Semilinear elliptic equations admitting similarity transformations

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 $\rho (x)=\text{dist}(x,\partial \Omega )$, $\alpha >0$ 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 $q>1$, $\beta >-2$.

30 pages