Nonnegative solutions of an elliptic equation and Harnack ends of Riemannian manifolds (Q1589739)

The Liouville property for positive solutions of a nonlinear elliptic operator \(\mathcal A\) on a complete Riemannian manifold is studied and a nonnegative (bounded, respectively) solution which takes the given data at infinity of each end constructed. According to \textit{I. Holopainen} [Proc. Lond. Math. Soc. (3) 65, No.~3, 651-672 (1992; Zbl 0782.53030)], \(\mathcal A\) takes the form \(-\text{div }{\mathcal A}_x(\nabla u) = 0\), where \({\mathcal A}_x\) maps the tangent space of \(M\) at \(x\) to itself such that continuous in \(x\), \(\langle{\mathcal A}_x(\xi),\xi\rangle\geq C_1|\xi|^p\), \(|{\mathcal A}_x(\xi)|\leq C_2|\xi|^{p-1}\), \(\langle{\mathcal A}_x(\xi_1)- {\mathcal A}_x(\xi_2),\xi_1-\xi_2\rangle > 0\) and \({\mathcal A}_x(\lambda\xi) = \lambda|\lambda|^{p-2}{\mathcal A}_x(\xi)\). \(u\) is said to be an \({\mathcal A}\)-harmonic function. Introducing the notion of Harnack end with respect to \({\mathcal A}\)-harmonic functions (the exact definition is given in section 2 of this paper), Holopainen showed the existence of limits of a nonnegative \({\mathcal A}\)-harmonic function along the Harnack end (reviewed as Lemma 1 in this paper). Applying this Lemma, the authors give characterizations of \(p\)-parabolic and nonparabolic ends (Lemma 3 and Corollary 3 of Lemma 1) (\(M\) is said to be \(p\)-parabolic if \(\inf_u\int_M|\nabla u|^p = 0\), for any compact set \(K\), where \(u\) is a compact support smooth function such that \(u = 1\) on \(K\). Otherwise, \(M\) is said to be \(p\)-nonparabolic). By using these results, the existence of nonnegative \(A\)-harmonic functions on a complete Riemannian manifold \(M\) is shown such that its limit along a \(p\)-nonparabolic end takes a given finite value and along a \(p\)-parabolic value diverges to \(\infty\), if \(M\) has finitely many ends each of which is a Harnack end (Theorem 1 in Introduction and Theorem 4, in section 3). It is known that ends satisfying the volume doubling condition \((D)_\infty\), the Poincaré inequality \((P)_\infty\), and the finite covering condition \((FC)\) are Harnack ends [\textit{I. Holopainen}, Math. Z. 217, No.~3, 459-477 (1994; Zbl 0833.58036)]. Applying the notion of rough isometry introduced by \textit{M. Kanai} [Lect. Notes Math. 1201, 122-137 (1986; Zbl 0593.53026)] rough isometric ends satisfying these three conditions are shown to be Harnack. Every complete Riemannian manifold with nonnegative Ricci curvature outside a compact set and finite first Betti number satisfies the conditions \((D)_\infty\), \((P)_\infty\) and \((FC)\) [\textit{P. Li} and \textit{L.-F. Tam}, J. Differ. Geom. 41, No.~2, 277-318 (1995; Zbl 0827.53033)], and the above existence theorem holds for complete Riemannian manifolds rough isometric to a complete Riemannian manifold with nonnegative Ricci curvature outside a compact set and finite first Betti number (Corollary 1 of Theorem 2 in Introduction and Theorem 6 in section 4). Applying this result to harmonic functions, the following generalization of Li-Tam's result (P. Li and L. M. Tam, loc. cit) is given: Theorem. Let \(M\) be a complete Riemannian manifold rough isometric to a manifold satisfying \((D)_\infty\), \((P)_\infty\), \((FC)\) with \(l \geq 1\) nonparabolic ends and \(s\geq 0\) parabolic ends. \({\mathcal H}^+(M)\) and \({\mathcal H}{\mathcal B}(M)\) be the spaces spanned by positive and bounded harmonic functions on \(M\), respectively. Then \(\dim {\mathcal H}^+(M)=l+s\) and \(\dim {\mathcal H}{\mathcal B}(M)=l\). (Corollary 2 in Introduction and Corollary 8 in section 4).