Elliptic Weighted Problem with Indefinite Asymptotically Linear Nonlinearity

The objective of this paper is the study the following nonlinear elliptic problem involving a weight function:

-div(a(x)∇υ) = f(x, u) in Ω and u = 0 on ∂Ω (P)

where, Ω is a regular bounded subset  and ℝ^N ≥ 2, a(x) is a nonnegative function and f(x, t) is allowed to be sign-changing. We employ variational techniques to prove the existence of a nontrivial solution for the problem (P), under some suitable assumptions, when the nonlinearity is asymptotically linear. Then, we prove by the same method the existence of positive solution when the function f is superlinear and subcritical at infinity.