Qualitative properties of positive solutions of quasilinear equations with Hardy terms

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 \(0 \leq -a\frac{(n+a)(p-1)}{n-p\beta}\), 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.