Existence and Nonexistence of Entire Positive Solutions for a Class of Singular p-Laplacian Elliptic System

Daojin Lei,Zuodong Yang

doi:10.4236/apm.2011.16063

Abstract

In this paper, we show the existence and nonexistence of entire positive solutions for a class of singular elliptic system We have that entire large positive solutions fail to exist if f and g are sublinear and b and d have fast decay at infinity. However, if f and g satisfy some growth conditions at infinity, and b, d are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.

Highlights

In this paper, we mainly consider the existence and nonexistence of positive solutions for the following singular p-laplacian elliptic system: div x ap u p 2 u = b x f u, v, x RN,=d x g u,v, x RN, (1)When a = 0, p = q = 2, the following semi-linear elliptic system: above system becomes u = b x v, x RN, v = d xu, x RN, for which existence results for boundary blow-up positive solution can be found in a recent paper by Lair and Wood [5]
U, x RN, for which existence results for boundary blow-up positive solution can be found in a recent paper by Lair and Wood [5]
The authors established that all positive entire radial solutions of system above are boundary blow-up provided that

Summary

Introduction

We mainly consider the existence and nonexistence of positive solutions for the following singular p-laplacian elliptic system: div x ap u p 2 u = b x f u, v , x RN ,. The authors established that all positive entire radial solutions of system above are boundary blow-up provided that 0 tb t. Ds, there exists an entire positive radial solution, and in addition, the function b, d satisfy (H3). 1d s ds q 1 dt = , all entire positive radial solutions are large. 1d s ds q 1 dt < , all entire positive radial solutions are bounded. Where m = min{ p, q} , problem (1) has no positive entire radial large solution. S 0 the system (1) has infinitely many positive entire bounded solutions

Proof of Theorem 1

U r V r m 1 p 1 r r0

U r V r 2 1 U r V r m 1

Proofs of Theorem 2 and Theorem 3

Proof of Theorem 4