Large solutions of degenerate and/or singular quasilinear elliptic equations in a ball

TL;DR

This paper studies the boundary blow-up solutions of certain degenerate or singular quasilinear elliptic equations in a ball, analyzing how their asymptotic behavior varies with degeneracy, singularity, and the nonlinearity's growth rate.

Contribution

It provides new insights into the asymptotic boundary behavior of large solutions for degenerate and/or singular quasilinear elliptic equations, including second order blow-up rates.

Findings

Characterization of boundary blow-up rates for solutions

Analysis of how degeneracy and singularity affect asymptotic behavior

Derivation of second order blow-up rates for semilinear cases

Abstract

We consider local weak large solutions with its blow-up rate near the boundary to certain class of degenerate and/or singular quasilinear elliptic equation\\ ${\rm div}(d^{\alpha}(x,\partial{}B)\Phi_p(\nabla u)) = b(x)f(u)$ in a ball B, where $f$ is normalized regularly varying at infinity with index $\sigma+1>p-1,\ p>1$. In particular, how the asymptotic behavior of the solution changes over the varying index and degeneracy and/ or singularity present in the equation. We also include the second order blow-up rate for the corresponding semilinear problem.