A Brief overview on the existence of solution to a class of quasilinear integro-differential equations

In this paper, we are concerned with a class of integro-differential equations whose prototype is given by $${aligned} \{ {array}{lcl} -div(a(u)∇ u)+η ψ (x)∫ Ω φ u = f(x,u), in Ω ,\\ u=0, on ∂ Ω , {array} . {aligned}$$ - d i v ( a ( u ) ∇ u ) + η ψ ( x ) ∫ Ω φ u = f ( x , u ) , in Ω , u = 0 , on ∂ Ω , for which we establish the existence of nontrivial solutions for certain classes of nonlinearities f . Here $$ Ω ⊂ R^N$$ Ω ⊂ R N , $$ N≥ 1$$ N ≥ 1 , is a bounded smooth domain, $$ η ∈ R$$ η ∈ R is a real parameter, and the functions a , $$ ψ $$ ψ , $$ φ $$ φ , and f satisfy assumptions that will be specified in due course.