Singular solutions of some elliptic equations involving mixed absorption-reaction

TL;DR

This paper investigates the properties and behaviors of nonnegative solutions with isolated singularities to a class of elliptic equations involving mixed absorption-reaction terms, focusing on how solutions depend on parameters p, q, and M.

Contribution

It provides a detailed analysis of the existence and behavior of solutions to a specific elliptic PDE with mixed terms, highlighting the influence of parameters p, q, and M, especially at critical exponent values.

Findings

Existence of solutions depends on the exponents p and q and the parameter M.

Solution behaviors vary significantly based on the sign of q - 2p/(p+1).

Radial solutions analysis is key to understanding the solution properties.

Abstract

We study properties of nonnegative functions satisfying (E)$\;-\Delta u+u^p-M|\nabla u|^q=0$ is a domain of $\mathbb{R}^N$ when $p>1$, $M>0$ and $1<q<p$. We concentrate our analysis on the solutions of (E) with an isolated singularity, or in an exterior domain, or in the whole space. The existence of such solutions and their behaviours depend strongly on the values of the exponents $p$ and $q$ and in particular according to the sign of $q-\frac{2p}{p+1}$, and when $q=\frac{2p}{p+1}$, also on the value of the parameter $M$ which becomes a key element. The description of the different behaviours is made possible by a sharp analysis of the radial solutions of (E).