Hopf reduction and orbit concentrating solutions for a class of superlinear elliptic equations (Q2123110)

Consider the singularly perturbed elliptic problem \[ \left\{\begin{array}{ll} \varepsilon^2\Delta u-u+u^p=0, \ \ \ &\text{in} \ \ A,\\ u>0 &\text{in} \ \ A,\\ u=0 &\text{on} \ \ \partial A, \end{array}\right. \ \ \hspace{3cm} (1) \] where \(\varepsilon>0\), \(p>1\), and \(A=\{x\in \mathbb{R}^N: a<|x|<b\}\). It is known that for \(N=4\) problem \((1)\) admits solutions concentrating on manifolds of any dimension between 0 and 3, as \(\varepsilon \rightarrow 0\), while for \(N=3\) only solutions concentrating on a single point or on a \(2d\)-manifold are known to exist. In this paper, the possible existence of solutions concentrating on \(1d\)-manifolds for \(N=3\) is investigated. The authors give a positive answer to this question by proving the existence of solutions concentrating on a circle which converges to the inner boundary of the annulus \(A\), as \(\varepsilon\rightarrow 0\). The proof is performed into two steps: the authors first prove the existence of solutions concentrating on a Clifford-torus in the case \(N=4\) for the equation \(\varepsilon^2\Delta u+|x|^2(-u+u^p)=0\) via a Lyapunov reduction. Then they use this result to show, via a Hopf-reduction, that for \(N=3\) problem \((1)\) admits solutions concentrating on a circle.