Let $\pi:X\longrightarrow Y$ be a non degenerate holomorphic mapping between compact Kähler manifolds such that $n:=dim_{\mathbb{C}}X\geq m:=dim_{\mathbb{C}}Y$. Let $\omega_{X}$, $\omega_{Y}$ Kähler forms on $X$ and $Y$ respectively. Let $F:X\longrightarrow\mathbb{R}^{+}$ be a non negative function such that $F\in L^{p}(X)$ for some $p>1$.

Set $\omega_{t}:=\pi^{*}(\omega_{Y})+t\omega_{X}$, $t>0$. We consider the following family of complex Monge-Ampère equations

${(\star )}_t$

$$\left\{\begin{array}[]{ll}(\omega_{t}+dd^{c}\varphi_{t})^{n}&=\ c_{t}t^{n-m}F\omega_{X}^{n}\\ \max_{X}\varphi_{t}=0&=\ 0\\ \end{array}\right.$$

where $\varphi_{t}$ is $\omega_{t}-$plurisubharmonic on $X$ and $c_{t}>0$ is a constant given by

$$c_{t}t^{n-m}\int_{X}F\omega_{X}^{n}=\int_{X}\omega_{t}^{n}.$$

It follows from the seminal work of S.T. Yau [Y] and S. Kolodziej [K 1], [K 2] that the equation $(\star)_{t}$ admits a unique continuous solution. (Observe that for $t\in]0,1]$, $\omega_{t}$ is a Kähler form).

Our aim here is to understand what happens when $t\to 0^{+}$, motivated by recent geometrical developpments [ST], [KT]. When $n=m$, the cohomology class ${\omega_{0}}$ is big and semi-ample and this problem has been adressed by several authors recently (see [CN], [EGZ], [TZ], [To]).

We focus here on the case $m<n$. This situation is motivated by the study of the Kähler-Ricci flow on manifolds $X$ of intermediate Kodaira dimension $1\leq kod(X)\leq n-1$. When $n=2$ this has been studied by J.Song and G.Tian [ST].

In a very recent and interesting paper [KT], S. Kolodziej and G. Tian were able to show, under a technical geometric assumption on the fibration $\pi$, that the solutions $(\varphi_{t})$ are uniformly bounded on $X$ when $t\searrow 0^{+}$.

The purpose of this note is to (re)prove this result without any technical assumption and with a different method: we actually follow the strategy introduced by S. Kolodziej in [K] and further developped in [EGZ], [BGZ].

It follows from our result that Theorems 1 and 2 in [KT] hold without any technical assumption on the fibration (see condition 0.2 in [KT]).

This result has been announced by J-P. Demailly and N. Pali [DP].