In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the problem given by where \(D= \left( \Omega \cup {\Pi _2} \cup (\partial \Omega \cap \overline{\Pi _2})\right) \) and \(D^c\) is the complement of D, \(\Omega \subseteq {\mathbb {R}}^n\) is a non empty bounded open set with sufficiently smooth boundary \(\partial \Omega \) , say of class \(C^1\) . \(\Pi _{1}\) , \(\Pi _{2}\) are open subsets of \({\mathbb {R}}^n\setminus {{{\bar{\Omega }}} }\) such that \(\overline{{\Pi _{1}} \cup {\Pi _{{2}}}}= {\mathbb {R}}^n\setminus {\Omega }\) , \(\Pi _{1} \cap \Pi _{{2}}= \emptyset \) , \(\partial \Omega \cap \overline{\Pi _2}\ne \emptyset \) and \(\Omega \cup \Pi _2\) is a bounded set with sufficiently smooth boundary, \(\lambda >0\) is a real parameter, \( 0< q< 1<p \) , \(n>2\) and \({\mathcal {L}}= -\Delta +(-\Delta )^{s},~ \text {for}~s \in (0, 1).\) We first present a functional setting to study any problem involving \({\mathcal {L}}\) under mixed boundary conditions in the presence of concave-convex power nonlinearity, for a suitable range of \(\lambda \) , q and p. Our article also contains results related to Picone’s identity, strong maximum principles and comparison principles. We have extended the results of [1] to problems admitting mixed type operator as well as mixed boundary conditions.
