We consider a problem in a bounded domain, with Dirichlet condition on one part of its boundary and on other parts non-linear slip conditions governed by Coulomb friction law and Fourier law. We assume that the problem is also governed by a particular constitutive law of elasticity system with a strongly nonlinear strain tensor given by \(\sigma _{ij} = \sum _{k,h=1}^{3} a_{ijkh} \, E_{hk}(\nabla u)\) where u is a displacement of a substance, \((a_{ijkh})_{1\le i, j, k,h\le 3}\) are the coefficients of elasticity and \(E_{hk}\) are the components of the nonlinear deformation tensor of St Venant \(E(\nabla u) = \frac{1}{2}\left( ^{T}\nabla u +\nabla u + {^{T}\nabla } u \nabla u\right) \) . The functional framework leads to use Sobolev spaces with variable exponent. The formulation of the problem leads to a variational inequality, for which we prove, by Schauder fixed point theorem, an existence solution.
