Radial solutions of truncated Laplacian equations in punctured balls

TL;DR

This paper investigates radial solutions of truncated Laplacian equations with singular potentials in punctured balls, analyzing eigenvalues and solution classification based on asymptotic behavior, revealing distinct outcomes for different operator types.

Contribution

It provides new insights into eigenvalue problems and solution classifications for truncated Laplacian equations, highlighting differences between maximising and minimising operators.

Findings

Results for Pk+ are analogous to classical Laplacian cases.

Degeneracy in Pk- leads to fundamentally different solution behaviors.

Asymptotic analysis near the origin characterizes solution types.

Abstract

We consider equations involving the truncated laplacians and having lower order terms with singular potentials posed in punctured balls. We study both the principal eigenvalue problem and the problem of classification of solutions, in dependence of their asymptotic behaviour near the origin, for equations having also superlinear absorbing lower order terms. In the case of the maximising truncated Laplacian "Pk+", owing to the mild degeneracy of the operator, we obtain results which are analogous to the results for the Laplacian in dimension k. On the other hand, for minimising operator "Pk-" we show that the strong degeneracy in ellipticity of the operator produces radically different results.