Separable $p$-harmonic functions in a cone and related quasilinear equations on manifolds

Abstract

In considering a class of quasilinear elliptic equations on a Riemannian manifold with nonnegative Ricci curvature, we give a new proof of Tolksdorf's result on the construction of separable p-harmonic functions in a cone. 1991 Mathematics Subject Classification. 35K60 .

The spherical p-harmonic eigenvalue problem in non-smooth domains

“…Proof of Theorem C. Since S c has a non-empty interior, the existence of a sequence {ω ′ k } corresponding to solutions of (1.1) in S ′ k with β = β S ′ k is the consequence of [13]. The fact that {β S ′ k } is increasing follows from Proposition 2.2.…”

On p-harmonic measures in half-spaces

“…(1) Our proof of the existence part in Theorem 1.2 is based on topological shooting techniques; it is entirely different from the previous proofs, relying on PDE methods or complicated techniques from Harmonic Analysis and PDE's on manifolds ( [15], [23], [22]). (2) This shooting approach is conceptually simpler and has the added important advantage that it is readily applicable to settings in which solutions do not remain positive and can change sign [24].…”

“…He also indicated how to get uniqueness from the boundary Harnack inequality. In [22] Tolksdorff's method is extended to cover positive singular solutions.…”

Estimates and asymptotic expansions for condenser p-capacities. The anisotropic case of segments

“…As for p-Laplace problems, most available results address the case of isolated singularities and of radial solutions ( [17,37]). Anisotropic solutions of the form u(x) = |x| λ ω(x/ |x|), where λ ∈ R and ω is defined on the unit sphere S N −1 , were studied for quasilinear equations with Dirichlet conditions in domains with conical boundary points ( [34,27]). To our best knowledge, the anisotropic effect caused by a segment in the p-Laplace equation has not been studied yet.…”