Branches of positive solutions for some semilinear Schrödinger equations

Branches of solutions to semilinear elliptic equations on $\mathbb{R}^N$

“…Recently there has been a resurgence of interest in the case of elliptic partial differential equations like (1.1) under various assumptions on f , [2] and [8]. For example in [8] it is shown (under appropriate conditions on f ) that the branch of positive solutions covers the interval (Λ, α) where Λ is the lowest eigenvalue of the linearization.…”

“…In [2] this difficulty is overcome by approximating (1.1) with Dirichlet boundary value problems on balls, and then by showing that branches of the approximate problems converge to a branch of positive solutions of (1.1).…”

Refuge versus dispersion in the logistic equation

“…We will prove that λ * = λ 0,1 . To this aim, we follow the arguments of Lemma 2.4 of [20]. First note that |w n | 2 → ∞, where | · | 2 stands for the L 2 (Ω) norm.…”

Global bifurcation for quasilinear elliptic equations on $\mathbb{R}^{N}$

“…1 We always have ρ > 0 but, after D u F ∞ (λ, 0) has been reduced to the form −∆ + ∂ ξ 0 b(x, 0, λ) by a linear change of variable, we obtain ρ = 1. Note also that…”

“…This fact is intimately related to the presence of an essential spectrum for the linear operator −∆ + V where V is a bounded potential. There are various ways of circumventing this difficulty, including approximation by problems on bounded domains and the use of weighted Sobolev spaces, [4], [1], [12], [21], [36], [6] but we prefer to use an extension of the Leray-Schauder degree since it seems to yield the most general results under natural hypotheses. For ordinary differential equations on [0, ∞) this approach was first adopted in [33], [34] and [35] using respectively the degree for k−set contractions and Galerkin maps; and it was subsequently developed in various ways.…”