K.J. Brown

ON PRINCIPAL EIGENVALUES FOR BOUNDARY VALUE PROBLEMS WITH INDEFINITE WEIGHT AND ROBIN BOUNDARY CONDITIONS

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem -Au(x) = Ag(x)u(x) onD; , (x) +au(x) 0on D a (x) + au(x) = 0 on AD, where D is a bounded region in RN with smooth boundary, g: D -R is a smooth function which changes sign on D, and a E R. Such problems have been studied in recent years because of associated nonlinear problems arising in the study of population genetics (see [3]). The study of the linear ordinary differential equation case, however, goes back to Picone and Bocher (see [2]). Attention has been confined mainly to the cases of Dirichlet (a = oc) and Neumann boundary conditions. In the case of Dirichlet boundary conditions it is well known (see [4]) that there exists a double sequence of eigenvalues for (1)> * **\> 0); in the case where fD g(x) = 0 there are no positive and no negative principal eigenvalues. We shall investigate how the principal eigenvalues of (1), depend on a, obtaining new results for the case where a 0, probably because it is more natural that the flux across the boundary should be outwards if there is a positive concentration at the Received by the editors April 30, 1997. 1991 Mathematics Subject Classification. Primary 35J15, 35J25.

The existence of positive solutions for a class of indefinite weight semilinear elliptic boundary value problems

where λ and α are real parameters and Ω is an open bounded region of R , N ≥ 2 with smooth boundary ∂Ω. We shall suppose that α ≤ 1; thus α = 0 corresponds to the Neumann problem, α = 1 to the Dirichlet problem and 0 < α < 1 to the usual Robin problem. We shall assume throughout that g : Ω → R is a smooth function which changes sign on Ω. Equation (I λ ) arises in population genetics with f(u) = u(1 − u) (see [1]). In this setting (I λ ) is a reaction-diffusion equation where the real parameter λ > 0 corresponds to the reciprocal of the diffusion coefficient and the unknown function u represents a relative frequency so that there is interest only in solutions satisfying 0 ≤ u ≤ 1. In this paper we shall study the structure of the set of positive solutions of (I λ ) in the cases where f(u) = u(1 − |u| ) and f(u) = u(1 + |u|), p > 0. In order to obtain a better understanding of this structure we no longer impose the restrictions that λ > 0 or that u ≤ 1. We obtain new existence results by using a variational method based on the properties of eigencurves, i.e., properties of the map λ→ μ(λ) where μ(λ) denotes the principal eigenvalue of the linear problem

Nontrivial solutions of predator-prey systems with small diffusion

On considere le systeme elliptique non lineaire -Δu(x)=au-bu 2 -cuv pour xeD, -dΔv(x)=ev -fv 2 +guv, D region bornee de R n . On etudie comment la famille des solutions non negatives de ce systeme change quand les parametres du systeme changent

On the construction of super and subsolutions for elliptic equations on all of R N

We prove existence of positive solutions for the semilinear elliptic equation ?u(x) = g(x)f(u(x)) for x 2 IR N ; which arises in population genetics, in the case where N 3 and g or g + is small at innnity, e.g., in L N=2 (IR N). Existence is proved by the construction of appropriate sub and supersolutions.

Curves of Solutions through Points of Neutral Stability

The structure of the set of positive solutions (λ,u) of the semilinear elliptic boundary value problem % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

S-shaped bifurcation curves

- \bigtriangleup u(x)=\lambda f(u(x))\ {\rm for}\ x\epsilon D;\qquad u(x)=0\ {\rm on}\ \partial

Global bifurcation in the Brusselator system

where D is a bounded region is investigated. Sufficient conditions are given to ensure that if (λ, u) is such that u is a positive neutrally stable solution then all solutions in a neighbourhood of (λ, u) lie on a single curve in the (gl, u) plane.