Existence and non-existence results for variational higher order elliptic systems

Delia Schiera

doi:10.3934/dcds.2018227

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October 2018, 38(10): 5145-5161. doi: 10.3934/dcds.2018227

Existence and non-existence results for variational higher order elliptic systems

Delia Schiera

Università degli Studi dell'Insubria, Dipartimento di Scienza e Alta Tecnologia, Via Valleggio 11, Como, 22100, Italy

Received January 2018 Revised May 2018 Published July 2018

Let

$α ∈ \mathbb{N}$

$α ≥ 1$

and

$(-Δ)^{α} = -Δ((-Δ)^{α-1})$

be the polyharmonic operator. We prove existence and non-existence results for the following Hamiltonian systems of polyharmonic equations under Dirichlet boundary conditions

$\begin{cases}\begin{aligned}(-Δ)^{α} u = H_v(u, v) \\(-Δ)^{α} v = H_u(u, v) \\\end{aligned} \text{ in } Ω \subset \mathbb{R}^N \\\frac{\partial^{r} u}{\partial ν^{r}} = 0, \, r = 0, \dots, α-1, \text{ on } \partial Ω \\\frac{\partial^{r} v}{\partial ν^{r}} = 0, \, r = 0, \dots, α-1, \text{ on } \partial Ω\end{cases}$

where

$Ω$

is a sufficiently smooth bounded domain,

$N >2α$

$ν$

is the outward pointing normal to

$\partial Ω$

and the Hamiltonian

$H ∈ C^1 (\mathbb{R}^2; \mathbb{R})$

satisfies suitable growth assumptions.

Keywords: Polyharmonic operators, Lane-Emden systems, higher order Dirichlet boundary conditions, Hamiltonian systems, topological methods.

Mathematics Subject Classification: 35J48, 35J58, 35J50.

Citation: Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227

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.

Figure 1. The light grey region represents values of

$p, q$

for which we prove existence of solutions to 14, whereas the dark grey region is the domain of non-existence given by Corollary 2. The curve

$l_1$

is $\frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2\alpha}{N}$, whereas

$l_2$

is given by

$pq = 1$

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