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Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term
Nonexistence results on the space or the half space of $ -\Delta u+\lambda u = |u|^{p-1}u $ via the Morse index
1. | Department of Mathematics, Northern Border university, Arar, Saudi Arabia, Université de Tunis, Département de Mathématiques, , Faculté des Sciences de Bizerte, Zarzouna, 7021 Bizerte, Tunisia |
2. | Department of Mathematics, Northern Border university, Arar, Saudi Arabia, Université de Kairouan, Département de Mathématiques, , Institut Superieur des Mathématiques Appliquées et de l'Informatique |
3. | Faculté des Sciences, Département de Mathématiques, , B.P 1171 Sfax 3000, Université de Sfax, Tunisia |
$ \begin{equation*} -\Delta u+\lambda u = |u|^{p-1}u\quad\mbox{in}\,\, \mathcal{O}, \tag{P} \end{equation*} $ |
$ \mathcal{O} = \mathbb{R}^N $ |
$ \mathcal{O} = \mathbb{R}^N_+ = \{x = (x',x_N),\, x'\in \mathbb{R}^{N-1},x_N>0\} $ |
$ N\geq2 $ |
$ p>1 $ |
$ \lambda $ |
$ \lambda u $ |
$ p> 1 $ |
$ p\geq \frac{N+2}{N-2} $ |
$ 1<p<\frac{N+2}{N-2} $ |
$ |u|^{p-1}<\frac{\lambda (p+1)}{2} $ |
$ \mathcal{O} = \mathbb{R}^N $ |
$ (P) $ |
$ p\in (1,\infty) $ |
$ N = 2 $ |
$ p\in(1,\frac{N+2}{N-2}) $ |
$ N\geq3. $ |
References:
[1] |
L. Ambrosio and X. Cabré,
Entire solutions of semilinear elliptic equations in $ \mathbb{R}^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739.
doi: 10.1090/S0894-0347-00-00345-3. |
[2] |
A. Bahri and P. L. Lions,
Solutions of superlinear elliptic equations and their Morse indices, Commun. Pure. App. Math., 45 (1992), 1205-1215.
doi: 10.1002/cpa.3160450908. |
[3] |
M. Ben Ayed, H. Fourti and A. Selmi,
Harmonic functions with nonlinear Neumann boundary condition and their Morse indices, Nonlinear Anal. Real World Appl., 38 (2017), 96-112.
doi: 10.1016/j.nonrwa.2017.04.012. |
[4] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, Part Ⅱ, Existence of infinitly many solutios groud, Ration. Mech. Anal., 82 (1982), 347-369.
doi: 10.1007/BF00250556. |
[5] |
E. N. Dancer,
Some notes on the method of moving planes, Bull. Austral. Math. Soc. angew. Math., 46 (1992), 425-434.
doi: 10.1017/S0004972700012089. |
[6] |
J. Dávila, L. Dupaigne, K. Wang and J. Wei,
A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.
doi: 10.1016/j.aim.2014.02.034. |
[7] |
D. G. de Figueiredo and J. Yang,
On a semilinear elliptic problem without (PS) condition, J. Differ. Equ., 187 (2003), 412-428.
doi: 10.1016/S0022-0396(02)00055-4. |
[8] |
B. Devyver,
On the finiteness of the Morse index for Schröinger operators, Manuscr. Math., 139 (2012), 249-271.
doi: 10.1007/s00229-011-0522-1. |
[9] |
L. Dupaigne and A. Harrabi,
The Lane-Emden Equation in Strips, Proc. R. Soc. Edinb. Sect. A Math., 148 (2018), 51-62.
doi: 10.1017/S0308210517000142. |
[10] |
M. J. Esteban and P. L. Lions,
Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. R. Soc. Edinb. Sect. A Math., 93 (1982), 1-14.
doi: 10.1017/S0308210500031607. |
[11] |
A. Farina,
On the classification of solutions of the Lane-Emden equation on unbounded domains of $ \mathbb{R}^N $, J. Math.Pures Appl., 87 (2007), 537-561.
doi: 10.1016/j.matpur.2007.03.001. |
[12] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ., 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[13] |
B. Gidas and J. Spruck,
Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[14] |
D. Gilbarg et Neil S Trudinger, Elliptic Partial Differntial Equations of Second Order, Grundlehren Math. Wiss., Vol. 224, Springer-Verlag, New York, 1977. |
[15] |
A. Harrabi, M. Ahmadou, S. Rebhi and A. Selmi,
A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case, Manuscr. Math., 137 (2012), 525-544.
doi: 10.1007/s00229-011-0488-z. |
[16] |
A. Harrabi and B. Rahal,
On the sixth-order Joseph-Lundgren exponent, Ann. Henri Poincare, 18 (2017), 1055-1094.
doi: 10.1007/s00023-016-0522-5. |
[17] |
A. Harrabi, B. Rahal, Liouville results for m-Laplace equations in half-space and strips with mixed boundary value conditions and Finite Morse index, J. Dyn. Differ. Equ., 30 (2018), 1161-1185.
doi: 10.1007/s10884-017-9593-3. |
[18] |
A. Harrabi, S. Rebhi and A. Selmi,
Solutions of superlinear equations and their Morse indices, Ⅰ, Duke. Math. J., 94 (1998), 141-157.
doi: 10.1215/S0012-7094-98-09407-8. |
[19] |
A. Harrabi, S. Rebhi and A. Selmi,
Solutions of superlinear equations and their Morse indices, Ⅱ, Duke. Math. J., 94 (1998), 159-179.
doi: 10.1215/S0012-7094-98-09407-8. |
[20] |
W. F. Moss and J. Piepenbrink,
Positive solutions of elliptic equations, Pac. J. Math., 75 (1978), 219-226.
|
[21] |
M. Ramos and P. Rodrigues,
On a fourth order superlinear elliptic problem, Electron. J. Differ. Equ. Conf., 06 (2001), 243-255.
|
[22] |
M. Ramos, S. Terracini and C. Troestler,
Superlinear indefinite elliptic problems and Phozaev type identities, J. Funct. Anal., 159 (1998), 596-628.
doi: 10.1006/jfan.1998.3332. |
[23] |
F. Pacard,
Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscr. Math., 79 (1993), 161-172.
doi: 10.1007/BF02568335. |
[24] |
S. I. Pohozaev, Eigenfunctions of $\Delta u+lf\left( u \right)=0$, Soviet Math. Dokl., 6 (1965), 1408-1411. Google Scholar |
[25] |
P. Polácik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems. Ⅰ. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[26] |
S. Solimini,
Morse index estimates in min-max theorems, Manuscr. Math., 63 (1989), 421-453.
doi: 10.1007/BF01171757. |
[27] |
X. Wang, X. Zheng, Liouville theorem for elliptic equations with mixed boundary valu conditions and finite Morse indices, J. Inequal. Appl., (2015), 860–871.
doi: 10.1186/s13660-015-0867-1. |
[28] |
X. Yu,
Solution of mixed boundary problems and their Morse indices, Nonlinear Anal., 96 (2014), 146-153.
doi: 10.1016/j.na.2013.11.011. |
[29] |
X. Yu,
Liouville theorem for elliptic equations with nonlinear boundary value conditions and finite Morse indices, J. Math. Anal. Appl., 421 (2015), 436-443.
doi: 10.1016/j.jmaa.2014.07.010. |
[30] |
X. Yu,
Solution of fractional Laplacian equations and their Morse indices, J. Differ. Equ., 260 (2016), 860-871.
doi: 10.1016/j.jde.2015.09.010. |
show all references
References:
[1] |
L. Ambrosio and X. Cabré,
Entire solutions of semilinear elliptic equations in $ \mathbb{R}^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739.
doi: 10.1090/S0894-0347-00-00345-3. |
[2] |
A. Bahri and P. L. Lions,
Solutions of superlinear elliptic equations and their Morse indices, Commun. Pure. App. Math., 45 (1992), 1205-1215.
doi: 10.1002/cpa.3160450908. |
[3] |
M. Ben Ayed, H. Fourti and A. Selmi,
Harmonic functions with nonlinear Neumann boundary condition and their Morse indices, Nonlinear Anal. Real World Appl., 38 (2017), 96-112.
doi: 10.1016/j.nonrwa.2017.04.012. |
[4] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, Part Ⅱ, Existence of infinitly many solutios groud, Ration. Mech. Anal., 82 (1982), 347-369.
doi: 10.1007/BF00250556. |
[5] |
E. N. Dancer,
Some notes on the method of moving planes, Bull. Austral. Math. Soc. angew. Math., 46 (1992), 425-434.
doi: 10.1017/S0004972700012089. |
[6] |
J. Dávila, L. Dupaigne, K. Wang and J. Wei,
A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.
doi: 10.1016/j.aim.2014.02.034. |
[7] |
D. G. de Figueiredo and J. Yang,
On a semilinear elliptic problem without (PS) condition, J. Differ. Equ., 187 (2003), 412-428.
doi: 10.1016/S0022-0396(02)00055-4. |
[8] |
B. Devyver,
On the finiteness of the Morse index for Schröinger operators, Manuscr. Math., 139 (2012), 249-271.
doi: 10.1007/s00229-011-0522-1. |
[9] |
L. Dupaigne and A. Harrabi,
The Lane-Emden Equation in Strips, Proc. R. Soc. Edinb. Sect. A Math., 148 (2018), 51-62.
doi: 10.1017/S0308210517000142. |
[10] |
M. J. Esteban and P. L. Lions,
Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. R. Soc. Edinb. Sect. A Math., 93 (1982), 1-14.
doi: 10.1017/S0308210500031607. |
[11] |
A. Farina,
On the classification of solutions of the Lane-Emden equation on unbounded domains of $ \mathbb{R}^N $, J. Math.Pures Appl., 87 (2007), 537-561.
doi: 10.1016/j.matpur.2007.03.001. |
[12] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ., 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[13] |
B. Gidas and J. Spruck,
Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[14] |
D. Gilbarg et Neil S Trudinger, Elliptic Partial Differntial Equations of Second Order, Grundlehren Math. Wiss., Vol. 224, Springer-Verlag, New York, 1977. |
[15] |
A. Harrabi, M. Ahmadou, S. Rebhi and A. Selmi,
A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case, Manuscr. Math., 137 (2012), 525-544.
doi: 10.1007/s00229-011-0488-z. |
[16] |
A. Harrabi and B. Rahal,
On the sixth-order Joseph-Lundgren exponent, Ann. Henri Poincare, 18 (2017), 1055-1094.
doi: 10.1007/s00023-016-0522-5. |
[17] |
A. Harrabi, B. Rahal, Liouville results for m-Laplace equations in half-space and strips with mixed boundary value conditions and Finite Morse index, J. Dyn. Differ. Equ., 30 (2018), 1161-1185.
doi: 10.1007/s10884-017-9593-3. |
[18] |
A. Harrabi, S. Rebhi and A. Selmi,
Solutions of superlinear equations and their Morse indices, Ⅰ, Duke. Math. J., 94 (1998), 141-157.
doi: 10.1215/S0012-7094-98-09407-8. |
[19] |
A. Harrabi, S. Rebhi and A. Selmi,
Solutions of superlinear equations and their Morse indices, Ⅱ, Duke. Math. J., 94 (1998), 159-179.
doi: 10.1215/S0012-7094-98-09407-8. |
[20] |
W. F. Moss and J. Piepenbrink,
Positive solutions of elliptic equations, Pac. J. Math., 75 (1978), 219-226.
|
[21] |
M. Ramos and P. Rodrigues,
On a fourth order superlinear elliptic problem, Electron. J. Differ. Equ. Conf., 06 (2001), 243-255.
|
[22] |
M. Ramos, S. Terracini and C. Troestler,
Superlinear indefinite elliptic problems and Phozaev type identities, J. Funct. Anal., 159 (1998), 596-628.
doi: 10.1006/jfan.1998.3332. |
[23] |
F. Pacard,
Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscr. Math., 79 (1993), 161-172.
doi: 10.1007/BF02568335. |
[24] |
S. I. Pohozaev, Eigenfunctions of $\Delta u+lf\left( u \right)=0$, Soviet Math. Dokl., 6 (1965), 1408-1411. Google Scholar |
[25] |
P. Polácik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems. Ⅰ. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[26] |
S. Solimini,
Morse index estimates in min-max theorems, Manuscr. Math., 63 (1989), 421-453.
doi: 10.1007/BF01171757. |
[27] |
X. Wang, X. Zheng, Liouville theorem for elliptic equations with mixed boundary valu conditions and finite Morse indices, J. Inequal. Appl., (2015), 860–871.
doi: 10.1186/s13660-015-0867-1. |
[28] |
X. Yu,
Solution of mixed boundary problems and their Morse indices, Nonlinear Anal., 96 (2014), 146-153.
doi: 10.1016/j.na.2013.11.011. |
[29] |
X. Yu,
Liouville theorem for elliptic equations with nonlinear boundary value conditions and finite Morse indices, J. Math. Anal. Appl., 421 (2015), 436-443.
doi: 10.1016/j.jmaa.2014.07.010. |
[30] |
X. Yu,
Solution of fractional Laplacian equations and their Morse indices, J. Differ. Equ., 260 (2016), 860-871.
doi: 10.1016/j.jde.2015.09.010. |
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