"Boundary blowup" type sub-solutions to semilinear elliptic equations with Hardy potential

TL;DR

This paper investigates boundary blowup solutions to semilinear elliptic equations perturbed by Hardy potentials, establishing existence and nonexistence results based on potential size and boundary behavior classifications.

Contribution

It introduces a new definition of large solutions in the presence of Hardy potentials and provides comprehensive existence and nonexistence results using boundary classification techniques.

Findings

Existence of large solutions depends on the Hardy potential size.

Nonexistence results are derived for small Hardy potentials.

Solutions are classified by their boundary behavior using a Phragmen-Lindelof theorem.

Abstract

Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential. The size of this potential effects the existence of a certain type of solutions (large solutions): if the potential is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a Phragmen-Lindelof type theorem which enables us to classify the solutions and sub-solutions according to their behavior near the boundary. Nonexistence follows from this principle together with the Keller-Osserman upper bound. The existence proofs rely on sub- and super-solution techniques and on estimates for the Hardy constant derived in Marcus, Mizel and Pinchover.