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 p-torsion problem) and Formula: see text (the principal frequency of the Formula: see text-Laplacian).
Bocea et al. (Fri,) studied this question.