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doi: 10.3934/eect.2020081

On finite Morse index solutions of higher order fractional elliptic equations

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  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].
Citation: Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations & Control Theory, 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. CowanP. 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. DavilaL. DupaigneK. 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. CowanP. 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. DavilaL. DupaigneK. 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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