POSITIVE LIOUVILLE THEOREMS AND ASYMPTOTIC BEHAVIOR FOR p-LAPLACIAN TYPE ELLIPTIC EQUATIONS WITH A FUCHSIAN POTENTIAL

Abstract

We study positive Liouville theorems and the asymptotic behavior of positive solutions of p-Laplacian type elliptic equations of the formwhere X is a domain in R d , d ≥ 2, and 1 < p < ∞. We assume that the potential V has a Fuchsian type singularity at a point ζ, where either ζ = ∞ and X is a truncated C 2 -cone, or ζ = 0 and ζ is either an isolated point of ∂X or belongs to a C 2 -portion of ∂X.

Isolated singularities of positive solutions of p-Laplacian type equations inRd

“…In the nonclassical cases, the integrability of V near ζ clearly implies that V satisfies conditions (C1) and (C2) near ζ . The present paper is a continuation of our recent paper [6]. In that paper we proved ratio limit theorems for quotients of two positive solutions of the equation Q V (u) = 0 near ζ if V satisfies condition (C1) or (C2) near ζ .…”

“…Then even in the linear case (p = 2) positive solutions of Q (u) = 0 in Ω \ {ζ } might not admit a limit at ζ (see [19,Example 9.1]), albeit, a ratio limit theorem holds true near any Fuchsian singular point if p = 2 [19]. Such a ratio limit theorem should also be true for p = 2 if (1.8) is satisfied (see [6] for partial results).…”

“…In [6] we studied the asymptotic behavior of positive We continue with two new general theorems concerning removable singularity. The first result deals with the existence of the limit.…”

On positive solutions of the (p,A)-Laplacian with potential in Morrey space

“…To see this for example in the case p < n, observe that V R M q (A ′ ) = R p−n/q V M q (AR) and by our assumptions on q we have that the exponent on R is nonnegative (it is in fact positive). Now from (5.6) we may readily deduce [15,16,39] and the references therein for partial results).…”

“…We follow the argument in [15] (for a bit different argument see [44, p. 278]). Without loss of generality, we assume that x 0 = 0 and B 1 (0) ⋐ Ω.…”