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

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

(1.1)

where is a domain in , 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 , 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 [5–7] 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

(1.2)

where , and . First we consider (1.2) as a perturbation of the nonlocal system and prove that if and grow linearly, then (1.2) has a solution provided is not too large. Afterwards, assuming that and are monotone, we will give estimates on the growth of these functions in terms of the parameter 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 as the Hilbert space of the Lebesgue measurable functions such that with the usual inner product

(2.1)

We also define

(2.2)

with the inner product

(2.3)

If and are both Hilbert spaces, we will consider the Hilbert product space with the inner product

(2.4)

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

Lemma 2.1.

The linear operator , defined by

(2.5)

is invertible if and only if .

Moreover, and are both continuous for .

Proof.

Let . The equation is equivalent to

(2.6)

We denote , , , and , where

(2.7)

is the Green's function associated to , . Notice that , are the solutions of , and , , respectively.

Now it is easy to see that is a solution of (2.6) if and only if

(2.8)

for some such that

(2.9)

Clearly this linear system is uniquely solvable for each pair of functions , if and only if .

In order to prove the continuity of it is easy to show that there exists such that

(2.10)

By the open mapping theorem we deduce that , , is continuous too.

In view of the previous lemma we will assume

.

Lemma 2.2.

Assume .Then the operator is compact and self-adjoint, where is the inclusion and .

Proof.

Since the inclusion is compact (see [8, Theorem ]) and and are continuous we obtain the compactness of . On the other hand an easy computation shows that

(2.11)

so is a self-adjoint operator.

3. An Existence Result of Perturbative Type

Let us introduce the basic assumption

and are continuous functions,

and set

(3.1)

Theorem 3.1.

Assume , , and .

Then problem (1.2) has a solution.

Proof.

Consider the homotopy for all , where is the Nemitskii operator given by

(3.2)

It is easy to check that if and only if is a solution of problem

(3.3)

We are going to prove that the possible solutions of are bounded independently of . By our assumptions, there exist , and such that

(3.4)

Multiplying the first equation of (3.3) by , the second one by , integrating between and and adding both equations we obtain

(3.5)

On the other hand, by the Poincaré inequality (see [9, Chapter 2])

(3.6)

so we have

(3.7)

and since we obtain that and 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 which is our problem (1.2).

Remark 3.2.

Notice that when the solution given by Theorem 3.1 may be the trivial one . However, under our assumptions if moreover or we obtain a proper solution.

4. Monotone Nonlinearities

In the following lemma we give some estimates for the minimum eigenvalue of .

Lemma 4.1.

Assume . If one denotes by the minimum of the eigenvalues of , one has , where is the maximum value between and the greater positive solution of the equation

(4.1)

More precisely, if one denotes by

(4.2)

one obtains that

(i),

(ii),

(iii),

(iv).

Proof.

By Lemma 2.2 the operator is compact, so its set of eigenvalues is bounded and nonempty (see [8, Theorem ]). Moreover we have that is a negative eigenvalue of if and only if there exists a pair , , such that

(D)

Differentiating twice on the first equation of (D) and replacing on the second one, we arrive at the following equality:

(4.3)

In consequence

(4.4)

Analogously, differentiating twice on the second equation of ( ) and replacing on the first one, we arrive at

(4.5)

Now, by means of the expression

(4.6)

we deduce that

(4.7)

and thus

(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

(4.9)

Therefore, we arrive at the following six-dimensional homogeneous linear system:

(4.10)

In consequence, the values of for which there exist nontrivial solutions of system (D) coincide with the zeroes of the determinant of the matrix

(4.11)

that is

(4.12)

where

(4.13)

We notice that for all we have

(4.14)

and for all , with odd,

(4.15)

Hence, is the greatest zero among the sequence . On the other hand, since , is solution of (4.15) if and only if and the remaining solutions are the zeroes of the last two factors on (4.15). A careful study shows that function

(4.16)

is such that is strictly decreasing on . Moreover

(4.17)

In consequence, there is a (unique) solution greater than of the equation if and only if . Moreover the greatest zero of function belongs to the interval if and only if .

On the other hand, function

(4.18)

satisfies that is strictly decreasing on its domain , and

(4.19)

So, there is a (unique) solution greater than of the equation if and only if . Moreover, since

(4.20)

it has its greatest zero between and if and only if .

Let denote the class of strictly increasing homeomorphisms from onto . We introduce the following assumption:

Let us define the functional given by

(4.21)

where and for all .

Notice that and are the Fenchel transform of and (see [11]).

Theorem 4.2.

Assume .Let satisfy and in addition

(4.22)

Then attains a minimum at some point .

Moreover, is a solution of (1.2), where we put .

Proof.

Claim 1 ( attains a minimum at some point ).

The space is reflexive, and by our assumptions is weakly sequentially lower semicontinuous. In fact, 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 ). So, in order to prove that has a minimum, it is enough to show that is coercive. By (4.22) we take such that

(4.23)

So, there exists such that

(4.24)

Thus, for every , there exists such that we have

(4.25)

On the other hand (see [8, Proposition ]),

(4.26)

Taking such that , we have

(4.27)

and therefore is coercive.

Claim 2.

If we denote then is a solution of (1.2).

Since is a critical point of then for all we have

(4.28)

which implies that and for a.e. , where we put . Then is a solution of (1.2).

Remark 4.3.

Under the more restrictive assumption

(4.29)

it follows that is a strictly monotone operator (see [11]). Hence, when (4.29) holds, 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 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, . If moreover

(4.30)

we claim that the solution given by the theorem is not the trivial one . In fact let be a normalized eigenvector associated to . The properties of eigenvectors imply that and are in fact continuous functions. Since (4.30) implies and for some and small, an easy computation implies that

(4.31)

for sufficiently small. Hence the minimum of is not attained at .

Remark 4.5.

If and or and , we have that is a lower solution. Moreover if and

(4.32)

then we can take an upper solution of the form with and then apply the monotone method.