Mediterr. J. Math. (2020) 17:129 https://doi.org/10.1007/s00009-020-01564-w c  Springer Nature Switzerland AG 2020 Existence of Solution for Nonlocal Heterogeneous Elliptic Problems Mahmoud Bousselsal and Elmehdi Zaouche Abstract. We consider a class of a nonlocal heterogeneous elliptic prob- lem of type −M(|u| q q ) div(a(x)∇u)= g(x, u) with a homogeneous Dirichlet boundary condition. Under different as- sumptions on the function g, we establish two existence theorems for this problem by using, respectively, the Schauder and Tychonoff fixed point theorems. Also, we give an example for each theorem. Mathematics Subject Classification. 35J60, 35J75, 47H10. Keywords. Nonlocal heterogeneous elliptic problem, Homogeneous Dirichlet boundary condition, Schauder and Tychonoff fixed point the- orems, Existence. 1. Introduction Let Ω be a bounded domain in R n (n ≥ 1). We consider the following weak formulation of the nonlocal heterogeneous elliptic problem (P) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Find u ∈ H 1 0 (Ω) such that : M (|u| q q )  Ω a(x)∇u ·∇ξ dx =  Ω g(x, u)ξ dx ∀ξ ∈ H 1 0 (Ω), where 1 ≤ q ≤ 2, for a.e. x ∈ Ω, a(x)=(a ij (x)) ij is an n × n matrix satisfying for two positive constants λ, Λ: ∀ξ ∈ R n : λ|ξ | 2 ≤ a(x)ξ · ξ a.e. x ∈ Ω, (1.1) ∀ξ ∈ R n : |a(x)ξ |≤ Λ|ξ | a.e. x ∈ Ω, (1.2) and M : R → R is a continuous function such that, for some constant m 0 > 0, we have M (t) ≥ m 0 ∀t ∈ R. (1.3) 0123456789().: V,-vol