely standard due to the presence of the singularity at 0, is given in the Appendix. This solution will be proved to be defined on ]0, 1], thanks to the linear growth of the functionsf andg, but not necessarily at t = 0. However, if t 0 = 0, the solution is defined on the whole interval [0, 1]. We denote by D ⊆ R 4 the domain of the flow Φ = Φ(t; t 0 , σ, τ).
The continuity of the flow Φ follows from the continuous dependence of the solutions of ( S) on the initial data. Again, the proof of this fact is given in the Appendix. Let us fix U 0 > 0 such that The following proposition localises the curve C .

Proof of Theorem 4
To prove our second result, we shall apply a double shooting argument, with the aim of finding σ ∈ R such that the solution (x, y) of the Cauchy problem (17) also satisfies y(1) = 0.
To define the flow associated with system ( S) under assumptions (A1), (A2) and (A3) , the set of possible initial data is now Let us fix U 0 > 0 such that For any σ 0 , σ 1 ∈ [−U 0 , U 0 ], we consider the the initial value problem x =f (t, y), a(t)y =g(t, x), and the final value problem x =f (t, y), a(t)y =g(t, x), We use a shooting argument to find a solution (x σ0 , y σ0 ) of (21) (defined on [0, 1[ ), and a solution (x σ1 , y σ1 ) of (22) (defined on ]0, 1]), satisfying will be the solution of ( P ) we are looking for. Let us define two continuous curves C 0 , The following statement describes some localisation properties of the curves C 0 and C 1 .
Proposition 19. Let C 0 and C 1 be the curves defined by (24). Then, the following properties hold: . The proof can be adapted from that of Propositions 18. We prove now that the two curves have a common value.
Proposition 20. Let C 0 and C 1 be the curves defined by (24). Then, there are Proof. We shall consider the restriction of C 0 and C 1 on some intervals [σ 0 , σ 0 r ] and [σ 1 , σ 1 r ], respectively, so that α( To this aim, we set Moreover, . Since the curves are continuous, they must cross each other at some point The parameters σ 0 , σ 1 obtained in Proposition 20 permit us to define the solution (x, y) of problem ( P ) as in (23). In particular, we have

Examples and Final Remarks
In this section, we provide some possible applications of our theorems.
In (4), we have considered for simplicity a differential equation ruled by a weighted p-Laplacian. In a similar way, we can consider a double-weighted φ-Laplace equation, in the unitary ball, of the type which can be written as a planar system of the form where ω(t) = 1/m(t), which is a special case of (3). Theorem 2 may be applied to study the boundary value problem Notice that all the regularity assumptions required in Theorem 2 immediately hold. So, if we are able to provide a well-ordered couple of lower/upper solutions for problem (27), then we can successfully apply it. The following statement describes a possible example of application in the case of constant lower and upper solutions.
, and assume the existence of some constants Proof. It is easy to verify that the constant functions α and β fulfill the conditions in Definitions 6 and 7 with the choice y α = y β ≡ 0. Then, Theorem 2 applies, thus completing the proof.
In particular, the previous corollary permits us to find an existence result for Eq. (26) with Dirichlet or Neumann boundary conditions on the unitary ball B.

Corollary 22. Assume the existence of some constants
Corollary 23. Assume the existence of some constants α ≤ 0 ≤ β such that h(r, α) ≤ 0 ≤ h(r, β) for every r ∈ [0, 1]. Then, problem Example 24. Let us consider two locally Lipschitz continuous functions f, g : R → R satisfying yf (y) > 0, when y = 0, and lim inf with θ ∈ [0, π 2 ], has a solution by applying Corollary 21. As particular cases, choosing with N ≥ 1, the problems in the unitary ball of R N have at least one radial solution. We, thus, recover some well-known classical results.
Analogous considerations allow to generalize problem (9), providing further applications of Theorem 4.
Example 25. Let g(x) and e(t) be as in the previous example. We can apply Theorem 4 so to find a solution of the problem Remark 26. In this paper, we have treated the case when the pair of lower and upper solutions is well ordered. The non-well-ordered case is left as an open problem.
Author contributions All authors contributed at the same level to the results in the paper.
Funding Open access funding provided by Università degli Studi di Trieste within the CRUI-CARE Agreement. International Science Program (ISP), Uppsala University.

Conflict of interest
There are no competing interests.

Ethical approval Not applicable.
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Appendix
In this appendix, we prove the continuity of the flow Φ : D ⊆ [0, 1]×X → R 2 , introduced in Sect. 3.1, associated with system ( S). Then, we will provide the corresponding result for the situation treated in Sect. 3.2.
Because of the presence of the singularity at t = 0, we provide a proof of existence, uniqueness and continuous dependence on initial data properties of the solutions of the Cauchy problems (17). Similar properties have been studied for second-order differential equations presenting a singularity. See e.g. [9,Appendix] or [2][3][4]12].
We recall that, under the assumptions of Theorem 2, the function a : [0, 1] → R is of class C 1 , positive in the interval ]0, 1], increasing in [0, ρ 0 ] ⊆ [0, 1], and satisfies a(0) = 0. To simplify the notation, we consider the Cauchy problem u = F (t, y), (a(t)y) = a(t)G(t, x), u(0) = u 0 , y(0) = 0, wheref (t, y) = F (t, y) andg(t, x) = a(t)G(t, x). Since, by construction, both F and G are locally Lipschitz continuous with respect to the second variable and they have an at most linear growth, we can assume that there exists A > 0 such that for every t ∈ [0, 1] and x, y ∈ R. At first we notice that, for t ∈]0, 1], the differential system in (28) can be rewritten as thus obtaining a planar system for which we can easily verify local existence and uniqueness of the solutions for the Cauchy problems. Moreover, since (29) holds, such solutions are globally defined on ]0, 1].
Hence, in what follows, we focus our attention on Cauchy problems of the form x = F (t, y), (a(t)y) = a(t)G(t, x), where x 0 ∈ R. In particular, it will be sufficient to prove existence, uniqueness and continuous dependence on initial data for such Cauchy problems only in a right neighborhood of 0. Since the functions F and G satisfy (29), we can then easily recover these properties in the whole interval [0, 1]. We start by stating the local existence and uniqueness theorem. Notice that T (x, y) ∈ B, for every (x, y) ∈ B. Indeed, for every t ∈]0, τ], we have both Let us prove that the function T is a contraction. We set κ = τ L < 1. Then, given any (x 1 , y 1 ), (x 2 , y 2 ) ∈ B and for every t ∈ [0, τ], we have both T 1 (x 1 , y 1 )[t] − T 1 (x 2 , y 2 )[t] = t 0 F (s, y 1 (s)) − F (s, y 2 (s))ds ≤ tL y 1 − y 2 ∞ < κ (x 1 , y 1 ) − (x 2 , y 2 ) X ,