Advanced Search
Article Contents
Article Contents

On finite Morse index solutions of higher order fractional elliptic equations

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • 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].

    Mathematics Subject Classification: 35B33, 35B35, 35B45, 35J30, 35B53.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
    [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.
    [3] J. Case and Sun-Yung Alice Chang, On fractional GJMS operators, preprint, arXiv: 1406.1846. doi: 10.1002/cpa.21564.
    [4] W. Chen, X. Cui, R. Zhuo and Z. Yuan, A Liouville theorem for the fractional laplacian, arXiv: 1401.7402.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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
    [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
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [29] D. Yafaev, Sharp Constants in the Hardy-Rellich Inequalities, Journal of Functional Analysis 168, 121-144, 1999. doi: 10.1006/jfan.1999.3462.
    [30] R. Yang, On higher order extensions for the fractional Laplacian, preprint., arXiv$\sharp$ preprint arXiv: 1302.4413, 2013.
  • 加载中

Article Metrics

HTML views(1742) PDF downloads(272) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint