On the solutions of a singular elliptic equation concentrating on a circle

Abstract

Let A={x∈ℝ2N+2:0 0 in A, ∂u∂ν=0$\frac{\partial u}{\partial \nu } = 0$ on ∂A$\partial A$, where 1<p<2*-1${1<p<2^*-1}$. We shall show that there exists a positive solution uε${u_\varepsilon }$ concentrating on an S1 orbit as ε→0${\varepsilon \rightarrow 0}$. We prove this by reducing the problem to a lower-dimensional one and analyzing a single point concentrating solution in the lower-dimensional space. We make precise how the single peak concentration depends on the parameter α.

Keywords

• hopf fibration
• s1 concentration
• neumann data
• 35b09
• 35j20