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

Abstract

Abstract In this paper, we are concerned with a class of integro-differential equations whose prototype is given by $$\begin{aligned} \left\{ \begin{array}{lcl} -div(a(u)\nabla u)+\eta \psi (x)\int _{\Omega } \varphi u = f(x,u), \text { in } \Omega ,\\ u=0, \text { on } \partial \Omega , \end{array} \right. \end{aligned}$$ - 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 $$\displaystyle \Omega \subset \mathbb {R}^N$$ Ω ⊂ R N , $$\displaystyle N\ge 1$$ N ≥ 1 , is a bounded smooth domain, $$\displaystyle \eta \in \mathbb {R}$$ η ∈ R is a real parameter, and the functions a , $$\displaystyle \psi $$ ψ , $$\displaystyle \varphi $$ φ , and f satisfy assumptions that will be specified in due course.