The authors show that if \(f\in L^p(\Omega),\) then under some certain hypotheses, the problem \[ \begin{cases} -\Delta_{p}u+\left| u(x)\right| ^{p-2}u(x) +g(x,u(x)) =f(x) &\text{ a.e.\;on }\Omega,\\ -\langle\nu,\left| \nabla u\right| ^{p-2}\nabla u\rangle\in\beta_x ( u(x)) &\text{ a.e.\;on }\partial\Omega, \end{cases} \] has a solutions in \(L^p(\Omega),\) where \(\frac{2N} {N+1}