Bifurcation of stable equilibria under nonlinear flux boundary condition with null average weight

Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

“…This observation suggests that a nonconstant positive solution u λ is always unique if it exists, and one obtains the following limiting behaviors of u λ both as λ → ∞ and as λ ց 0. In every case of r 0 , r 1 , the limiting function of u λ as λ → ∞ is given by u ∞ (x) = x, which is consistent with [22,Theorem 4.1] and [23,Theorem 1.3]. On the other side, in the case when r 0 + r 1 > 0, the limiting function of λu λ as λ ց 0 is given by…”

“…A motivation for our study of (4.1) arises in population genetics ( [15,9]). For previous works on the boundary version, we refer to [21,22,18,23]. Clearly, u ≡ 0, 1 satisfies (4.1) for all λ > 0, which are called constant solutions, and…”

“…In this section, we discuss the existence and uniqueness of nonconstant positive solutions u of (4.1), which have been well studied for u ≤ 1 in Ω ( [21,22,18,23]). In turn, our objective is to discuss the case of u ≤ 1 in Ω, i.e., the case that u > 1 somewhere in Ω.…”

“…Known results for positive solutions u ≤ 1. In this subsection, we focus our consideration on nonnegative solutions u of (4.1) such that u ≤ 1 in Ω, and we summarize known results for them from [22,23], as illustrated by Figure 5. Then, by applying the SMP and BPL, a nonconstant positive solution u implies that 0 < u < 1 in Ω.…”