Bol. Soc. Paran. Mat. (3s.) v. ???? (??) : 1–11. c SPM –ISSN-2175-1188 on line ISSN-0037-8712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.44144 Multiple Solutions for a Class of Bi-nonlocal Problems with Nonlinear Neumann Boundary Conditions Ghasem A. Afrouzi, Z. Naghizadeh and N. T. Chung abstract: In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using (S + ) mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provided that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities. Key Words: Bi-nonlocal problems, Nonlinear Neumann boundary conditions, Mountain pass theo- rem. Contents 1 Introduction 1 2 Proofs of the main results 5 1. Introduction In this paper, we are interested in a class of Kirchhoff type problems with nonlinear Neumann bound- ary conditions of the form  M 1 (L 1 (u))  − div( ϕ(x, ∇u)) + |u| p−2 u  = λM 2 (L 2 (u)) f (x,u),x ∈ Ω, M 1 (L 1 (u)) ϕ(x, ∇u).ν = µg(x,u),x ∈ ∂ Ω, (1.1) where Ω is a smooth bounded domain in R N (N ≥ 3), ν is the outward normal vector on the boundary ∂ Ω, 2 ≤ p<N , λ,µ are parameters, L 1 (u)= 1 p  Ω (H (|∇u| p )+ |u| p )dx, H (t)=  t 0 h(s)ds for all t ∈ R, ϕ(x,v)= h(|v| p )|v| p−2 v with increasing continuous functions h from R into R, L 2 (u)=  Ω F (x,u)dx, where F (x,u)=  u 0 f (x,s)ds and f :Ω × R → R, g : ∂ Ω × R → R satisfy the Carath´ eodory condition. Moreover, M 1 : R + 0 = [0, +∞) → R and M 2 : R + 0 → R are assumed to be continuous functions. It should be noticed that if h(t) ≡ 1, problem (1.1) becomes a nonlocal Kirchhoff type equation with nonlinear boundary condition        M 1  1 p  Ω (|∇u| p + |u| p ) dx  − div(|∇u| p−2 ∇u)+ |u| p−2 u  = λM 2 (  Ω F (x,u) dx)f (x,u),x ∈ Ω, M 1  1 p  Ω (|∇u| p + |u| p ) dx  |∇u| p−2 ∂u ∂ν = µg(x,u),x ∈ ∂ Ω. (1.2) Since the first equation in (1.2) contains an integral over Ω, it is no longer a pointwise identity; therefore it is often called nonlocal problem. The interest of such problems comes from the fact that Kirchhoff type problems usually model several physical and biological systems, where u describes a process which depends on the average of itself, such as the population density. Moreover, problem (1.2) is related to the stationary version of Kirchhoff equation ρ ∂ 2 u ∂t 2 −  p 0 h + E 2L  L 0     ∂u ∂x     2 dx  ∂ 2 u ∂x 2 =0 (1.3) 2010 Mathematics Subject Classification: 35D30, 35J20, 35J66, 35J60. Submitted August 15, 2018. Published January 02, 2019 1 Typeset by B S P M style. c  Soc. Paran. de Mat.