Radial solutions for a nonlocal boundary value problem
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We consider the boundary value problem for the nonlinear Poisson equation with a nonlocal term −Δu = f (u, U g(u)), u|∂U = 0. We prove the existence of a positive radial solution when f grows linearly in u, using Krasnoselskii’s fixed point theorem together with eigenvalue theory. In presence of upper and lower solutions, we consider monotone approximation to solutions. Copyright © 2006 R. Enguic¸a and L. Sanchez. 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. 1. Introduction Let us consider the following nonlocal BVP in a ball U = B(0,R) of Rn : −Δu = f u,
U
g(u) , (1.1)
u|∂U = 0, where f and g are continuous functions. For simplicity we shall take R = 1. We want to study the existence of positive radial solutions
u(x) = v x ,
(1.2)
of (1.1). This may be seen as the stationary problem corresponding to a class of nonlocal evolution (parabolic) boundary value problems related to relevant phenomena in engineering and physics. The literature dealing with such problems has been growing in the last decade. The reader may find some hints on the motivation for the study of this mathematical model, for example, in the paper by Bebernes and Lacey [1]. For more recent developments, see [2] and the references therein.
Hindawi Publishing Corporation Boundary Value Problems Volume 2006, Article ID 32950, Pages 1–18 DOI 10.1155/BVP/2006/32950
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Radial solutions for a nonlocal boundary value problem
Here we are considering a nonlocal term inserted in the right-hand side of the equation. Note, however, that it is also of interest to study boundary value problems where the nonlocal expression appears in a boundary condition. We refer the reader to the recent paper by Yang [13] and its references. When dealing with a nonlinear term with rather general dependence on the nonlocal functional as in (1.1) new difficulties arise with respect to the treatment of standard boundary value problems. Differences of behaviour which are met in general elliptic and parabolic problems are already present in simple models as those we shall analyse in this paper. For instance, the use of the powerful lower and upper solution method (good accounts of which can be consulted in the monographs of Pao [10] and De Coster and Habets [3]) is limited by the absence of general maximum principles. Even for linear problems with nonlocal terms the issue of positivity is far from trivial and may require a detailed study via the analysis of the Green’s operator, as in Freitas and Sweers [6]. The purpose of this paper is twofold. First, we want to improve a quite recent result of Fijałkowski and Przeradzki [5]: these authors have obtained existence of positive radial solutions of (1.1) by using Krasnoselskii’s fixed point theorem in cones; the main assumption is that f may grow at most like Au + B, the bound on A being computed by means of a Green’s function. By using a similar th
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