Qualitative properties of positive solutions of quasilinear equations with Hardy terms

Abstract

Abstract In this paper, we are concerned with the following quasilinear PDE with a weight: - div ⁡ A ⁢ ( x , ∇ ⁡ u ) = | x | a ⁢ u q ⁢ ( x ) , u > 0 in ⁢ ℝ n , -\operatorname{div}A(x,\nabla u)=|x|^{a}u^{q}(x),\qquad u>0\quad\text{in }% \mathbb{R}^{n}, where n ≥ 1 {n\geq 1} , q > p - 1 {q>p-1} with p ∈ ( 1 , 2 ] {p\in(1,2]} and a ≤ 0 {a\leq 0} . The positive weak solution u of the quasilinear PDE is 𝒜 {\mathcal{A}} -superharmonic. We also consider an integral equation involving the Wolff potential u ⁢ ( x ) = R ⁢ ( x ) ⁢ W β , p ⁢ ( | y | a ⁢ u q ⁢ ( y ) ) ⁢ ( x ) , u > 0 in ⁢ ℝ n , u(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 β > 0 {\beta>0} and p ⁢ β < n {p\beta<n} . When - a > p ⁢ β {-a>p\beta} or 0 < q ≤ ( n + a ) ⁢ ( p - 1 ) n - p ⁢ β {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 ≤ - a < p ⁢ β {0\leq-a<p\beta} and q > ( n + a ) ⁢ ( p - 1 ) n - p ⁢ β {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 n - p ⁢ β p - 1 {\frac{n-p\beta}{p-1}} if and only if it is integrable (i.e., it belongs to L n ⁢ ( q - p + 1 ) p ⁢ β + a ⁢ ( ℝ n ) {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 p ⁢ β + a q - p + 1 {\frac{p\beta+a}{q-p+1}} . In addition, we also discuss the case of - a = p ⁢ β {-a=p\beta} . Thus, all the properties above are still true for the quasilinear PDE.