Bifurcation from essential spectrum for an elliptic equation with general nonlinearity

TL;DR

This paper demonstrates that the zero point in the essential spectrum acts as a bifurcation point for a superlinear elliptic equation under local conditions, extending previous results on an open problem from 1983.

Contribution

It generalizes earlier bifurcation results for elliptic equations by establishing the zero essential spectrum as a bifurcation point with minimal assumptions.

Findings

Zero spectrum is a bifurcation point for the elliptic equation.

Generalizes previous bifurcation results to broader conditions.

Addresses an open problem from 1983.

Abstract

In this paper, based on some prior estimates, we show that the essential spectrum $\lambda=0$ is a bifurcation point for an superlinear elliptic equation with only local conditions, which generalizes a series of earlier results on an open problem proposed by C. A. Stuart in 1983 [Lecture Notes in Mathematics, 1017].