Existence of entire explosive positive radial solutions of quasilinear elliptic systems

Yang Zuodong

doi:10.1155/s0161171203212394

Abstract

Our main purpose is to establish that entire explosive positive radial solutions exist for quasilinear elliptic systems. The main results of the present paper are new and extend previous results.

Highlights

Existence and nonexistence of solutions of the quasilinear elliptic system− div |∇u|p−2∇u = f (u, v), x ∈ RN, − div |∇v|q−2∇v = g(u, v), x ∈ RN, (1.1)have received much attention recently
We study the existence of entire explosive positive solutions of the system div |∇u|p−2∇u = m |x| vα, x ∈ RN, div |∇v|q−2∇v = n |x| uβ, x ∈ RN, (1.4)
Let f (u) satisfy the following conditions: (i) f (s) is a single-value real continuous function defined for all real values of s and there exists a positive nondecreasing continuous function F (s) such that f (s) ≥ F (s) and

Summary

Existence and nonexistence of solutions of the quasilinear elliptic system

We study the existence of entire explosive positive solutions of the system div |∇u|p−2∇u = m |x| vα, x ∈ RN , div |∇v|q−2∇v = n |x| uβ, x ∈ RN ,. As far as the author knows, there are no results that contain existence criteria of entire explosive positive solutions to the elliptic system (1.4) Motivated by this fact, we will study mainly this problem here. Let f (u) satisfy the following conditions: (i) f (s) is a single-value real continuous function defined for all real values of s and there exists a positive nondecreasing continuous function F (s) such that f (s) ≥ F (s) and. The existence and uniqueness of a positive solution v of (2.15) are assured because F1 is a nondecreasing function.

This implies

Methods