Supercritical Lane-Emden equation on a cone with an inhomogeneous Dirichlet boundary condition

TL;DR

This paper studies the supercritical Lane-Emden equation on an infinite cone with inhomogeneous boundary conditions, classifying solution existence and multiplicity based on boundary data and nonlinearity, and extends results to Hardy-Hénon equations.

Contribution

It provides a complete classification of solution existence and nonexistence, and demonstrates multiple solutions via bifurcation theory for the supercritical Lane-Emden equation on cones.

Findings

Classified existence and nonexistence of solutions based on boundary data.

Established multiple solutions through bifurcation analysis.

Extended results to Hardy-Hénon equations on cones.

Abstract

We consider the Lane-Emden equation with a supercritical nonlinearity with an inhomogeneous Dirichlet boundary condition on an infinite cone. Under suitable conditions for the boundary data and the exponent of nonlinearity, we give a complete classification of the existence/nonexistence of a solution with respect to the size of boundary data. Moreover, we give a result on the multiple existence of solutions via bifurcation theory. We also state results on Hardy-H\'enon equations on infinite cones as a generalization.