Bifurcation for indefinite weighted $p$-laplacian problems with slightly subcritical nonlinearity

TL;DR

This paper extends bifurcation analysis for indefinite weighted p-Laplacian problems to slightly subcritical nonlinearities, using Orlicz spaces to handle compactness issues, and identifies bifurcation points at principal eigenvalues.

Contribution

It generalizes Drabek's bifurcation result to include slightly subcritical nonlinearities in indefinite weighted p-Laplacian problems.

Findings

Bifurcation points occur at principal eigenvalues.

Extension of bifurcation results to subcritical nonlinearities.

Use of Orlicz spaces to address compactness challenges.

Abstract

We study a superlinear elliptic boundary value problem involving the $p$-laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the corresponding linear weighted boundary value problem. The two principal eigenvalues are bifurcation points from the trivial solution set to positive solutions. Drabek's bifurcation result applies when the nonlinearity is of power growth. We extend Drabek's bifurcation result to {\it slightly subcritical} nonlinearities. Compactness in this setting is a delicate issue obtained via Orlicz spaces.