Semilinear elliptic equations admitting similarity transformations

In this paper we study the equation −Δu+ρ−(α+2)h(ραu)=0 in a smooth bounded domain Ω where ρ(x)=dist(x,∂Ω), α>0 and h is a nondecreasing 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 [13] treating equations of the form −Δu+ρβuq=0 with q>1, β>−2.