Remark on closedness of perturbation set and regularity of associated extremal solution Original Research Article

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 View the MathML sourceg↦u¯(g) are established. Moreover, we show a counter example for power case View the MathML sourcef(u)=eu when n>9n>9.