On the solutions of a singular elliptic equation concentrating on two orthogonal spheres

Abstract

Let $A=\{x\in {\mathbb{R}}^{2m}:0<a<|x|<b\}$ be an annulus. Consider the following singularly perturbed elliptic problem on $A$ \begin{equation} \begin{array}{lll} -\eps^2{\De u} + |x|^ηu = |x|^ηu^p, &\mbox{\qquad in} A \notag u>0 &\mbox{\qquad in} A u = 0 &\mbox{\qquad on} \partial A \end{array} %\label{a1} \end{equation} $1<p<2^{\ast }-1$. We shall prove the existence of a positive solution $u_\eps$ which concentrates on two different orthogonal spheres of dimension $(m-1)$ as $\eps\to 0$. We achieve this by studying a reduced problem on an annular domain in ${\mathbb{R}}^{m+1}$ and analyzing the profile of a two point concentrating solution in this domain.

13 pages

References in corpus (1)

• Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents