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On the solvability of a boundary value problem on the real line

Giovanni Cupini1, Cristina Marcelli2 and Francesca Papalini2*

Author Affiliations

1 Dipartimento di Matematica - Università di Bologna, Piazza di Porta S.Donato 5, 40126 Bologna, Italy

2 Dipartimento di Scienze Matematiche - Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy

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Boundary Value Problems 2011, 2011:26  doi:10.1186/1687-2770-2011-26


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/26


Received:22 January 2011
Accepted:23 September 2011
Published:23 September 2011

© 2011 Cupini et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M1">View MathML</a>

governed by a nonlinear differential operator Φ extending the classical p-Laplacian, with right-hand side f having the critical rate of decay -1 as |t| → +∞, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M2">View MathML</a>. We prove general existence and non-existence results, as well as some simple criteria useful for right-hand side having the product structure f(t, x, x') = b(t, x)c(x, x').

Mathematical subject classification: Primary: 34B40; 34C37; Secondary: 34B15; 34L30.

Keywords:
boundary value problems; unbounded domains; heteroclinic solutions; nonlinear differential operators; p-Laplacian operator; Φ-Laplacian operator

1. Introduction

Differential equations governed by nonlinear differential operators have been extensively studied in the last decade, due to their several applications in various sciences. The most famous differential operator is the well-known p-Laplacian and its generalization to the generic Φ-Laplacian operator (an increasing homeomorphism of ℝ with Φ(0) = 0). Many articles have been devoted to the study of differential equations of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M3">View MathML</a>

for Φ-Laplacian operators, and recently also the study of singular or non-surjective differential operators has become object of an increasing interest (see, i.e., [1-10]).

On the other hand, in many applications the dynamic is described by a differential operator also depending on the state variable, like (a(x)x')' for some sufficiently regular function a(x), which can be everywhere positive [non-negative] (as in the diffusion [degenerate] processes), or a changing sign function, as in the diffusion-aggregation models (see [7], [11-13]).

So, it naturally arises the interest for mixed nonlinear differential operators of the type (a(x)Φ(x'))'. In this context, in [11] we studied boundary value problems on the whole real line

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M4">View MathML</a>

obtaining results on both existence and non-existence of heteroclinic solutions. Such criteria are based on the comparison between the behavior of the right-hand side f(t, x, x') as |t| → +∞ and x' → 0, combined to the infinitesimal order of the differential operator Φ(x') as x' → 0. Rather surprisingly, the presence of the state variable x inside the right-hand side and the differential operator does not influence in any way the existence or the non-existence of solutions, but it only entails a more technical proof and a sligthly stronger set of assumptions on the operator Φ. Roughly speaking, if a(x) is positive and f(t, x, x') = g(t, x')h(x) for some positive continuous function h, then the solvability of the boundary value problem depends neither on a, nor on h. Moreover, even the prescribed boundary values ν1, ν2 are not involved on the existence of solutions.

A crucial assumption in [11] is a limitation on the rate of the possible decay of f(·, x, x') as |t| → +∞; precisely, we assumed that f(t, x, x') ≈ |t|δ for some δ > -1 (possibly positive).

In the present article we focus our attention on right-hand sides having the critical rate of decay δ = -1 and show that, contrary to the situation studied in [11], now the solvability of the boundary value problem is influenced by the behavior of the right-hand side and of the differential operator with respect to the state variable x. For instance, when f(t, x, x') = g(x)h(t, x') the existence of solutions depends on the amplitude of the range of the values assumed by the functions a and g in the interval [ν1, ν2] determined by the prescribed boundary values.

In Section 2 we study the existence/non-existence of solutions for general right-hand sides f(t, x(t), x'(t)) (see Theorems 2.3-2.5); more operative criteria are stated in the subsequent section for f of product type.

We conclude the article with some examples (see Examples 3.8-3.10), useful to have a quick glance on the role played by the behavior with respect to x.

The study of the solvability of the boundary value problem for rates of decay δ < -1 is still open.

2. Existence and non-existence theorems

Let us consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M5">View MathML</a>

(2.1)

where a : ℝ → ℝ is a positive continuous function, and f : ℝ3 → ℝ is a given Carathéodory function. From now on we will take into consideration increasing homeomorphisms Φ : ℝ → ℝ, with Φ(0) = 0.

Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions, according to the following definition.

Definition 2.1. A lower [upper] solution to equation (2.1) is a bounded function α C1(ℝ) such that (a α)(Φ ○ α') ∈ W1,1(ℝ) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M6">View MathML</a>

Throughout this section we will assume the existence of an ordered pair of lower and upper solutions α, β, i.e., satisfying α(t) ≤ β(t) for every t ∈ ℝ, and we will adopt the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M7">View MathML</a>

Note that the value d is well-defined, in fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M8">View MathML</a>, since (a α)(Φ ○ α'), (a β)(Φ ○ β') belong to W1,1(ℝ) and m > 0.

Moreover, in what follows [x]+ and [x]- will respectively denote the positive and negative part of the real number x, and we set x y := min{x, y}, x y := max{x, y}.

The next result proved in [11] concerns the convergence of sequences of functions correlated to solutions of the previous equation.

Lemma 2.2. For all n ∈ ℕ let In := [-n, n] and let un C1(In) be such that: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M9">View MathML</a>, the sequences (un(0))n and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M10">View MathML</a>are bounded and finally

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M11">View MathML</a>

Assume that there exist two functions H, γ L1(ℝ) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M12">View MathML</a>

Then, the sequence (xn)n C1(ℝ) defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M13">View MathML</a>

admits a subsequence uniformly convergent in to a function x C1(ℝ), with (a x) (Φ ○ x') ∈ W1,1(ℝ), solution to equation (2.1).

Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M14">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M15">View MathML</a>, then we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M16">View MathML</a>

The first existence result concerns differential operators growing at most linearly at infinity.

Theorem 2.3. Assume that there exists a pair of lower and upper solutions α, β C1(ℝ) of the equation (2.1), satisfying α(t) ≤ β(t), for every t ∈ ℝ, with α increasing in (-∞, -L), β increasing in (L, +∞), for some L > 0.

Let Φ be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M17">View MathML</a>

(2.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M18">View MathML</a>

(2.3)

for some positive constant μ.

Assume that there exist a constant H > 0, a continuous function θ : ℝ+ → ℝ+ and a function λ Lq([-L, L]), with 1 ≤ q ≤ ∞, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M19">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M20">View MathML</a>

(2.5)

(with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M21">View MathML</a>if q = +∞).

Finally, suppose that for every C > 0 there exist a function ηC L1(ℝ) and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M22">View MathML</a>, null in [0, L] and strictly increasing in [L, +∞),

such that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M23">View MathML</a>

(2.6)

and put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M24">View MathML</a>

(2.7)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M25">View MathML</a>

(2.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M26">View MathML</a>

(2.9)

Then, there exists a function x C1(ℝ), with (a x)(Φ ○ x') ∈ W1,1(ℝ), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M27">View MathML</a>

Proof. In some parts the proof is similar to that of Theorem 3.2 [11]. So, we provide here only the arguments which differ from those used in that proof.

By (2.2), without loss of generality we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M28">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M29">View MathML</a>

(2.10)

for some constant K > 0.

Moreover, by (2.5), there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M30">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M31">View MathML</a>

(2.11)

Fix n ∈ ℕ, n > L, and put In := [-n, n].

Let us consider the following auxiliary boundary value problem on the compact interval In:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M32">View MathML</a>

where T : W1,1(In) → W 1,1(In) is the truncation operator defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M33">View MathML</a>

and finally w : ℝ2 → ℝ is the penalty function defined by w(t, x) := [x - β(t)]+ - [x -α(t)]-.

By the same argument used in the proof of Theorem 3.2 [11], one can show, using only assumption (2.9), that for every n > L problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M34">View MathML</a> admits a solution un such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M35">View MathML</a>

(2.12)

hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M36">View MathML</a> and w(t, un(t)) ≡ 0. Moreover, it is possible to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M37">View MathML</a>

(2.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M38">View MathML</a>

(2.14)

(see Steps 3 and 4 in the proof of Theorem 3.2 [11]).

Now our goal is to prove an a priori bound for the derivatives, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M39">View MathML</a> for a.e. t In. We split this part into two steps.

Step 1. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M40">View MathML</a> for every t ∈ [-L, L].

Indeed, since un C1(In) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M41">View MathML</a>, we can apply Lagrange Theorem to deduce that for some τ0 ∈ [-L, L] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M42">View MathML</a>

Assume, by contradiction, the existence of an interval (τ1, τ2) ⊂ (-L, L) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M43">View MathML</a> in (τ1, τ2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M45">View MathML</a> or viceversa.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M46">View MathML</a> for every t ∈ (τ1, τ2), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M47">View MathML</a> for every t ∈ (τ1, τ2). Then, by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M34">View MathML</a> and assumption (2.4), for a.e. t ∈ (τ1, τ2) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M48">View MathML</a>

Therefore, using a change of variable and the Hölder inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M49">View MathML</a>

(2.15)

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M50">View MathML</a> has constant sign in (τ1, τ2), using (2.12) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M51">View MathML</a>

Therefore, by (2.10), from the previous chain of inequalities we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M52">View MathML</a>

(2.16)

in contradiction with (2.11). Thus, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M53">View MathML</a> for every t ∈ [-L, L] and the claim is proved.

Step 2. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M54">View MathML</a>(t) for every t In \ [-L, L].

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M55">View MathML</a>, and assume by contradiction that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M56">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M57">View MathML</a> and by (2.13), (2.14) we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M58">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M59">View MathML</a>. Moreover, by (2.12) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M60">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M61">View MathML</a>

so, by (2.8) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M62">View MathML</a>

Then, recalling that KC (L) = 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M58">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M63">View MathML</a>, we infer

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M64">View MathML</a>

implying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M65">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M66">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M67">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M63">View MathML</a>, in contradiction with the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M68">View MathML</a>. The same argument works in the interval [-n, -L] and the claim is proved.

Summarizing, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M39">View MathML</a> for every t In, by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M60">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M69">View MathML</a>

Observe now that condition (2.3) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M70">View MathML</a>. Hence, by assumption (2.6) we get NC L1(ℝ) and applying Lemma 2.2 with H(t) = NC(t) and γ(t) = ηC(t) we deduce the existence of a solution x to problem (P). □

In order to deal with differential operators having superlinear growth at infinity, we need to strengthen condition (2.5), taking a Nagumo function with sublinear growth at infinity, as in the statement of the following result.

Theorem 2.4. Suppose that all the assumptions of Theorem 2.3 are satisfied, with the exception of (2.2), and with (2.5) replaced by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M71">View MathML</a>

(2.17)

Then, the assertion of Theorem 2.3 follows.

Proof. The proof is quite similar to that of the previous Theorem. Indeed, notice that assumptions (2.2) and (2.5) of Theorem 2.3 have been used only in the choice of the constant C (see (2.11)) and in the proof of Step 1. Hence, we now present only the proof of this part, the rest being the same.

Notice that by assumption (2.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M72">View MathML</a>

hence, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M30">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M73">View MathML</a>

(2.18)

With this choice of the constant C, the proof proceeds as in Theorem 2.3. The only modification concerns formula (2.16), which becomes, taking (2.15) into account:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M74">View MathML</a>

in contradiction with (2.18). From here on, the proof proceeds in the same way. □

In the particular case of p-Laplacian operators, one can use the positive homogeneity for weakening assumption (2.17) of Theorem 2.4 and widening the class of the admissible Nagumo functions, as we show in the following result.

Theorem 2.5. Let Φ : ℝ → ℝ, Φ(y) = |y|p-2y, and assume that there exists a pair of lower and upper solutions α, β C1(ℝ) to equation (2.1), satisfying α(t) ≤ β(t), for every t ∈ ℝ, with α increasing in (-∞, -L), β increasing in (L, +∞), for some constant L > 0.

Moreover, assume that there exist a positive constant H, a continuous function

θ : ℝ+ → ℝ+ and a function λ Lq([-L, L]), with 1 ≤ q ≤ +∞, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M75">View MathML</a>

(2.19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M76">View MathML</a>

(2.20)

Finally, suppose that for every C > 0 there exist a function ηC L1(ℝ) and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M22">View MathML</a>, null in [0, L] and strictly increasing in [L, +∞), such that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M77">View MathML</a>

(2.21)

and put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M78">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M79">View MathML</a>

(2.22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M80">View MathML</a>

(2.23)

Then, there exists a function x C1(ℝ), with (a x)(Φ ○ x') ∈ W1,1(ℝ), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M81">View MathML</a>

Proof. The proof is quite similar to that of Theorem 2.3. Indeed, notice that the present statement has the same assumptions of Theorem 2.3, written for Φ(y) = |y|p-2y, with the exception of conditions (2.2) and (2.5), which were used only in the proof of Step 1. Hence, as in the proof of the previous Theorem 2.4, we now provide only the proof of Step 1, the rest being the same.

At the beginning of the proof, without loss of generality we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M28">View MathML</a> and we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M82">View MathML</a>, in such a way that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M83">View MathML</a>

(2.24)

The proof of Step 1 begins as previously, determining an interval J = (τ1, τ2) ⊂ (-L, L) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M42">View MathML</a> in J, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M45">View MathML</a> or vice versa. Then, as in the proof of Theorem 2.3, assumption (2.19) implies that for a.e. t J we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M84">View MathML</a>

Therefore, put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M85">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M86">View MathML</a>

in contradiction with (2.24). Thus, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M53">View MathML</a> for every t ∈ [-L, L] and Step 1 is proved. □

As we mentioned in Section 1, the assumptions of the previous existence Theorems are not improvable in the sense that if conditions (2.3) and (2.8) are satisfied with the reversed inequalities and the summability condition (2.6) [respectively (2.21) for the case of p-Laplacian] does not hold, then problem (P) does not admit solutions, as the following results state.

Theorem 2.6. Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M87">View MathML</a>

(2.25)

for some positive constant μ. Moreover, assume that there exist two constants L ≥ 0, ρ > 0 and a positive strictly increasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M88">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M89">View MathML</a>

(2.26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M90">View MathML</a>, such that one of the following pair of conditions holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M91">View MathML</a>

(2.27)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M92">View MathML</a>

(2.28)

Moreover, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M93">View MathML</a>

(2.29)

Then, problem (P) can only admit solutions which are constant in [L, +∞) (when (2.27) holds) or constant in (-∞, -L] (when (2.28) holds). Therefore, if both (2.27) and (2.28) hold and L = 0, then problem (P) does not admit solutions. More precisely, no function x C1(ℝ), with (ax)(Φ○x') almost everywhere differentiable, exists satisfying the boundary conditions and the differential equation in (P).

Proof. Suppose that (2.27) holds (the proof is the same if (2.28) holds).

Let x C1(ℝ), with (a x)(Φ○x') almost everywhere differentiable (not necessarily belonging to W1,1(ℝ)), be a solution of problem (P). First of all, let us prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M94">View MathML</a>.

Indeed, since x(+∞) = ν+ ∈ ℝ, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M96">View MathML</a>.

Taking into account that Φ is an increasing homeomorphism with Φ(0) = 0, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M97">View MathML</a>, then there exists an interval [t1, t2] ⊂ [L, +∞) such that -ρ < Φ (x'(t)) < 0 in [t1, t2], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M98">View MathML</a>. But by virtue of assumption (2.29)

we deduce that a(x(t))Φ(x'(t)) is decreasing in [t1, t2] and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M99">View MathML</a>

a contradiction. Hence, necessarily <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M100">View MathML</a>. We can prove in a similar way that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M101">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M102">View MathML</a> and we can define t* := inf{t L : |x'(τ)| < ρ in [t, +∞)}.

We claim that x'(t) ≥ 0 for every t t*. Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M103">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M104">View MathML</a>, since a(x(t))Φ(x'(t)) is decreasing in [t*, +∞) by (2.29), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M105">View MathML</a>

(2.30)

Since a is positive, then Φ(x'(t)) < 0 for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M106">View MathML</a>. Hence, from (2.30) we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M107">View MathML</a>, and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M108">View MathML</a>

in contradiction with the boundedness of x. Thus, the claim is proved.

Let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M109">View MathML</a>. We now prove that x'(t) = 0 for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M110">View MathML</a>.

Let us assume by contradiction that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M111">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M112">View MathML</a>. Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M113">View MathML</a>; we claim that T = +∞. Indeed, if T < +∞, since 0 < x'(t) < ρ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M114">View MathML</a>, by (2.27) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M115">View MathML</a>

(2.31)

So, assuming without loss of generality ρ ≤ 1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M116">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M117">View MathML</a>. Then, integrating in [t, T] with t < T we obtain (taking into account that x'(T) = 0)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M118">View MathML</a>

so by the Gronwall's inequality we deduce a(x(t))Φ(x'(t)) ≤ 0, i.e. x'(t) ≤ 0 in the same interval, in contradiction with the definition of T. Hence T = +∞.

Therefore, since 0 < x'(t) < ρ and ν - x(t) ≤ ν+ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M119">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M120">View MathML</a>

for a.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M121">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M122">View MathML</a>. The above inequalities imply that for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M121">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M123">View MathML</a>

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M124">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M125">View MathML</a>. By virtue of (2.25) and (2.26), since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M126">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M127">View MathML</a>, in contradiction with the boundedness of x.

Therefore, x'(t) ≡ 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M128">View MathML</a> and by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M129">View MathML</a> this implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M130">View MathML</a>. So, x'(t) ≡ 0 in [t*, +∞) and by the definition of t* this implies t* = L. □

Remark 2.7. In view of what observed in Remark 6 [13], if the sign condition in (2.29) is satisfied with the reverse inequality, i.e., if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M131">View MathML</a>

(2.32)

then it is possible to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M132">View MathML</a> and x'(t) ≤ 0 for |t| ≥ L. So, since ν - < ν+, when L = 0 problem (P) does not admit solutions.

3. Criteria for right-hand side of the type f(t, x, y) = b(t, x)c(x, y)

In this section we present some operative criteria useful when the right-hand side has the following product structure

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M133">View MathML</a>

As we will show, there is a strict link between the local behaviors of c(x, ·) at y = 0 and of b(·, x) at infinity which plays a key role for the existence or non-existence of solutions.

In what follows we assume that b is a Carathéodory function and c is a continuous function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M134">View MathML</a>

Notice that in this framework, the constant functions α(t) :≡ ν- and β(t) :≡ ν+ are a pair of well-ordered, monotone, lower and upper solutions. Consequently, according to the notations given after Definition 2.1, in this case we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M135">View MathML</a>

and again

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M136">View MathML</a>

According to the results of the previous section, the first three results provide sufficient conditions for the existence of solutions for our special f split in the product of b and c. Then we will deal with sufficient conditions for the non-existence of solutions.

Theorem 3.1. Let there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M137">View MathML</a>, 1 ≤ q ≤ +∞, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M138">View MathML</a>

(3.1)

Suppose that there exist positive constants h1, h2, k1, k2, ρ, H, L, ε, with ε ≤ 1, and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M139">View MathML</a> , such that for every x ∈ [ν-, ν+] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M140">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M141">View MathML</a>

(3.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M142">View MathML</a>

(3.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M143">View MathML</a>

(3.5)

Finally, let conditions (2.2) and (2.3) hold with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M144">View MathML</a>.

Then, problem (P) admits solutions.

Proof. Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M145">View MathML</a> for r > 0, from (3.1) and (3.5) it is immediate to verify the validity of conditions (2.4) and (2.5). Let us now fix a constant C > 0 and put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M146">View MathML</a>

Since c(x, y) > 0 for y ≠ 0, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M147">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M148">View MathML</a>. Finally, put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M149">View MathML</a>

Consider the following functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M150">View MathML</a>

Observe that by assumption (3.3) we have γ(t) > 0 for a.e. t L and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M151">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M152">View MathML</a>. So, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M153">View MathML</a> such that MC(t) ≤ ρ whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M154">View MathML</a>. Let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M155">View MathML</a>

and let NC be the function defined in (2.7).

By the positivity of the function γ, KC is strictly increasing for t L. Moreover, by condition (3.1), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M22">View MathML</a>. Further, since ψC k1 we have HC(t) ≤ KC(t) for every t ≥ 0 and then NC(t) ≤ MC(t) for every t ∈ ℝ.

Observe that by (3.2) and the definition of ψC, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M156">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M157">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M158">View MathML</a>, every x ∈ [ν-,ν+] and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M159">View MathML</a>. Similarly, by (3.4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M160">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M161">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M162">View MathML</a>, every x ∈ [ν-,ν+] and every |y| ≤ NC(t) ≤ MC(t) ≤ ρ. Then, condition (2.8) of Theorem 2.3 holds.

Now, from (3.3) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M163">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M162">View MathML</a>. As a consequence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M164">View MathML</a>

(3.6)

Then, by the upper bound on the exponent μ we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M165">View MathML</a>

and condition (2.6) follows.

Finally, let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M166">View MathML</a>

By (3.3) and (3.4), for every y ∈ ℝ such that |y| ≤ NC(t) for a.e. t ∈ ℝ and every x ∈ [ν-,ν+], it results

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M167">View MathML</a>

that is condition (2.9), so it remains to prove that ηC L1(ℝ). To this purpose, notice that by (3.1) and the continuity of c we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M168">View MathML</a>. Moreover, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M169">View MathML</a>, by (3.6) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M170">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M171">View MathML</a>, we get ηC L1(ℝ). Therefore, Theorem 2.3 applies and guarantees the assertion of the present result. □

For differential operators having superlinear growth at infinity, the following result can be applied, whose proof is a consequence of Theorem 2.4.

Theorem 3.2. Let all the assumptions of Theorem 3.1 be satisfied with the exception of (2.2) and with (3.5) replaced by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M172">View MathML</a>

(3.7)

Then, if (2.3) holds true with a positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M173">View MathML</a>, problem (P) admits solutions.

Proof. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M174">View MathML</a>

Observe that θ is a continuous function on [0, +∞), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M175">View MathML</a>

hence (2.4) holds. Moreover, by (3.7), for every ε > 0 there exists a real cε such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M176">View MathML</a>

Hence, for every s M max{Φ(cε), -Φ(-cε)} we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M177">View MathML</a>, that is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M178">View MathML</a>

Hence, the proof proceeds as that of Theorem 3.1, applying Theorem 2.4 instead of Theorem 2.3. □

Finally, in the case of p-Laplacian operators, the following result holds, as a consequence of Theorem 2.5, by the same proof of Theorem 3.1.

Theorem 3.3. Consider Φ(y) = |y|p-2y, p > 1, and let all the assumptions of Theorem 3.1 be satisfied with the exception of (2.2) and condition (3.5) replaced by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M179">View MathML</a>

(3.8)

Then, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M180">View MathML</a> , problem (P) admits solutions.

Proof. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M181">View MathML</a>. Easy computations allow to verify that assumptions (2.19) and (2.20) are satisfied. The conclusion follows as in the proof of Theorem 3.1, now applying Theorem 2.5. In fact, observe that in this case (2.3) is satisfied for μ = p - 1 and, defined KC and ηC as in the proof of Theorem 3.1, conditions (2.22) and (2.23) hold true. Notice that they are the rewriting of conditions (2.8) and (2.9), respectively, in the case of p-Laplacian operators. □

In the previous results the requirement <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M173">View MathML</a> is not merely technical, but it is essential, as it will be clarified by the following non-existence result.

Theorem 3.4. Suppose that (3.2) holds for a.e. t ∈ ℝ and let there exist a real constant Λ > 0 and a positive function λ L1(0, Λ) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M182">View MathML</a>

(3.9)

Moreover, assume that there exist positive constants h, k, ρ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M183">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M184">View MathML</a>

(3.11)

If (2.25) holds with a positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M185">View MathML</a> , then problem (P) does not have solutions.

Proof. Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M186">View MathML</a> for t ∈ [0, Λ] and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M187">View MathML</a> for t ≥ Λ. Note that assumptions (2.27) and (2.28) are satisfied for L = 0. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M188">View MathML</a>

by the lower bound on the exponent μ. So, the assertion follows from Theorem 2.6. □

The following results are immediate consequences of Theorems 3.1, 3.2, and 3.4.

Corollary 3.5. Let f(t, x, y) = h(t)g(x)c(y), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M189">View MathML</a>, for 1 ≤ q ≤ +∞, c continuous in and g continuous and positive in [ν-,ν+].

Assume that t · h(t) ≤ 0 for every t ∈ ℝ and c(y) > 0 for every y ≠ 0. Moreover, suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M190">View MathML</a>

Let (2.2) holds and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M191">View MathML</a>

(3.12)

Then, if (2.3) holds with an exponent μ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M192">View MathML</a> , problem (P) admits solutions; instead if (2.25) holds with an exponent μ satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M193">View MathML</a>, (P) does not admit solutions.

Corollary 3.6. Let all the assumptions of Corollary 3.5 be satisfied, apart (2.2) and with (3.12) replaced by the following condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M194">View MathML</a>

(3.13)

Then, the same conclusions of Corollary 3.5 hold.

Finally, for the p-Laplacian operator we can state the following criterium, consequence of Theorems 3.3 and 3.4.

Corollary 3.7. Let f(t, x, y) = h(t)g(x)c(y), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M189">View MathML</a>, for 1 ≤ q ≤ +∞, b continuous in ℝ, g continuous and positive in [ν-, ν+]. Let Φ(y) = |y|p-2 y, for p > 1.

Assume that t · h(t) ≤ 0 for every t ∈ ℝ and c(y) > 0 for every y ≠ 0. Moreover, suppose that there exist

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M195">View MathML</a>

for a positive constant h1.

Then, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M196">View MathML</a> , problem (P) admits solutions; instead if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M197">View MathML</a>, (P) does not have solutions.

We conclude with some examples in which the previous corollaries apply.

Example 3.8. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M198">View MathML</a> where h is a positive constant and g is a generic continuous function, positive in [ν-,ν+]. Suppose that Φ(y) = y|y|μ-2| arctan y| with μ ≥ 1 for every y ∈ ℝ and a(x) ≡ 1 for every x ∈ ℝ.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M199">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M200">View MathML</a>, it is immediate to check that all the assumptions of Corollary 3.5 are satisfied for q := +∞, h1 := h, k1 := 1. Then, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M201">View MathML</a> problem (P) has solutions, instead if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M202">View MathML</a> then problem (P) does not have solutions.

Example 3.9. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M203">View MathML</a>, where h is a positive constant and g is a generic continuous function, positive in [ν-,ν+]. Let Φ(y) := y|y|β-1 e|y| and a(x) ≡ 1. Then condition (3.13) is satisfied for every β > 0 and all the assumptions of Corollary 3.6 hold with h1 := h and k1 := 1. Then, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M204">View MathML</a> problem (P) has solutions, instead if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M205">View MathML</a> then problem (P) does not have solutions.

Example 3.10. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M206">View MathML</a>, where h is a positive constant and g is a generic continuous function, positive in [ν-,ν+]. Let Φ(y) := y|y|p-2 and a(x) ≡ 1. Then all the assumptions of Corollary 3.7 are satisfied, for q := +∞, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M207">View MathML</a>, h2 := 1. Then, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M208">View MathML</a> problem (P) has solutions, instead if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M209">View MathML</a> then problem (P) does not have solutions.

Remark 3.11. Note that in [11] the existence of heteroclinic solutions was proved when in assumption (2.8) one has Φ(|y|)γ, instead of Φ(|y|), for some γ > 1. Of course, for small |y| we have Φ(|y|)γ < Φ(|y|) for each γ > 1, hence the present condition (2.8) implies the validity of the analogous condition with γ > 1, assumed in [11] (see condition (8)). But, on the other hand, taking γ > 1 one can lose the summability of the function KC required in assumption (7) of [11]. In fact, in the following example the present Theorems 2.3 and 2.4 are applicable, whereas the results established in [11] do not work.

Consider the problem, already discussed in Example 4 [7]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M210">View MathML</a>

where a(x) ≡ 1 and m : ℝ → ℝ is the function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/26/mathml/M211">View MathML</a>

for some α > 0. As it easy to check, the best function KC satisfying condition (8) in [11] is KC(t) := [αlog t]+, but condition (7) of [11] does not hold, whatever γ > 1 may be. Hence, the existence results proved in [11] are not applicable. Instead, notice that condition (2.6) herein considered holds whenever α > μ (see (2.3)) and Theorem 2.3 (or Theorem 2.4) applies, provided that the operator Φ also satisfies the other required assumptions. Similar considerations can be done for the p-Laplacian operator too, using Theorem 2.5.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors wrote this article in collaboration and with same responsibility. All authors read and approved the final manuscript.

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