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

TL;DR

This paper investigates the existence of solutions for a non-linear eigenvalue problem with variable weights, establishing conditions under which minimizers exist based on the parameters and the behavior of the weight function.

Contribution

It extends previous work by showing minimizer existence for certain parameter ranges where the weight function's growth dominates the non-linear effects.

Findings

Minimizers exist for eta > kn/q + 2 when 0<λ≤αλ₁(Ω), 0≤k≤q-2.

Existence depends on the balance between linear and non-linear terms.

Previous non-existence results are complemented by new existence results in the specified parameter regime.

Abstract

We study the non-linear minimization problem on $H^1_0(\Omega)\subset L^q$ with $q=\frac{2n}{n-2}$, $\alpha>0$ and $n\geq4$~: \[\inf_{\substack{u\in H^1_0(\Omega) \|u\|_{L^q}=1}}\int_\Omega a(x,u)|\nabla u|^2 - \lambda \int_{\Omega} |u|^2.\] 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 \[\inf_{\substack{u\in H^1_0(\Omega) \|u\|_{L^q}=1}}\int_\Omega (\alpha+|x|^\beta |u|^k)|\nabla u|^2 - \lambda \int_{\Omega} |u|^2.\] 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 $\beta\geq kn/q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda\leq \alpha\lambda_1(\Omega)$, $0\leq k\leq q-2$ and $\beta > kn/q + 2$, minimizers do exist.