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Existence results for classes of infinite semipositone problems

Jerome Goddard1, Eun Kyoung Lee2, Lakshmi Sankar3 and R Shivaji4*

Author Affiliations

1 Department of Mathematics, Auburn University Montgomery, Montgomery, AL, 36124, USA

2 Department of Mathematics Education, Pusan National University, Busan, 609-735, Korea

3 Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS, 39762, USA

4 Department of Mathematics & Statistics, University of North Carolina at Greensboro, Greensboro, NC, 27412, USA

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Boundary Value Problems 2013, 2013:97  doi:10.1186/1687-2770-2013-97


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/97


Received:23 October 2012
Accepted:5 April 2013
Published:19 April 2013

© 2013 Goddard II 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 consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3">View MathML</a>, Ω is a smooth bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M9">View MathML</a>. Given a, b, γ and α, we establish the existence of a positive solution for small values of c. These results are also extended to corresponding exterior domain problems. Also, a bifurcation result for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M10">View MathML</a> is presented.

1 Introduction

Consider the nonsingular boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M11">View MathML</a>

(1)

where Ω is a smooth bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M16">View MathML</a> is the Laplacian of u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M17">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M18">View MathML</a> function satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M19">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M23">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M24">View MathML</a>. Existence of positive solutions of problem (1) was studied in [1]. In particular, it was proved that given an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6">View MathML</a> there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M27">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M28">View MathML</a> (1) has positive solutions. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M29">View MathML</a> is the first eigenvalue of −Δ with Dirichlet boundary conditions. Nonexistence of a positive solution was also proved when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M30">View MathML</a>. Later in [2], these results were extended to the case of the p-Laplacian operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M31">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3">View MathML</a>. Boundary value problems of the form (1) are known as semipositone problems since the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M34">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M35">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M36">View MathML</a>. See [3-9] for some existence results for semipositone problems.

In this paper, we study positive solutions to the singular boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M37">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3">View MathML</a>, Ω is a smooth bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M46">View MathML</a>. In the literature, problems of the form (2) are referred to as infinite semipositone problems as the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M47">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M48">View MathML</a>. One can refer to [10-14], and [15-17] for some recent existence results of infinite semipositone problems. We establish the following theorem.

Theorem 1.1Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M50">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44">View MathML</a>, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M52">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M53">View MathML</a>, (2) has a positive solution.

Remark 1.1 In the nonsingular case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M54">View MathML</a>), positive solutions exist only when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M25">View MathML</a> (the principal eigenvalue) (see [1,2]). But in the singular case, we establish the existence of a positive solution for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5">View MathML</a>.

Next, we study positive radial solutions to the problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M57">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M58">View MathML</a> is an exterior domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M66">View MathML</a> belongs to a class of continuous functions such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M67">View MathML</a>. By using the transformation: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M69">View MathML</a>, we reduce (3) to the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M70">View MathML</a>

(4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M71">View MathML</a>. We assume:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M72">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M73">View MathML</a> and satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M74">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M75">View MathML</a>, and for some θ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M76">View MathML</a>.

With the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M72">View MathML</a>), h satisfies:

(5)

We note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M79">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M80">View MathML</a> is nonsingular at 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M81">View MathML</a>. In this case, problem (4) can be studied using ideas in the proof of Theorem 1.1. Hence, our focus is on the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M82">View MathML</a> in which, h may be singular at 0. Note that in this case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M83">View MathML</a>.

Remark 1.2 Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M84">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M85">View MathML</a>.

We then establish the following theorem.

Theorem 1.2Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44">View MathML</a>, and assume (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M72">View MathML</a>) holds. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M90">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M91">View MathML</a>, (3) has a positive radial solution.

Finally, we prove a bifurcation result for the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M92">View MathML</a>

(6)

where Ω is a smooth bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M4">View MathML</a>, a is a positive parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M46">View MathML</a>. We prove the following.

Theorem 1.3The boundary value problem (6) has a branch of positive solutions bifurcating from the trivial branch of solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M97">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M98">View MathML</a> (as shown in Figure 1).

thumbnailFigure 1. Bifurcation diagram,avs.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M99">View MathML</a>for (6).

Our results are obtained via the method of sub-super solutions. By a subsolution of (2), we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M100">View MathML</a> that satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M101">View MathML</a>

and by a supersolution we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M102">View MathML</a> that satisfies:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M103">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M104">View MathML</a>. The following lemma was established in [13].

Lemma 1.4 (see [13,18])

Letψbe a subsolution of (2) andZbe a supersolution of (2) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M105">View MathML</a>in Ω. Then (2) has a solutionusuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M106">View MathML</a>in Ω.

Finding a positive subsolution, ψ, for such infinite semipositone problems is quite challenging since we need to construct ψ in such a way that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M108">View MathML</a> in a large part of the interior. In this paper, we achieve this by constructing subsolutions of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M109">View MathML</a>, where k is an appropriate positive constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M111">View MathML</a> is the eigenfunction corresponding to the first eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M112">View MathML</a> in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M113">View MathML</a> on Ω.

In Sections 2, 3, and 4, we provide proofs of our results. Section 5 is concerned with providing some exact bifurcation diagrams of positive solutions of (2) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M114">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M115">View MathML</a>.

2 Proof of Theorem 1.1

We first construct a subsolution. Consider the eigenvalue problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M116">View MathML</a> in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M113">View MathML</a> on Ω. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M111">View MathML</a> be an eigenfunction corresponding to the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M29">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M120">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M121">View MathML</a>. Also, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M122">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M123">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M125">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M126">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M127">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M128">View MathML</a> be fixed. Here, note that since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M130">View MathML</a>. Choose a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M131">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M132">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M133">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M134">View MathML</a> by the choice of k and β. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M135">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M136">View MathML</a>

To prove ψ is a subsolution, we need to establish:

(7)

in Ω if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M53">View MathML</a>. To achieve this, we split the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M139">View MathML</a> into three, namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M140">View MathML</a>

Now to prove (7) holds in Ω, it is enough to show the following three inequalities:

(8)

(9)

(10)

From the choice of k, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M144">View MathML</a>, hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M145">View MathML</a>

(11)

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M146">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M148">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M149">View MathML</a>

(12)

Finally, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M150">View MathML</a>, in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M124">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M152">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M153">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M154">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M155">View MathML</a>

(13)

From (11), (12) and (13) we see that equation (7) holds in Ω, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M53">View MathML</a>. Next, we construct a supersolution. Let e be the solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M157">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M158">View MathML</a> on Ω. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M159">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M160">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M162">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M163">View MathML</a>. Then Z is a supersolution of (2). Thus, Theorem 1.1 is proven.

3 Proof of Theorem 1.2

We begin the proof by constructing a subsolution. Consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M164">View MathML</a>

(14)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M111">View MathML</a> be an eigenfunction corresponding to the first eigenvalue of (14) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M120">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M167">View MathML</a>. Then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M168">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M169">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M170">View MathML</a>. Also, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M172">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M173">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M125">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M176">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M177">View MathML</a> be fixed and choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M131">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M179">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M180">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M181">View MathML</a> by the choice of k and β. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M109">View MathML</a>. This implies that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M183">View MathML</a>

To prove ψ is a subsolution, we need to establish:

(15)

Here, we note that the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M185">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M186">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M187">View MathML</a>. Now to prove (15) holds in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M188">View MathML</a>, it is enough to show the following three inequalities:

(16)

(17)

(18)

From the choice of k, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M192">View MathML</a>, hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M193">View MathML</a>

(19)

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M146">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M196">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M197">View MathML</a>

(20)

Next, we prove (18) holds in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M198">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M199">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M200">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M201">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M202">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M198">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M204">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M205">View MathML</a>

(21)

Proving (18) holds in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M206">View MathML</a> is straightforward since h is not singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M207">View MathML</a>. Thus, from equations (19), (20) and (21), we see that (15) holds in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M188">View MathML</a>. Hence, ψ is a subsolution. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M163">View MathML</a> where e satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M210">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M213">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M214">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M162">View MathML</a>. Then Z is a supersolution of (4) and there exists a solution u of (4) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M217">View MathML</a>. Thus, Theorem 1.2 is proven.

4 Proof of Theorem 1.3

We first prove (6) has a positive solution for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5">View MathML</a>. We begin by constructing a subsolution. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M111">View MathML</a> be as in the proof of Theorem 1.1 (see Section 2). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M110">View MathML</a>, and choose a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M131">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M222">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M109">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M224">View MathML</a>

To prove ψ is a subsolution, we will establish:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M225">View MathML</a>

(22)

in Ω. To achieve this, we rewrite the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M139">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M227">View MathML</a>. Now to prove (22) holds in Ω, it is enough to show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M228">View MathML</a>. From the choice of k, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M229">View MathML</a>, hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M230">View MathML</a>

Thus, ψ is a subsolution. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M231">View MathML</a> is a supersolution of (6). Since k, can be chosen small enough, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M105">View MathML</a>. Thus, (6) has a positive solution for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M5">View MathML</a>. Also, all positive solutions are bounded above by Z. Hence, when a is close to 0, every positive solution of (6) approaches 0. Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M234">View MathML</a> is a solution for every a. This implies we have a branch of positive solutions bifurcating from the trivial branch of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M97">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M98">View MathML</a>.

5 Numerical results

Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M237">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M44">View MathML</a>. Using the quadrature method (see [19]), the bifurcation diagram of positive solutions of (23) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M241">View MathML</a>

(24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M242">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M243">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M244">View MathML</a>. We plot the exact bifurcation diagram of positive solutions of (23) using Mathematica. Figure 2 shows bifurcation diagrams of positive solutions of (23) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M245">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M246">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M247">View MathML</a> for different values of α.

thumbnailFigure 2. Bifurcation diagrams,cvs.ρfor (23) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M248">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M249">View MathML</a>.

Bifurcation diagrams of positive solutions of (23) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M250">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M251">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M247">View MathML</a> for different values of α is shown in Figure 3.

thumbnailFigure 3. Bifurcation diagrams,cvs.ρfor (23) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M253">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M249">View MathML</a>.

Finally, we provide the exact bifurcation diagram for (6) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M115">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M114">View MathML</a>. Consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M257">View MathML</a>

(25)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M258">View MathML</a>. The bifurcation diagram of positive solutions of (25) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M259">View MathML</a>

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M260">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M261">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M244">View MathML</a>. The bifurcation diagram of positive solutions of (25) as well as the trivial solution branch are shown in Figure 4 when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M263">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M247">View MathML</a>.

thumbnailFigure 4. Bifurcation diagram,avs.ρfor (25) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M265">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/97/mathml/M249">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Equal contributions from all authors.

Acknowledgements

EK Lee was supported by 2-year Research Grant of Pusan National University.

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