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On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions
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 |
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.
References:
| [1] |
A. Bahri, Critical Point at Infinity in Some Variational Problem, Pitman Res. Notes math, Ser 182, Longman Sci. Tech. Harlow, 1989. |
| [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. |
| [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. |
| [4] |
A. Bahri and Y. Xu, Recent Pregress in Conformal Geometry, Advan. Imperial College Press, Londres, 2007.
doi: 10.1142/9781860948602.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [4] |
A. Bahri and Y. Xu, Recent Pregress in Conformal Geometry, Advan. Imperial College Press, Londres, 2007.
doi: 10.1142/9781860948602.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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