Please use this identifier to cite or link to this item:
http://hdl.handle.net/10077/16205
Title: | Positive and nodal single-layered solutions to supercritical elliptic problems above the higher critical exponents | Authors: | Clapp, Mónica Rizzi, Matteo |
Keywords: | Supercritical elliptic problem; positive solutions; nodal solutions; blow up; higher critical exponents | Issue Date: | 2017 | Publisher: | EUT Edizioni Università di Trieste | Source: | Mónica Clapp, Matteo Rizzi, "Positive and nodal single-layered solutions to supercritical elliptic problems above the higher critical exponents", in: "Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics", 49 (2017), Trieste, EUT Edizioni Università di Trieste, 2017, pp. 53-71 | Journal: | Rendiconti dell’Istituto di matematica dell’Università di Trieste: an International Journal of Mathematics | Part of: | 49 (2017) | Abstract: | We study the problem −Δv + λv = |u|p−2 u in Ω, u= 0 on ∂Ω, for λ ∈ R and supercritical exponents p, in domains of the form Ω := {(y, z) ∈ RN−m−1 x Rm+1 : (y, |z|) ∈ Θ}, where m ≥ 1, N − m ≥ 3, and Θ is a bounded domain in RN−m whose closure is contained in RN−m−1 x (0,∞). Under some symmetry assumptions on Θ, we show that this problem has infinitely many solutions for every λ in an interval which contains [0,∞) and p > 2 up to some number which is larger than the (m+1)st critical exponent 2∗N,m := 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 Ω, as the hole shrinks and p → 2∗N,m 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 −Δu = |u|2∗n−2 u, u ∈ D1,2(Rn), where 2∗n := 2n n−2 is the critical exponent in dimension n. |
Type: | Article | URI: | http://hdl.handle.net/10077/16205 | ISSN: | 0049-4704 | eISSN: | 2464-8728 | DOI: | 10.13137/2464-8728/16205 | Rights: | Attribution-NonCommercial-NoDerivatives 4.0 Internazionale |
Appears in Collections: | Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.49 (2017) |
Files in This Item:
File | Description | Size | Format | |
---|---|---|---|---|
05_RIMUT_ClappRizzi.pdf | 347.05 kB | Adobe PDF | ![]() View/Open |
CORE Recommender
Page view(s)
235
checked on Jul 4, 2022
Download(s)
71
checked on Jul 4, 2022
Google ScholarTM
Check
Altmetric
Altmetric
This item is licensed under a Creative Commons License