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Existence and non-existence results for variational higher order elliptic systems
Università degli Studi dell'Insubria, Dipartimento di Scienza e Alta Tecnologia, Via Valleggio 11, Como, 22100, Italy |
$α ∈ \mathbb{N}$ |
$α ≥ 1$ |
$(-Δ)^{α} = -Δ((-Δ)^{α-1})$ |
$\begin{cases}\begin{aligned}(-Δ)^{α} u = H_v(u, v) \\(-Δ)^{α} v = H_u(u, v) \\\end{aligned} \text{ in } Ω \subset \mathbb{R}^N \\\frac{\partial^{r} u}{\partial ν^{r}} = 0, \, r = 0, \dots, α-1, \text{ on } \partial Ω \\\frac{\partial^{r} v}{\partial ν^{r}} = 0, \, r = 0, \dots, α-1, \text{ on } \partial Ω\end{cases}$ |
$Ω$ |
$N >2α$ |
$ν$ |
$\partial Ω$ |
$H ∈ C^1 (\mathbb{R}^2; \mathbb{R})$ |
References:
[1] |
R. Adams and J. Fournier,
Sobolev Spaces, 2$^{nd}$ edition, Pure and Applied Mathematics 140, Elsevier/Academic Press, Amsterdam, 2003, xiv+305 pp. |
[2] |
F. Arthur and X. Yan,
A Liouville-type theorem for higher order elliptic systems of Hénon-Lane-Emden type, Commun. Pure Appl. Anal., 15 (2016), 807-830.
doi: 10.3934/cpaa.2016.15.807. |
[3] |
V. Benci and P. H. Rabinowitz,
Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.
doi: 10.1007/BF01389883. |
[4] |
D. Bonheure, E. M. dos Santos and H. Tavares,
Hamiltonian elliptic systems: A guide to variational frameworks, Port. Math, 71 (2014), 301-395.
doi: 10.4171/PM/1954. |
[5] |
G. Caristi, L. D'Ambrosio and E. Mitidieri,
Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.
doi: 10.1007/s00032-008-0090-3. |
[6] |
D. Cassani, B. Kaltenbacher and A. Lorenzi, Direct and inverse problems related to MEMS Inverse Problems, 25 (2009), 105002 (22 pp.)
doi: 10.1088/0266-5611/25/10/105002. |
[7] |
D. Cassani and C. Tarsi,
Existence of solitary waves for supercritical Schrödinger systems in dimension two, Calc. Var. Partial Differential Equations, 54 (2015), 1673-1704.
doi: 10.1007/s00526-015-0840-3. |
[8] |
D. Cassani and J. Zhang,
A priori estimates and positivity for semiclassical ground states for systems of critical Schrödinger equations in dimension two, Comm. Partial Differential Equations, 42 (2017), 655-702.
doi: 10.1080/03605302.2017.1295062. |
[9] |
P. Clément, P. L. Felmer and E. Mitidieri,
Homoclinic orbits for a class of infinite-dimensional Hamiltonian systems, Scuola Norm. Sup. Pisa Cl. Sci.(4), 24 (1997), 367-393.
|
[10] |
L. C. Evans,
Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, 2010, xxii+749 pp.
doi: 10.1090/gsm/019. |
[11] |
P. L. Felmer,
Periodic solutions of "superquadratic" Hamiltonian systems, J. Differential Equations, 102 (1993), 188-207.
doi: 10.1006/jdeq.1993.1027. |
[12] |
D. G. de Figueiredo and P. L. Felmer,
On superquadratic elliptic systems, Trans. Amer. Math. Soc., 343 (1994), 99-116.
doi: 10.1090/S0002-9947-1994-1214781-2. |
[13] |
F. Gazzola, H. C. Grunau, and G. Sweers,
Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Springer-Verlag, Berlin, 2010, xviii+423 pp.
doi: 10.1007/978-3-642-12245-3. |
[14] |
J. Hulshof and R. Van der Vorst,
Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58.
doi: 10.1006/jfan.1993.1062. |
[15] |
J. L. Lions,
Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs, J. Math. Soc. Japan, 14 (1962), 233-241.
doi: 10.2969/jmsj/01420233. |
[16] |
J. L. Lions and E. Magenes,
Non-homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York-Heidelberg, 1972, xvi+357 pp. |
[17] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[18] |
J. Liu, Y. Guo and Y. Zhang,
Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270.
|
[19] |
J. Liu, Y. Guo and Y. Zhang,
Liouville-type theorems for polyharmonic systems in $\mathbb{R}^N$, J. Differential Equations, 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[20] |
E. Mitidieri,
A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[21] |
E. Mitidieri and S. I. Pohozaev,
A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
|
[22] |
A. Persson,
Compact linear mappings between interpolation spaces, Ark. Mat., 5 (1964), 215-219.
doi: 10.1007/BF02591123. |
[23] |
P. Poláčik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems, Ⅰ: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[24] |
P. Pucci and J. Serrin,
A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[25] |
B. Ruf, Lorentz spaces and nonlinear elliptic systems, in Contributions to Nonlinear Analysis,
Vol. 66, Birkhäuser, Basel, (2006), 471–489.
doi: 10.1007/3-7643-7401-2_32. |
[26] |
B. Ruf, Superlinear elliptic equations and systems, in Handbook of Differential Equations:
Stationary Partial Differential Equations, Vol. 5, Elsevier, (2008), 211–276.
doi: 10.1016/S1874-5733(08)80010-1. |
[27] |
E. M. dos Santos,
Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation, J. Math. Anal. Appl., 342 (2008), 277-297.
doi: 10.1016/j.jmaa.2007.11.056. |
[28] |
D. Schiera,
Existence of solutions to higher order Lane-Emden type systems, Nonlinear Anal., 168 (2018), 130-153.
doi: 10.1016/j.na.2017.11.011. |
[29] |
B. Sirakov,
Existence results and a priori bounds for higher order elliptic equations and systems, J. Math. Pures Appl.(9), 89 (2008), 114-133.
doi: 10.1016/j.matpur.2007.10.003. |
[30] |
B. Sirakov,
On the existence of solutions of Hamiltonian elliptic systems in $\mathbb{R}^N$, Adv. Differential Equations, 5 (2000), 1445-1464.
|
[31] |
P. Souplet,
The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[32] |
M. Struwe,
Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4$^{th}$ edition, Springer-Verlag, Berlin, 2008, xx+302 pp. |
[33] |
Y. Zhang,
A Liouville type theorem for polyharmonic elliptic systems, J. Math. Anal. Appl., 326 (2007), 677-690.
doi: 10.1016/j.jmaa.2006.03.027. |
show all references
References:
[1] |
R. Adams and J. Fournier,
Sobolev Spaces, 2$^{nd}$ edition, Pure and Applied Mathematics 140, Elsevier/Academic Press, Amsterdam, 2003, xiv+305 pp. |
[2] |
F. Arthur and X. Yan,
A Liouville-type theorem for higher order elliptic systems of Hénon-Lane-Emden type, Commun. Pure Appl. Anal., 15 (2016), 807-830.
doi: 10.3934/cpaa.2016.15.807. |
[3] |
V. Benci and P. H. Rabinowitz,
Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.
doi: 10.1007/BF01389883. |
[4] |
D. Bonheure, E. M. dos Santos and H. Tavares,
Hamiltonian elliptic systems: A guide to variational frameworks, Port. Math, 71 (2014), 301-395.
doi: 10.4171/PM/1954. |
[5] |
G. Caristi, L. D'Ambrosio and E. Mitidieri,
Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.
doi: 10.1007/s00032-008-0090-3. |
[6] |
D. Cassani, B. Kaltenbacher and A. Lorenzi, Direct and inverse problems related to MEMS Inverse Problems, 25 (2009), 105002 (22 pp.)
doi: 10.1088/0266-5611/25/10/105002. |
[7] |
D. Cassani and C. Tarsi,
Existence of solitary waves for supercritical Schrödinger systems in dimension two, Calc. Var. Partial Differential Equations, 54 (2015), 1673-1704.
doi: 10.1007/s00526-015-0840-3. |
[8] |
D. Cassani and J. Zhang,
A priori estimates and positivity for semiclassical ground states for systems of critical Schrödinger equations in dimension two, Comm. Partial Differential Equations, 42 (2017), 655-702.
doi: 10.1080/03605302.2017.1295062. |
[9] |
P. Clément, P. L. Felmer and E. Mitidieri,
Homoclinic orbits for a class of infinite-dimensional Hamiltonian systems, Scuola Norm. Sup. Pisa Cl. Sci.(4), 24 (1997), 367-393.
|
[10] |
L. C. Evans,
Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, Providence, 2010, xxii+749 pp.
doi: 10.1090/gsm/019. |
[11] |
P. L. Felmer,
Periodic solutions of "superquadratic" Hamiltonian systems, J. Differential Equations, 102 (1993), 188-207.
doi: 10.1006/jdeq.1993.1027. |
[12] |
D. G. de Figueiredo and P. L. Felmer,
On superquadratic elliptic systems, Trans. Amer. Math. Soc., 343 (1994), 99-116.
doi: 10.1090/S0002-9947-1994-1214781-2. |
[13] |
F. Gazzola, H. C. Grunau, and G. Sweers,
Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Springer-Verlag, Berlin, 2010, xviii+423 pp.
doi: 10.1007/978-3-642-12245-3. |
[14] |
J. Hulshof and R. Van der Vorst,
Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58.
doi: 10.1006/jfan.1993.1062. |
[15] |
J. L. Lions,
Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs, J. Math. Soc. Japan, 14 (1962), 233-241.
doi: 10.2969/jmsj/01420233. |
[16] |
J. L. Lions and E. Magenes,
Non-homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York-Heidelberg, 1972, xvi+357 pp. |
[17] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[18] |
J. Liu, Y. Guo and Y. Zhang,
Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270.
|
[19] |
J. Liu, Y. Guo and Y. Zhang,
Liouville-type theorems for polyharmonic systems in $\mathbb{R}^N$, J. Differential Equations, 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[20] |
E. Mitidieri,
A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[21] |
E. Mitidieri and S. I. Pohozaev,
A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
|
[22] |
A. Persson,
Compact linear mappings between interpolation spaces, Ark. Mat., 5 (1964), 215-219.
doi: 10.1007/BF02591123. |
[23] |
P. Poláčik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems, Ⅰ: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[24] |
P. Pucci and J. Serrin,
A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[25] |
B. Ruf, Lorentz spaces and nonlinear elliptic systems, in Contributions to Nonlinear Analysis,
Vol. 66, Birkhäuser, Basel, (2006), 471–489.
doi: 10.1007/3-7643-7401-2_32. |
[26] |
B. Ruf, Superlinear elliptic equations and systems, in Handbook of Differential Equations:
Stationary Partial Differential Equations, Vol. 5, Elsevier, (2008), 211–276.
doi: 10.1016/S1874-5733(08)80010-1. |
[27] |
E. M. dos Santos,
Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation, J. Math. Anal. Appl., 342 (2008), 277-297.
doi: 10.1016/j.jmaa.2007.11.056. |
[28] |
D. Schiera,
Existence of solutions to higher order Lane-Emden type systems, Nonlinear Anal., 168 (2018), 130-153.
doi: 10.1016/j.na.2017.11.011. |
[29] |
B. Sirakov,
Existence results and a priori bounds for higher order elliptic equations and systems, J. Math. Pures Appl.(9), 89 (2008), 114-133.
doi: 10.1016/j.matpur.2007.10.003. |
[30] |
B. Sirakov,
On the existence of solutions of Hamiltonian elliptic systems in $\mathbb{R}^N$, Adv. Differential Equations, 5 (2000), 1445-1464.
|
[31] |
P. Souplet,
The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[32] |
M. Struwe,
Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4$^{th}$ edition, Springer-Verlag, Berlin, 2008, xx+302 pp. |
[33] |
Y. Zhang,
A Liouville type theorem for polyharmonic elliptic systems, J. Math. Anal. Appl., 326 (2007), 677-690.
doi: 10.1016/j.jmaa.2006.03.027. |

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