Large solutions of elliptic semilinear equations non-degenerate near the boundary

TL;DR

This paper investigates the uniqueness and boundary behavior of large solutions to elliptic semilinear equations with non-zero sources, emphasizing the importance of uniform ellipticity and Keller-Osserman conditions near the boundary.

Contribution

It establishes conditions under which large solutions are unique and characterizes their boundary blow-up behavior, including cases where solutions only exist as large solutions.

Findings

Large solutions are unique under certain ellipticity and Keller-Osserman conditions.

Solutions exhibit boundary blow-up consistent with Keller-Osserman integral.

Some PDEs admit only large solutions due to boundary explosion conditions.

Abstract

In this paper we study the so-called large solutions of elliptic semilinear equations with non null sources term, thus solutions blowing up on the boundary of the domain for which reason they are greater than any other solution whenever Weak Maximum Principle holds. The main topic about large solutions is uniqueness results and their behavior near the boundary. It is much less than being simple. The structure of the semilinear equations considered includes the well known Keller-Osserman integral and an assumption on the ellipticity of the leading part of the differential operator. In our study an uniform ellipticity near the boundary is required. We consider source terms in the PDE whose boundary explosion is consistent with the Keller-Osserman condition. Extra Keller-Osserman explosions on the source are also studied, showing in particular that in some cases the PDE only admits large solutions.