Radial singular solutions for the N-Laplace Equation with exponential nonlinearities

TL;DR

This paper studies radial solutions of the N-Laplace equation with exponential nonlinearities, establishing conditions for extending solutions and analyzing singular solutions' growth near the origin.

Contribution

It provides sharp criteria for extending radial solutions and characterizes the behavior of singular solutions for exponential nonlinearities in the N-Laplace equation.

Findings

Sharp conditions for solution extension to the entire domain.

Existence of singular solutions with specific growth rates.

Upper and lower bounds on singularity behavior.

Abstract

In this paper, we consider radial distributional solutions of the quasilinear equation $-\Delta_N u=f(u)$ in the punctured open ball $ B_R\backslash\{0\}\subset \RR^N$, $N \geq 2$. We obtain sharp conditions on the nonlinearity $f$ for extending such solutions to the whole domain $B_R$ by preserving the regularity. For a certain class of noninearity $f$ we obtain the existence of singular solutions and deduce upper and lower estimates on the growth rate near the singularity.