Boundary singular solutions of a class of equations with mixed absorption-reaction

TL;DR

This paper investigates boundary singular solutions of a class of nonlinear PDEs with mixed absorption-reaction terms, focusing on solutions with isolated boundary singularities, their construction, and conditions for removability of boundary sets.

Contribution

It introduces new methods to construct positive solutions with boundary singularities and analyzes conditions for their removability in equations with mixed absorption-reaction.

Findings

Constructed various boundary singular solutions.

Identified conditions for boundary set removability.

Analyzed the Dirichlet problem with measure boundary data.

Abstract

We study properties of positive functions satisfying (E) --$\Delta$u + u p -- M |$\nabla$u| q = 0 is a domain $\Omega$ or in R N + when p > 1 and 1 < q < min{p, 2}. We concentrate our research on the solutions of (E) vanishing on the boundary except at one point. This analysis depends on the existence of separable solutions in R N +. We consruct various types of positive solutions with an isolated singularity on the boundary. We also study conditions for the removability of compact boundary sets and the Dirichlet problem associated to (E) with a measure for boundary data.