In this paper, we study the following singular problem associated with mixed operators (the combination of the classical Laplace operator and the fractional Laplace operator) under mixed boundary conditions \(\begin{aligned} \left\{ \begin{aligned} {\mathcal {L}}u&= g(u), \quad u > 0 \quad \text {in} \quad \Omega ,\\ u&= 0 \quad \text {in} \quad U^c,\\ {\mathcal {N}}_s(u)&= 0 \quad \text {in} \quad {\mathcal {N}},\\ \frac{\partial u}{\partial \nu }&= 0 \quad \text {in} \quad \partial \Omega \cap \overline{{\mathcal {N}}}, \end{aligned} \right. \qquad \qquad \qquad \qquad \qquad {(P_\lambda )} \end{aligned}\) where \(U= (\Omega \cup {{\mathcal {N}}} \cup (\partial \Omega \cap \overline{{\mathcal {N}}}))\) , \(\Omega \subseteq \mathbb {R}^N\) is a non empty open set, \({\mathcal {D}}\) , \({\mathcal {N}}\) are open subsets of \(\mathbb {R}^N\setminus {\bar{\Omega }}\) such that \({{\mathcal {D}}} \cup {{\mathcal {N}}}= \mathbb {R}^N{\setminus }{\bar{\Omega }}\) , \({\mathcal {D}} \cap {{\mathcal {N}}}= \emptyset \) and \(\Omega \cup {\mathcal {N}}\) is a bounded set with smooth boundary, \(\lambda >0\) is a real parameter and \({\mathcal {L}}= -\Delta +(-\Delta )^{s},~ \text {for}~s \in (0, 1).\) Here \(g(u)=u^{-q}\) or \(g(u)= \lambda u^{-q}+ u^p\) with \(0<q<1<p\le 2^*-1\) . We study \((P_\lambda )\) to derive the existence of weak solutions along with its \(L^\infty \) -regularity. Moreover, some Sobolev-type variational inequalities associated with these weak solutions are established.
