American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation
  • CPAA Home
  • This Issue
  • Next Article
    A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain
December  2020, 19(12): 5413-5436. doi: 10.3934/cpaa.2020246

Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $

Mathew Gluck

Department of Mathematics, Towson University, Towson, MD 21252, USA

Received  October 2019 Revised  August 2020 Published  December 2020 Early access  October 2020

This work concerns the distributional solutions of a conformally invariant system of $ n^{\rm th} $-order elliptic equations on $ \mathbb R^n $ having exponential type nonlinearity. The system in question is a natural generalization of the constant $ Q $-curvature equation on $ \mathbb R^n $. Under an $ L^1 $-finiteness assumption and some assumptions on the coupling coefficients, an asymptotic estimate for solutions as $ \left|x\right|\to \infty $ is obtained. Under a growth constraint and further $ L^1 $-norm assumptions the method of moving spheres is used to show that, up to an additive polynomial of low degree, each of the unknown functions is a standard bubble with common center and scale parameters.

Keywords: conformally invariant, elliptic system, higher-order fractional Laplacian.
Mathematics Subject Classification: 35C05, 35G50, 35J48.
Citation: Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246
References:
[1]

Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville's equation in dimension $4$, J. Eur. Math. Soc., 8 (2006), 171–180. doi: 10.4171/jems/44.  Google Scholar

[2]

H. Brézis and F. Merle, Uniform estimates and blow–up behavior for solutions of $\Delta u = V(x)e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.  Google Scholar

[3]

S. Y. A. Chang and W. Chen, A note on a class of higher order conformally covariant equations, Discrete Contin. Dyn. Syst., 63 (2001), 275-281.  doi: 10.3934/dcds.2001.7.275.  Google Scholar

[4]

S. Chanillo and M. K. H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal., 5 (1995), 924-947.  doi: 10.1007/BF01902215.  Google Scholar

[5]

W. Chen and C. Li, Classification of solutions to some nonlinear elliptic equations, Duke Math. J., 3 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[6]

W. Chen and R. Zhang, Classification of solutions and nonlocal curvatures on conformally flat manifolds, preprint. Google Scholar

[7]

M. Chipot, I. Shafrir and G. Wolansky, On the solutions of Liouville systems, J. Differ. Equ., 140 (1997), 59-105.  doi: 10.1006/jdeq.1997.3316.  Google Scholar

[8]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[9]

M. Gluck, Subcritical approach to conformally invariant extension operators on the upper half space, J. Funct. Anal. 278 (2020), 46pp. doi: 10.1016/j.jfa.2018.08.012.  Google Scholar

[10]

M. Gluck and L. Zhang, Classification of solutions to a system of critically nonlinear elliptic equations on Euclidean half space, J. Partial Differ. Equ., 28 (2015), 74-94.   Google Scholar

[11]

E. A. Gorin, Asymptotic properties of polynomials and algebraic functions of several variables, Russ. Math. Surv., 16 (1961), 93-119.   Google Scholar

[12]

L. Grafakos, Classical Fourier Analysis, 2$^nd$ edition, Springer, 2008.  Google Scholar

[13]

X. Huang and D. Ye, Conformal metrics in $ \mathbb R^2m$ with constant $Q$-curvature and arbitrary volume, Calc. Var. Partial Differ. Equ., 54 (2016), 3373-3348.  doi: 10.1007/s00526-015-0907-1.  Google Scholar

[14]

A. Hyder, Existence of entire solutions to a fractional Liouville equation in $ \mathbb R^n$, Rend. Lincei. Mat. Appl., 27 (2016), 1-14.  doi: 10.4171/RLM/718.  Google Scholar

[15]

A. Hyder, Conformally Euclidean metrics on $ \mathbb R^n$ with arbitrary total $Q$-curvature, Anal. PDE, 10 (2017) 635–652. doi: 10.2140/apde.2017.10.635.  Google Scholar

[16]

A. Hyder, Structure of conformal metrics on $\mathbb R^n$ with constant $Q$-curvature, Differ. Integral Equ., 32 (2019), 423-454.   Google Scholar

[17]

A. Hyder and L. Martinazzi, Conformal metrics on $ \mathbb R^2m$ with constant $Q$-curvature, prescribed volume and asymptotic behavior, Discrete Contin. Dynam. Sys. A, 35 (2015), 283-299.  doi: 10.3934/dcds.2015.35.283.  Google Scholar

[18]

A. Hyder, G. Mancini and L. Martinazzi, Local and nonlocal singular Liouville equations in Euclidean spaces, arXiv: 1808.03624. Google Scholar

[19]

T. Jin, A. Maalaoui, L. Martinazzi and J. Xiong, Existence and asymptotics for solutions of a non-local $Q$-curvature equation in dimension three, Calc. Var., 52 (2015), 469-488.  doi: 10.1007/s00526-014-0718-9.  Google Scholar

[20]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.   Google Scholar

[21]

Y. Li and I. Shafrir, Blow-up analysis for solutions of $-\Delta u = V e^u$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270.  doi: 10.1512/iumj.1994.43.43054.  Google Scholar

[22]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[23]

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

[24]

C. S. Lin and L. Zhang, A Profile of bubbling solutions to a Liouville system, Ann. I. H. Poincaré, 27 (2010), 117–143. doi: 10.1016/j.anihpc.2009.09.001.  Google Scholar

[25]

C. S. Lin and L. Zhang, A topological degree counting for some Liouville systems of mean field equations, Commun. Pure Appl. Math., 64 (2011), 556-590.  doi: 10.1002/cpa.20355.  Google Scholar

[26]

A. Malchiodi, Compactness of solutions to some geometric fourth-order equations, J. Reine Agnew. Math., 594 (2006), 137-174.  doi: 10.1515/CRELLE.2006.038.  Google Scholar

[27]

L. Martinazzi, Classification of solutions to the higher order Liouville's equation on $\mathbb R^2m$, Math. Z., 263 (2008), 307-329.  doi: 10.1007/s00209-008-0419-1.  Google Scholar

[28]

L. Martinazzi, Concentration-compactness phenomena in higher order Liouville's equation, J. Funct. Anal., 265 (2009), 3743-3771.  doi: 10.1016/j.jfa.2009.02.017.  Google Scholar

[29]

L. Martinazzi, Conformal metrics on $ \mathbb R^2m$ with constant $Q$-curvature and large volume, Ann. I. H. Poincare Anal., 30 (2013), 969-982.  doi: 10.1016/j.anihpc.2012.12.007.  Google Scholar

[30]

F. Robert and M. Struwe, Asymptotic profile for a fourth order PDE with critical exponential growth in dimension four, Adv. Nonlinear Stud., 4 (2004), 397-415.  doi: 10.1515/ans-2004-0403.  Google Scholar

[31]

J. Wei, Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Commun. Partial Differ. Equ., 21 (1996), 1451-1467.  doi: 10.1080/03605309608821234.  Google Scholar

[32]

J. Wei and D. Ye, Nonradial solutions for a conformally invariant fourth order equation in $ \mathbb R^4$, Calc. Var. Partial Differ. Equ., 32 (2008), 373-386. doi: 10.1007/s00526-007-0145-2.  Google Scholar

show all references

References:
[1]

Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville's equation in dimension $4$, J. Eur. Math. Soc., 8 (2006), 171–180. doi: 10.4171/jems/44.  Google Scholar

[2]

H. Brézis and F. Merle, Uniform estimates and blow–up behavior for solutions of $\Delta u = V(x)e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.  Google Scholar

[3]

S. Y. A. Chang and W. Chen, A note on a class of higher order conformally covariant equations, Discrete Contin. Dyn. Syst., 63 (2001), 275-281.  doi: 10.3934/dcds.2001.7.275.  Google Scholar

[4]

S. Chanillo and M. K. H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal., 5 (1995), 924-947.  doi: 10.1007/BF01902215.  Google Scholar

[5]

W. Chen and C. Li, Classification of solutions to some nonlinear elliptic equations, Duke Math. J., 3 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[6]

W. Chen and R. Zhang, Classification of solutions and nonlocal curvatures on conformally flat manifolds, preprint. Google Scholar

[7]

M. Chipot, I. Shafrir and G. Wolansky, On the solutions of Liouville systems, J. Differ. Equ., 140 (1997), 59-105.  doi: 10.1006/jdeq.1997.3316.  Google Scholar

[8]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[9]

M. Gluck, Subcritical approach to conformally invariant extension operators on the upper half space, J. Funct. Anal. 278 (2020), 46pp. doi: 10.1016/j.jfa.2018.08.012.  Google Scholar

[10]

M. Gluck and L. Zhang, Classification of solutions to a system of critically nonlinear elliptic equations on Euclidean half space, J. Partial Differ. Equ., 28 (2015), 74-94.   Google Scholar

[11]

E. A. Gorin, Asymptotic properties of polynomials and algebraic functions of several variables, Russ. Math. Surv., 16 (1961), 93-119.   Google Scholar

[12]

L. Grafakos, Classical Fourier Analysis, 2$^nd$ edition, Springer, 2008.  Google Scholar

[13]

X. Huang and D. Ye, Conformal metrics in $ \mathbb R^2m$ with constant $Q$-curvature and arbitrary volume, Calc. Var. Partial Differ. Equ., 54 (2016), 3373-3348.  doi: 10.1007/s00526-015-0907-1.  Google Scholar

[14]

A. Hyder, Existence of entire solutions to a fractional Liouville equation in $ \mathbb R^n$, Rend. Lincei. Mat. Appl., 27 (2016), 1-14.  doi: 10.4171/RLM/718.  Google Scholar

[15]

A. Hyder, Conformally Euclidean metrics on $ \mathbb R^n$ with arbitrary total $Q$-curvature, Anal. PDE, 10 (2017) 635–652. doi: 10.2140/apde.2017.10.635.  Google Scholar

[16]

A. Hyder, Structure of conformal metrics on $\mathbb R^n$ with constant $Q$-curvature, Differ. Integral Equ., 32 (2019), 423-454.   Google Scholar

[17]

A. Hyder and L. Martinazzi, Conformal metrics on $ \mathbb R^2m$ with constant $Q$-curvature, prescribed volume and asymptotic behavior, Discrete Contin. Dynam. Sys. A, 35 (2015), 283-299.  doi: 10.3934/dcds.2015.35.283.  Google Scholar

[18]

A. Hyder, G. Mancini and L. Martinazzi, Local and nonlocal singular Liouville equations in Euclidean spaces, arXiv: 1808.03624. Google Scholar

[19]

T. Jin, A. Maalaoui, L. Martinazzi and J. Xiong, Existence and asymptotics for solutions of a non-local $Q$-curvature equation in dimension three, Calc. Var., 52 (2015), 469-488.  doi: 10.1007/s00526-014-0718-9.  Google Scholar

[20]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.   Google Scholar

[21]

Y. Li and I. Shafrir, Blow-up analysis for solutions of $-\Delta u = V e^u$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270.  doi: 10.1512/iumj.1994.43.43054.  Google Scholar

[22]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[23]

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

[24]

C. S. Lin and L. Zhang, A Profile of bubbling solutions to a Liouville system, Ann. I. H. Poincaré, 27 (2010), 117–143. doi: 10.1016/j.anihpc.2009.09.001.  Google Scholar

[25]

C. S. Lin and L. Zhang, A topological degree counting for some Liouville systems of mean field equations, Commun. Pure Appl. Math., 64 (2011), 556-590.  doi: 10.1002/cpa.20355.  Google Scholar

[26]

A. Malchiodi, Compactness of solutions to some geometric fourth-order equations, J. Reine Agnew. Math., 594 (2006), 137-174.  doi: 10.1515/CRELLE.2006.038.  Google Scholar

[27]

L. Martinazzi, Classification of solutions to the higher order Liouville's equation on $\mathbb R^2m$, Math. Z., 263 (2008), 307-329.  doi: 10.1007/s00209-008-0419-1.  Google Scholar

[28]

L. Martinazzi, Concentration-compactness phenomena in higher order Liouville's equation, J. Funct. Anal., 265 (2009), 3743-3771.  doi: 10.1016/j.jfa.2009.02.017.  Google Scholar

[29]

L. Martinazzi, Conformal metrics on $ \mathbb R^2m$ with constant $Q$-curvature and large volume, Ann. I. H. Poincare Anal., 30 (2013), 969-982.  doi: 10.1016/j.anihpc.2012.12.007.  Google Scholar

[30]

F. Robert and M. Struwe, Asymptotic profile for a fourth order PDE with critical exponential growth in dimension four, Adv. Nonlinear Stud., 4 (2004), 397-415.  doi: 10.1515/ans-2004-0403.  Google Scholar

[31]

J. Wei, Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Commun. Partial Differ. Equ., 21 (1996), 1451-1467.  doi: 10.1080/03605309608821234.  Google Scholar

[32]

J. Wei and D. Ye, Nonradial solutions for a conformally invariant fourth order equation in $ \mathbb R^4$, Calc. Var. Partial Differ. Equ., 32 (2008), 373-386. doi: 10.1007/s00526-007-0145-2.  Google Scholar

[1]

Mei Yu, Xia Zhang, Binlin Zhang. Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3597-3612. doi: 10.3934/cpaa.2020157
[2]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062
[3]

Ali Hyder, Juncheng Wei. Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2757-2764. doi: 10.3934/cpaa.2019123
[4]

Daomin Cao, Guolin Qin. Liouville type theorems for fractional and higher-order fractional systems. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2269-2283. doi: 10.3934/dcds.2020361
[5]

Sun-Yung Alice Chang, Wenxiong Chen. A note on a class of higher order conformally covariant equations. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 275-281. doi: 10.3934/dcds.2001.7.275
[6]

George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure & Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035
[7]

Robert Jankowski, Barbara Łupińska, Magdalena Nockowska-Rosiak, Ewa Schmeidel. Monotonic solutions of a higher-order neutral difference system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 253-261. doi: 10.3934/dcdsb.2018017
[8]

Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134
[9]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81
[10]

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
[11]

Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183
[12]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990
[13]

Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial & Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385
[14]

Pedro D. Prieto-Martínez, Narciso Román-Roy. Higher-order mechanics: Variational principles and other topics. Journal of Geometric Mechanics, 2013, 5 (4) : 493-510. doi: 10.3934/jgm.2013.5.493
[15]

Feliz Minhós, Hugo Carrasco. Solvability of higher-order BVPs in the half-line with unbounded nonlinearities. Conference Publications, 2015, 2015 (special) : 841-850. doi: 10.3934/proc.2015.0841
[16]

Clesh Deseskel Elion Ekohela, Daniel Moukoko. On higher-order anisotropic perturbed Caginalp phase field systems. Electronic Research Announcements, 2019, 26: 36-53. doi: 10.3934/era.2019.26.004
[17]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[18]

Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181
[19]

Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381
[20]

Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1873-1884. doi: 10.3934/jimo.2019033

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (156)
  • HTML views (59)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy