Positive bounded solutions for semilinear elliptic equations in smooth domains

We are concerned with the following semilinear elliptic equation Delta u = lambda f (x,u) in D, subject to some Dirichlet conditions, where lambda >= 0 is a parameter and D is a smooth domain in R-n (n >= 3). Under some appropriate assumptions on the nonnegative nonlinearity term f (x,u), we show the existence of a positive bounded solution for the above semilinear elliptic equation. Our approach is based on Schauder's fixed point Theorem.