Existence and multiplicity of solutions for a locally coercive elliptic equation

For a bounded domain Ω , we establish existence and multiplicity of nontrivial solutions for the semilinear elliptic problem {[ -Δ u = g(u) - h(x) f(u), in Ω; u = 0, on ∂Ω ,; ]. where h∈ L^∞ (Ω ) is nonnegative and nontrivial, g is asymptotically linear, f is superlinear and g(0)=f(0)=0 . We also study the existence of solutions for the problem {[ -Δ u = g(u) - h(x)f(u)+k(x), in Ω; u = 0, on ∂Ω ,; ]. when k∈ L^2(Ω ) .