Entire solutions of superlinear problems with indefinite weights and Hardy potentials

TL;DR

This paper analyzes the structure of entire solutions for superlinear elliptic equations with indefinite weights and Hardy potentials, using dynamical systems methods to classify solutions with various decay properties.

Contribution

It extends the analysis of elliptic equations to include indefinite weights and Hardy potentials, providing a detailed dynamical systems approach to classify solutions.

Findings

Classification of solutions with different decay rates

Extension to equations with Hardy potentials

Construction of invariant manifolds for non-global solutions

Abstract

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an in- definite weight on the nonlinearity f (u, r). In particular we are interested in the case where f is positive in a ball and negative outside, or in the re- versed situation. We extend the approach to elliptic equations in presence of Hardy potentials. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured.