
-
Previous Article
Vector-valued Schrödinger operators in Lp-spaces
- DCDS-S Home
- This Issue
-
Next Article
Bifurcation revisited along footprints of Jürgen Scheurle
The hypoelliptic Robin problem for quasilinear elliptic equations
Institute of Mathematics, University of Tsukuba, Tsukuba 305–8571, Japan |
This paper is devoted to the study of a hypoelliptic Robin boundary value problem for quasilinear, second-order elliptic differential equations depending nonlinearly on the gradient. More precisely, we prove an existence and uniqueness theorem for the quasilinear hypoelliptic Robin problem in the framework of Hölder spaces under the quadratic gradient growth condition on the nonlinear term. The proof is based on the comparison principle for quasilinear problems and the Leray–Schauder fixed point theorem. Our result extends earlier theorems due to Nagumo, Akô and Schmitt to the hypoelliptic Robin case which includes as particular cases the Dirichlet, Neumann and regular Robin problems.
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Second edition, Elsevier/Academic Press, Amsterdam, 2003.
![]() |
[2] |
K. Akô,
On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan, 13 (1961), 45-62.
doi: 10.2969/jmsj/01310045. |
[3] |
H. Amann,
Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295.
doi: 10.1007/BF01221152. |
[4] |
H. Amann and M. G. Crandall,
On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J., 27 (1978), 779-790.
doi: 10.1512/iumj.1978.27.27050. |
[5] |
J.-M. Bony,
Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris, Sér. A., 265 (1967), 333-336.
|
[6] |
K. C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005. |
[7] |
M. A. del Pino,
Positive solutions of a semilinear equation on a compact manifold, Nonlinear Analysis TMA, 22 (1994), 1423-1430.
doi: 10.1016/0362-546X(94)90121-X. |
[8] |
P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Second edition, Birkhäuser/Springer Basel AG, Basel, 2013.
doi: 10.1007/978-3-0348-0387-8. |
[9] |
A. Friedman, Partial Differential Equations, Dover Publications Inc., Mineola, New York, 1969/2008. Google Scholar |
[10] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Reprint of the 1998 edition, Springer-Verlag, New York Berlin Heidelberg Tokyo, 2001. |
[11] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc., Academic Press, New York London, 1968.
![]() |
[12] |
J. M. Lee and T. H. Parker,
The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91.
doi: 10.1090/S0273-0979-1987-15514-5. |
[13] |
M. Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J., 6 (1954), 207–229. https://projecteuclid.org/euclid.ojm/1200688553. |
[14] |
T.-C. Ouyang,
On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^{p} = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527.
doi: 10.2307/2154124. |
[15] |
T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 3, Walter de Gruyter & Co., Berlin New York, 1996.
doi: 10.1515/9783110812411. |
[16] |
K. Schmitt,
Boundary value problems for quasilinear second-order elliptic equations, Nonlinear Anal. TMA, 2 (1978), 263-309.
doi: 10.1016/0362-546X(78)90019-6. |
[17] |
K. Taira,
The Yamabe problem and nonlinear boundary value problems, J. Differential Equations, 122 (1995), 316-372.
doi: 10.1006/jdeq.1995.1151. |
[18] |
K. Taira,
Boundary value problems for elliptic integro-differential operators, Math. Z., 222 (1996), 305-327.
doi: 10.1007/BF02621868. |
[19] |
K. Taira,
Existence and uniqueness theorems for semilinear elliptic boundary value problems, Adv. Differential Equations, 2 (1997), 509-534.
|
[20] |
K. Taira,
Bifurcation theory for semilinear elliptic boundary value problems, Hiroshima Math. J., 28 (1998), 261-308.
doi: 10.32917/hmj/1206126761. |
[21] |
K. Taira, Semigroups, Boundary Value Problems and Markov Processes, Springer Monogr. Math., Second edition, Springer-Verlag, Berlin Heidelberg New York, 2014.
doi: 10.1007/978-3-662-43696-7. |
[22] |
K. Taira, Analytic Semigroups and Semilinear Initial-Boundary Value Problems, London Mathematical Society Lecture Note Series, Vol. 434, Second edition, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316729755.![]() ![]() |
[23] |
K. Taira, D. K. Palagachev and P. R. Popivanov,
A degenerate Neumann problem for quasilinear elliptic equations, Tokyo J. Math., 23 (2000), 227-234.
doi: 10.3836/tjm/1255958817. |
[24] |
F. Tomi,
Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 111 (1969), 350-366.
doi: 10.1007/BF01110746. |
[25] |
G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, The University Series in Mathematics, Plenum Press, New York, 1987.
doi: 10.1007/978-1-4899-3614-1.![]() ![]() |
show all references
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Second edition, Elsevier/Academic Press, Amsterdam, 2003.
![]() |
[2] |
K. Akô,
On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan, 13 (1961), 45-62.
doi: 10.2969/jmsj/01310045. |
[3] |
H. Amann,
Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295.
doi: 10.1007/BF01221152. |
[4] |
H. Amann and M. G. Crandall,
On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J., 27 (1978), 779-790.
doi: 10.1512/iumj.1978.27.27050. |
[5] |
J.-M. Bony,
Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris, Sér. A., 265 (1967), 333-336.
|
[6] |
K. C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005. |
[7] |
M. A. del Pino,
Positive solutions of a semilinear equation on a compact manifold, Nonlinear Analysis TMA, 22 (1994), 1423-1430.
doi: 10.1016/0362-546X(94)90121-X. |
[8] |
P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Second edition, Birkhäuser/Springer Basel AG, Basel, 2013.
doi: 10.1007/978-3-0348-0387-8. |
[9] |
A. Friedman, Partial Differential Equations, Dover Publications Inc., Mineola, New York, 1969/2008. Google Scholar |
[10] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Reprint of the 1998 edition, Springer-Verlag, New York Berlin Heidelberg Tokyo, 2001. |
[11] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc., Academic Press, New York London, 1968.
![]() |
[12] |
J. M. Lee and T. H. Parker,
The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91.
doi: 10.1090/S0273-0979-1987-15514-5. |
[13] |
M. Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J., 6 (1954), 207–229. https://projecteuclid.org/euclid.ojm/1200688553. |
[14] |
T.-C. Ouyang,
On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^{p} = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527.
doi: 10.2307/2154124. |
[15] |
T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 3, Walter de Gruyter & Co., Berlin New York, 1996.
doi: 10.1515/9783110812411. |
[16] |
K. Schmitt,
Boundary value problems for quasilinear second-order elliptic equations, Nonlinear Anal. TMA, 2 (1978), 263-309.
doi: 10.1016/0362-546X(78)90019-6. |
[17] |
K. Taira,
The Yamabe problem and nonlinear boundary value problems, J. Differential Equations, 122 (1995), 316-372.
doi: 10.1006/jdeq.1995.1151. |
[18] |
K. Taira,
Boundary value problems for elliptic integro-differential operators, Math. Z., 222 (1996), 305-327.
doi: 10.1007/BF02621868. |
[19] |
K. Taira,
Existence and uniqueness theorems for semilinear elliptic boundary value problems, Adv. Differential Equations, 2 (1997), 509-534.
|
[20] |
K. Taira,
Bifurcation theory for semilinear elliptic boundary value problems, Hiroshima Math. J., 28 (1998), 261-308.
doi: 10.32917/hmj/1206126761. |
[21] |
K. Taira, Semigroups, Boundary Value Problems and Markov Processes, Springer Monogr. Math., Second edition, Springer-Verlag, Berlin Heidelberg New York, 2014.
doi: 10.1007/978-3-662-43696-7. |
[22] |
K. Taira, Analytic Semigroups and Semilinear Initial-Boundary Value Problems, London Mathematical Society Lecture Note Series, Vol. 434, Second edition, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316729755.![]() ![]() |
[23] |
K. Taira, D. K. Palagachev and P. R. Popivanov,
A degenerate Neumann problem for quasilinear elliptic equations, Tokyo J. Math., 23 (2000), 227-234.
doi: 10.3836/tjm/1255958817. |
[24] |
F. Tomi,
Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 111 (1969), 350-366.
doi: 10.1007/BF01110746. |
[25] |
G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, The University Series in Mathematics, Plenum Press, New York, 1987.
doi: 10.1007/978-1-4899-3614-1.![]() ![]() |

[1] |
Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747 |
[2] |
Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775 |
[3] |
Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 |
[4] |
Xiaowei Tang, Xilin Fu. New comparison principle with Razumikhin condition for impulsive infinite delay differential systems. Conference Publications, 2009, 2009 (Special) : 739-743. doi: 10.3934/proc.2009.2009.739 |
[5] |
Shigeaki Koike, Takahiro Kosugi. Remarks on the comparison principle for quasilinear PDE with no zeroth order terms. Communications on Pure & Applied Analysis, 2015, 14 (1) : 133-142. doi: 10.3934/cpaa.2015.14.133 |
[6] |
Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381 |
[7] |
Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks & Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361 |
[8] |
Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897 |
[9] |
Giuseppe Riey. Regularity and weak comparison principles for double phase quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4863-4873. doi: 10.3934/dcds.2019198 |
[10] |
Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026 |
[11] |
VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 |
[12] |
Y. Kabeya. Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 117-134. doi: 10.3934/dcds.2006.14.117 |
[13] |
Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393 |
[14] |
Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363 |
[15] |
Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 |
[16] |
Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 |
[17] |
Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549 |
[18] |
Raffaela Capitanelli. Robin boundary condition on scale irregular fractals. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1221-1234. doi: 10.3934/cpaa.2010.9.1221 |
[19] |
Adriana C. Briozzo, María F. Natale, Domingo A. Tarzia. The Stefan problem with temperature-dependent thermal conductivity and a convective term with a convective condition at the fixed face. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1209-1220. doi: 10.3934/cpaa.2010.9.1209 |
[20] |
Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297 |
2018 Impact Factor: 0.545
Tools
Metrics
Other articles
by authors
[Back to Top]