Asymptotic Dirichlet Problem for $\mathcal {A}$-Harmonic Functions on Manifolds with Pinched Curvature

Abstract

We study the asymptotic Dirichlet problem for $\mathcal {A}$-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound

$K(P)\le - \frac {1+\varepsilon }{r(x)^{2} \log r(x)}$

and a pointwise pinching condition

$| K(P) |\le C_{K}| K(P^{\prime }) |$

for some constants ε > 0 and C _K ≥ 1, where P and $P^{\prime }$ are any 2-dimensional subspaces of T _x M containing the (radial) vector ∇r(x) and r(x) = d(o, x) is the distance to a fixed point o ∈ M. We solve the asymptotic Dirichlet problem with any continuous boundary data $f\in C(\partial _{\infty } M)$. The results apply also to the Laplacian and p-Laplacian, $1<p<\infty ,$ as special cases.