A quasilinear elliptic equation with absorption term and Hardy potential

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 $p>2$ .