On n-superharmonic functions and some geometric applications

In this paper we study asymptotic behaviors of n-superharmonic functions at singularity using the Wolff potential and capacity estimates in nonlinear potential theory. Our results are inspired by and extend [6] of Arsove–Huber and [63] of Taliaferro in 2 dimensions. To study n-superharmonic functions we use a new notion of thinness in terms of n-capacity motivated by a type of Wiener criterion in [6]. To extend [63], we employ the Adams–Moser–Trudinger’s type inequality for the Wolff potential, which is inspired by the inequality used in [15] of Brezis–Merle. For geometric applications, we study the asymptotic end behaviors of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. These geometric applications seem to elevate the importance of n-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions.

Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms

We study quasilinear elliptic equations of the type - Δ p ⁢ u = σ ⁢ u q + μ {-\Delta_{p}u=\sigma u^{q}+\mu} in ℝ n {\mathbb{R}^{n}} in the case 0 < q < p - 1 {0<q<p-1} , where μ and σ are nonnegative measurable functions, or locally finite measures, and Δ p ⁢ u = div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ) {\Delta_{p}u=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian. Similar equations with more general local and nonlocal operators in place of Δ p {\Delta_{p}} are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions u: u ⁢ ( x ) ≈ ( 𝐖 p ⁢ σ ⁢ ( x ) ) p - q p - q - 1 + 𝐊 p , q ⁢ σ ⁢ ( x ) + 𝐖 p ⁢ μ ⁢ ( x ) , x ∈ ℝ n , u(x)\approx({\mathbf{W}}_{p}\sigma(x))^{\frac{p-q}{p-q-1}}+{\mathbf{K}}_{p,q}% \sigma(x)+{\mathbf{W}}_{p}\mu(x),\quad x\in\mathbb{R}^{n}, where 𝐖 p {{\mathbf{W}}_{p}} and 𝐊 p , q {{\mathbf{K}}_{p,q}} are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on p, q, and n. The contributions of μ and σ in these pointwise estimates are totally separated, which is a new phenomenon even when p = 2 {p=2} .

Qualitative properties of positive solutions of quasilinear equations with Hardy terms

In 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.