  We study the asymptotic profile, as $\hbar\rightarrow 0$, of positive
solutions to $$-\hbar^2\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta
u^2=K(x)|u|^{p-2}u,\ \ x\in \mathbb{R}^N $$ where $\gamma\geq 0$ is a parameter
with relevant physical interpretations, $V$ and $K$ are given potentials and
$N\geq 5$. We investigate the concentrating behavior of solutions when
$\gamma>0$ and, differently form the case $\gamma=0$ where the leading
potential is $V$, the concentration is here localized by the source potential
$K$. Moreover, surprisingly for $\gamma>0$ we find a different concentration
behavior of solutions in the case $p=\frac{2N}{N-2}$ and when
$\frac{2N}{N-2}<p<\frac{4N}{N-2}$. This phenomenon does not occur when
$\gamma=0$.
