Existence of Solutions for Elliptic Systems with Nonlocal Terms in One Dimension
© Alberto Cabada et al. 2011
Received: 2 June 2010
Accepted: 26 August 2010
Published: 2 September 2010
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.
where is a domain in , a survey was given by De Figueiredo in . 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  and by Lee , 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 . 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.
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.
We first study the invertibility of the linear part of (1.2).
is invertible if and only if .
Moreover, and are both continuous for .
is the Green's function associated to , . Notice that , are the solutions of , and , , respectively.
Clearly this linear system is uniquely solvable for each pair of functions , if and only if .
By the open mapping theorem we deduce that , , is continuous too.
In view of the previous lemma we will assume
Assume .Then the operator is compact and self-adjoint, where is the inclusion and .
so is a self-adjoint operator.
3. An Existence Result of Perturbative Type
Let us introduce the basic assumption
and are continuous functions,
Assume , , and .
Then problem (1.2) has a solution.
and since we obtain that and are bounded.
Thus we may invoke the properties of the Leray-Schauder degree (see, e.g., ) to deduce the existence of a solution for (3.3) with which is our problem (1.2).
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 .
one obtains that
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 .
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:
where and for all .
Notice that and are the Fenchel transform of and (see ).
Then attains a minimum at some point .
Moreover, is a solution of (1.2), where we put .
Claim 1 ( attains a minimum at some point ).
and therefore is coercive.
If we denote then is a solution of (1.2).
which implies that and for a.e. , where we put . Then is a solution of (1.2).
it follows that is a strictly monotone operator (see ). 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.
for sufficiently small. Hence the minimum of is not attained at .
then we can take an upper solution of the form with and then apply the monotone method.
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.
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