Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity

Abstract

We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation producing solutions that bifurcate from infinity. We study the bifurcation branches, characterize when they are sub- or supercritical and analyse the stability type of the solutions. Furthermore, we apply these results and techniques to obtain Landesman–Lazer-type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti-maximum principle.

Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary

“…In [40] a similar result was proved when the nonlinearities are asymptotically linear, and in [7] when the nonlinearity and the bifurcation parameter appear on the boundary (see also [21]). We omit the proof here and refer to those papers for the details.…”

Infinite Resonant Solutions and Turning Points in a Problem With Unbounded Bifurcation

“…Note that, as in [1,2], solutions of (1.1) are determined and estimated in terms of their boundary values. Therefore, we can look at (1.1) as a problem posed in a space of functions defined on ∂Ω.…”

“…This problem has already been studied in [1,2] where we analyzed the existence of unbounded sets of solutions as well as their stability and some of the dynamical properties of the associated parabolic problem. This analysis was carried over assuming that the nonlinear term is sublinear at infinity, which roughly speaking means that…”

Bifurcation from infinity for reaction–diffusion equations under nonlinear boundary conditions

“…In § 3, we show the existence of unbounded branches of solutions bifurcating from the generalized eigenvalues of odd multiplicity. The proof is based on the global bifurcation results of Rabinowitz [20,21] (see also [5][6][7]). Section 4 is devoted to the sub-and supercritical bifurcations from infinity.…”