| Abstract: |
| Many authors studied in the past the following
boundary value problem:
\\begin{equation}
\\label{problem.0}
\\left\\{
\\begin{array}{llll}
\\Delta u +u ^p = 0 & \\hbox{ in } & A, \\\\
u > 0 & \\textnormal{in} & A, \\\\
u = 0 & \\textnormal{on} & \\partial A, \\\\
\\end{array}
\\right.
\\end{equation}
where $A \\subset \\R^n,$ $n \\geq 2,$ is an annulus, that is
$A=\\{x \\in \\R^n: R_1 < r(x)< R_2\\},$
with $r(x)$ equal to the distance to the origin.
The radial solution always exists for any $p>1$,
it is unique and radially non-degenerate.
If $A$ is replaced by an expanding domain
in $\\R^n$ which is diffeomorphic to an annulus, then
it is known (see \\cite{CW},\\cite{DY}) that there exists
an increasing number of solutions as the domain expands.
Furthermore in \\cite{DY} it is shown
such solutions are not close to the radial one,
indeed they exhibit a finite number of bumps.
In \\cite{BCGP} Bartsch, Clapp, Grossi, Pacella
show instead the existence of a positive solution
to the problem \\eqref{problem.0}
on an expanding annular domain $\\Omega_R$ diffeomorphic to
$A_R=\\{x \\in \\R^n: R < r(x)< R+1\\}$,
which is close to the radial
solution to the corresponding problem on the annulus
$A_R$.
In our work \\cite{M} we show the result of \\cite{BCGP} holds true
for an unbounded
Riemannian manifold $M$ of dimension $n \\geq 2$ with metric
$g:=dr^2+S^2(r)g_{\\esse^{n-1}},$ where $g_{\\esse^{n-1}}$ denotes the standard metric of the
$(n-1)$-dimensional unit sphere $\\esse^{n-1};$ $r\\in [0,+\\infty)$ is the geodesic distance measured from a point $O.$
\\begin{thebibliography}{9}
\\bibitem{BCGP} T. Bartsch, M. Clapp, M. Grossi,
F. Pacella, {\\it Asymptotically radial solutions in
expanding annular domains}, Math. Annalen, 352, 485-515, 2012.
\\bibitem{CW} F. Catrina, Z. Q. Wang,
{\\it Nonlinear elliptic equations on expanding symmetric
domains}, J. Differential Equations, 1999, 156, 153-181.
\\bibitem{DY} E. N. Dancer, S. Yan
{\\it Multibump solutions for an elliptic problem
in expanding domains}, Comm. Partial Diff. Equations 2002, 27,
23-55.
\\bibitem{M} F. Morabito, {\\it Asymptotically radial solutions
to an elliptic problem on expanding
annular domains in Riemannian manifolds with radial symmetry},
Boundary Value Problems, art. 124, 2016.
\\end{thebibliography} |
|