| In the study of interacting populations one might consider a competitive system in which the population densities have spatial dependence. In this paper we are interested in studying the existence of positive solutions to the following elliptic system.(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}eqalign{lcub}{lcub}-{rcub}Delta u&= uM(u,v)cr{lcub}-{rcub}Delta v&= vN(u,v),cr{rcub}eqno(1){dollar}{dollar}(TABLE/EQUATION ENDS)with homogeneous Dirichlet boundary conditions on a bounded open region {dollar}Omega subset IRsp{lcub}n{rcub}{dollar}. We assume that the functions {dollar}M,N{dollar} are such that system (1) models a competitive system. If in addition {dollar}M(0,0) > lambdasb1{dollar}, {dollar}N(0,0) > lambdasb1{dollar} (where {dollar}lambdasb1{dollar} is the principal eigenvalue of the Lapalcian with Dirichlet boundary conditions) then we have two trivial solutions ({dollar}usb0{dollar},0) and (0,{dollar}v sb0{dollar}). We then linearize system (1) about these two solutions to obtain two Schrodinger type operators, {dollar}Delta + M(0,vsb0)I{dollar} and {dollar}Delta + N(usb0,0)I.{dollar} We then have the main result.; Theorem. (i) If {dollar}M(0,0) lambdasb1{dollar} and {dollar}N(0,0) > lambdasb1{dollar} and the principal eigenvalues {dollar}lambdasb1 (Delta + M(0,v sb0) I){dollar}, {dollar}lambdasb1 (Delta + N(usb0,0) I){dollar} have the same sign. Then system (1) has a strictly positive solution ({dollar}u,v {dollar}).; The results obtained by Cosner-Lazer, by McKenna-Walter, Pao, and by Leung, {dollar}et al.{dollar} are special cases of the main theorem. |
