Elliptic Eigenvalue Problems and Unbounded Continua of Positive Solutions of a Semilinear Elliptic Equation

Abstract

We derive a result on the limit of certain sequences of principal eigenvalues associated with some elliptic eigenvalue problems. This result is then used to give a complete description of the global structure of the curves of positive steady states of a parameter dependent diffusive version of the classical logistic equation. In particular, we characterize the bifurcation values from infinity to positive steady states. The stability of the positive steady states as well as the asymptotic behaviour of positive solutions is also discussed.

Increasing Powers in a Degenerate Parabolic Logistic Equation

“…Without such regularity assumption, we would obtain convergence in each set of type Ω ′ × (t 1 , t 2 ) with Ω ′ ⋐ Ω, 0 < t 1 < t 2 . Now, in view of Theorem 1.3, we want to prove that in this case u solves (8). By Lemma 3.1, we know that u p−1…”

“…[8,Theorem 3.7] or [7,Theorem 2.2]) that u p (t, x) converges to the unique positive solution of (2) whenever a ∈ (λ 1 (Ω), λ 1 (Ω 0 )). Hence in this situation, if we combine all this information together with the results obtained in this paper, then we can conclude that the following diagram commutes:…”

“…x ∈ Ω. Fix a > λ 1 (Ω) and let u be the unique solution of (8) and take w the unique solution of (3). Then, as t → +∞,…”

Pointwise Growth and Uniqueness of Positive Solutions for a Class of Sublinear Elliptic Problems where Bifurcation from Infinity Occurs

“…Quite surprisingly, the general problem when the species u is free from crowding effects on some subdomain of , i.e., when a(x) vanishes on some subdomain of , has not been tackled until very recently (cf. [3,17,7] and the references therein), although there is a huge amount of literature dealing with the case when a(x) is positive and bounded away from zero. In order to summarize what it is known for (1.1), (1.2) and give our main results, we need to introduce some notation.…”

“…Throughout this work, given an open subset 1 ⊂ with a finite number of components and an elliptic operator L in , σ 1 1 [L] stands for the principal eigenvalue of L in 1 subject to homogeneous Dirichlet boundary conditions on ∂ 1 (the minimum of the principal eigenvalues of L on each of the components of 1 separately). From the relatively recent results of [3,17,7], it readily follows that problem (1.2) possesses a positive solution if and only if σ 1 [− ] < λ < σ 0 1 [− ]. Moreover, it is unique, if it exists, and its L ∞ -norm decays to zero as λ ↓ σ 1 [− ], and grows to infinity as λ ↑ σ 0 1 [− ].…”