Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain with the boundary \(\partial\Omega= \Gamma_0\cup\Gamma_1\) (\(\Gamma_0\) being the interior boundary and \(\Gamma_1\neq\emptyset\) the outer boundary with \(\Gamma_0\cap\Gamma_1= \emptyset\)). For simplicity, we assume \(\Gamma_0,\Gamma_1\in C^\infty\). In this paper, we consider the following nonlinear boundary-value problem with equivalued surface: \[ Au\equiv -\sum{\partial\over\partial x_i} \Biggl(a_{ij}(x,u){\partial u\over\partial x_j}\Biggr)= b(x)\quad\text{in }\Omega,\tag{1} \]
\[ u|_{\Gamma_1}= 0,\tag{2} \]
\[ u|_{\Gamma_0}= C\quad(\text{a constant to be determined}),\tag{3} \]
\[ \int_{\Gamma_0} {\partial u\over\partial n_A} ds= A_0\quad(\text{a known constant}),\tag{4} \] where \[ {\partial u\over\partial n_A}= \sum a_{ij}(x,u) {\partial u\over\partial x_j} n_i. \] We prove the existence of a weak solution to problem (1)–(4) by means of the pseudomonotone operator theory and the existence of \(C^{2,\alpha}(\overline\Omega)\) solution to problem (1)–(4) by Leray-Schauder’s fixed point theorem.