Linear theory for a mixed operator with Neumann conditions

We consider here a new type of mixed local and nonlocal equation under suitable Neumann conditions. discuss the spectral properties associated to weighted eigenvalue problem present global bound for subsolutions. The condition that we take into account comprises, as particular case, one has been recently introduced in (Rev. Mat. Iberoam. 33(2) (2017), 377–416). Also, results find natural application logistic motivated by biological problems considered (Dipierro, Proietti Lippi Valdinoci (2021)).