Lyubanova∗ Institute of Space and Information Technology Siberian Federal University Kirenskogo, 26, Krasnoyarsk, 660026 Russia Received 12.11.2014, received in revised form 03.12.2014, accepted 20.12.2014 The identification of an unknown constant coefficient in the main term of the partial differential equation −kMψ(u)+g(x)u = f(x) with the Dirichlet boundary condition is investigated. <...> Here ψ(u) is a nonlinear increasing function of u, M is a linear self-adjoint elliptic operator of the second order. <...> The coefficient k is recovered on the base of additional integral boundary data. <...> The existence and uniqueness of the solution to the inverse problem involving a function u and a positive real number k is proved. <...> Keywords: inverse problem, boundary value problem, second-order elliptic equations, existence and uniqueness theorem, filtration. <...> This paper proceeds the investigation started in [5,6] and is concerned with an inverse problem for the second order differential equation −div(k(x,u)∇ψ(u))+γ(x,u) = f with the Dirichlet boundary data u ∂Ω = β(x) (0.2) where k(x,u) is a matrix of functions, ψ(u) and γ(x,u) are scalar functions. <...> Of special interest is the problem of finding the leading coefficients of (0.1) given the additional boundary data on ∂Ω or a part of ∂Ω. <...> In this paper we study the problem of the identification of the constant coefficient k in equation −k{div(M(x)∇ψ(u))+m(x)ψ(u)}+g(x)u = f(x), x ∈ Ω, (0.3) under the boundary data (0.2). <...> As an additional data for the recovery of the coefficient k we take the condition of overdetermination k ∂Ω ∂ψ(u) ∂N ω ds = ϕ (0.4) where ∂u/∂N ≡ (M(x)∇u,n)R is the conormal derivative, n is a unit outward normal to ∂Ω, ω is a given function, ϕ is a given real number. <...> Lyubanova On an Inverse Problem for Quasi-Linear Elliptic Equation conditions for elliptic equations were considered in [5,6]. <...> The inverse problem for linear elliptic equation (0.3) with ψ(ρ) = ρ is discussed in [5]. <...> The goal of our paper is to establish the existence and uniqueness of a solution to the nonlinear problem (0.2), (0.3), (0.4). <...> The preliminaries We start with preliminary results for the direct <...>