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A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain
Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $
Department of Mathematics, Towson University, Towson, MD 21252, USA |
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.
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
W. Chen and R. Zhang, Classification of solutions and nonlocal curvatures on conformally flat manifolds, preprint. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [11] |
E. A. Gorin,
Asymptotic properties of polynomials and algebraic functions of several variables, Russ. Math. Surv., 16 (1961), 93-119.
|
| [12] |
L. Grafakos, Classical Fourier Analysis, 2$^nd$ edition, Springer, 2008. |
| [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. |
| [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. |
| [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. |
| [16] |
A. Hyder,
Structure of conformal metrics on $\mathbb R^n$ with constant $Q$-curvature, Differ. Integral Equ., 32 (2019), 423-454.
|
| [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. |
| [18] |
A. Hyder, G. Mancini and L. Martinazzi, Local and nonlocal singular Liouville equations in Euclidean spaces, arXiv: 1808.03624. |
| [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. |
| [20] |
Y. Y. Li,
Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [31] |
J. Wei,
Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Commun. Partial Differ. Equ., 21 (1996), 1451-1467.
doi: 10.1080/03605309608821234. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
W. Chen and R. Zhang, Classification of solutions and nonlocal curvatures on conformally flat manifolds, preprint. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [11] |
E. A. Gorin,
Asymptotic properties of polynomials and algebraic functions of several variables, Russ. Math. Surv., 16 (1961), 93-119.
|
| [12] |
L. Grafakos, Classical Fourier Analysis, 2$^nd$ edition, Springer, 2008. |
| [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. |
| [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. |
| [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. |
| [16] |
A. Hyder,
Structure of conformal metrics on $\mathbb R^n$ with constant $Q$-curvature, Differ. Integral Equ., 32 (2019), 423-454.
|
| [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. |
| [18] |
A. Hyder, G. Mancini and L. Martinazzi, Local and nonlocal singular Liouville equations in Euclidean spaces, arXiv: 1808.03624. |
| [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. |
| [20] |
Y. Y. Li,
Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [31] |
J. Wei,
Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Commun. Partial Differ. Equ., 21 (1996), 1451-1467.
doi: 10.1080/03605309608821234. |
| [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. |
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