A Class Of Semilinear Elliptic Equations Disturbance Sets And Regional Dimension Of Research

Let Ω is a bounded smooth domain of Rn(n > 2). g ∈ Ca (Ω), 0 < α < 1, we are concerned with the following problem:where f satisfies (H): f is a C1 nonnegativo strictly convex function on ( - ∞,∞) , and increasing on [0, ∞) such thatSet K={g ∈ Cα(Ω), such that a solution of (1.1) exists }. In this paper, we mainly conside the effect of dimension on closeness of the set K and the continuous of the map g → u(g) on K. In 1990, Shixiao Wang considcd the properties of the map: 9 → u(g)(Cα(Ω) →C2,α 0(Ω)) (see [W]), and proved that in IntK this map is continuous. Moreover, for the special case f(u) = |u|p, 1 < p <∞, the following result holds:If n < 10, f(u) =|u|p, with 1 < p < ∞, then K is closed and the map g → u(g) is continuous in K.In this work, we investigate the effect of dimension on closeness of K and obtain some simple results. Moreover, we show a counter example for power case. f(u) = eu when n > 9. Our main results are shown as following:Theorem Suppose that f verifies (H) and is twice differentiate. Assume that , then K is closed and the map g t→u(g) is continuous in K.Theorem Set f = eu, a) if n < 9, then K is closed and the map g →u(g) is continuous in K; b) if n > 9 ,then K need not be closed.