In this paper we prove the existence and uniqueness of renormalised solutions to the nonlinear elliptic problem defined as follows \(\begin{aligned} \left\{ \begin{array}{lll} \displaystyle -\text{ div }(a(x,|\nabla u|)\nabla u)+ |u|^{p(x)-2}u \displaystyle = f(x)-\text{ div }(\phi (x,u))& \text{ in } & \Omega \\ \displaystyle \lambda u+(a(x,|\nabla u|)\nabla u-\phi (x,u)).\eta \displaystyle = g & \text{ on } & \partial \Omega , \end{array}\right. \end{aligned}\) in the setting of Sobolev spaces with variable exponents, where \(\Omega \) is a bounded open subset in \(I\!\!R^{N}\) ( \(N\ge 3\) ) with Lipschitz boundary \(\partial \Omega \) , \(\eta \) is the outer unit normal vector on \(\partial \Omega \) , and \(\lambda \) is a strictly positive real constant. The nonlinear term \(\phi (x,u)\) verifies some growth condition, the right hand-side f belongs to \(L^{1}(\Omega ) \) and \(g \in L^{1}( \partial \Omega )\) .
