On the structure of positive solutions to an elliptic problem arising in thin film equations

We consider the structure of positive radial solutions of Δ u + u − α − u − β = 0 in B R , u = 0 on ∂ B R , where B R is a ball in R N with radius R. When 0 < α < β < 1 , we show that there exists R ∗ > 0 such that when R > R ∗ , the Dirichlet problem has exactly two radial solutions; when R = R ∗ , the solution is unique and there is no solution for R < R ∗ . When 0 < β < α < 1 , we show that for any R > 0 , the radial solution exists and is unique.