Type: Article

The spectrum in R and R 2 of nonlinear elliptic equations with positive parameters

Trejo, Imelda; Felipe, Raúl;

Abstract:

In this article, the authors consider the boundary value problem {Lu(x)+?f(x,u(x))+?g(x,u(x))=0u=0in ?,on ??, where ? is a bounded, convex domain in Rm, m?2, L is an elliptic differential operator with essential properties identical to those of the negative Laplacian, and the nonlinear term F(x,u):=?f(x,u)+?g(x,u) satisfies reasonable assumptions of Hölder continuity and of monotonicity in the u variable. The authors develop a generalized method of sub- and super-solutions, alternatively a Picard-iteration method, and then apply that method to prove two-parameter existence theorems that generalize some well-known one-parameter existence results for positone (F(0)>0) problems and semipositone (F(0)<0) problems. The exposition is clear and the results should be of interest to a broad audience of mathematicians who study PDEs and nonlinear analysis.