Non-Linear Effects in a Yamabe-Type Problem with Quasi-Linear Weight

TL;DR

This paper investigates a quasi-linear minimization problem involving a weight function with non-linear effects, establishing conditions for the existence of minimizers based on the dominance of non-linear terms.

Contribution

It identifies the precise range of parameters where minimizers exist, highlighting the transition between non-linear dominance and linear influence.

Findings

Minimizers exist only when eta<kn/q.

Existence is prevented when eta kn/q due to linear influence.

The critical threshold eta=kn/q marks the transition point.

Abstract

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.