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Positive solution for a class of coupled (p,q)-Laplacian nonlinear systems

Eder M Martins* and Wenderson M Ferreira

Author Affiliations

Departamento de Matemática, Universidade Federal de Ouro Preto, Ouro Preto, MG, 35.400-000, Brazil

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Boundary Value Problems 2014, 2014:21  doi:10.1186/1687-2770-2014-21

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/21


Received:21 May 2013
Accepted:20 November 2013
Published:20 January 2014

© 2014 Martins and Ferreira; 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

In this article, we prove the existence of a nontrivial positive solution for the elliptic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M2">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M3">View MathML</a> denotes the p-Laplacian operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M4">View MathML</a> and Ω is a smooth bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M6">View MathML</a>). The weight functions ω and ρ are continuous, nonnegative and not identically null in Ω, and the nonlinearities f and g are continuous and satisfy simple hypotheses of local behavior, without involving monotonicity hypotheses or conditions at ∞. We apply the fixed point theorem in a cone to obtain our result.

MSC: 35B09, 35J47, 58J20.

1 Introduction

Coupled systems involving quasilinear operators as the p-Laplacian have been a theme of interest for researchers of partial differential equations. In this paper we prove the existence of a nontrivial positive solution for the elliptic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M7">View MathML</a>

(P)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M3">View MathML</a> denotes the p-Laplacian operator defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M4">View MathML</a> and Ω denotes a smooth bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M6">View MathML</a>). In other words, we will prove the existence of a pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M13">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14">View MathML</a> satisfies (P), with u and v strictly positive in Ω. The weight functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M15">View MathML</a> are continuous, nonnegative in Ω and positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M16">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M17">View MathML</a>. The nonlinearities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M18">View MathML</a> are continuous, g is positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M19">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M20">View MathML</a>, and both satisfy simple hypotheses of local behavior.

We suppose that the nonlinearity f is superlinear at origin and f, g are allowed to be sub- or superlinear at ∞. Moreover, there is no monotonicity hypotheses on these nonlinearities. We suppose the existence of positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M21">View MathML</a> such that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M23">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M24">View MathML</a>,

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M25">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M26">View MathML</a>,

where the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28">View MathML</a> depend only on ω, ρ and Ω (see Figure 1). These constants will be defined later on in this paper and, as proved in [1], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M29">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M30">View MathML</a> is the first eigenvalue of the p-Laplacian operator.

thumbnailFigure 1. The nonlinearitiesfandgsatisfy (H1) and (H2).

Elliptic problems concerning the existence of positive solutions for equations and systems of equations related to Dirichlet problems have been studied in several papers during the last decades. In this way, many existence results for systems involving the p-Laplacian operator in general bounded domains in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M5">View MathML</a> have been considered in recent articles. In particular, systems as (P) have been studied in articles in [2-6] for example.

The main interest of this paper is studying systems whose nonlinearities present some kind of coupling as (P). A paper which deals with this sort of problem is [2], where problem (P) is considered under the assumptions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M33">View MathML</a>. In this paper, among other hypotheses, the nonlinearities f and g are supposed to be at least continuous if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M34">View MathML</a> and locally Holder continuous with exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M35">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M6">View MathML</a>. Moreover, both are supposed to be nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M37">View MathML</a>, nonnegative at origin and satisfying (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M38">View MathML</a>) the fundamental condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M39">View MathML</a>

(1)

Schauder’s fixed point theorem, the Leray-Schauder degree and a variant of Krasnoselskii’s method are applied to guarantee the existence of a positive solution for (P).

The studies of [2] were extended by Hai and Shivaji in [4] (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M40">View MathML</a>), [5] (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M32">View MathML</a>) and by Hai in [3] (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M4">View MathML</a>). In these papers, the authors deal with problem (P), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M44">View MathML</a> (in [4] and [5], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M45">View MathML</a>), with no sign conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M46">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M47">View MathML</a> and without monotonicity conditions on f or g. In this way, semipositone cases were also considered in these papers (for more details about semipositone problems, see [7] and the references therein).

In [4], the nonlinearities f and g in (P) are supposed to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M48">View MathML</a>, monotone and satisfying the following conditions at ∞

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M49">View MathML</a>

(2)

in addition to condition (1). The existence of a positive solution is guaranteed for large λ by applying the sub-supersolution method.

The paper [5] deals with problem (P) in the particular case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M38">View MathML</a> and a positive solution is guaranteed by applying the sub- and supersolution method and Schauder’s fixed point theorem. The nonlinearities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M51">View MathML</a> considered are continuous and there exist positive numbers L, K such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M53">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M54">View MathML</a>. Moreover, the authors considered, as it has been done in [2], condition (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M38">View MathML</a>.

In [3], the author obtains necessary and sufficient conditions for the existence of positive solutions for problem (P). The nonlinearities f and g are positive, continuous and nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M56">View MathML</a>, with g strictly positive for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M57">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M58">View MathML</a>

(3)

sublinear at 0 and ∞. In this paper, the maximum principle and fixed point arguments are applied to guarantee the existence of a solution.

Another paper dealing with the existence of a positive solution for a class of coupled systems is [6]. In this paper, the authors studied problem (P), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M60">View MathML</a>, in which λ is a positive parameter. The existence of a solution is guaranteed via the method of sub- and supersolution if, among other assumptions, the functions a and b considered are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M48">View MathML</a> sign-changing functions that may be negative near the boundary and the positive nonlinearities f and g are supposed to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M48">View MathML</a> and nondecreasing.

Recently, many articles have applied fixed point results to prove the existence of positive solutions of partial differential equations or systems (see, for example, [1,8-12]). In this paper we study problem (P) in general domains, assuming that (H1) and (H2) hold. As system (P) has no variational structure, our main arguments are based on fixed-point index and comparison theorems, following the ideas of [8,10] and [11]. In particular, our assumptions on the nonlinearities do not involve monotonicity hypotheses or sublinearity conditions at ∞.

Our strategy is as follows. At first, we show an existence result for the radial case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M63">View MathML</a>, applying a fixed point theorem in a cone. Afterwards, we utilize this result to prove our main existence result for (P), when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M64">View MathML</a> is a bounded smooth domain. In this case, we do a symmetrization of weigh functions and combine comparison theorems with a new application of the fixed point theorem.

For completeness, we will consider concrete examples of coupled systems for which it is possible to apply our method to guarantee the existence of at least one positive solution. It will be clear in some of these examples that conditions (1), (2) and (3) are not required in our method.

2 The radial case

Let us consider the radial version of problem (P)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M66">View MathML</a> and the weight functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M67">View MathML</a> are radial, continuous, nonnegative and not identically null functions. The positive functions f and g are supposed to be continuous and satisfying local conditions that depend on the positive constants defined in (6) and (7).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M68">View MathML</a> (the inverse of the well-known function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M69">View MathML</a>), and let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M70">View MathML</a>

(4)

Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M71">View MathML</a> the real number such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M72">View MathML</a>

(5)

Moreover, define the positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M75">View MathML</a>

(6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M76">View MathML</a>

(7)

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M77">View MathML</a>. In fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M78">View MathML</a>

Remark 1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M79">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M80">View MathML</a>) is the unitary ball and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M81">View MathML</a> are the weight functions in problem (P), it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28">View MathML</a> are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M84">View MathML</a>

(8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M85">View MathML</a>

(9)

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M87">View MathML</a> are the conjugate exponents of p and q, respectively.

Finally we assume that the nonlinearities f and g satisfy the local conditions.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M89">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M90">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M91">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M25">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M93">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M94">View MathML</a>.

Now, we are in a position to state the main result of this section: the existence of a positive radial solution for (Pr).

Theorem 2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M95">View MathML</a>are positive, continuous nonlinearities satisfying (H1) and (H2) (with constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28">View MathML</a>defined in (6) and (7)), and let the radial weight functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M98">View MathML</a>be continuous, nonnegative and not identically null. Then problem (Pr) has at least a nontrivial positive solution. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14">View MathML</a>is a positive solution for (Pr), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M100">View MathML</a>

To prove the last theorem, we will apply a well-known result of the fixed-point index theory, known as a fixed point cone theorem (see, for example, [13]).

Lemma 3LetEbe a Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M101">View MathML</a>be its norm. LetKbe a cone inE. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M102">View MathML</a>, define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M103">View MathML</a>and denote its boundary by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M104">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M105">View MathML</a>. Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M106">View MathML</a>

is completely continuous.

(i) If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M107">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M108">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M109">View MathML</a>

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M110">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M111">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M112">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M111">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M114">View MathML</a>

In what follows, we will consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M115">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M116">View MathML</a>

and the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M117">View MathML</a>.

Proof of Theorem 2 It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14">View MathML</a> is a solution of (Pr) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14">View MathML</a> is a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M120">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M121">View MathML</a>

(10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M122">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M123">View MathML</a>

Moreover, it is straightforward that the operator T is completely continuous.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M124">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M125">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M126">View MathML</a>

In fact, by (H1) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M127">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M128">View MathML</a>

As a consequence of Lemma 3, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M129">View MathML</a>

Now, we will prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M130">View MathML</a>. In order to prove that, we will show that exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M131">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M132">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M134">View MathML</a>, be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M135">View MathML</a>

and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M136">View MathML</a>

We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M137">View MathML</a>

In fact, let us suppose that there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M139">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M140">View MathML</a>

(11)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M141">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M142">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M143">View MathML</a> be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M144">View MathML</a>

(12)

(Note that it is immediate that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M145">View MathML</a>. In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M146">View MathML</a> and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M147">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M148">View MathML</a>, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M149">View MathML</a>.)

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M150">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M151">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M152">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M153">View MathML</a>, by (11) and (H2) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M154">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M155">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M156">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M157">View MathML</a>

which contradicts (12).

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M158">View MathML</a>

and, consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M159">View MathML</a>

Thus, T has a nontrivial fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M160">View MathML</a>. The regularity of the solution follows from classical results of Lieberman and Tolksdorf (see [14] and [15]). □

3 The general case

Now we will establish the main result of this paper: the existence of a nontrivial positive solution for (P) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M161">View MathML</a> is a smooth bounded domain.

3.1 The constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28">View MathML</a> in Ω

For the general problem

we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M120">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M166">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M167">View MathML</a>

(13)

It is well known in p-Laplacian operator theory that T is completely continuous. Moreover, a simple maximum principle argument guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M168">View MathML</a> in Ω for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M169">View MathML</a>.

As it has been done in the radial case, in order to obtain a result of existence for (P), we will apply Lemma 3.

Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M170">View MathML</a> the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M171">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M172">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M173">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M174">View MathML</a>. As ρ and w satisfy the same hypotheses as h, we can define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M175">View MathML</a>

(14)

Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M176">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M177">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M102">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M179">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M180">View MathML</a>

(15)

Furthermore, let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M182">View MathML</a>

(16)

(where ξ is defined similarly as it has been done in (4) and (5)).

3.2 Main theorem

Theorem 4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M95">View MathML</a>are positive, continuous nonlinearities satisfying (H1) and (H2) (with constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M27">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M28">View MathML</a>defined in (14) and (16), respectively), and let the weight functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M186">View MathML</a>be continuous, nonnegative and not identically null. Then problem (P) has at least a nontrivial positive solution. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14">View MathML</a>is a positive solution for (P), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M188">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M124">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M125">View MathML</a>. It follows from (H1) and (13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M191">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M192">View MathML</a>

Therefore, by the maximum principle, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M193">View MathML</a>

and, consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M194">View MathML</a>

In the same way, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M195">View MathML</a>

and, as a consequence of the last two inequalities, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M196">View MathML</a>

It follows from Lemma 3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M197">View MathML</a>

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M130">View MathML</a>. To prove our claim, we will show, as it has been done in the radial case, that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M199">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M200">View MathML</a>

For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M176">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M202">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M203">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M204">View MathML</a>

Consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M205">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M206">View MathML</a>.

We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M207">View MathML</a>

In fact, suppose that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M139">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M210">View MathML</a>

(17)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M211">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M141">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M213">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M214">View MathML</a> be the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M215">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M216">View MathML</a>

(18)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M217">View MathML</a>

Note that given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M218">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M219">View MathML</a>

and by the maximum principle, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M220">View MathML</a>

Thus, by (17) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M221">View MathML</a>

Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M222">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M223">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M224">View MathML</a>, we conclude from (18) that there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M225">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M226">View MathML</a>

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M227">View MathML</a> and considering

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M228">View MathML</a>

(19)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M229">View MathML</a>

by the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M230">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M225">View MathML</a>. Repeating the same ideas of the radial case, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M232">View MathML</a>

which contradicts (19). By the additivity of index, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M233">View MathML</a>

proving the theorem. As in the radial case, the regularity follows from [14] and [15]. □

Remark 5 Due to the hypotheses on the nonlinearities and on the weight functions, simple applications of the maximum principle allow us to guarantee that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14">View MathML</a> is a solution of problem (P), then both u and v are strictly positive in Ω.

4 Examples

Example 6 Let us consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M235">View MathML</a>

(20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M64">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M80">View MathML</a>) is a smooth domain. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M240">View MathML</a>, problem (20) has at least one positive solution.

Considering constants (8) and (9), it is enough to consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M241">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M242">View MathML</a>

(21)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M243">View MathML</a>

(22)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M244">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M245">View MathML</a>.

Choosing δ and M as above, hypotheses (H1) and (H2) are verified and, as a consequence of Theorem 2, we guarantee the existence of a positive solution for coupled system (20). Furthermore, according to Theorem 4, it is easy to see that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M14">View MathML</a> is the considered positive solution, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M247">View MathML</a> as large as α is.

One of the advantages of our method is that conditions (1), (2) and (3) are not required. Let us see examples of these situations.

Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M248">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M64">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M80">View MathML</a>) is a smooth domain and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M252">View MathML</a> is a nonlinearity satisfying (H1) and (H2). Examples of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M252">View MathML</a> will be presented in the following examples.

Example 7 Consider the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M252">View MathML</a> in problem (23) given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M255">View MathML</a>

(24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M257">View MathML</a> and M is the positive constant whose existence is guaranteed in Example 6. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M258">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M259">View MathML</a>, the same arguments as those applied in Example 6 can guarantee the existence of a positive solution to (23).

It is clear that according to the constant k, condition (1) is not satisfied by the nonlinearities of problem (23). In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M261">View MathML</a>, simple calculations show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M262">View MathML</a>

and condition (1) does not hold.

Example 8 Now, let us consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M252">View MathML</a> given by (24) as the nonlinearity of problem (23). In this case, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M264">View MathML</a>

In this way, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M265">View MathML</a> can be either sub- or superlinear at +∞ according to the constant k. Therefore, we have an example in which we guarantee the existence of a positive solution even if condition (3) is not satisfied.

Example 9 Finally, let us consider problem (23), with the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M252">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M267">View MathML</a>

in which M is the positive constant whose existence is guaranteed in Example 6, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M258">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M259">View MathML</a>. As a consequence of previous examples, it is straightforward to guarantee the existence of a positive solution to this problem. Furthermore, it is clear that condition (2) does not hold.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

The authors thank for the support of FAPEMIG and Universidade Federal de Ouro Preto.

References

  1. Bueno, H, Ercole, G, Zumpano, A: Positive solutions for the p-Laplacian and bounds for its first eigenvalue. Adv. Nonlinear Stud.. 9, 313–338 (2009)

  2. Dalmasso, R: Existence and uniqueness of positive solutions for some quasilinear elliptic systems. Nonlinear Anal.. 39, 559–568 (2000). Publisher Full Text OpenURL

  3. Hai, DD: Existence and uniqueness of solutions for quasilinear elliptic systems. Proc. Am. Math. Soc.. 133(1), 223–228 (2004)

  4. Hai, DD, Shivaji, R: An existence result on positive solutions for a class of p-Laplacian systems. Nonlinear Anal.. 56, 1007–1010 (2004). Publisher Full Text OpenURL

  5. Hai, DD, Shivaji, R: An existence result on positive solutions for a class of semilinear elliptic systems. Proc. R. Soc. Edinb. A. 134, 137–141 (2004). Publisher Full Text OpenURL

  6. Rasouli, SH, Halimi, Z, Mashhadban, Z: A remark on the existence of positive weak solution for a class of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/21/mathml/M270">View MathML</a>-Laplacian nonlinear system with sign-changing weight. Nonlinear Anal.. 73, 385–389 (2010). Publisher Full Text OpenURL

  7. Castro, A, Maya, C, Shivaji, R: Nonlinear eigenvalue problems with semipositone structure. Electron. J. Differ. Equ.. 5, 33–49 (2000)

  8. Bueno, H, Ercole, G, Ferreira, W, Zumpano, A: Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient. J. Math. Anal. Appl.. 343, 151–158 (2008). Publisher Full Text OpenURL

  9. Hai, DD, Wang, H: Nontrivial solutions for p-Laplacian systems. J. Math. Anal. Appl.. 330, 186–194 (2007). Publisher Full Text OpenURL

  10. O’Regan, D, Wang, H: Positive radial solutions for p-Laplacian systems. Aequ. Math.. 75, 43–50 (2008). Publisher Full Text OpenURL

  11. Wang, H: An existence theorem for quasilinear systems. Proc. Edinb. Math. Soc.. 49, 505–511 (2006). Publisher Full Text OpenURL

  12. Wang, H: Existence and nonexistence of positive radial solutions for quasilinear systems. Discrete Contin. Dyn. Syst.. 2009, 810–817 (2009) suppl.

  13. Deimling, K: Nonlinear Functional Analysis, Springer, Berlin (1985)

  14. Lieberman, GM: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal.. 12, 1203–1219 (1988). Publisher Full Text OpenURL

  15. Tolksdorf, P: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ.. 51, 126–150 (1984). Publisher Full Text OpenURL