Existence of solutions of integro-differential semilinear elliptic equations

In this paper, we focus our attention on the equation (1) −div(a(x)∇u(x))+ηψ(x)∫Ωφu=f(x,u)inΩ,u=0on∂Ω(1) for which we get nontrivial solutions for some classes of functions f, where Ω⊂RN, N≥1, is a bounded smooth domain and a, ψ, φ, f are functions whose properties will be timely introduced.