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

Abstract

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.

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

“…We may then look at (P λ ) as the limit problem of (P λ,ǫ ) when ǫ → 0 + . This procedure has been already used in [25], where a regularized version of a nonlinear boundary condition is studied. Note that the mapping t → |t + ǫ| q−2 t is analytic at t = 0 and any nontrivial non-negative solution of (P λ,ǫ ) is positive on Ω.…”

The logistic elliptic equation with the nonlinear boundary condition arising from coast fishery harvesting

“…The sublinear nonlinearity u q with 0 < q < 1 appearing in (1.1) induces the absorption effect on ∂Ω. Sublinear boundary conditions were explored in [12,10,27,11,5,19,22]. The case of sublinear incoming flux on ∂Ω, the mixed case of sublinear absorption, and incoming flux on ∂Ω, and the sublinear nonlinear term u q multiplied by indefinite weight were studied in [12,27,11], in [10], and in [5,19,22], respectively.…”

Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting II

“…The sublinear nonlinearity (−u q ) that appears in (1.1) induces the absorption effect on ∂Ω. Sublinear boundary conditions of the u q type were explored in [16,14,15,28,29]. The case of an incoming flux on ∂Ω was studied in [16,15].…”