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

Abstract

Let $A=\{x\in {\mathbb{R}}^{2N+2}:0<a<|x|<b\}$ be an annulus. Consider the following singularly perturbed elliptic problem on $A$ \begin{equation} \begin{array}{lll} -\eps^2{\De u} + |x|^αu = |x|^αu^p, &\mbox{\qquad in} A \notag u>0 &\mbox{\qquad in} A \frac{\partial u}{\partialν} = 0 &\mbox{\qquad on} \partial A \end{array} %\label{a1} \end{equation} $1<p<2^{\ast }-1$. We shall show that there exists a positive solution $u_\eps$ concentrating on an $S^1$ orbit as $\eps\to 0$. 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 $\alpha$.

24 pages