Large solutions of coupled sublinear/superlinear elliptic equations

Abstract

We show that large positive solutions exist for the semilinear elliptic equation Δu = p(x)u ^α + q(x)v ^β on bounded domains in R ⁿ , n ≥ 3, for the superlinear case 0 < α ≤ β, β > 1, but not the sublinear case 0 < α ≤ β ≤ 1. We also show that entire large positive solutions exist for both the superlinear and sublinear cases provided the nonnegative continuous functions p and q satisfy certain decay conditions at infinity. Existence and nonexistence of entire bounded solutions are established as well.

Keywords:

• large solution
• semilinear elliptic equation

AMS Subject Classifications:

• Primary 35J25
• 35J60

Acknowledgements

The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defence, or the US Government. The authors thank Dr Alan Lair for suggesting the problem and the referee for helpful suggestions.