Date of Degree
Dissertation - Open Access
Doctor of Philosophy
College of Arts and Sciences
Department of Mathematics and Statistics
We study positive solutions to classes of nonlinear elliptic singular problems of the form: -Δpu = λ g(u) uα in Ω u = 0 on δΩ where Ω is a bounded domain in ℝN, N ≥ 1 with smooth boundary δΩ, &lambda¸ is a positive parameter, α ∈(0; 1), Δpu := div(⌊∇u⌋p-2 ∇u); p > 1 is the p-Laplacian operator, and g is a smooth function. Such elliptic problems naturally arise in the study of steady state reaction diffusion processes. In particular, we will be interested in the challenging new class of problems when g(0) < 0 (hence lims→0+g(s) sα = - ∞ which we refer to as infinite semipositone problems. Our focus is on existence results. We obtain results for the single equation case as well as to the case of systems. We use the method of sub-super solutions to prove our results. The results in this dissertation provide a solid foundation for the analysis of such infinite semipositone problems.
Ye, Jinglong, "Infinite semipositone systems" (2009). Theses and Dissertations MSU. 2701.