Open Access

Existence of Solutions for Elliptic Systems with Nonlocal Terms in One Dimension

Boundary Value Problems20102011:518431

DOI: 10.1155/2011/518431

Received: 2 June 2010

Accepted: 26 August 2010

Published: 2 September 2010

Abstract

We study the solvability of a system of second-order differential equations with Dirichlet boundary conditions and non-local terms depending upon a parameter. The main tools used are a dual variational method and the topological degree.

1. Introduction

In the past decade there has been a lot of interest on boundary value problems for elliptic systems. For general systems of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq1_HTML.gif is a domain in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq2_HTML.gif , a survey was given by De Figueiredo in [1]. The specific case of one-dimensional systems, motivated by the problem of finding radial solutions to an elliptic system on an annulus of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq3_HTML.gif , has been considered by Dunninger and Wang [2] and by Lee [3], who have obtained conditions under which such a system may possess multiple positive solutions.

On the other hand, systems of two equations that include non-local terms have also been considered recently. These are of importance because they appear in the applied sciences, for example, as models for ignition of a compressible gas, or general physical phenomena where temperature has a central role in triggering a reaction. In fact their interest ranges from physics and engineering to population dynamics. See for instance [4]. The related parabolic problems are also of great interest in reaction-diffusion theory; see [57] where the approach to existence and blow-up for evolution systems with integral terms may be found.

In this paper we are interested in a simple one-dimensional model: the two-point boundary value problem for the system of second order differential equations with a linear integral term
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq6_HTML.gif . First we consider (1.2) as a perturbation of the nonlocal system and prove that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq8_HTML.gif grow linearly, then (1.2) has a solution provided https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq9_HTML.gif is not too large. Afterwards, assuming that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq11_HTML.gif are monotone, we will give estimates on the growth of these functions in terms of the parameter https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq12_HTML.gif to ensure solvability. This will be done on the basis of some spectral analysis for the linear part and a dual variational setting.

2. Preliminaries

Let us introduce some notation: we define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq13_HTML.gif as the Hilbert space of the Lebesgue measurable functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq14_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq15_HTML.gif with the usual inner product
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ3_HTML.gif
(2.1)
We also define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ4_HTML.gif
(2.2)
with the inner product
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ5_HTML.gif
(2.3)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq17_HTML.gif are both Hilbert spaces, we will consider the Hilbert product space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq18_HTML.gif with the inner product
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ6_HTML.gif
(2.4)

We first study the invertibility of the linear part of (1.2).

Lemma 2.1.

The linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq19_HTML.gif , defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ7_HTML.gif
(2.5)

is invertible if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq20_HTML.gif .

Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq22_HTML.gif are both continuous for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq23_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq24_HTML.gif . The equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq25_HTML.gif is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ8_HTML.gif
(2.6)
We denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq26_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq27_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq28_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq29_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ9_HTML.gif
(2.7)

is the Green's function associated to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq30_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq31_HTML.gif . Notice that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq32_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq33_HTML.gif are the solutions of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq37_HTML.gif , respectively.

Now it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq38_HTML.gif is a solution of (2.6) if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ10_HTML.gif
(2.8)
for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq39_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ11_HTML.gif
(2.9)

Clearly this linear system is uniquely solvable for each pair of functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq41_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq42_HTML.gif .

In order to prove the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq43_HTML.gif it is easy to show that there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq44_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ12_HTML.gif
(2.10)

By the open mapping theorem we deduce that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq46_HTML.gif , is continuous too.

In view of the previous lemma we will assume

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq48_HTML.gif .

Lemma 2.2.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq49_HTML.gif .Then the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq50_HTML.gif is compact and self-adjoint, where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq51_HTML.gif is the inclusion and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq52_HTML.gif .

Proof.

Since the inclusion https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq53_HTML.gif is compact (see [8, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq54_HTML.gif ]) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq56_HTML.gif are continuous we obtain the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq57_HTML.gif . On the other hand an easy computation shows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ13_HTML.gif
(2.11)

so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq58_HTML.gif is a self-adjoint operator.

3. An Existence Result of Perturbative Type

Let us introduce the basic assumption

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq61_HTML.gif are continuous functions,

and set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ14_HTML.gif
(3.1)

Theorem 3.1.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq63_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq64_HTML.gif .

Then problem (1.2) has a solution.

Proof.

Consider the homotopy https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq65_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq66_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq67_HTML.gif is the Nemitskii operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq68_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ15_HTML.gif
(3.2)
It is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq69_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq70_HTML.gif is a solution of problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ16_HTML.gif
(3.3)
We are going to prove that the possible solutions of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq71_HTML.gif are bounded independently of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq72_HTML.gif . By our assumptions, there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq73_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq75_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ17_HTML.gif
(3.4)
Multiplying the first equation of (3.3) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq76_HTML.gif , the second one by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq77_HTML.gif , integrating between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq79_HTML.gif and adding both equations we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ18_HTML.gif
(3.5)
On the other hand, by the Poincaré inequality (see [9, Chapter 2])
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ19_HTML.gif
(3.6)
so we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ20_HTML.gif
(3.7)

and since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq80_HTML.gif we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq82_HTML.gif are bounded.

Thus we may invoke the properties of the Leray-Schauder degree (see, e.g., [10]) to deduce the existence of a solution for (3.3) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq83_HTML.gif which is our problem (1.2).

Remark 3.2.

Notice that when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq84_HTML.gif the solution given by Theorem 3.1 may be the trivial one https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq85_HTML.gif . However, under our assumptions if moreover https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq86_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq87_HTML.gif we obtain a proper solution.

4. Monotone Nonlinearities

In the following lemma we give some estimates for the minimum eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq88_HTML.gif .

Lemma 4.1.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq89_HTML.gif . If one denotes by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq90_HTML.gif the minimum of the eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq91_HTML.gif , one has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq92_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq93_HTML.gif is the maximum value between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq94_HTML.gif and the greater positive solution of the equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ21_HTML.gif
(4.1)
More precisely, if one denotes by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ22_HTML.gif
(4.2)

one obtains that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq95_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq96_HTML.gif ,

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq97_HTML.gif ,

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq98_HTML.gif .

Proof.

By Lemma 2.2 the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq99_HTML.gif is compact, so its set of eigenvalues is bounded and nonempty (see [8, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq100_HTML.gif ]). Moreover we have that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq101_HTML.gif is a negative eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq102_HTML.gif if and only if there exists a pair https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq103_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq104_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ23_HTML.gif
(D)
Differentiating twice on the first equation of (D) and replacing on the second one, we arrive at the following equality:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ24_HTML.gif
(4.3)
In consequence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ25_HTML.gif
(4.4)
Analogously, differentiating twice on the second equation of ( ) and replacing on the first one, we arrive at
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ26_HTML.gif
(4.5)
Now, by means of the expression
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ27_HTML.gif
(4.6)
we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ28_HTML.gif
(4.7)
and thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ29_HTML.gif
(4.8)
So, we have that in the expression of the solutions of the two equations on system (D) six real parameters are involved. Now, to fix the value of such parameters, we use the four boundary value conditions imposed on problem(D) together with the fact that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ30_HTML.gif
(4.9)
Therefore, we arrive at the following six-dimensional homogeneous linear system:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ31_HTML.gif
(4.10)
In consequence, the values of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq109_HTML.gif for which there exist nontrivial solutions of system (D) coincide with the zeroes of the determinant of the matrix
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ32_HTML.gif
(4.11)
that is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ33_HTML.gif
(4.12)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ34_HTML.gif
(4.13)
We notice that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq111_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ35_HTML.gif
(4.14)
and for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq112_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq113_HTML.gif odd,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ36_HTML.gif
(4.15)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq114_HTML.gif is the greatest zero among the sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq115_HTML.gif . On the other hand, since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq116_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq117_HTML.gif is solution of (4.15) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq118_HTML.gif and the remaining solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq119_HTML.gif are the zeroes of the last two factors on (4.15). A careful study shows that function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ37_HTML.gif
(4.16)
is such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq120_HTML.gif is strictly decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq121_HTML.gif . Moreover
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ38_HTML.gif
(4.17)

In consequence, there is a (unique) solution greater than https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq122_HTML.gif of the equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq123_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq124_HTML.gif . Moreover the greatest zero of function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq125_HTML.gif belongs to the interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq126_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq127_HTML.gif .

On the other hand, function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ39_HTML.gif
(4.18)
satisfies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq128_HTML.gif is strictly decreasing on its domain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq129_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ40_HTML.gif
(4.19)
So, there is a (unique) solution greater than https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq130_HTML.gif of the equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq131_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq132_HTML.gif . Moreover, since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ41_HTML.gif
(4.20)

it has its greatest zero between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq134_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq135_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq136_HTML.gif denote the class of strictly increasing homeomorphisms from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq137_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq138_HTML.gif . We introduce the following assumption:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq140_HTML.gif

Let us define the functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq141_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ42_HTML.gif
(4.21)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq142_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq143_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq144_HTML.gif .

Notice that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq146_HTML.gif are the Fenchel transform of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq147_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq148_HTML.gif (see [11]).

Theorem 4.2.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq149_HTML.gif .Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq150_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq151_HTML.gif and in addition
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ43_HTML.gif
(4.22)

Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq152_HTML.gif attains a minimum at some point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq153_HTML.gif .

Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq154_HTML.gif is a solution of (1.2), where we put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq155_HTML.gif .

Proof.

Claim 1 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq156_HTML.gif attains a minimum at some point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq157_HTML.gif ).

The space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq158_HTML.gif is reflexive, and by our assumptions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq159_HTML.gif is weakly sequentially lower semicontinuous. In fact, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq160_HTML.gif is the sum of a convex continuous functional (corresponding to the two last summands in the integrand) with a weakly sequentially continuous functional (because of the compactness of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq161_HTML.gif ). So, in order to prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq162_HTML.gif has a minimum, it is enough to show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq163_HTML.gif is coercive. By (4.22) we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq164_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ44_HTML.gif
(4.23)
So, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq165_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ45_HTML.gif
(4.24)
Thus, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq166_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq167_HTML.gif such that we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ46_HTML.gif
(4.25)
On the other hand (see [8, Proposition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq168_HTML.gif ]),
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ47_HTML.gif
(4.26)
Taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq169_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq170_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ48_HTML.gif
(4.27)

and therefore https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq171_HTML.gif is coercive.

Claim 2.

If we denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq172_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq173_HTML.gif is a solution of (1.2).

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq174_HTML.gif is a critical point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq175_HTML.gif then for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq176_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ49_HTML.gif
(4.28)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq178_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq179_HTML.gif , where we put https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq180_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq181_HTML.gif is a solution of (1.2).

Remark 4.3.

Under the more restrictive assumption
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ50_HTML.gif
(4.29)

it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq182_HTML.gif is a strictly monotone operator (see [11]). Hence, when (4.29) holds, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq183_HTML.gif has a unique critical point. The argument of Claim 2 in previous theorem shows that there is a one-to-one correspondence between critical points of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq184_HTML.gif and the solutions to (1.2). In consequence, the solution of problem (1.2) is unique.

Remark 4.4.

Suppose that under the conditions of the theorem, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq185_HTML.gif . If moreover
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ51_HTML.gif
(4.30)
we claim that the solution given by the theorem is not the trivial one https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq186_HTML.gif . In fact let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq187_HTML.gif be a normalized eigenvector associated to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq188_HTML.gif . The properties of eigenvectors imply that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq189_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq190_HTML.gif are in fact continuous functions. Since (4.30) implies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq191_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq192_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq193_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq194_HTML.gif small, an easy computation implies that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ52_HTML.gif
(4.31)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq195_HTML.gif sufficiently small. Hence the minimum of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq196_HTML.gif is not attained at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq197_HTML.gif .

Remark 4.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq198_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq199_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq200_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq201_HTML.gif , we have that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq202_HTML.gif is a lower solution. Moreover if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq203_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ53_HTML.gif
(4.32)

then we can take an upper solution of the form https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq204_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq205_HTML.gif and then apply the monotone method.

Declarations

Acknowledgments

The authors are indebted to the anonymous referees for useful hints to improve the presentation of the paper. The first and the second authors were partially supported by Ministerio de Educación y Ciencia, Spain, Project MTM2007-61724. The third author was supported by FCT, Financiamento Base 2009.

Authors’ Affiliations

(1)
Departamento de Análise Matemática, Facultade de Matemáticas, Campus Sur, Universidade de Santiago de Compostela
(2)
Departamento de Matemáticas, Universidad de Jaén
(3)
Faculdade de Ciências da Universidade de Lisboa, Centro de Matemática e Aplicações Fundamentais

References

  1. de Figueiredo DG: Nonlinear elliptic systems. Anais da Academia Brasileira de Ciências 2000, 72(4):453-469. 10.1590/S0001-37652000000400002View ArticleMathSciNetGoogle Scholar
  2. Dunninger DR, Wang H: Multiplicity of positive radial solutions for an elliptic system on an annulus. Nonlinear Analysis: Theory, Methods & Applications 2000, 42(5):803-811. 10.1016/S0362-546X(99)00125-XView ArticleMathSciNetGoogle Scholar
  3. Lee Y-H: Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus. Journal of Differential Equations 2001, 174(2):420-441. 10.1006/jdeq.2000.3915View ArticleMathSciNetGoogle Scholar
  4. Corrêa FJSA, Lopes FPM: Positive solutions for a class of nonlocal elliptic systems. Communications on Applied Nonlinear Analysis 2007, 14(2):67-77.MathSciNetGoogle Scholar
  5. Deng W, Li Y, Xie C: Blow-up and global existence for a nonlocal degenerate parabolic system. Journal of Mathematical Analysis and Applications 2003, 277(1):199-217. 10.1016/S0022-247X(02)00533-4View ArticleMathSciNetGoogle Scholar
  6. Zhang R, Yang Z: Global existence and blow-up solutions and blow-up estimates for a non-local quasilinear degenerate parabolic system. Applied Mathematics and Computation 2008, 200(1):267-282. 10.1016/j.amc.2007.11.012View ArticleMathSciNetGoogle Scholar
  7. Li F, Chen Y, Xie C: Asymptotic behavior of solution for nonlocal reaction-diffusion system. Acta Mathematica Scientia. Series B 2003, 23(2):261-273.MathSciNetGoogle Scholar
  8. Brezis H: Analyse Fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, France; 1983:xiv+234. Théorie et applicationsGoogle Scholar
  9. Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications. Volume 53. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.View ArticleGoogle Scholar
  10. Zeidler E: Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems. Springer, New York; 1986:xxi+897.View ArticleGoogle Scholar
  11. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989:xiv+277.View ArticleGoogle Scholar

Copyright

© Alberto Cabada et al. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.