Entire large solutions for semilinear elliptic equations

TL;DR

This paper studies entire large solutions to a class of semilinear elliptic equations in high-dimensional space, establishing existence, asymptotic behavior, and symmetry properties under certain growth and decay conditions.

Contribution

It proves the existence of entire large solutions for equations with Keller-Osserman growth and decaying coefficients, and analyzes their asymptotic behavior, uniqueness, and symmetry.

Findings

Existence of entire large solutions under specified conditions

Asymptotic behavior of solutions at infinity characterized

Conditions for uniqueness and symmetry discussed

Abstract

We analyze the semilinear elliptic equation $\Delta u=\rho(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that $\lim_{|x|\rightarrow +\infty}u(x)=+\infty$. Assuming that $f$ satisfies the Keller-Osserman growth assumption and that $\rho$ decays at infinity in a suitable sense, we prove the existence of entire large solutions. We then discuss the more delicate questions of asymptotic behavior at infinity, uniqueness and symmetry of solutions.