  We consider here a new type of mixed local and nonlocal equation under
suitable Neumann conditions. We discuss the spectral properties associated to a
weighted eigenvalue problem and present a global bound for subsolutions. The
Neumann condition that we take into account comprises, as a particular case,
the one that has been recently introduced in [S. Dipierro, X. Ros-Oton, E.
Valdinoci, Rev. Mat. Iberoam. (2017)]. Also, the results that we present here
find a natural application to a logistic equation motivated by biological
problems that has been recently considered in [S. Dipierro, E. Proietti Lippi,
E. Valdinoci, preprint (2020)].
