Qualitative properties of positive solutions of quasilinear equations with Hardy terms

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.