In this paper, we prove the existence of weak, very weak and duality solutions to a class of elliptic problems involving singularity and measure data which is given by: \(-\Delta u+(-\Delta )^s u = \frac{f(x)}{u^\gamma } +\mu \) in \(\Omega \) with the zero Dirichlet boundary data \(u=0\) in \({\mathbb {R}}^N \setminus \Omega \) . The existence of weak solutions is obtained by approximating a sequence of problems for \(0<\gamma \le 1\) and \(\gamma >1\) . We employ Schauder’s fixed point theorem and embeddings of Marcinkiewicz spaces. The novelty of our work is that we prove the existence of a duality solution and its equivalence with weak solutions to the problem \({\mathcal {L}}u=\mu \) . Moreover, we prove a very weak maximum principle and a Kato-type inequality for the mixed local-nonlocal operator \({\mathcal {L}}=-\Delta +(-\Delta )^s\) without using the Green’s function, which are crucial tools to guarantee the existence of very weak solutions to the problem. Using a Kato-type inequality, maximum principle together with sub-super solution method, we prove the existence of very weak solution for \(0<\gamma <1\) . Our work extends the studies due to Oliva and Petitta [ESAIM Control Optim. Calc. Var., 22(1):289–308, 2016.] and Petitta [Adv. Nonlinear Stud., 16(1):115-124, 2016.] for the mixed local-nonlocal operator.
