Existence and global asymptotic behavior of positive solutions for combined second-order differential equations on the half-line

1 Introduction

Problems of the type

where A is a positive and differentiable function on (0,∞) and φ is a measurable function on (0,∞)×(0,∞) that satisfies some further hypotheses, have been extensively studied in the literature (see, for example, [1, 2, 3, 18, 19, 26, 28] and the references therein).

In [28], Zhao considered (1.1) with A⁢(t)≡1, where φ is a measurable function on (0,∞)×(0,∞) dominated by a convex positive function. Then, he showed that there exists b>0 such that for each μ∈(0,b], there exists a positive continuous solution u of (1.1) satisfying limt→∞⁡u⁢(t)/t=μ.

In [18], Mâagli and Masmoudi generalized the result of Zhao to the more general boundary value problem

where A is a positive and differentiable function on (0,∞) and f is a measurable function on (0,∞)×(0,∞)×(0,∞) which may change sign and is dominated by a regular function. Then, they proved the existence of a constant b>0 such that for each μ∈(0,b], (1.2) has a continuous solution u satisfying

where

On the other hand, in [2], the first author studied (1.1) with A satisfying some appropriate conditions and where φ:(0,∞)×(0,∞)→[0,∞) is a continuous function, nonincreasing with respect to the second variable such that for each c>0, φ⁢(⋅,c)>0 and

He proved the existence of a unique positive solution u in C⁢([0,∞))∩C2⁢((0,∞)) to (1.1) satisfying

Recently, in [3], we have proved the existence, uniqueness and the global asymptotic behavior for a positive solution to the boundary value problem

where σ<1 and where A is a continuous function on [0,∞) which is positive and differentiable on (0,∞) such that

with

The nonnegative continuous function a is required to satisfy some appropriate assumptions related to the Karamata regular variation theory.

In this paper, we deal with the boundary value problem

where σ1,σ2<1 and A is a continuous function on [0,∞) which is positive and differentiable on (0,∞) such that

with

Our goal is to study (1.4) especially when the nonlinearity is the sum of a singular and a sublinear term. Under appropriate assumptions on a1 and a2 related to the Karamata classes 𝒦 and 𝒦∞ (see Definitions 1.1 and 1.2), we prove the existence, uniqueness and the global asymptotic behavior of a positive continuous solution to (1.4). In particular, we extend the results obtained in [3].

Observe that in the case A⁢(t)=tn-1, n≥2, the operator

is the radially symmetric Laplacian Δ⁢u in ℝn, where u is a function that depends only on t=|x|, x∈ℝn (see [26]). For various existence, uniqueness and asymptotic behavior results for such a problem, we refer the reader to [4, 6, 7, 8, 9, 10, 11, 15, 14, 13, 12, 17, 21, 22, 23, 24, 26, 27] and the references therein. However, we emphasize that in (1.4), the function A⁢(t)=tn-1, n≥2, is not allowed and the potential functions a1 and a2 may be singular at t=0.

Throughout this paper and without loss of generality, we assume that

To state our result, we need some definitions and notation.

Note that the functions belonging to the classes 𝒦 and 𝒦∞ are in particular slowly varying functions. The theory of such functions was initiated by Karamata in the fundamental paper [16]. We also point out that the first use of the Karamata theory in the study of the growth rate of solutions near the boundary is done in the paper of Cirstea and Rădulescu [10].

It is easy to verify the following result.

As a typical example of a function belonging to the classes 𝒦 and 𝒦∞ (see [5, 20, 25]), we quote

where

and ξk∈ℝ, τk∈(0,1) and where ω is a sufficiently large positive real number such that L (resp. L~) is defined and positive on (0,η] for η>1 (resp. on [1,∞)).

In the sequel, we will denote by B+⁢((0,∞)) the set of nonnegative Borel measurable functions in (0,∞) and by C0⁢([0,∞)) the set of continuous functions v on [0,∞) such that limt→∞⁡v⁢(t)=0. It is known that C0⁢([0,∞)) is a Banach space with the uniform norm ∥v∥∞=supt>0⁡|v⁢(t)|.

For s,t∈ℝ, we write min⁡(s,t)=s∧t and max⁡(s,t)=s∨t. For two nonnegative functions f and g defined on a set S, the notation f⁢(t)≈g⁢(t), t∈S, means that there exists c>0 such that

for all t∈S. Furthermore, we denote by G⁢(t,s)=A⁢(s)⁢ρ⁢(t∧s) the Green’s function of the operator

on (0,∞) with the Dirichlet conditions

For f∈B+⁢((0,∞)), we put

We point out that if the map

is continuous and integrable on (0,∞), then Vf is the nonnegative continuous solution of the boundary value problem

To simplify our statement, we introduce the following. Let λ≤2, σ<1 and μ≥1+σ. Let also L∈𝒦 be defined on (0,η] for η>1 and L~∈𝒦∞ such that

Define the function ΨL,λ,σ⁢(t) for t∈(0,η) by

and Ψ~L~,μ,σ⁢(t) for t∈[1,∞) by

Throughout this paper we will make the following assumption.

1. For i∈{1,2}, let σi<1 and let ai be a nonnegative continuous function on (0,∞) such that

   (1.8)ai⁢(t)≈1(A⁢(t))2⁢(ρ⁢(t))-λi⁢(1+ρ⁢(t))λi-μi⁢Li⁢(ρ⁢(1∧t))⁢L~i⁢(ρ⁢(1∨t)),t>0,

   where λi≤2, μi≥1+σi and with Li∈𝒦, L~i∈𝒦∞ satisfying

   (1.9)∫0ηLi⁢(s)sλi-1⁢𝑑s<∞,∫1∞L~i⁢(s)sμi-σi⁢𝑑s<∞.

As it will be seen, for i∈{1,2}, the numbers

will play an important role in the study of the asymptotic behavior of the solution. From here on and without loss of generality, we may assume that

and we introduce the function θ defined on (0,∞) by

For t∈(0,1], observe that

and for t≥1,

Our main results are the following two theorems.

The content of this paper is organized as follows. In Section 2, we present some fundamental properties of the Karamata classes 𝒦 and 𝒦∞. In particular, we recall some sharp estimates on some convenient potential functions. In Section 3, exploiting the results of the previous section, we first prove Theorem 1.4, which allows us to prove Theorem 1.5 by means of the sub-supersolution method. We conclude this section by studying the uniqueness of solutions and giving an example to illustrate our existence result.

2 Properties of the Karamata classes 𝒦 and 𝒦∞

We collect in this paragraph some properties of functions belonging to the Karamata classes 𝒦 and 𝒦∞.

The proof of the next lemma can be found in [9].

In the next lemma, we have similar properties related to the class 𝒦∞. For the proof we refer to [8].

Next, we recall the following key sharp estimates.

3 Proof of the main results

We recall that for i∈{1,2}, σi<1 and ai is a nonnegative continuous function on (0,∞) such that

where λi≤2, μi≥1+σi, Li∈𝒦 defined on (0,η] and L~i∈𝒦∞ satisfying

First, we will give sharp estimates of the function θ on (0,1] and on [1,∞), respectively. To this end, let L, M and N be the nonnegative functions defined on (0,η) by

and L~, M~ and N~ the nonnegative functions defined on [1,∞) by

We recall that for i∈{1,2}, we have λi≤2, μi≥1+σi and

Since ν1<ν2 is equivalent to