A quasilinear elliptic equation with absorption term and Hardy potential

TL;DR

This paper analyzes positive solutions of a quasilinear elliptic equation with Hardy potential and absorption term, providing a complete classification of their existence, asymptotic behavior, and symmetry properties.

Contribution

It offers a comprehensive description of solutions' existence and behavior, extending previous results and introducing new phenomena related to nonuniqueness and constant solutions.

Findings

Global solutions are radial and explicitly characterized.

Existence depends on the Hardy coefficient relative to a critical value.

New phenomena of nonuniqueness and constant solutions are identified.

Abstract

Here we study the positive solutions of the equation \begin{equation*} -\Delta _{p}u+\mu \frac{u^{p-1}}{\left\vert x\right\vert ^{p}}+\left\vert x\right\vert ^{\theta }u^{q}=0,\qquad x\in \mathbb{R}^{N}\backslash \left\{ 0\right\} \end{equation*}% where $\Delta _{p}u={div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u) $ and $1<p<N,q>p-1,\mu ,\theta \in \mathbb{R}.$ We give a complete description of the existence and the asymptotic behaviour of the solutions near the singularity $0,$ or in an exterior domain. We show that the global solutions $\mathbb{R}^{N}\backslash \left\{ 0\right\} $ are radial and give their expression according to the position of the Hardy coefficient $\mu $ with respect to the critical exponent $\mu _{0}=-(\frac{N-p}{p})^{p}.$ Our method consists into proving that any nonradial solution can be compared to a radial one, then making exhaustive radial study by phase-plane techniques. Our results are optimal, extending the known results when $\mu =0$ or $p=2$, with new simpler proofs.They make in evidence interesting phenomena of nonuniqueness when $\theta +p=0$, and of existence of locally constant solutions when moreover $p>2$ .