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doi: 10.3934/cpaa.2021191
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Blowing-up solutions for a supercritical elliptic equation

Yessine Dammak

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.

Keywords: Supercritical exponent, non simple blow-up point, finite dimensional reduction, critical points.
Mathematics Subject Classification: Primary 35J20; Secondary 35J60.
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. Bahri, Y. 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 Ayed, K. El Mehdi, M. 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. Castro, J. 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 Pino, P. 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. Bahri, Y. 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 Ayed, K. El Mehdi, M. 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. Castro, J. 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 Pino, P. 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

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