Local and global bifurcation results for a semilinear boundary value problem

2017

Isr. J. Math.

Abstract. We investigate the problemwhere Ω is a bounded smooth domain in IR N (N ≥ 2), 1 < q < 2 < p, λ ∈ IR, and a, b ∈ C α (Ω) with 0 < α < 1. Under some indefinite type conditions on a and b we prove the existence of two nontrivial non-negative solutions for |λ| small. We characterize then the asymptotic profiles of these solutions as λ → 0, which implies in some cases the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type subcontinuum in the non-negative solutions set. We prove in some cases the existence of such subcontinuum via a bifurcation and topological analysis of a regularized version of (P λ ).

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%

An indefinite concave-convex equation under a Neumann boundary condition I

Quoirin

2017

Isr. J. Math.

“…In view of this difficulty, our second purpose is to show that nontrivial solutions lying on C 0 satisfy u ≫ 0. with Ω bounded (under different boundary conditions) or Ω = R N , have been studied by several authors, see e.g. [4,5,8,6,22,7,20,19]. According to [6,7,20], a bounded subcontinuum linking two different points on (λ, 0) is called a mushroom, one that meets a single point on (λ, 0) is called a loop, and one that does not touch (λ, 0) is called an isola.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

Kaufmann

Humberto

Advanced Nonlinear Studies

2018

We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a non-regular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn's topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved in [15], and extend previous results established in the powerlike case.2010 Mathematics Subject Classification. 35J25, 35J61, 35B32.

“…Based on this result, we shall carry out a bifurcation analysis for (P λ ) by a topological method proposed by Whyburn [29]. For similar works where a topological technique is used to obtain bifurcating nontrivial solutions from trivial lines, we refer to Hetzer [14] for ordinary differential equations, to Arcoya, Diaz and Tello [3] for degenerate multivalued equations, to Colorado and Peral [10] for Dirichlet and Neumann mixed boundary condition, to Brown [5] for indefinite superlinear elliptic problems, and to Umezu [28] for the case q > 2.…”

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%

Bifurcation for a logistic elliptic equation with nonlinear boundary conditions: A limiting case

Quoirin

Journal of Mathematical Analysis and Applications

2015

We investigate bifurcation from the zero solution for a logistic elliptic equation with a sign-definite nonlinear boundary condition. In view of the lack of regularity of the term on the boundary, the abstract theory on bifurcation from simple eigenvalues due to Crandall and Rabinowitz does not apply. A regularization procedure and a topological method due to Whyburn are used to prove the existence and the global behavior at infinity of a subcontinuum of nontrivial non-negative weak solutions. The direction of the bifurcation component at zero is also investigated. This paper treats a limiting case of our previous work [19], where the case of sign-changing nonlinear boundary conditions is considered.