Semilinear elliptic equations admitting similarity transformations

TL;DR

This paper investigates a class of semilinear elliptic equations with boundary behavior characterized by distance functions, establishing conditions for subcriticality and well-posedness, extending previous results to more general nonlinearities.

Contribution

It introduces a new condition on the nonlinearity h that characterizes subcriticality and boundary value problem solvability for a broad class of semilinear elliptic equations.

Findings

Established a criterion for subcriticality based on h.

Proved the boundary value problem is well-posed under certain conditions.

Extended previous results to more general nonlinear functions h.

Abstract

In this paper we study the equation $-\Delta u+\rho^{-(\alpha+2)}h(\rho^{\alpha}u)=0$ in a smooth bounded domain $\Omega$ where $\rho(x)=\textrm{dist}\,(x,\partial \Omega)$, $\alpha>0$ and $h$ is a non-decreasing function which satisfies Keller-Osserman condition. We introduce a condition on $h$ which implies that the equation is subcritical, i.e. the corresponding boundary value problem is well posed with respect to data given by finite measures. Under additional assumptions on $h$ we show that this condition is necessary as well as sufficient. We also discuss b.v. problems with data given by positive unbounded measures. Our results extend results of \cite{MV1} treating equations of the form $-\Delta u+\rho^\beta u^q=0$ with $q>1$, $\beta>-2$.