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The identification of an unknown constant coefficient in the main term of the second order differential equation-  k  M  (  |  u   |   p  -  2    u  )  +  g   (  x  )   u  =  f   (  x  )${ - kM(|u|^{p-2} u) + g(x) u = f(x) }$    with Dirichlet boundary condition is considered. The elliptic operator  M  is self-adjoint,p  ≥  2    ${p \ge 2}$   . The identification of  k  here is based on an integral boundary data. The local existence and uniqueness theorems for the inverse problem is proved in the class of the pairs involving a function  u  such that|  u  |    p  -  2    u  ∈   W  2  2    (  Ω  )${|u|^{p-2} u \in W^2_2 (\Omega )}$    and a positive real number  k .

Identification of a constant coefficient in a quasi-linear elliptic equation