Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity
José M. Arrieta, Rosa Pardo San Gil, Anı́bal Rodriguez-Bernal · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 2007
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.