Qualitative properties of positive solutions of quasilinear equations with Hardy terms

TL;DR

This paper investigates positive solutions of a class of quasilinear PDEs with Hardy weights, establishing existence, decay rates, and integrability conditions, and translating these properties into an associated integral equation involving the Wolff potential.

Contribution

It introduces a new integral equation framework for analyzing quasilinear PDEs with Hardy terms and characterizes the decay and integrability properties of solutions.

Findings

Existence of nontrivial solutions depends on the exponent q.

Bounded solutions decay at a specific fast rate if integrable.

Non-integrable solutions decay at a slower rate.

Abstract

In this paper, we are concerned with the quasilinear PDE with weight $$ -div A(x,\nabla u)=|x|^a u^q(x), \quad u>0 \quad \textrm{in} \quad R^n, $$ where $n \geq 3$, $q>p-1$ with $p \in (1,2]$ and $a \in (-n,0]$. The positive weak solution $u$ of the quasilinear PDE is $\mathcal{A}$-superharmonic and satisfies $\inf_{R^n}u=0$. We can introduce an integral equation involving the wolff potential $$ u(x)=R(x) W_{\beta,p}(|y|^au^q(y))(x), \quad u>0 \quad \textrm{in} \quad R^n, $$ which the positive solution $u$ of the quasilinear PDE satisfies. Here $p \in (1,2]$, $q>p-1$, $\beta>0$ and $0 \leq -a<p\beta<n$. When $0<q \leq \frac{(n+a)(p-1)}{n-p\beta}$, there does not exist any positive solution to this integral equation. When $q>\frac{(n+a)(p-1)}{n-p\beta}$, the positive solution $u$ of the integral equation is bounded and decays with the fast rate $\frac{n-p\beta}{p-1}$ if and only if it is integrable (i.e. it belongs to $L^{\frac{n(q-p+1)}{p\beta+a}}(R^n)$). On the other hand, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one $\frac{p\beta+a}{q-p+1}$. Thus, all the properties above are still true for the quasilinear PDE. Finally, several qualitative properties for this PDE are discussed.