Positive solutions of elliptic systems with superlinear nonlinearities on the boundary

TL;DR

This paper investigates the existence and bifurcation behavior of positive solutions in elliptic systems with superlinear boundary nonlinearities, revealing connected solution branches and their dependence on a bifurcation parameter.

Contribution

It introduces a novel combination of rescaling, degree theory, and elliptic regularity to analyze bifurcations from infinity and zero in elliptic systems with superlinear boundary conditions.

Findings

Existence of a connected branch bifurcating from infinity as the parameter approaches zero.

Global bifurcation branch of positive solutions bifurcates from zero under certain conditions.

Analysis of the number of positive solutions depending on the bifurcation parameter.

Abstract

We consider elliptic systems with superlinear and subcritical boundary conditions and a bifurcation parameter as a multiplicative factor. By combining the rescaling method with degree theory and elliptic regularity theory, we prove the existence of a connected branch of positive weak solutions that bifurcates from infinity as the parameter approaches zero. Furthermore, under additional conditions on the nonlinearities near zero, we obtain a global connected branch of positive solutions bifurcating from zero, which possesses a unique bifurcation point from infinity when the parameter is zero. Finally, we analyze the behavior of this branch and discuss the number of positive weak solutions with respect to the parameter using bifurcation theory, degree theory, and sub- and super-solution methods.