ABSTRACT
By construction, Ωq0 = Ω \ Ω¯0 = Ω−. (4.5)
Throughout the rest of this book, we will adopt the notation
σq0+1 :=∞. By the results of Chapter 3, f should satisfy (KO) so that (4.2) can admit a positive solution. These solutions will regulate the dynamics of the parabolic model (1.1) according to the different ranges of values of the parameter λ ∈ R, which explains our interest in analyzing (4.2). Precisely, the solutions of (4.1) regulate the dynamics of (1.1) for λ < dσ1, while the solutions of (4.2) regulate the dynamics of (1.1) for dσj ≤ λ < dσj+1, if 1 ≤ j ≤ q0−1, and for λ ≥ dσq0 if j = q0. The solutions of (4.2) will be referred to as the large solutions of order j of
−d∆u = λu+ a(x)f(x, u)u in Ωj .
