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September  2021, 10(3): 575-597. doi: 10.3934/eect.2020081

On finite Morse index solutions of higher order fractional elliptic equations

Belgacem Rahal 1, and  Cherif Zaidi 2,

1. 

Institut Supérieur des Sciences Appliquées et de Technologie de Kairouan, Avenue Beit El Hikma, 3100 Kairouan - Tunisia
2. 

Faculté des Sciences, Département de Mathématiques, B.P 1171 Sfax 3000, Université de Sfax, Tunisia

Received  December 2019 Revised  May 2020 Published  September 2021 Early access  July 2020

We consider sign-changing solutions of the equation
$ (-{\Delta})^s u+\lambda u = |u|^{p-1}u \; \mbox{in}\; \mathbb R^n, $
where
$ n\geq 1 $
,
$ \lambda>0 $
,
$ p>1 $
 and
$ 1<s\leq2 $
. The main goal of this work is to analyze the influence of the linear term
$ \lambda u $
, in order to classify stable solutions possibly unbounded and sign-changing. We prove Liouville type theorems for stable solutions or solutions which are stable outside a compact set of
$ \mathbb R^n $
. We first derive a monotonicity formula for our equation. After that, we provide integral estimate from stability which combined with Pohozaev-type identity to obtain nonexistence results in the subcritical case with the restrictive condition
$ |u|_{L^{\infty}( \mathbb R^n)}^{p-1}< \frac{\lambda (p+1) }{2} $
. The supercritical case needs more involved analysis, motivated by the monotonicity formula, we then reduce the nonexistence of nontrivial entire solutions which are stable outside a compact set of
$ \mathbb R^n $
. Through this approach we give a complete classification of stable solutions for all
$ p>1 $
. Moreover, for the case
$ 0<s\leq1 $
, finite Morse index solutions are classified in [19,25].
Keywords: Morse index, Liouville-type theorems, Pohozaev identity, monotonicity formula, stable solutions.
Mathematics Subject Classification: 35B33, 35B35, 35B45, 35J30, 35B53.
Citation: Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations & Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081
References:
[1]

A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992) 1205–1215. doi: 10.1002/cpa.3160450908.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[3]

J. Case and Sun-Yung Alice Chang, On fractional GJMS operators, preprint, arXiv: 1406.1846. doi: 10.1002/cpa.21564.  Google Scholar

[4]

W. Chen, X. Cui, R. Zhuo and Z. Yuan, A Liouville theorem for the fractional laplacian, arXiv: 1401.7402. Google Scholar

[5]

C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigevalue problems on general domains, DCDS-A, 28 (2010), 1033-1050.  doi: 10.3934/dcds.2010.28.1033.  Google Scholar

[6]

C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigevalue problems on general domains, to appear Cal. Var. PDE DOI 10.1007/s00526-012-0582-4. doi: 10.1007/s00526-012-0582-4.  Google Scholar

[7]

L. Damascelli and F. Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations, De Gruyter Series in Nonlinear Analysis and Applications, 30, 2019. doi: 10.1515/9783110538243.  Google Scholar

[8]

E. N. Dancer, Superlinear problems on domains with holes of asymptotic shape and exterior problems, Math. Z., 229 (1998), 475-491.  doi: 10.1007/PL00004666.  Google Scholar

[9]

J. Davila, L. Dupaigne, K. Wang and J. Wei, A monotonicity formula and a Liouville-type Theorem for a fourth order supercritical problem, Advances in Mathematicas, 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.  Google Scholar

[10]

J. Davila, L. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087–6104. Nonlocal time porous medium equation with fractional time derivative. Djida, J.-D., Nieto, J.J., Area, I. Revista Matematica Complutense 32 (2019), 273-304. doi: 10.1090/tran/6872.  Google Scholar

[11]

J.D. Djida, J.J. Nieto, I., Area, Nonlocal time porous medium equation with fractional time derivative, Revista Matematica Complutense 32 (2019), 273-304. doi: 10.1007/s13163-018-0287-0.  Google Scholar

[12]

X. L. Ding, J.J. Nieto, Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions, Communications in Nonlinear Science and Numerical Simulation 52 (2017), 165-176. doi: 10.1016/j.cnsns.2017.04.020.  Google Scholar

[13]

E. B. Fabes, C. E. Kenig, and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. doi: 10.1080/03605308208820218.  Google Scholar

[14]

A. Farina, On the classification of soultions of the Lane-Emden equation on unbounded domains of $ \mathbb R^n$, J. Math. Pures Appl., 87 (9), no. 5,537-561, 2007. doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[15]

M. Fazly, Regularity of extremal solutions of nonlocal elliptic systems, M. Discrete and Continuous Dynamical Systems- Series A 40 (2020), 107-131. doi: 10.3934/dcds.2020005.  Google Scholar

[16]

M. Fazly, J. Wei, On finite Morse index solutions of higher order fractional Lane-Emden equations, American Journal of Mathematics, 139, (2), 433-460, 2017. doi: 10.1353/ajm.2017.0011.  Google Scholar

[17]

F. Gazzola, H. C. Grunau, Radial entire solutions of supercritical biharmonic equations, Math. Annal., 334, 905-936, 2006. doi: 10.1007/s00208-005-0748-x.  Google Scholar

[18]

Z.M. Guo, J. Wei, Qualitative properties of entire radial solutions for a biharmonic equation with supcritical nonlinearity, Proc. American Math. Soc. 138, no.11, 3957–3964, 2010. doi: 10.1090/S0002-9939-10-10374-8.  Google Scholar

[19]

A.Selmi, A.Harrabi, and C.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 19 (5) (2020): 2839. Google Scholar

[20]

A. Harrabi and B. Rahal, On the Sixth-order JosephLundgren Exponent, Journal of Annales Henri Poincaré 18 (3), 1055-1094, 2017. doi: 10.1007/s00023-016-0522-5.  Google Scholar

[21]

A. Harrabi and B. Rahal, Liouville type theorems for elliptic equations in half-space with mixed boundary value conditions, J. Advances in Nonlinear Analysis, doi:10.1515/anona-2016-0168  Google Scholar

[22]

A. Harrabi and B. Rahal, Liouville results for $m$-Laplace equations in half-space and strips with mixed boundary value conditions and finite Morse index, Journal of Dynamics and Differential Equations, doi: 10.1007/s10884-017-9593-3  Google Scholar

[23]

Ira W. Herbst, Spectral theory of the operator $(p^2+m^2)^\frac{1}{2}-Ze^2/r$, , Comm. Math. Phys. 53, no. 3,285-294, 1977.  Google Scholar

[24]

C. S. Lin, A classification of soluitions of a conformally invariant fourth order equation in $R^n$, Comment. Math. Helv., 73, 206-231, 1998. doi: 10.1007/s000140050052.  Google Scholar

[25]

B. Rahal, C. Zaidi, On the Classification of Stable Solutions of the Fractional Equation, Journal of Potential Anal, (4) 50 (2018): 565-579.. doi: 10.1007/s11118-018-9694-6.  Google Scholar

[26]

B. Rahal, C. Zaidi, Liouville results for elliptic equations in strips with finite Morse index, Journal of Annali di Matematica https://doi.org/10.1007/s10231-018-0743-y. doi: 10.1007/s10231-018-0743-y.  Google Scholar

[27]

X. Ros-Oton, J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213, no. 2,587–628, 2014. doi: 10.1007/s00205-014-0740-2.  Google Scholar

[28]

J. Wei, X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313, no. 2,207-228, 1999. doi: 10.1007/s002080050258.  Google Scholar

[29]

D. Yafaev, Sharp Constants in the Hardy-Rellich Inequalities, Journal of Functional Analysis 168, 121-144, 1999. doi: 10.1006/jfan.1999.3462.  Google Scholar

[30]

R. Yang, On higher order extensions for the fractional Laplacian, preprint., arXiv$\sharp$ preprint arXiv: 1302.4413, 2013. Google Scholar

show all references

References:
[1]

A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992) 1205–1215. doi: 10.1002/cpa.3160450908.  Google Scholar

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[3]

J. Case and Sun-Yung Alice Chang, On fractional GJMS operators, preprint, arXiv: 1406.1846. doi: 10.1002/cpa.21564.  Google Scholar

[4]

W. Chen, X. Cui, R. Zhuo and Z. Yuan, A Liouville theorem for the fractional laplacian, arXiv: 1401.7402. Google Scholar

[5]

C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigevalue problems on general domains, DCDS-A, 28 (2010), 1033-1050.  doi: 10.3934/dcds.2010.28.1033.  Google Scholar

[6]

C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigevalue problems on general domains, to appear Cal. Var. PDE DOI 10.1007/s00526-012-0582-4. doi: 10.1007/s00526-012-0582-4.  Google Scholar

[7]

L. Damascelli and F. Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations, De Gruyter Series in Nonlinear Analysis and Applications, 30, 2019. doi: 10.1515/9783110538243.  Google Scholar

[8]

E. N. Dancer, Superlinear problems on domains with holes of asymptotic shape and exterior problems, Math. Z., 229 (1998), 475-491.  doi: 10.1007/PL00004666.  Google Scholar

[9]

J. Davila, L. Dupaigne, K. Wang and J. Wei, A monotonicity formula and a Liouville-type Theorem for a fourth order supercritical problem, Advances in Mathematicas, 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.  Google Scholar

[10]

J. Davila, L. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087–6104. Nonlocal time porous medium equation with fractional time derivative. Djida, J.-D., Nieto, J.J., Area, I. Revista Matematica Complutense 32 (2019), 273-304. doi: 10.1090/tran/6872.  Google Scholar

[11]

J.D. Djida, J.J. Nieto, I., Area, Nonlocal time porous medium equation with fractional time derivative, Revista Matematica Complutense 32 (2019), 273-304. doi: 10.1007/s13163-018-0287-0.  Google Scholar

[12]

X. L. Ding, J.J. Nieto, Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions, Communications in Nonlinear Science and Numerical Simulation 52 (2017), 165-176. doi: 10.1016/j.cnsns.2017.04.020.  Google Scholar

[13]

E. B. Fabes, C. E. Kenig, and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. doi: 10.1080/03605308208820218.  Google Scholar

[14]

A. Farina, On the classification of soultions of the Lane-Emden equation on unbounded domains of $ \mathbb R^n$, J. Math. Pures Appl., 87 (9), no. 5,537-561, 2007. doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[15]

M. Fazly, Regularity of extremal solutions of nonlocal elliptic systems, M. Discrete and Continuous Dynamical Systems- Series A 40 (2020), 107-131. doi: 10.3934/dcds.2020005.  Google Scholar

[16]

M. Fazly, J. Wei, On finite Morse index solutions of higher order fractional Lane-Emden equations, American Journal of Mathematics, 139, (2), 433-460, 2017. doi: 10.1353/ajm.2017.0011.  Google Scholar

[17]

F. Gazzola, H. C. Grunau, Radial entire solutions of supercritical biharmonic equations, Math. Annal., 334, 905-936, 2006. doi: 10.1007/s00208-005-0748-x.  Google Scholar

[18]

Z.M. Guo, J. Wei, Qualitative properties of entire radial solutions for a biharmonic equation with supcritical nonlinearity, Proc. American Math. Soc. 138, no.11, 3957–3964, 2010. doi: 10.1090/S0002-9939-10-10374-8.  Google Scholar

[19]

A.Selmi, A.Harrabi, and C.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 19 (5) (2020): 2839. Google Scholar

[20]

A. Harrabi and B. Rahal, On the Sixth-order JosephLundgren Exponent, Journal of Annales Henri Poincaré 18 (3), 1055-1094, 2017. doi: 10.1007/s00023-016-0522-5.  Google Scholar

[21]

A. Harrabi and B. Rahal, Liouville type theorems for elliptic equations in half-space with mixed boundary value conditions, J. Advances in Nonlinear Analysis, doi:10.1515/anona-2016-0168  Google Scholar

[22]

A. Harrabi and B. Rahal, Liouville results for $m$-Laplace equations in half-space and strips with mixed boundary value conditions and finite Morse index, Journal of Dynamics and Differential Equations, doi: 10.1007/s10884-017-9593-3  Google Scholar

[23]

Ira W. Herbst, Spectral theory of the operator $(p^2+m^2)^\frac{1}{2}-Ze^2/r$, , Comm. Math. Phys. 53, no. 3,285-294, 1977.  Google Scholar

[24]

C. S. Lin, A classification of soluitions of a conformally invariant fourth order equation in $R^n$, Comment. Math. Helv., 73, 206-231, 1998. doi: 10.1007/s000140050052.  Google Scholar

[25]

B. Rahal, C. Zaidi, On the Classification of Stable Solutions of the Fractional Equation, Journal of Potential Anal, (4) 50 (2018): 565-579.. doi: 10.1007/s11118-018-9694-6.  Google Scholar

[26]

B. Rahal, C. Zaidi, Liouville results for elliptic equations in strips with finite Morse index, Journal of Annali di Matematica https://doi.org/10.1007/s10231-018-0743-y. doi: 10.1007/s10231-018-0743-y.  Google Scholar

[27]

X. Ros-Oton, J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213, no. 2,587–628, 2014. doi: 10.1007/s00205-014-0740-2.  Google Scholar

[28]

J. Wei, X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313, no. 2,207-228, 1999. doi: 10.1007/s002080050258.  Google Scholar

[29]

D. Yafaev, Sharp Constants in the Hardy-Rellich Inequalities, Journal of Functional Analysis 168, 121-144, 1999. doi: 10.1006/jfan.1999.3462.  Google Scholar

[30]

R. Yang, On higher order extensions for the fractional Laplacian, preprint., arXiv$\sharp$ preprint arXiv: 1302.4413, 2013. Google Scholar

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