On an abstract bifurcation result concerning homogeneous potential operators with applications to PDEs

Abstract

We study an abstract equation in a reflexive Banach space, depending on a real parameter $\lambda$. The equation is composed by homogeneous potential operators. By analyzing the Nehari sets, we prove a bifurcation result. In some particular cases we describe the full bifurcation diagram, and in general, we estimate the parameter ${\lambda }_b$ for which the problem does not have non-zero solution when $\lambda >{\lambda }_b$. We give many applications to partial differential equations: Kirchhoff type equations, Schrödinger equations coupled with the electromagnetic field, Chern-Simons-Schrödinger systems and a nonlinear eigenvalue problem.