A Study of Singular Solutions for a $p$-Laplacian Equation with Combined Power-Type Nonlinearity

Equation with Mixed Powers Term

Abstract

This paper is devoted to the analysis of the existence and asymptotic behavior of singular positive radial solutions of the equation \[ \Delta_p v + v^q - v^{-\delta} = 0, \quad \quad \textrm{in} \;\; \mathbb{R}^N, \] where $N>p > 2$, $q>1$ and $\delta>0$. We investigate radial solutions to specific initial conditions, with a particular focus on singular solutions that vanish at a finite radius. By employing scaling techniques and comparison principles, we derive constraints on the parameters $q$ and $\delta$ ensuring the existence of such singular solutions. Furthermore, we characterize the asymptotic behavior of these solutions.