Abstract
AbstractIn this paper, we consider semilinear elliptic problems in a bounded domain $$\Omega $$
Ω
contained in a given unbounded Lipschitz domain $${\mathcal {C}} \subset {\mathbb {R}}^N$$
C
⊂
R
N
. Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain $$\Omega $$
Ω
inside $${\mathcal {C}}$$
C
. Once a rigorous variational approach to this question is set, we focus on the cases when $${\mathcal {C}}$$
C
is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems.
Funder
Università degli Studi di Roma La Sapienza
Publisher
Springer Science and Business Media LLC
Reference23 articles.
1. Afonso, D.G., Iacopetti, A., Pacella, F.: Overdetermined problems and relative Cheeger sets in unbounded domains. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 34(2), 531–546 (2023)
2. Alkhutov, Y., Maz’ya, V.G.: $${L}^{1, p}$$-coercivity and estimates of the Green function of the Neumann problem in a convex domain. J. Math. Sci. 196 (2014)
3. Amadori, A.L., Gladiali, F.: On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDEs. Nonlinear Anal.: Real World Appl. 55, 103–133 (2020)
4. Ambrosetti, A., Malchiodi, A.: Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge University Press, Cambridge (2007)
5. Baer, E., Figalli, A.: Characterization of isoperimetric sets inside almost-convex cones. Discrete Contin. Dyn. Syst.—A 37 (2017)