Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations

2.1 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 half-right 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 noncontinuable 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 stepwise coefficient a(t); indeed, we enter the setting of the result in [30] 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.

In this section, we focus our attention on the Neumann boundary value problem associated with system (2.2). Our goal is to prove that

there exists a unique initial condition ( x ∗ , y ∗ ) ∈ ] 0 , + ∞ [ × R such that (x(t; τ, x _∗, y _∗), y(t; τ, x _∗, y _∗)) is a solution of system (2.2) defined in [0, T], x(t; τ, x _∗, y _∗) > 0 for all t ∈ [0, T], and

(2.3) y ( 0 ; τ , x ∗ , y ∗ ) = 0 , y ( T ; τ , x ∗ , y ∗ ) = 0 .

Setting h = ϕ ⁻¹, we can exploit (1.4) and so define the two autonomous systems

Let us assume that there is a hypothetical solution

of (2.2) defined in [0, T], with x(t) > 0 for all t ∈ [0, T], and satisfying the boundary conditions (2.3). For future convenience, we set

Clearly, α > 0 and β > 0. From ( S + ) and y(0) = 0, we deduce that

and thus x ′ ( t ) = h ( y ( t ) ) < 0 for every t ∈ ]0, τ[. Hence, x(t) is strictly monotone decreasing in [0, τ] and x(t) < α for every t ∈ ]0, τ]. Analogously, from ( S − ) and y(T) = 0, we have that y(t) < 0 for every t ∈  [τ, T[, so x(t) is strictly monotone decreasing in [τ, T] and so x(t) > β for every t ∈  [τ, T[. We conclude that

so the solution (x(t), y(t)) is in the fourth quadrant ]0, + ∞ [ × ]− ∞ , 0]. Moreover, x(t) is strictly monotone decreasing in [0, T], and thus

Let H and G be primitives of h and g, respectively. For convenience, we suppose that H(0) = 0. When g can be continuously extended to zero, we shall also assume G(0) = 0. We denote G 0 ≔ lim x → 0 G ( x ) and G + ∞ ≔ lim x → + ∞ G ( x ) . In general, for an arbitrary function g: ]0, + ∞ [ → ]0, + ∞ [the following four cases are possible:

1. G 0 ∈ R and G + ∞  = + ∞ ; in this situation, without loss of generality, we can suppose that G: ]0, + ∞ [ →]0, + ∞ [ is a strictly monotone increasing surjective function;

2. G 0 ∈ R and G + ∞ ∈ R ; in this situation, without loss of generality, we can suppose that G: ]0, + ∞ [ →]0, Λ[ is a strictly monotone increasing surjective function, with Λ ≔ ∫ 0 + ∞ g ( u ) d u ;

3. G ₀ = − ∞ and G + ∞  = + ∞ ; in this situation, without loss of generality, we can suppose that G: ] 0 , + ∞ [ → R is a strictly monotone increasing surjective function;

4. G ₀ = − ∞ and G + ∞ ∈ R ; in this situation, without loss of generality, we can suppose that G: ]0, + ∞ [ →]− ∞ , 0[ is a strictly monotone increasing surjective function.

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 l − 1 = − H r − 1 and, therefore, ℒ h = − h ∘ H r − 1 .

Recalling (2.5), from (2.7), ( S + ) and ( S − ) , we infer that

Recalling (2.6), by an integration in [0, τ[ and in [τ, T[, we obtain

respectively. Conversely, if α, β, x _∗, with 0 < β < x _∗ < α, are such that the aforementioned relations hold, we infer that the solution (x(t; τ, x _∗, y _∗), y(t; τ, x _∗, y _∗)) of the Cauchy problem is defined in [0, T] and is such that (x(0; τ, x _∗, y _∗), y(0; τ, x _∗, y _∗)) = (α, 0) and (x(T; τ, x _∗, y _∗), y(T; τ, x _∗, y _∗)) = (β, 0). Hence, by the choice of h = ϕ ⁻¹, we deduce that x(t) is a positive decreasing solution of the Neumann boundary value problem associated with (2.1).

Let us perform the change of variable

in (2.8), obtaining

and

From (2.7) with (x, y) = (x _∗, y _∗) one can deduce that

thus G(x _∗) is a convex combination of G(α) and G(β).

Next, we set

Accordingly, (2.9) and (2.10) take the simplified form

and

Let O ≔ { ( ω , σ ) ∈ G ( ] 0 , + ∞ [ ) × G ( ] 0 , + ∞ [ ) : ω > σ } and we introduce the functions ℳ I , ℳ II : O → ] 0 , + ∞ [ defined as

From the above discussion, the following result holds true.

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 case (iii) in the aforementioned list.

We introduce the new variable

Thus, we have

By performing the change of variable ϑ = G(α)ξ = ω ξ, formulas (2.11) and (2.12) are as follows:

respectively.

Let

and define the functions ℱ I , ℱ II : D → ] 0 , + ∞ [ as follows:

From the above discussion, we have the following uniqueness result.

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

and integrations in [0, τ[ and, respectively, in [τ, T[ give

The following result holds true.

2.2 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 weight 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 the 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.

Claim 1. Let (x(t), y(t)) be a solution of (2.2) defined in the interval τ 2 , T + τ 2 with x(t) > 0 for all t ∈ τ 2 , T + τ 2 and satisfying the Neumann condition at the boundary, that is

Let also ( x ˆ ( t ) , y ˆ ( t ) ) be the T-periodic extension of (x(t), y(t)) symmetric with respect to t = τ/2, namely,

Then, ( x ˆ ( t ) , y ˆ ( t ) ) is a T-periodic solution of (2.2) with x ˆ ( t ) > 0 for all t ∈ R .

Indeed, by construction, ( x ˆ ( t ) , y ˆ ( 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 ˆ ( t ) , y ˆ ( 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 ˆ ( t ) , y ˆ ( t ) ) is a T-periodic extension of (2.18), we immediately conclude that ( x ˆ ( t ) , y ˆ ( t ) ) solves (2.2) and is such that x ˆ ( t ) > 0 for all t ∈ R , and, by construction, ( x ˆ ( 0 ) , y ˆ ( 0 ) ) = ( x ˆ ( T ) , y ˆ ( T ) ) . Then, Claim 1 is proved.

Claim 2. Let (x(t), y(t)) be a T-periodic solution of (2.2) with x(t) > 0 for all t ∈ R . Then, the restriction of (x(t), y(t)) to the interval τ 2 , T + τ 2 is a solution of (2.2) with x(t) > 0 for all t ∈ τ 2 , T + τ 2 and satisfying the Neumann boundary condition (2.17).

Indeed, for (x(t), y(t)) as in the assumption and by the special form (1.4) of the weight function a(t), we have that y ′ ( t ) < 0 in ]0, τ[ and y ′ ( t ) > 0 in ]τ, T[, and so y(t) is strictly monotone decreasing in [0, τ] and strictly monotone increasing in [τ, T]. Since h is strictly monotone increasing, we also deduce that x ′ ( t ) is strictly monotone decreasing in [0, τ] and strictly monotone increasing in [τ, T].

We note that if x ′ ( 0 ) = 0 , then we have x ′ ( t ) < 0 in ]0, τ[ and, since x ′ ( T ) = h ( y ( T ) ) = h ( y ( 0 ) ) = x ′ ( 0 ) = 0 , we also have x ′ ( t ) < 0 in ]τ, T[; thus x(t) is strictly monotone decreasing in [0, T], a contradiction with x(0) = x(T). Moreover, if x ′ ( τ ) = 0 , then x ′ ( t ) > 0 in ]0, τ[ and x ′ ( t ) > 0 in ]τ, T[, a contradiction with x(0) =x(T), as before. Therefore, from the above discussion and Rolle’s theorem, we conclude that there exist exactly one critical point in ]0, τ[ and exactly one critical point in ]τ, T[. Let t ˆ ∈ ] 0 , τ [ and t ˇ ∈ ] τ , T [ be such that x ′ ( t ˆ ) = x ′ ( t ˇ ) = 0 . We claim that t ˆ and t ˇ are the maximum point and, respectively, the minimum point of x(t) in [0, T]. This is obvious since x ( t ) = x ( t ˆ ) + ∫ t ˆ t x ′ ( ξ ) d ξ < x ( t ˆ ) in every neighbourhood of t ˆ (contained in ]0, τ[), while x ( t ) = x ( t ˇ ) + ∫ t ˇ t x ′ ( ξ ) d ξ > x ( t ˇ ) in every neighbourhood of t ˇ (contained in ]τ, T[).

We remark that the function (x(t), y(t)) is such that y ( t ˆ ) = y ( t ˇ ) = 0 , y(t) < 0 in ] t ˆ , t ˇ [ and x(t) is strictly monotone decreasing in [ t ˆ , t ˇ ] , y(t) > 0 in ] 0 , t ˆ [ ∪ ] t ˇ , T [ and x(t) is strictly monotone increasing in [ 0 , t ˆ ] ∪ [ t ˇ , T ] .

We are going to prove that t ˆ = τ 2 and t ˇ = 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 [16]. 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 ˆ ) and β = x ( t ˇ ) . The trajectory on the time interval [ 0 , t ˆ ] satisfies the relation

which describes a strictly monotone decreasing curve. Analogously, using the fact that H l − 1 = − H r − 1 , the trajectory on the time interval [ t ˆ , τ ] satisfies the relation

which describes a strictly monotone increasing curve. On the other hand, the trajectory on the time interval [ τ , t ˇ ] satisfies the relation

which describes a strictly monotone decreasing curve, while the trajectory on the time interval [ t ˇ , 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 (2.21) and (2.24) in ]α, β[ × ]0, + ∞ [, and likewise at most one intersection point between (2.22) and (2.23) in ]α, β[ × ]− ∞ , 0[. Actually, these intersections exist and are uniquely determined as (x ₊, y ₊) = (x(0), y(0)) = (x(T), y(T)) (in the first quadrant) and (x ₋, y ₋) = (x(τ), y(τ)) (in the fourth quadrant). Due to the symmetry of H, it follows that x ₊ = x ₋ and y ₊ = −y ₋. Moreover, from the Abelian-type integrals representing the time mappings, we find that the time necessary to connect (along the level line (2.19)) (x ₊, y ₊) to (α, 0) coincides with the time necessary to connect (along the same level line) (α, 0) to (x ₋, y ₋). Hence, t ˆ = τ 2 . The same argument, with respect to the level line (2.20), shows that the times necessary to connect (x ₋, y ₋) to (β, 0) and (β, 0) to (x ₊, y ₊) are equal. Then, t ˇ = 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 stepwise 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 the 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 also due to the elementary fact that the ratio 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.

3.1 The case ϕ(s) = s

We deal first with the simpler case

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

which leads to

Moreover, concerning the nonlinear term (3.1), for γ ∈ R ⧹ { − 1 } , we deduce that

and thus

Incidentally, note that in this situation case (iii) listed in Section 2.1 does not hold, so we are allowed to use Corollary 2.1.

We study the unique solvability of system (2.15) for ( ω , ρ ) ∈ D , where D is defined in (2.14). In this special framework, we have

and

For simplicity in notation, we introduce the functions I 1 , I 2 : ] 0 , + ∞ [ → R defined as