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

Abstract

We study the non-linear minimization problem on $H_0^1(\Omega )\subset L^q$ with $q=\frac{2n}{n-2}$, $\alpha >0$ and $n\ge 4$~: \[\inf_{\substack{u\in H^1_0(Ω) \|u\|_{L^q}=1}}\int_Ωa(x,u)|\nabla u|^2 - λ\int_Ω |u|^2.\] where $a(x,s)$ presents a global minimum $\alpha$ at $(x_0,0)$ with $x_0\in \Omega$. 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 \[\inf_{\substack{u\in H^1_0(Ω) \|u\|_{L^q}=1}}\int_Ω(α+|x|^β|u|^k)|\nabla u|^2 - λ\int_Ω |u|^2.\] In a previous paper dedicated to the same problem with $\lambda =0$, we showed 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 $\beta \ge kn/q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda \le \alpha {\lambda }_1(\Omega )$, $0\le k\le q-2$ and $\beta >kn/q+2$, minimizers do exist.

21 pages