-
Previous Article
The study on solutions to Camassa-Holm equation with weak dissipation
- CPAA Home
- This Issue
-
Next Article
Stable determination of a semilinear term in a parabolic equation
Multiple bubbling for the exponential nonlinearity in the slightly supercritical case
| 1. | Departamento de Ingeneria Matematica F.C.F.M., Casilla 170 Correro 3, Santiago, Chile |
| 2. | Ceremade (UMR CNRS no. 7534), Université Paris Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16 |
| 3. | Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino |
$ -\Delta_{2-\varepsilon} u=\lambdae^u$ in $B\quad u =0 $ on $ \partial B,$
for small $\varepsilon>0$. We show that if $k\ge 1$ is given and $\lambda>0$ is fixed and small, then there is a family of radial solutions exhibiting multiple blow-up as $\varepsilon\to 0$ in the form of a superposition of $k$ bubbles of different blow-up orders and shapes. Similar phenomena is found for the same problem involving the operator $\Delta_{N-\varepsilon}$ in $\mathbb R^N$, $N\ge 3$.
| [1] |
Francesca Colasuonno, Fausto Ferrari. The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem. Communications on Pure & Applied Analysis, 2020, 19 (2) : 983-1000. doi: 10.3934/cpaa.2020045 |
| [2] |
Po-Chun Huang, Shin-Hwa Wang, Tzung-Shin Yeh. Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2297-2318. doi: 10.3934/cpaa.2013.12.2297 |
| [3] |
Jie Xiao. On the variational $p$-capacity problem in the plane. Communications on Pure & Applied Analysis, 2015, 14 (3) : 959-968. doi: 10.3934/cpaa.2015.14.959 |
| [4] |
Hideo Ikeda, Koji Kondo, Hisashi Okamoto, Shoji Yotsutani. On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows. Communications on Pure & Applied Analysis, 2003, 2 (3) : 381-390. doi: 10.3934/cpaa.2003.2.381 |
| [5] |
Guowei Dai. Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5323-5345. doi: 10.3934/dcds.2016034 |
| [6] |
Guowei Dai, Ruyun Ma, Haiyan Wang. Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2839-2872. doi: 10.3934/cpaa.2013.12.2839 |
| [7] |
Marta García-Huidobro, Raul Manásevich, J. R. Ward. Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 299-321. doi: 10.3934/dcds.2007.19.299 |
| [8] |
Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 |
| [9] |
Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130 |
| [10] |
Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075 |
| [11] |
Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063 |
| [12] |
Guowei Dai, Ruyun Ma. Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 99-116. doi: 10.3934/dcds.2015.35.99 |
| [13] |
Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems & Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49 |
| [14] |
Futoshi Takahashi. Singular extremal solutions to a Liouville-Gelfand type problem with exponential nonlinearity. Conference Publications, 2015, 2015 (special) : 1025-1033. doi: 10.3934/proc.2015.1025 |
| [15] |
Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171 |
| [16] |
Marek Galewski, Renata Wieteska. Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2535-2547. doi: 10.3934/dcdsb.2014.19.2535 |
| [17] |
Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51 |
| [18] |
Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595 |
| [19] |
Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675 |
| [20] |
Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure & Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1 |
2018 Impact Factor: 0.925
Tools
Metrics
Other articles
by authors
[Back to Top]