Nonradial solutions of a Neumann Hénon equation on a ball
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- by Craig Cowan;
- Proc. Amer. Math. Soc. 152 (2024), 3955-3969
- DOI: https://doi.org/10.1090/proc/16897
- Published electronically: July 29, 2024
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Abstract:
In this work we examine the existence of positive classical solutions of \begin{equation*} \begin {cases} -\Delta u +u = |x|^\alpha u^{p-1} & \text { in } B_1, \\ u>0 & \text { in } B_1, \\ \partial _\nu u= 0 & \text { on } \partial B_1, \end{cases} \end{equation*} where $p>1$, $\alpha >0$ and $B_1$ is the unit ball in ${\mathbb {R}}^N$ where $N \ge 4$ and is even. Of particular interest is the existence of nonradial position classical solutions. We show that under suitable conditions on $p,\alpha$ and $N$ there are positive classical nonradial solutions. Our approach is to utilize a variational approach on suitable convex cones.References
- Claudianor O. Alves and Abbas Moameni, Super-critical Neumann problems on unbounded domains, Nonlinearity 33 (2020), no. 9, 4568–4589. MR 4127789, DOI 10.1088/1361-6544/ab8bac
- Vivina Barutello, Simone Secchi, and Enrico Serra, A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl. 341 (2008), no. 1, 720–728. MR 2394119, DOI 10.1016/j.jmaa.2007.10.052
- Denis Bonheure, Massimo Grossi, Benedetta Noris, and Susanna Terracini, Multi-layer radial solutions for a supercritical Neumann problem, J. Differential Equations 261 (2016), no. 1, 455–504. MR 3487266, DOI 10.1016/j.jde.2016.03.016
- Denis Bonheure, Benedetta Noris, and Tobias Weth, Increasing radial solutions for Neumann problems without growth restrictions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 29 (2012), no. 4, 573–588. MR 2948289, DOI 10.1016/j.anihpc.2012.02.002
- Denis Bonheure and Enrico Serra, Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth, NoDEA Nonlinear Differential Equations Appl. 18 (2011), no. 2, 217–235. MR 2788329, DOI 10.1007/s00030-010-0092-z
- Francesca Colasuonno and Benedetta Noris, A $p$-Laplacian supercritical Neumann problem, Discrete Contin. Dyn. Syst. 37 (2017), no. 6, 3025–3057. MR 3622073, DOI 10.3934/dcds.2017130
- Craig Cowan and Abbas Moameni, On supercritical elliptic problems: existence, multiplicity of positive and symmetry breaking solutions, Math. Ann. 389 (2024), no. 2, 1731–1794.
- Craig Cowan and Abbas Moameni, A new variational principle, convexity, and supercritical Neumann problems, Trans. Amer. Math. Soc. 371 (2019), no. 9, 5993–6023. MR 3937316, DOI 10.1090/tran/7250
- Craig Cowan, Abbas Moameni, and Leila Salimi, Existence of solutions to supercritical Neumann problems via a new variational principle, Electron. J. Differential Equations (2017), Paper No. 213, 19. MR 3711166
- Manuel del Pino, Monica Musso, and Angela Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 1, 45–82 (English, with English and French summaries). MR 2114411, DOI 10.1016/j.anihpc.2004.05.001
- Changfeng Gui and Nassif Ghoussoub, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z. 229 (1998), no. 3, 443–474. MR 1658569, DOI 10.1007/PL00004663
- Massimo Grossi, A class of solutions for the Neumann problem $-\Delta u+\lambda u=u^{(N+2)/(N-2)}$, Duke Math. J. 79 (1995), no. 2, 309–334. MR 1344764, DOI 10.1215/S0012-7094-95-07908-3
- Massimo Grossi and Benedetta Noris, Positive constrained minimizers for supercritical problems in the ball, Proc. Amer. Math. Soc. 140 (2012), no. 6, 2141–2154. MR 2888200, DOI 10.1090/S0002-9939-2011-11133-X
- Changfeng Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996), no. 3, 739–769. MR 1408543, DOI 10.1215/S0012-7094-96-08423-9
- Changfeng Gui and Chang-Shou Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem, J. Reine Angew. Math. 546 (2002), 201–235. MR 1900999, DOI 10.1515/crll.2002.044
- Changfeng Gui and Juncheng Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 (1999), no. 1, 1–27. MR 1721719, DOI 10.1016/S0022-0396(99)80016-3
- Abbas Moameni, Critical point theory on convex subsets with applications in differential equations and analysis, J. Math. Pures Appl. (9) 141 (2020), 266–315 (English, with English and French summaries). MR 4134457, DOI 10.1016/j.matpur.2020.05.005
- Ruyun Ma, Tianlan Chen, and Yanqiong Lu, On the Bonheure-Noris-Weth conjecture in the case of linearly bounded nonlinearities, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 8, 2649–2662. MR 3555134, DOI 10.3934/dcdsb.2016066
- Olivier Rey and Juncheng Wei, Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. I. $N=3$, J. Funct. Anal. 212 (2004), no. 2, 472–499. MR 2064935, DOI 10.1016/j.jfa.2003.06.006
- Wei Ming Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J. 31 (1982), no. 6, 801–807. MR 674869, DOI 10.1512/iumj.1982.31.31056
- Enrico Serra and Paolo Tilli, Monotonicity constraints and supercritical Neumann problems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 28 (2011), no. 1, 63–74 (English, with English and French summaries). MR 2765510, DOI 10.1016/j.anihpc.2010.10.003
- Andrzej Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 2, 77–109 (English, with French summary). MR 837231, DOI 10.1016/s0294-1449(16)30389-4
- Juncheng Wei, On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differential Equations 134 (1997), no. 1, 104–133. MR 1429093, DOI 10.1006/jdeq.1996.3218
Bibliographic Information
- Craig Cowan
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada
- MR Author ID: 815665
- Email: craig.cowan@umanitoba.ca
- Received by editor(s): August 16, 2023
- Received by editor(s) in revised form: March 31, 2024
- Published electronically: July 29, 2024
- Communicated by: Ryan Hynd
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3955-3969
- MSC (2020): Primary 35J15, 35J20, 35J60
- DOI: https://doi.org/10.1090/proc/16897
- MathSciNet review: 4781987