Liouville-Type Theorems for Higher Order Elliptic Systems

Authors

Frank Arthur, University of Connecticut - StorrsFollow

Date of Completion

8-16-2016

Embargo Period

8-8-2016

Major Advisor

Dr. Xiaodong Yan

Associate Advisor

Dr. Yung-Sze Choi

Associate Advisor

Dr. Damin Wu

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

ABSTRACT

We study positive solutions of the following higher order elliptic system

(-∆)^m u =|x|^a v^p and

(-∆)^m v =|x|^b u^q in R^N.

Here p ≥ 1, q ≥ 1, (p, q) ≠ (1, 1). Henon-Lane-Emden conjecture says (1.0.1) admits no positive solutions if (1+a/N)/(p+1) + (1+b/N)/(q+1) >1-2m/N.

When a = b = 0, we solve the conjecture under the additional assumption

max ((2m(p+1)+a+bp)/(pq-1) , (2m(q+1)+aq+b)/(pq-1)) > N − 2m − 1.

In particular, when N = 2m + 1 or N = 2m + 2, the conjecture hold true.

When a > 0, b > 0, we prove the conjecture under the additional assumptions

max ((2m(p+1)+a+bp)/(pq-1) , (2m(q+1)+aq+b)/(pq-1)) > N − 2m − 1.

Recommended Citation

Arthur, Frank, "Liouville-Type Theorems for Higher Order Elliptic Systems" (2016). Doctoral Dissertations. 1238.
https://digitalcommons.lib.uconn.edu/dissertations/1238