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

We study the quasi-linear minimization problem on $H^1_0(Ω)\subset L^q$ with $q=\frac{2n}{n-2}$~: $\inf_{\|u\|_{L^q}=1}\int_Ω(1+|x|^β|u|^k)|\nabla u|^2.$ We show that minimizers exist only in the range $β<kn/q$ which corresponds to a dominant non-linear term. On the contrary, the linear influence for $β\geq kn/q$ prevents their existence.

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