In the present paper, we study a class of quasilinear equation with mixed boundary conditions: \(\begin{aligned}\left\{ \begin{array}{ll} \displaystyle - \text{ div }(a(x,u_{1},Du_{1})) + \mathcal {K}(x,u_{1}) = f(x) \qquad & \text{ in } \Omega _{1},\\ \displaystyle - \text{ div }(a(x,u_{2},Du_{2})) + \mathcal {K}(x,u_{2}) = f(x) \qquad & \text{ in } \Omega _{2},\\ \displaystyle u_{1} = 0 & \text{ on } \partial \Omega ,\\ \displaystyle a(x,u_{1},Du_{1})\cdot \nu _{1} + \lambda u_{1} = a(x,u_{2},Du_{2})\cdot \nu _{1} + \lambda u_{2} & \text{ on } \Gamma ,\\ \displaystyle a(x,u_{1},Du_{1})\cdot \nu _{1} +\lambda u_{1} = - h(x)|u_{1}-u_{2}|^{p-2}(u_{1}-u_{2}) + G(x) & \text{ on } \Gamma . \end{array} \right. \end{aligned}\) Here, \(\Omega \) is a connected, open and bounded subset of \(\mathbb {R}^{N}\) \((N\ge 2)\) with a Lipschitz boundary \(\partial \Omega .\) The set \(\Omega \) can be decomposed as \(\Omega =\Omega _{1}\cup \Omega _{2}\cup \Gamma \) , where \(\Omega _{2}\) is an open subset such that \(\overline{\Omega _{2}}\) is included in \(\Omega ,\) and \(\Omega _1=\Omega \smallsetminus \overline{\Omega }_{2} \) with a Lipschitz boundary \(\Gamma = \partial \Omega _{2}\) . We will prove the existence of renormalized solutions for this class of equation and we will conclude some regularity results.
