Qualitative properties of positive solutions of quasilinear equations with Hardy terms

Abstract

In this paper, we are concerned with the quasilinear PDE with weight $$-divA(x,\nabla u)=|x{|}^a{u}^q(x),u>0\text{in}{R}^n,$$ where $n\ge 3$, $q>p-1$ 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 $$u(x)=R(x)W_{\beta ,p}(|y{|}^a{u}^q(y))(x),u>0\text{in}{R}^n,$$ which the positive solution $u$ of the quasilinear PDE satisfies. Here $p\in (1,2]$, $q>p-1$, $\beta >0$ and $0\le -a<p\beta <n$. When $0<q\le \frac{(n+a)(p-1)}{n-p\beta }$, 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.

arXiv comment27 pages

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