Boundary-layers for a Neumann problem at higher critical exponents

We consider the Neumann problem $$(P)\qquad - \Delta v + v= v^{q-1} \ \text{in }\ \mathcal{D}, \ v > 0 \ \text{in } \ \mathcal{D},\ \partial_\nu v = 0 \ \text{on } \partial\mathcal{D} ,$$ where $\mathcal{D}$ is an open bounded domain in $\mathbb{R}^N,$ $\nu$ is the unit inner normal at the boundary and $q>2.$ For any integer, $1\le h\le N-3,$ we show that, in some suitable domains $\mathcal D,$ problem $(P)$ has a solution which blows-up along a $h-$dimensional minimal submanifold of the boundary $\partial\mathcal D$ as $q$ approaches from either below or above the higher critical Sobolev exponent ${2(N-h)\over N-h-2}.$Comment: 13 page

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

We consider singularly perturbed equations of the form {ε2Δu−u+up=0 in A⊂RN,u>0 in A,u=0 on ∂A, where A is a annulus and p>1. It has been conjectured for a long time that such problems possess solutions having m-dimensional concentration sets for every 0≤m≤N−1. For N=3 solutions with 2-dimensional and 0-dimensional concentration sets are known, while no result was available for 1-dimensional concentration sets. We answer positively this conjecture, proving the existence of solutions which concentrate on circles S1. The proof relies on work by Santra-Wei who proved the existence of solutions concentrating on a Clifford torus S1×S1 for an annulus in R4. We extend this result to equations with weights. Then we use the Hopf fibration to show that these solutions give rise to the S1-concentrating solutions in R3. © 2022 Elsevier Inc