Let \(D\) be an \((N+1)\)-connected bounded domain in the complex plane \(C\) with the boundary \(\Gamma =\Gamma_0 \cup \Gamma_1 \cup \dots \cup \Gamma_N\), where \(\Gamma_0=\{ | z| =1\} \) and \(\Gamma_j =\{ | z-a_j| =r_j\} \), \(j=1,\dots ,N\). Put \(T=\{ z_1,\dots ,z_m\} \), where \(z_1,\dots ,z_m\) are distinct points on \(\Gamma \). It is studied the following problem: Find a solution \(u\) of \(u_{z \overline z}= F(z,u,u_z,u_{zz})\) in \(D\), which is continuous in \(\overline D\), whose partial derivatives \(u_x\), \(u_y\) in \(D^*=\overline D\setminus T\) are continuous, and satisfies the boundary conditions \(\partial u/\partial \nu +2a_1(z)u(z)=2a_2(z)\), \(z\in \Gamma^* =\Gamma \setminus T\), \(u(z_k)=b_k\). Here \(\nu \neq 0\) can be arbitrary at every point on \(\Gamma^*\). At first, the uniqueness of a solution of this boundary value problem is studied and a priory estimates of solutions are given. Then the existence of solutions of the problem is verified using Leray-Schauder theorem.