[Paper Review] Isolated singularities of positive solutions of p-Laplacian type equations in R^d

Martin Fraas, Yehuda Pinchover|arXiv (Cornell University)|Aug 23, 2010

Nonlinear Partial Differential Equations14 references3 citations

TL;DR

This paper investigates isolated singularities of positive solutions to p-Laplacian type equations in R^d, establishing removable singularity theorems under Kato-class potential conditions. It proves that when p > d and V is in a suitable Kato class near ζ = 0, or p < d near ζ = ∞, positive solutions extend continuously across the singularity, with precise asymptotic behavior derived via a new three-spheres inequality and Wolff potential estimates.

ABSTRACT

We study the behavior of positive solutions of p-Laplacian type elliptic equations of the form Q'(u) := -p-Laplacian(u) + V |u|^(p-2) u = 0 in Omega near an isolated singular point zeta, where 1 &lt; p &lt; inf, Omega is a domain in R^d with d &gt; 1, and zeta = 0 or zeta = inf. We obtain removable singularity theorems for positive solutions near zeta. In particular, using a new three-spheres theorems for certain solutions of the above equation near zeta we prove that if V belongs to a certain Kato class near zeta and p&gt;d (respectively, p

Motivation & Objective

• To analyze the behavior of positive solutions to p-Laplacian type equations near isolated singularities in R^d.
• To establish conditions under which isolated singularities are removable for positive solutions.
• To characterize the asymptotic behavior of such solutions near singular points ζ = 0 or ζ = ∞.
• To extend classical results on p-harmonic functions to equations with potential V via Kato-class and Fuchsian-type conditions.
• To prove a positive Liouville theorem for p < d under criticality and integrability assumptions on V.

Proposed method

• Introduce a new three-spheres inequality for solutions near isolated singularities, generalizing classical subharmonic estimates.
• Use the Wolff potential to characterize the integrability and decay properties of solutions near singular points.
• Define Fuchsian and weak Fuchsian singularities via scaling limits of the potential V under rescaling R^n → 0 or ∞.
• Employ Kato-class conditions on V to control the potential's local behavior, ensuring regularity of solutions.
• Apply variational methods and criticality theory to analyze minimal growth solutions and uniqueness up to constants.
• Use radial test functions and explicit fundamental solutions to derive asymptotic profiles for u near ζ.

Experimental results

Research questions

• RQ1 Under what conditions on the potential V is an isolated singularity at ζ = 0 or ζ = ∞ removable for positive solutions of the p-Laplacian equation?
• RQ2 How does the asymptotic behavior of positive solutions depend on the relationship between p and d, especially in the classical vs. nonclassical cases?
• RQ3 What role does the Kato-class condition on V play in ensuring continuity or boundedness of solutions at isolated singularities?
• RQ4 Can a positive Liouville theorem be established for p < d when the associated quadratic form is non-negative and critical?
• RQ5 Is the germ of positive solutions near an isolated singularity structured into exactly two equivalence classes under asymptotic comparison?

Key findings

• If p > d and V belongs to the Kato class near ζ = 0, then any positive solution of Q'(u) = 0 in a punctured neighborhood of 0 extends continuously to 0.
• If p < d and V belongs to the Kato class near ζ = ∞, then any positive solution of Q'(u) = 0 in a punctured neighborhood of ∞ extends continuously to ∞.
• For p > d and V in the Kato class near 0, the solution satisfies lim_{x→0} u(x) = C < ∞, with C = 0 if the solution has minimal growth.
• For p < d and V in the Kato class near ∞, the solution satisfies lim_{x→∞} u(x) = C < ∞, and C = 0 if and only if the quadratic form Q is critical.
• When V is integrable near ∞ and p < d, the solution satisfies u(x) ∼ |x|^{(p−d)/(p−1)} as x → ∞ if and only if it has minimal growth.
• The conjecture that the germ of positive solutions near an isolated singularity has exactly two equivalence classes under asymptotic comparison is partially confirmed via criticality and regularity results.