Abstract: In the present paper, we consider the problem
$\begin{equation} \left\{\begin{array}{ll}\label{0001} -\Delta u=u^{\beta_1}|\nabla u|^{\beta_2}, &\ \ { in} \ \Omega,\\ u=0,&\ \ { on} \ \partial{\Omega},\\ u>0,&\ \ { in} \ {\Omega},\\ \end{array}\right. \end{equation}$ $\ \ \ \ \ $ (0.1)
where $\beta_1,\beta_2>0$ and $\beta_1+\beta_2<1$, and $\Omega$ is a convex domain in $\mathbb{R}^{n}$. The existence, uniqueness, regularity and $\frac{2-\beta_{2}}{1-\beta_1-\beta_2}$-concavity of the positive solutions of the problem (0.1) are proven.

Key words: Harmonic-Mappings type equation, positive solution, existence, regularity, $\alpha$-concavity