• Previous Article
    Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations
  • CPAA Home
  • This Issue
  • Next Article
    Controllability and stabilization of gravity-capillary surface water waves in a basin
doi: 10.3934/cpaa.2021191
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Blowing-up solutions for a supercritical elliptic equation

Sfax University, Sfax Business School, Tunisia, Ecole Superieure de Commerce de Sfax, BP 1081, 3018, Sfax, Tunisia

Received  May 2021 Revised  September 2021 Early access November 2021

This paper concerns the existence of solutions of the following supercritical PDE: $ (P_\varepsilon) $: $ -\Delta u = K|u|^{\frac{4}{n-2}+\varepsilon}u\; \mbox{ in }\Omega, \; u = 0 \mbox{ on }\partial\Omega, $ where $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n $, $ n\geq 3 $, $ K $ is a $ C^3 $ positive function and $ \varepsilon $ is a small positive real. Our method is inspired from the work of Bahri-Li-Rey. It consists to reduce the existence of a critical point to a finite dimensional system. Using a fixed-point theorem, we are able to construct positive solutions of $ (P_\varepsilon) $ having the form of two bubbles with non comparable speeds and which have only one blow-up point in $ \Omega $. That means that this blow-up point is non simple. This represents a new phenomenon compared with the subcritical case.

Citation: Yessine Dammak. Blowing-up solutions for a supercritical elliptic equation. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021191
References:
[1]

A. Bahri, Critical Point at Infinity in Some Variational Problem, Pitman Res. Notes math, Ser 182, Longman Sci. Tech. Harlow, 1989.  Google Scholar

[2]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effet of topology of the domain, Commun. pure Appl. Math., 41 (1988), 255-294.  doi: 10.1002/cpa.3160410302.  Google Scholar

[3]

A. Bahri and J. M. Coron, The scalar curvature problem on the standard three dimensional spheres, J. Funct. Anal., 95 (1991), 106-172.  doi: 10.1016/0022-1236(91)90026-2.  Google Scholar

[4] A. Bahri and Y. Xu, Recent Pregress in Conformal Geometry, Advan. Imperial College Press, Londres, 2007.  doi: 10.1142/9781860948602.  Google Scholar
[5]

A. BahriY. Y Li and O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Cal. Var. Partial Differ. Equ. V., 3 (1995), 67-93.  doi: 10.1007/BF01190892.  Google Scholar

[6]

T. Bartsch and T. Weth, A note on additonnal properties of sign-changing solutions to super-linear elliptic equations, Top. Metho. Nonlin. Anal., 22 (2003), 1-14.  doi: 10.12775/TMNA.2003.025.  Google Scholar

[7]

M. Ben Ayed, Finite dimensional reduction of a supercritical exponent equation, Tunisian J. Math., 2 (2020), 379-397.  doi: 10.2140/tunis.2020.2.379.  Google Scholar

[8]

M. Ben Ayed and Y. Dammak, Construction of solutions of a supercritical elliptic PDE in low dimensions, Ann. Matemat. Pura Appl., 200 (2021), 51-66.  doi: 10.1007/s10231-020-00982-7.  Google Scholar

[9]

M. Ben AyedK. El MehdiM. Grossi and O. Rey, A Nonexistence results of sigle peaked solutions to a supercritical nonlinear problem, Commun. Contemp. Math., 2 (2003), 179-195.  doi: 10.1142/S0219199703000951.  Google Scholar

[10]

M. Ben Ayed and K. Ould Bouh, Nonexistence results of sign-changing solutions to supercritical nonlinear problem, Commun. Pure Appl. Anal., 7 (2008), 1057-1075.  doi: 10.3934/cpaa.2008.7.1057.  Google Scholar

[11]

M. Ben Ayed and K. Ould Bouh, Construction of solutions of an elliptic PDE with a Supercritical exponent nonlinearity using the Finite Dimensional Reduction, Mon. Hefte. Math., 192 (2020), 49-63.  doi: 10.1007/s00605-020-01386-8.  Google Scholar

[12]

D. Cao and S. Peng, The asymptotic behavior of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17.  doi: 10.1016/S0022-247X(02)00292-5.  Google Scholar

[13]

A. CastroJ. Cossio and J. M. Newgerger, A sign-changing solution for a supercritical Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053.  doi: 10.1216/rmjm/1181071858.  Google Scholar

[14]

Y. Dammak and R. Ghoudi, Sign-changing tower of bubbles to an elliptic subcritical equation, Commun. Contemp. Math., 21 (2019), 1850052.  doi: 10.1142/S0219199718500529.  Google Scholar

[15]

M. Del PinoP. Felmer and M. Musso, Two bubbles solutions in the supercritical Bahri-Coron's problem, Calc. Var. Partial Differ. Equ., 16 (2003), 113-145.  doi: 10.1007/s005260100142.  Google Scholar

[16]

W. Y. Ding, Positive solution of $\Delta u+u^{(n+2)/(n-2)} = 0$ on contractible domain, J. Partial Differ. Equ., 2 (1988), 83-88.   Google Scholar

[17]

S. Khenissi and O. Rey, A criterion for existence of solutions to the supercritical Bahri-Coron's problem, Houston J. Math., 30 (2004), 587-613.   Google Scholar

[18]

Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems, part Ⅰ, J. Differ. Equ. 120 (1995), 319-410. doi: 10.1006/jdeq.1995.1115.  Google Scholar

[19]

Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems, part Ⅱ: Existence and compactness, Commun. pure Appl. Math., 49 (1996), 541-597.  doi: 10.1006/jdeq.1995.1115.  Google Scholar

[20]

D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal., 114 (1993), 97-105.  doi: 10.1006/jfan.1993.1064.  Google Scholar

[21]

D. Passaseo, Nontrivial solutions of elliptic equations with supercritical exponent in contractible domains, Duke Math. J., 92 (1998), 429-457.  doi: 10.1215/S0012-7094-98-09213-4.  Google Scholar

[22]

A. Pistoia and T. Weth, Sign-changing bubble-tower solutions in a slightly subcritical semi-linear Dirichlet problem, Ann. Inst. H. Poincaré, Anal. nonlinéaire, 24 (2007), 325-340.  doi: 10.1016/j.anihpc.2006.03.002.  Google Scholar

show all references

References:
[1]

A. Bahri, Critical Point at Infinity in Some Variational Problem, Pitman Res. Notes math, Ser 182, Longman Sci. Tech. Harlow, 1989.  Google Scholar

[2]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effet of topology of the domain, Commun. pure Appl. Math., 41 (1988), 255-294.  doi: 10.1002/cpa.3160410302.  Google Scholar

[3]

A. Bahri and J. M. Coron, The scalar curvature problem on the standard three dimensional spheres, J. Funct. Anal., 95 (1991), 106-172.  doi: 10.1016/0022-1236(91)90026-2.  Google Scholar

[4] A. Bahri and Y. Xu, Recent Pregress in Conformal Geometry, Advan. Imperial College Press, Londres, 2007.  doi: 10.1142/9781860948602.  Google Scholar
[5]

A. BahriY. Y Li and O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Cal. Var. Partial Differ. Equ. V., 3 (1995), 67-93.  doi: 10.1007/BF01190892.  Google Scholar

[6]

T. Bartsch and T. Weth, A note on additonnal properties of sign-changing solutions to super-linear elliptic equations, Top. Metho. Nonlin. Anal., 22 (2003), 1-14.  doi: 10.12775/TMNA.2003.025.  Google Scholar

[7]

M. Ben Ayed, Finite dimensional reduction of a supercritical exponent equation, Tunisian J. Math., 2 (2020), 379-397.  doi: 10.2140/tunis.2020.2.379.  Google Scholar

[8]

M. Ben Ayed and Y. Dammak, Construction of solutions of a supercritical elliptic PDE in low dimensions, Ann. Matemat. Pura Appl., 200 (2021), 51-66.  doi: 10.1007/s10231-020-00982-7.  Google Scholar

[9]

M. Ben AyedK. El MehdiM. Grossi and O. Rey, A Nonexistence results of sigle peaked solutions to a supercritical nonlinear problem, Commun. Contemp. Math., 2 (2003), 179-195.  doi: 10.1142/S0219199703000951.  Google Scholar

[10]

M. Ben Ayed and K. Ould Bouh, Nonexistence results of sign-changing solutions to supercritical nonlinear problem, Commun. Pure Appl. Anal., 7 (2008), 1057-1075.  doi: 10.3934/cpaa.2008.7.1057.  Google Scholar

[11]

M. Ben Ayed and K. Ould Bouh, Construction of solutions of an elliptic PDE with a Supercritical exponent nonlinearity using the Finite Dimensional Reduction, Mon. Hefte. Math., 192 (2020), 49-63.  doi: 10.1007/s00605-020-01386-8.  Google Scholar

[12]

D. Cao and S. Peng, The asymptotic behavior of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17.  doi: 10.1016/S0022-247X(02)00292-5.  Google Scholar

[13]

A. CastroJ. Cossio and J. M. Newgerger, A sign-changing solution for a supercritical Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053.  doi: 10.1216/rmjm/1181071858.  Google Scholar

[14]

Y. Dammak and R. Ghoudi, Sign-changing tower of bubbles to an elliptic subcritical equation, Commun. Contemp. Math., 21 (2019), 1850052.  doi: 10.1142/S0219199718500529.  Google Scholar

[15]

M. Del PinoP. Felmer and M. Musso, Two bubbles solutions in the supercritical Bahri-Coron's problem, Calc. Var. Partial Differ. Equ., 16 (2003), 113-145.  doi: 10.1007/s005260100142.  Google Scholar

[16]

W. Y. Ding, Positive solution of $\Delta u+u^{(n+2)/(n-2)} = 0$ on contractible domain, J. Partial Differ. Equ., 2 (1988), 83-88.   Google Scholar

[17]

S. Khenissi and O. Rey, A criterion for existence of solutions to the supercritical Bahri-Coron's problem, Houston J. Math., 30 (2004), 587-613.   Google Scholar

[18]

Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems, part Ⅰ, J. Differ. Equ. 120 (1995), 319-410. doi: 10.1006/jdeq.1995.1115.  Google Scholar

[19]

Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems, part Ⅱ: Existence and compactness, Commun. pure Appl. Math., 49 (1996), 541-597.  doi: 10.1006/jdeq.1995.1115.  Google Scholar

[20]

D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal., 114 (1993), 97-105.  doi: 10.1006/jfan.1993.1064.  Google Scholar

[21]

D. Passaseo, Nontrivial solutions of elliptic equations with supercritical exponent in contractible domains, Duke Math. J., 92 (1998), 429-457.  doi: 10.1215/S0012-7094-98-09213-4.  Google Scholar

[22]

A. Pistoia and T. Weth, Sign-changing bubble-tower solutions in a slightly subcritical semi-linear Dirichlet problem, Ann. Inst. H. Poincaré, Anal. nonlinéaire, 24 (2007), 325-340.  doi: 10.1016/j.anihpc.2006.03.002.  Google Scholar

[1]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103

[2]

Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086

[3]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[4]

Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108

[5]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006

[6]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032

[7]

Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183

[8]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052

[9]

Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941

[10]

Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4847-4885. doi: 10.3934/dcds.2021060

[11]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065

[12]

Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715

[13]

Nejib Mahmoudi. Single-point blow-up for a multi-component reaction-diffusion system. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 209-230. doi: 10.3934/dcds.2018010

[14]

Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225

[15]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

[16]

Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.

[17]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[18]

Mikhaël Balabane, Mustapha Jazar, Philippe Souplet. Oscillatory blow-up in nonlinear second order ODE's: The critical case. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 577-584. doi: 10.3934/dcds.2003.9.577

[19]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[20]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134

2020 Impact Factor: 1.916

Article outline

[Back to Top]