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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

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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

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