Uniqueness of positive solutions for boundary value problems associated with indefinite $\phi$-Laplacian type equations

The paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the $\phi$-Laplacian equation \begin{equation*} \bigl{(} \phi(u') \bigr{)}' + a(t) g(u) = 0, \end{equation*} where $\phi$ is a homeomorphism with $\phi(0)=0$, $a(t)$ is a stepwise indefinite weight and $g(u)$ is a continuous function. When dealing with the $p$-Laplacian differential operator $\phi(s)=|s|^{p-2}s$ with $p>1$, and the nonlinear term $g(u)=u^{\gamma}$ with $\gamma\in\mathbb{R}$, we prove the existence of a unique positive solution when $\gamma\in\mathopen{]}-\infty,(1-2p)/(p-1)\mathclose{]} \cup \mathopen{]}p-1,+\infty\mathclose{[}$.


Introduction
This paper deals with the φ-Laplacian differential equation where φ is an increasing homeomorphism defined in an open interval including the origin, with φ(0) = 0, a(t) is a sign-changing L 1 -weight function and g(u) is a continuous function with g(u) > 0 for u > 0.
It is worth noticing that the φ-Laplacian operator appearing in equation (1.1) includes several classical differential operators, such as the linear operator φ(s) = s, or the p-Laplacian operator φ(s) = ϕ p (s) = |s| p−2 s with p > 1, or even the one-dimensional mean-curvature operator in Lorentz-Minkowski spaces φ(s) = s/ 1 − |s| 2 . Such differential operators have been widely investigated in the literature for their relevance in many mechanical and physical models (cf. [27]).
Starting from [26], it is common to refer to (1.1) as a nonlinear indefinite equation, due to the presence of a sign-changing weight function. The study of indefinite problems, both in the ODE and in the PDE setting, has shown an exceptional interest, from the pioneering works [1,2,5] till to the recent developments dealing with positive solutions of boundary value problems associated with (1.1) (we refer to [19] for a quite comprehensive list of references).
In this paper, we focus our attention on the Neumann and the periodic boundary value problems associated with (1.1) and we analyse both the power-type nonlinearity, that is g(u) = u q , q > 0, (1.2) and the singular nonlinearity In this framework, lot of work has been done concerning existence and multiplicity of positive solutions, see, for instance, [3,5,7,9,20,21,29,31] for the power-type nonlinearity (1.2), and [4,11,12,22,23,25,38,38,40] for the singularity (1.3). Looking in the above-mentioned contributions and, in general, in the literature, we notice that the natural question of uniqueness of solutions has received very few attention. More precisely, in the framework of indefinite problems, the uniqueness of positive solution is proved in [3,13] when dealing with a concave g(u), and in [12] when dealing with a singularity of the form (1.3) with κ = 3. An intermediate situation is studied in [24]; other types of special nonlinearities (convex-concave) are analysed in [33]. We highlight that all these results concerning the uniqueness of the solutions are obtained for the linear differential operator.
As is shown by [16,17,18], for convex or superlinear nonlinearities, the problem of uniqueness of positive solutions is of greater complexity even when the weight function a(t) is of constant positive sign and apparently it has not yet been completely solved for sign-changing weights. Indeed, for weight functions with multiple changes of the sign, uniqueness is not possible, in view of the results about the multiplicity of positive solutions obtained in [19,21].
The aim of this paper is twofold: on one side, we plan to produce a uniqueness result including both linear and nonlinear differential operators; on the other hand, we investigate a situation allowing the superlinear nonlinearities as a special case. Due to the fact that a weight function a(t) with more than one change of sign allows multiplicity of positive solutions, it is natural to consider a coefficient a(t) with a single change of sign. Similar coefficients have been considered in [11,12,23,29,30,35]. In particular, following [12,23], we will focus our attention on a stepwise weight function of the form where a + , a − > 0 and 0 < τ < T . This framework allows us to study the uniqueness question exploiting techniques typical of autonomous system. The general statement will be given in Section 2 (cf. Theorem 2.1); by now, we just present it for some special cases. When dealing with the linear differential operator, i.e. φ(s) = s, it yields the following. Theorem 1.1. Let a ∈ L ∞ (0, T ) be as in (1.4). Let γ ∈ ]−∞, −3] ∪ ]1, +∞[. Then, the Neumann and the periodic boundary value problems associated with equation have at most one positive solution. Moreover, there exists a unique positive solution if and only if γ · T 0 a(t) dt < 0. Notice that the case of a singularity with γ = −3, already solved in [12], is included in the above result. We mention that with our strategy of proof we can also deal with the linear case (i.e. γ = 1) so as to recover the existence of a simple principal eigenvalue (see Remark 3.1 and Remark 3.2).
As for the more general case of a p-Laplacian operator φ(s) = ϕ p (s) = |s| p−2 s with p > 1, our main contribution is the following.
Then, the Neumann and the periodic boundary value problems associated with equation have at most one positive solution. Moreover, there exists a unique positive solution if and only if γ · T 0 a(t) dt < 0. The plan of the paper is the following. In Section 2, we present our main abstract uniqueness result for equation (1.1) and, to prove Theorem 1.1 and Theorem 1.2, in Section 3 we apply it to the case φ(s) = s and φ(s) = ϕ p (s) = |s| p−2 s with p > 1. In Section 4, some remarks and open questions are presented, including a brief discussion for the Minkowski-curvature operator.

An abstract uniqueness result
In this section, we aim to present a method to deal with a general class of nonlinear differential problems. Accordingly, we deal with the second-order equation where, for Ω ⊆ R an open interval with 0 ∈ Ω, we assume that As a consequence, we immediately obtain a necessary condition for the existence of positive solutions of (2

The Neumann problem
Let us consider the planar system associated with equation (2.1), that is A solution of (2.2) is a couple (x, y) of absolutely continuous functions satisfying (2.2) for almost every t. Throughout the section, we confine ourselves in the halfright part ]0, +∞[ × R of the phase-plane. According to the assumptions on φ(s), a(t) and g(x), for every time t 0 ∈ [0, T [ and every initial condition (x 0 , y 0 ) ∈ ]0, +∞[ × R, system (2.2) admits a unique local non-continuable solution with x(t 0 ) = x 0 and y(t 0 ) = y 0 , denoted by The uniqueness of the solutions of the Cauchy problems is guaranteed by the special choice of the step-wise coefficient a(t), indeed we enter the setting of the result in [34] concerning planar Hamiltonian systems. Moreover, we remark that x(t; t 0 , x 0 , y 0 ) > 0 for all t in the maximal interval of existence, where the solution is defined.
We observe that the quantities H(y)+a + G(x) and H(y)−a − G(x) are constant for all (x, y) solving (S + ) and (S − ), respectively. In particular, due to (2.3), (2.4) and H(0) = 0, we have that the solution (x(t), y(t)) satisfies We notice that the functions H l := H| ]−∞,0] , H r := H| [0,+∞[ and G are invertible since strictly monotone. For sake of simplicity in the notation, we set We remark that, when h is odd, we find that H −1
From the above discussion, the following result holds true.

13)
if and only if the Neumann boundary value problem associated with equation (2.1) has a positive solution. Moreover, the unique solvability of (2.13) is equivalent to the uniqueness of the positive solution of the Neumann problem.
For our applications, we will consider a simplified but equivalent formulation of system (2.13) which can be obtained when ω = G(α) is of constant sign. This excludes only the case (iii) in the list above.
We introduce the new variable Thus, we have By performing the change of variable ϑ = G(α)ξ = ωξ, formulas (2.11) and (2.12) read as respectively. Let (2.14) and define the functions F I , F II : D → ]0, +∞[ as follows , .
From the above discussion, we have the following uniqueness result.

15)
if and only if the Neumann boundary value problem associated with equation (2.1) has a positive solution. Moreover, the unique solvability of (2.15) is equivalent to the uniqueness of the positive solution of the Neumann problem.
We conclude this section, by presenting an equivalent version of Theorem 2.1. Instead of (2.7), our new starting point is the fact that the hypothetical solution (x(t), y(t)) also satisfies

We immediately infer that
(2.16) The following result holds true.

The periodic problem
In this section, we deal with the periodic boundary value problem associated with (2.1) and we show that Theorem 2.1 holds true also in the periodic case. Following a procedure which is standard in this situation, we extend by T -periodicity the weigh a(t) as an L ∞ -stepwise function defined in the whole real line. In this framework, finding a solution of (2.1) satisfying u(0) = u(T ) and u ′ (0) = u ′ (T ) is equivalent to finding a T -periodic solution of (2.1) defined on R.
As in Section 2.1, we analyse the associated planar system (2.2) and we look for periodic solutions (x(t), y(t)) of (2.2) such that x(t) > 0 for all t ∈ R. Our purpose is to reduce the study of the periodic problem to the Neumann one, analysed in previous section. To this aim, we further assume that The next two claims relate the existence/uniqueness of positive solutions of the T -periodic problem to the corresponding one for the Neumann problem.
and satisfying the Neumann condition at the boundary, that is Let also (x(t),ŷ(t)) be the T -periodic extension of (x(t), y(t)) symmetric with respect to t = τ /2, namely (2.18) Indeed, by construction, (x(t),ŷ(t)) is symmetric with respect to τ 2 and, by direct inspection, one can easily check that it is a solution of system (2.2) on the interval [ T −τ 2 , T +τ 2 ]; this follows from the fact that the extension of a(t) by T -periodicity is symmetric with respect to τ 2 . Moreover, (x(t),ŷ(t)) satisfies the T -periodic condition at the boundary of [ T −τ 2 , T +τ 2 ], that is Since the weight a(t) has been extended by T -periodicity on the whole real line and (x(t),ŷ(t)) is a T -periodic extension of (2.18), we immediately conclude that (x(t),ŷ(t)) solves (2.2) and is such thatx(t) > 0 for all t ∈ R, and, by construction, (x(0),ŷ(0)) = (x(T ),ŷ(T )). Then, Claim 1 is proved.
Then, the restriction of (x(t), y(t)) to the interval [ τ 2 , T +τ 2 ] is a solution of (2. We remark that the function (x(t), y(t)) is such that We are going to prove thatt = τ 2 andť = T +τ 2 . Our approach is based on an analysis of the phase-portrait in the (x, y)-plane and is similar to the one performed in [12]. Due to the more general framework and in order to justify the additional hypothesis on h, we give here all the details.
Arguing as in Section 2.1 (cf. (2.7)), we deduce that the solution (x(t), y(t)) satisfies with α = x(t) and β = x(ť). The trajectory on the time interval [0,t] satisfies the relation which describes a strictly monotone decreasing curve. Analogously, using the fact that H −1 l = −H −1 r , the trajectory on the time interval [t, τ ] satisfies the relation which describes a strictly monotone increasing curve. On the other hand, the trajectory on the time interval [τ,ť] satisfies the relation which describes a strictly monotone decreasing curve, while the trajectory on the time interval [ť, T ] satisfies the relation which describes a strictly monotone increasing curve. By the strict monotonicity of the above curves, we infer that there is at most one intersection point between ) (x + , y + ) to (α, 0) coincides with the time necessary to connect (along the same level line) (α, 0) to (x − , y − ). Hence,t = τ 2 . The same argument, with respect to the level line (2.20) shows that that the times necessary to connect (x − , y − ) to (β, 0) and (β, 0) to (x + , y + ) are equal. Then,t = τ +T 2 . This concludes the proof of Claim 2.
From the above discussion, we deduce that we can reduce the problem of proving the existence and uniqueness of a positive solution of the T -periodic problem to the study of a Neumann problem defined in [ τ 2 , τ +T 2 ] for a step-wise function. Clearly, this latter problem is equivalent to the original problem studied in the previous section. In particular, observe also that systems (2.13) and (2.15) would be formally changed to corresponding new systems in which the target vector (τ, T −τ ) should be replaced by ( τ 2 , T −τ 2 ). With reference to the result obtained in the subsequent section, no relevant point has to be changed due also to the elementary fact that the ration of the two components of the target vector (τ, T −τ ) or, respectively, ( τ 2 , T −τ 2 ) remains unchanged (see the second equations in (3.5) and (3.9)). Finally, we conclude that Theorem 2.1 and Corollary 2.1 are both valid also in the periodic case, with the additional assumption that h is odd.

Proofs of Theorem 1.1 and Theorem 1.2
In this section, we apply Corollary 2.1 when and for φ(s) = s in Section 3.1, thus proving Theorem 1.1, while for φ(s) = |s| p−2 s with p > 1 in Section 3.2, thus proving Theorem 1.2. We focus our analysis on the Neumann boundary conditions; the result for the periodic problem follows from this as explained in Section 2.2 (indeed, the functions h = φ −1 will be odd).

The case φ(s) = s
We deal first with the simpler case φ(s) = s.
As a first step, we prove that the Neumann boundary value problem associated with (1.5) has at most one positive solution. As a second step, to conclude the proof, we show that there exists at least one positive solution.
In this special framework, we have Moreover, concerning the nonlinear term (3.1), for γ ∈ R \ {−1}, we deduce that Incidentally, notice that in this situation the case (iii) listed in Section 2.1 does not hold, so we are allowed to use Corollary 2.1.
Remark 3.1 (The case γ = 1). The proof of Theorem 1.1 in particular states that, for γ = 1, the function F 2 (ρ) is strictly monotone increasing. Arguing as in the second part of the proof, one can also show that (3.7) and (3.8) are still valid, using L'Hôpital's rule and, respectively, the fact that, when γ = 1, as ρ → 0 + , I 1 (ρ) tends to a positive constant, while I 2 (ρ) → +∞. We can conclude that the second equation in (3.5) is uniquely solved. Letρ be the solution, then the first equation in (3.5) reads as I 1 (ρ)/2 = τ , which either is not solvable, or holds for all ω ∈ ]0, +∞[. This is in agreement with the well known fact that the eigenspace of Neumann/periodic one-signed solutions of the linear equation u ′′ + a(t)u = 0 has dimension less than or equal to 1 (cf. [15]). ⊳ Remark 3.2 (The linear/nonlinear eigenvalue problem). Let us consider the nonlinear eigenvalue problem u ′′ + λa(t)u γ = 0 together with the Neumann/periodic boundary conditions. In this case, in the above analysis, system (3.5) should be modified to The study of the second equation of this system is the same as above and, for γ ∈ ]−∞, −3] ∪ [1, +∞[, it provides a unique solutionρ. By inserting this value in the first equation, for γ = 1, we obtain the curve which ensures the existence of a unique positive principal eigenvalue consistently with the classical linear theory (cf. [6,14,39] and, concerning the p-Laplacian, see [32] and the references therein).  , and, by Corollary 2.1, we infer that proving that system (2.15) admits a unique solution (ω, ρ) ∈ D is equivalent to prove that there exists a unique pair (ω, ρ) ∈ D which solves which is exactly system (3.5) when p = 2.
As above, we study the auxiliary function We compute

Moreover, we have
Reasoning as in the proof of Theorem 1.1, we conclude that F p (ρ) is strictly monotone if which is equivalent to Moreover, if γ = (1−2p)/(p−1) we obtain that the function I 1 (ρ)/I 2 (ρ) is constant and so F p is strictly monotone increasing, while if γ = p−1 then F p (ρ) = I 1 (ρ)/I 2 (ρ) is strictly monotone increasing. For the part concerning the existence of solutions, we can complete the proof of Theorem 1.2 by arguing exactly as in Section 3.1.

Final remarks
In this final section, we collect some complementary results and open questions motivated by our approach.
4.1. The "excluded" values of γ Even if Theorem 1.1 and Theorem 1.2 cover a wide range of values of the real exponent γ, some interesting cases are not investigated. In this section, we present some comments concerning these situations. In order to simplify the exposition, we focus our attention on the linear differential operator, namely φ(s) = s; similar analysis could be performed for the p-Laplacian operator.
By L'Hôpital's rule, we observe that Therefore, we deduce that On the other hand, the limit of F 2 (ρ) as ρ → (a − /a + ) − is the same as obtained in (3.7).
The fact that K 0 (γ) > 0 for all γ ∈ ]−1, 1[ implies that the function F 2 (ρ) is not surjective on ]0, a − /a + [. As a consequence, the condition γ ·ā < 0 is not sufficient for the existence of positive solutions. This fact is not surprising and actually it is consistent with the previous observation that in the interval where the weight is negative we have the possibility of solutions vanishing in finite time, due to the lack of uniqueness for the Cauchy problems, or hitting the singularity, due to the absence of the strong force condition in zero.
We are now ready to discuss the solvability of the boundary value problems. In the case γ ∈ ]0, 1[, the nonlinearity is concave and smooth in ]0, +∞[ and we can apply [3, Lemma 3.1] (see also [13]) to ensure the fact that there exists at most one positive solution of the problem. This implies that the function F 2 (ρ) is strictly monotone. More precisely, taking into account the monotonicity of K 0 (γ) and K 0 (0) = a − /a + , we have Therefore, F 2 (ρ) is strictly monotone increasing. Hence, taking into account the continuity of F 2 (ρ) on ]0, a − /a + [, we can state the following.
Moreover, the solution is unique.
In the case γ ∈ ]−1, 0[, by the previous argument as before, we have instead Even without information of the monotonicity, the continuity of F 2 (ρ) implies that its range covers the open interval ]a − /a + , K 0 (γ)[. Hence, we have the following.  To the best of our knowledge, there are no general uniqueness results available for this range of the exponent. Numerical simulations suggest that the function F 2 (ρ) is strictly monotone decreasing (see Figure 1). This would guarantee that the positive solution is unique also in this situation and consequently condition (4.3) would be sharp, likewise condition (4.2) is sharp in the case γ ∈ ]0, 1[. To compare our result with a similar case previously studied, we recall that in [22,Corollary 3] the authors investigate a singular equation with a stepwise indefinite function, and obtain a range for the existence of positive solutions, that for instance in the case γ = −1/2 and a − = a + would read like On the other hand, an easy computation shows that K 0 (γ) = 5 and thus (4.3) reads as 1 < τ T − τ < 5.
This shows that, at least in this case, the range provided by our result is better.

The Minkowski-curvature operator
In this paper, we have applied Theorem 2.1 to the case of homogeneous differential operators φ(s) and homogeneous nonlinearities g(u). This choice has the advantage to simplify (2.13) to (3.5) or (3.9), and the consequent analysis of the functions F 2 (ρ) or, respectively, F p (ρ). However, our technique appears useful to study more general situations, especially when the function L h can be computed explicitly. For example, in the case of the Minkowski-curvature equation we have φ(s) = s √ 1 − s 2 and therefore, for h = φ −1 , we find h(y) = y 1 + y 2 , H(y) = 1 + y 2 − 1, L h (ξ) = − ξ 2 + 2ξ 1 + ξ .
As a consequence, the analysis of system (2.13) could be simplified for some special choices of g(u). We do not pursue here this investigation which can be the topic of future researches. Recent works for the Neumann and periodic problems associated with (4.4) show that for g(u) = u γ with γ > 1 multiple positive solutions do exist also in the case of a weight with a single change of sign as in (1.4) (see [8,9]). On the other hand, numerical simulations suggest the possibility of uniqueness results when g(u) is a strictly increasing function with "super-exponential" growth at infinity (cf. [10]).