On an Asymptotically Linear Elliptic Dirichlet Problem

where Ω is a bounded domain in RN (N ≥ 1) with smooth boundary ∂Ω. The conditions imposed on f (x, t) are as follows: (f1) f ∈ C(Ω×R,R); f (x,0) = 0, for all x ∈Ω. (f2) lim|t|→0( f (x, t)/t) = μ, lim|t|→∞( f (x, t)/t) = uniformly in x ∈Ω. Since we assume (f2), problem (1.1) is called asymptotically linear at both zero and infinity. This kind of problems have captured great interest since the pioneer work of [1]. For more information, see [2, 3, 4, 5, 6, 7, 8, 11, 12] and the references therein. Obviously, the constant function u= 0 is a trivial solution of problem (1.1). Therefore, we are interested in finding nontrivial solutions. Let F(x,u) = ∫ u 0 f (x,s)ds. It follows from (f1) and (f2) that the functional J(u) = 1 2 ∫