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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  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

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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

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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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