Nonsolvability of the asymptotic Dirichlet problem for the \(p\)-Laplacian on Cartan-Hadamard manifolds (Q267817)

Let \((M,g)\) be a complete simply connected Riemannian manifold endowed with negative sectional curvature, denoted by \(\operatorname{Sect}(M)\), and let \(A\) be a suitable operator. We recall that the \(A\)-asymptotic Dirichlet problem states that given a continuous function \(h\) on \(M(\infty)\) (the sphere at infinity associated to \((M,g)\)), does exist a unique function \(u\) continuous on \(\widetilde{M}:=M\cup M(\infty)\) and \(A\)-harmonic on \(M\)? The \(A\)-asymptotic Dirichlet problem is solvable when there is a solution of the following boundary problem NEWLINE\[NEWLINE (*)_A:\;A(u)=0\text{ in }M\text{ and }u=h\text{ on }M(\infty).NEWLINE\]NEWLINE The purpose of the article is to show that \((*)_A\) is not solvable for \(A=-\operatorname{div}(|\nabla(\cdot)|^{p-2}\nabla(\cdot))\) for \(p>1\) and \(\operatorname{Sect}(M)\) is at most minus one, besides there are (non-constant) bounded \(A\)-harmonic functions in \(M\) (Theorem 1.5). Firstly, the author expresses \(\mathbb H^3\)-metric, the Riemannian metric associated to the hyperbolic space \(\mathbb H^3\), in Fermi normal coordinates, then by modifying the \(\mathbb H^3\)-metric in some direction, the author builds up \((M,g)\) where \(g\) takes an almost explicit form, see the seventh section. Curvature conditions bearing on \(g\) are treated in the sixth section. On the existence of a positive \(A\)-subsolution (resp. \(A\)-supersolution) and bounded by one in \(\widetilde{M}\). The constructions of these latter two functions are given in the eighth and ninth sections.