  We study the quasi-linear minimization problem on $H^1_0(\Omega)\subset L^q$
with $q=\frac{2n}{n-2}$~: $$\inf_{\|u\|_{L^q}=1}\int_\Omega (1+|x|^\beta
|u|^k)|\nabla u|^2.$$ We show that minimizers exist only in the range
$\beta<kn/q$ which corresponds to a dominant non-linear term. On the contrary,
the linear influence for $\beta\geq kn/q$ prevents their existence.

Analysis of PDEs