Qualitative properties of positive solutions of quasilinear equations with Hardy terms

Abstract

AbstractIn this paper, we are concerned with the following quasilinear PDE with a weight:-\operatorname{div}A(x,\nabla u)=|x|^{a}u^{q}(x),\qquad u>0\quad\text{in }% \mathbb{R}^{n},where {n\geq 1}, {q>p-1} with {p\in(1,2]} and {a\leq 0}. The positive weak solution u of the quasilinear PDE is {\mathcal{A}}-superharmonic. We also consider an integral equation involving the Wolff potentialu(x)=R(x)W_{\beta,p}(|y|^{a}u^{q}(y))(x),\qquad u>0\quad\text{in }\mathbb{R}^{% n},which the positive solution u of the quasilinear PDE satisfies. Here {\beta>0} and {p\beta<n}. When {-a>p\beta} or {0<q\leq\frac{(n+a)(p-1)}{n-p\beta}}, there does not exist any positive solution to this integral equation. On the other hand, when {0\leq-a<p\beta} and {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}}(\mathbb{R}^{n})}). However, 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}}. In addition, we also discuss the case of {-a=p\beta}. Thus, all the properties above are still true for the quasilinear PDE.