Existence of Solutions of a Non-Linear Eigenvalue Problem with a Variable Weight

We study the non-linear minimization problem on $H_0^1(\mathrm{\Omega })\subset L^q$ with $q=\frac{2n}{n-2}$, $\alpha >0$ and $n\ge 4$~: $$\underset{\begin{array}{c}u\in H_0^1(\mathrm{\Omega })\mathrm{\parallel }u{\mathrm{\parallel }}_{L^q}=1\end{array}}{\inf }{\int }_{\mathrm{\Omega }}a(x,u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2-\lambda {\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{.}$$ where $a(x,s)$ presents a global minimum $\alpha$ at $(x_0,0)$ with $x_0\in \mathrm{\Omega }$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behaviour of $a(x,s)$ with respect to $s$. The model case is $$\underset{\begin{array}{c}u\in H_0^1(\mathrm{\Omega })\mathrm{\parallel }u{\mathrm{\parallel }}_{L^q}=1\end{array}}{\inf }{\int }_{\mathrm{\Omega }}(\alpha +\mathrm{\mid }x{\mathrm{\mid }}^{\beta }\mathrm{\mid }u{\mathrm{\mid }}^k)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2-\lambda {\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{.}$$ In a previous paper dedicated to the same problem with $\lambda =0$, we showed that minimizers exist only in the range $\beta <kn\mathrm{/}q$, which corresponds to a dominant non-linear term. On the contrary, the linear influence for $\beta \ge kn\mathrm{/}q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda \le \alpha {\lambda }_1(\mathrm{\Omega })$, $0\le k\le q-2$ and $\beta >kn\mathrm{/}q+2$, minimizers do exist.