  The aim of this paper is to develop a basic framework of the $L_p$ theory for
the geometry of log-concave functions, which can be viewed as a functional
"lifting" of the $L_p$ Brunn-Minkowski theory for convex bodies. To fulfill
this goal, by combining the $L_p$ Asplund sum of log-concave functions for all
$p>1$ and the total mass, we obtain a Pr\'ekopa-Leindler type inequality and
propose a definition for the first variation of the total mass in the $L_p$
setting. Based on these, we further establish an $L_p$ Minkowski type
inequality related to the first variation of the total mass and derive a
variational formula which motivates the definition of our $L_p$ surface area
measure for log-concave functions. Consequently, the $L_p$ Minkowski problem
for log-concave functions, which aims to characterize the $L_p$ surface area
measure for log-concave functions, is introduced. The existence of solutions to
the $L_p$ Minkowski problem for log-concave functions is obtained for $p>1$
under some mild conditions on the pre-given Borel measures.
