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

Abstract

We study the quasi-linear minimization problem on $H_0^1(\Omega )\subset L^q$ with $q=\frac{2n}{n-2}$~: $$\underset{\Vert u{\Vert }_{L^q}=1}{\inf }{\int }_{\Omega }(1+|x{|}^{\beta }|u{|}^k)|\nabla u{|}^2.$$ 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 $\beta \ge kn/q$ prevents their existence.

13 pages