Title: Positive and nodal single-layered solutions to supercritical elliptic problems above the higher critical exponents,
Abstract: We study the problem% \[ -\Delta v+\lambda v=| v| ^{p-2}v\text{ in }\Omega
,\text{\qquad}v=0\text{ on $\partial\Omega$},\text{ }% \] for
$\lambda\in\mathbb{R}$ and supercritical exponents $p,$ in domains of the form%
\[ \Omega:=\{(y,z)\in\mathbb{R}^{N-m-1}\times\mathbb{R}^{m+1}:(y,| z|
)\in\Theta\}, \] where $m\geq1,$ $N-m\geq3,$ and $\Theta$ is a bounded domain
in $\mathbb{R}% ^{N-m}$ whose closure is contained in
$\mathbb{R}^{N-m-1}\times(0,\infty)$. Under some symmetry assumptions on
$\Theta$, we show that this problem has infinitely many solutions for every
$\lambda$ in an interval which contains $[0,\infty)$ and $p>2$ up to some
number which is larger than the $(m+1)^{st}$ critical exponent
$2_{N,m}^{\ast}:=\frac{2(N-m)}{N-m-2}$. We also exhibit domains with a
shrinking hole, in which there are a positive and a nodal solution which
concentrate on a sphere, developing a single layer that blows up at an
$m$-dimensional sphere contained in the boundary of $\Omega,$ as the hole
shrinks and $p\rightarrow2_{N,m}^{\ast}$ from above. The limit profile of the
positive solution, in the transversal direction to the sphere of concentration,
is a rescaling of the standard bubble, whereas that of the nodal solution is a
rescaling of a nonradial sign-changing solution to the problem% \[ -\Delta u=|
u| ^{2_{n}^{\ast}-2}u,\text{\qquad}u\in D^{1,2}(\mathbb{R}^{n}), \] where
$2_{n}^{\ast}:=\frac{2n}{n-2}$ is the critical exponent in dimension
$n.$\medskip