A supercritical elliptic equation in the annulus

Boscaggin, Alberto; Colasuonno, Francesca; Noris, Benedetta; Weth, Tobias

doi:10.4171/aihpc/38

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A supercritical elliptic equation in the annulus

Boscaggin, Alberto ; Colasuonno, Francesca ; Noris, Benedetta ; Weth, Tobias

Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 1, pp. 157-183

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Résumé

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation

- Δ u + u = {a (x) | u |}^{p - 2} u

in an annulus $A \subset ℝ^{N}$ ( $N \geq 3$ ). Here $p > 2$ is allowed to be supercritical and $a (x)$ is an axially symmetric but possibly nonradial function with additional symmetry and monotonicity properties, which are shared by the solution $u$ we construct. In the case where $a$ equals a positive constant, we detect conditions, only depending on the exponent $p$ and on the inner radius of the annulus, that ensure that the solution is nonradial.

Accepté le : 2021-11-03
Publié le : 2022-07-08

DOI : 10.4171/aihpc/38

Classification : 35J25, 35A15, 35A16, 35B07, 35J15
Mots-clés : variational and topological methods, invariant cones, high Morse index solutions, axially symmetric solutions, Supercritical elliptic equations

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     title = {A supercritical elliptic equation in the annulus},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     year = {2023},
     doi = {10.4171/aihpc/38},
     language = {EN},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/38/}
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Boscaggin, Alberto; Colasuonno, Francesca; Noris, Benedetta; Weth, Tobias. A supercritical elliptic equation in the annulus. Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 1, pp. 157-183. doi: 10.4171/aihpc/38

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