Uniqueness for local–nonlocal elliptic equations

We study mixed local and nonlocal elliptic equation with a variable coefficient [Formula: see text]. Under suitable assumptions on the behavior at infinity of [Formula: see text], we obtain uniqueness of solutions belonging to certain weighted Lebesgue spaces, with a weight depending on the coefficient [Formula: see text]. The hypothesis on [Formula: see text] is optimal; indeed, when it fails we get nonuniqueness of solutions. We also investigate the parabolic counterpart of such equation.