Semilinear elliptic equations admitting similarity transformations

In this paper we study the equation $-Δu+ρ^{-(α+2)}h(ρ^αu)=0$ in a smooth bounded domain $Ω$ where $ρ(x)=\textrm{dist}\,(x,\partial Ω)$, $α>0$ 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 $-Δu+ρ^βu^q=0$ with $q>1$, $β>-2$.

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