Minimization problems interpolating between the p-torsion problem and the principal frequency of the p-Laplacian

In this paper, we study the family of minimization problems [Formula: see text] where [Formula: see text] [Formula: see text] is an open and bounded set with smooth boundary and [Formula: see text] and [Formula: see text] are two real numbers. We analyze the existence of minimizers, their asymptotic behavior as [Formula: see text] when [Formula: see text] is fixed, and as [Formula: see text] and [Formula: see text] when [Formula: see text] is fixed. The case where [Formula: see text] satisfying [Formula: see text] is considered as well. Further, we investigate the monotonicity of the map [Formula: see text] when [Formula: see text], [Formula: see text] and [Formula: see text] are given. In particular, we recover the results available in the literature for the limiting cases where [Formula: see text] (related to the [Formula: see text]-torsion problem) and [Formula: see text] (the principal frequency of the [Formula: see text]-Laplacian).