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

  • Alberto Cabada1,

    Affiliated with

    • J. Ángel Cid2 and

      Affiliated with

      • Luís Sanchez3Email author

        Affiliated with

        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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq1_HTML.gif is a domain in http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ2_HTML.gif
        (1.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq5_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq7_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq10_HTML.gif and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq14_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq15_HTML.gif with the usual inner product
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ3_HTML.gif
        (2.1)
        We also define
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ4_HTML.gif
        (2.2)
        with the inner product
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ5_HTML.gif
        (2.3)
        If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq16_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq18_HTML.gif with the inner product
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq19_HTML.gif , defined by
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq20_HTML.gif .

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

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq24_HTML.gif . The equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq25_HTML.gif is equivalent to
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ8_HTML.gif
        (2.6)
        We denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq26_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq27_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq28_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq29_HTML.gif , where
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq31_HTML.gif . Notice that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq33_HTML.gif are the solutions of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq34_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq36_HTML.gif , http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ10_HTML.gif
        (2.8)
        for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq39_HTML.gif such that
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq41_HTML.gif if and only if http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq44_HTML.gif such that
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq45_HTML.gif , http://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

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

        Lemma 2.2.

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

        Proof.

        Since the inclusion http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq53_HTML.gif is compact (see [8, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq54_HTML.gif ]) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq55_HTML.gif and http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ13_HTML.gif
        (2.11)

        so http://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

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

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

        Theorem 3.1.

        Assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq62_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq63_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq65_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq66_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq67_HTML.gif is the Nemitskii operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq68_HTML.gif given by
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq69_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq70_HTML.gif is a solution of problem
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq71_HTML.gif are bounded independently of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq72_HTML.gif . By our assumptions, there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq73_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq75_HTML.gif such that
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq76_HTML.gif , the second one by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq77_HTML.gif , integrating between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq79_HTML.gif and adding both equations we obtain
        http://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])
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ19_HTML.gif
        (3.6)
        so we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ20_HTML.gif
        (3.7)

        and since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq80_HTML.gif we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq81_HTML.gif and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq85_HTML.gif . However, under our assumptions if moreover http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq86_HTML.gif or http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq88_HTML.gif .

        Lemma 4.1.

        Assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq89_HTML.gif . If one denotes by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq90_HTML.gif the minimum of the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq91_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq92_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq93_HTML.gif is the maximum value between http://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
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ22_HTML.gif
        (4.2)

        one obtains that

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

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

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

        (iv) http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq100_HTML.gif ]). Moreover we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq101_HTML.gif is a negative eigenvalue of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq103_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq104_HTML.gif , such that
        http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ24_HTML.gif
        (4.3)
        In consequence
        http://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
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ27_HTML.gif
        (4.6)
        we deduce that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ28_HTML.gif
        (4.7)
        and thus
        http://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
        http://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:
        http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ32_HTML.gif
        (4.11)
        that is
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ33_HTML.gif
        (4.12)
        where
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq111_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ35_HTML.gif
        (4.14)
        and for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq112_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq113_HTML.gif odd,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ36_HTML.gif
        (4.15)
        Hence, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq115_HTML.gif . On the other hand, since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq116_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq118_HTML.gif and the remaining solutions http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ37_HTML.gif
        (4.16)
        is such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq120_HTML.gif is strictly decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq121_HTML.gif . Moreover
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq122_HTML.gif of the equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq123_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq124_HTML.gif . Moreover the greatest zero of function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq125_HTML.gif belongs to the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq126_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq127_HTML.gif .

        On the other hand, function
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ39_HTML.gif
        (4.18)
        satisfies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq128_HTML.gif is strictly decreasing on its domain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq129_HTML.gif , and
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq130_HTML.gif of the equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq131_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq132_HTML.gif . Moreover, since
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq134_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq135_HTML.gif .

        Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq137_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq138_HTML.gif . We introduce the following assumption:

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

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

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

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

        Theorem 4.2.

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

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

        Moreover, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq155_HTML.gif .

        Proof.

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

        The space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq158_HTML.gif is reflexive, and by our assumptions http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq161_HTML.gif ). So, in order to prove that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq164_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ44_HTML.gif
        (4.23)
        So, there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq165_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ45_HTML.gif
        (4.24)
        Thus, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq166_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq167_HTML.gif such that we have
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq168_HTML.gif ]),
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ47_HTML.gif
        (4.26)
        Taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq169_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq170_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ48_HTML.gif
        (4.27)

        and therefore http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq172_HTML.gif then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq174_HTML.gif is a critical point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq175_HTML.gif then for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq176_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ49_HTML.gif
        (4.28)

        which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq178_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq179_HTML.gif , where we put http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq180_HTML.gif . Then http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ50_HTML.gif
        (4.29)

        it follows that http://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, http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq185_HTML.gif . If moreover
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq186_HTML.gif . In fact let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq187_HTML.gif be a normalized eigenvector associated to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq188_HTML.gif . The properties of eigenvectors imply that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq189_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq191_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq192_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq193_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq194_HTML.gif small, an easy computation implies that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_Equ52_HTML.gif
        (4.31)

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

        Remark 4.5.

        If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq198_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq199_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq200_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq201_HTML.gif , we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq202_HTML.gif is a lower solution. Moreover if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq203_HTML.gif and
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F518431/MediaObjects/13661_2010_Article_44_IEq204_HTML.gif with http://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

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