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May  2020, 19(5): 2839-2852. doi: 10.3934/cpaa.2020124

Nonexistence results on the space or the half space of $ -\Delta u+\lambda u = |u|^{p-1}u $ via the Morse index

Abdelbaki Selmi 1,, , Abdellaziz Harrabi 2, and  Cherif Zaidi 3,

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

*Corresponding author

Received  June 2019 Revised  October 2019 Published  March 2020

Fund Project: This work is supported by Deanship of the Scientific Research of Northern Border University. KSA, grant no. SCI-2018-3-9-F-7713

In this paper we consider the following semi-linear elliptic problem
$ \begin{equation*} -\Delta u+\lambda u = |u|^{p-1}u\quad\mbox{in}\,\, \mathcal{O}, \tag{P} \end{equation*} $
where
$ \mathcal{O} = \mathbb{R}^N $
; or
$ \mathcal{O} = \mathbb{R}^N_+ = \{x = (x',x_N),\, x'\in \mathbb{R}^{N-1},x_N>0\} $
 with Dirichlet boundary conditions. Here
$ N\geq2 $
,
$ p>1 $
 and
$ \lambda $
 is a positive real parameter. The main goal ofthis work is to analyze the influence of the linear term
$ \lambda u $
, in order to classify regular stable solutions possibly unbounded and sign-changing. Our analysis reveals the nonexistence of nontrivial stable solutions (respectively solutions which are stable outside a compact set) for all
$ p> 1 $
 (respectively for all
$ p\geq \frac{N+2}{N-2} $
, or
$ 1<p<\frac{N+2}{N-2} $
 and
$ |u|^{p-1}<\frac{\lambda (p+1)}{2} $
). Inspired by [6,9,16,23], we establish a monotonicity formula to discuss the supercritical case.
Regarding the case
$ \mathcal{O} = \mathbb{R}^N $
, we obtain a complete classification which states that problem
$ (P) $
 has regular solutions which are stable outside a compact set if and only if
$ p\in (1,\infty) $
 and
$ N = 2 $
; or
$ p\in(1,\frac{N+2}{N-2}) $
 and
$ N\geq3. $
Keywords: Elliptic equations, Liouville type theorems, stable solutions, Morse index, monotonicity formula.
Mathematics Subject Classification: Primary: 35J60, 35J65; Secondary: 58E05.
Citation: Abdelbaki Selmi, Abdellaziz Harrabi, Cherif Zaidi. Nonexistence results on the space or the half space of $ -\Delta u+\lambda u = |u|^{p-1}u $ via the Morse index. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2839-2852. doi: 10.3934/cpaa.2020124
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

D. Gilbarg et Neil S Trudinger, Elliptic Partial Differntial Equations of Second Order, Grundlehren Math. Wiss., Vol. 224, Springer-Verlag, New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. F. Moss and J. Piepenbrink, Positive solutions of elliptic equations, Pac. J. Math., 75 (1978), 219-226.   Google Scholar

[21]

M. Ramos and P. Rodrigues, On a fourth order superlinear elliptic problem, Electron. J. Differ. Equ. Conf., 06 (2001), 243-255.   Google Scholar

[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.  Google Scholar

[23]

F. Pacard, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscr. Math., 79 (1993), 161-172.  doi: 10.1007/BF02568335.  Google Scholar

[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.  Google Scholar

[26]

S. Solimini, Morse index estimates in min-max theorems, Manuscr. Math., 63 (1989), 421-453.  doi: 10.1007/BF01171757.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

D. Gilbarg et Neil S Trudinger, Elliptic Partial Differntial Equations of Second Order, Grundlehren Math. Wiss., Vol. 224, Springer-Verlag, New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. F. Moss and J. Piepenbrink, Positive solutions of elliptic equations, Pac. J. Math., 75 (1978), 219-226.   Google Scholar

[21]

M. Ramos and P. Rodrigues, On a fourth order superlinear elliptic problem, Electron. J. Differ. Equ. Conf., 06 (2001), 243-255.   Google Scholar

[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.  Google Scholar

[23]

F. Pacard, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscr. Math., 79 (1993), 161-172.  doi: 10.1007/BF02568335.  Google Scholar

[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.  Google Scholar

[26]

S. Solimini, Morse index estimates in min-max theorems, Manuscr. Math., 63 (1989), 421-453.  doi: 10.1007/BF01171757.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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