Remark on closedness of perturbation set and regularity of associated extremal solution

We investigate here the effect of dimension on closedness of KK—the set of perturbation term for some semi-linear elliptic equation −Δu=f(u)+g(x)−Δu=f(u)+g(x) posed on a bounded smooth domain in RnRn with Dirichlet boundary condition. Some closedness of KK and continuous of the map g↦u¯(g) are established. Moreover, we show a counter example for power case f(u)=eu when n>9n>9.