  For a finite dimensional algebra $\Lambda$ and a non-negative integer $n$, we
characterize when the set $\tilt_n\Lambda$ of additive equivalence classes of
tilting modules with projective dimension at most $n$ has a minimal (or
equivalently, minimum) element. This generalize results of Happel-Unger.
Moreover, for an $n$-Gorenstein algebra $\Lambda$ with $n\geq 1$, we construct
a minimal element in $\tilt_{n}\Lambda$.
  As a result, we give equivalent conditions for a $k$-Gorenstein algebra to be
Iwanaga-Gorenstein. Moreover, for an $1$-Gorenstein algebra $\Lambda$ and its
factor algebra $\Gamma=\Lambda/(e)$, we show that there is a bijection between
$\tilt_1\Lambda$ and the set $\sttilt\Gamma$ of isomorphism classes of basic
support $\tau$-tilting $\Gamma$-modules, where $e$ is an idempotent such that
$e\Lambda $ is the additive generator of projective-injective
$\Lambda$-modules.
