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May  2020, 13(5): 1601-1618. doi: 10.3934/dcdss.2020091

The hypoelliptic Robin problem for quasilinear elliptic equations

Kazuaki Taira

Institute of Mathematics, University of Tsukuba, Tsukuba 305–8571, Japan

Dedicated to Professor Angelo Favini on the occasion of his 70th birthday

Received  November 2017 Revised  May 2018 Published  June 2019

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.

Keywords: Quasilinear elliptic equation, hypoelliptic Robin problem, Nagumo condition, comparison principle, Leray–Schauder fixed point theorem.
Mathematics Subject Classification: Primary: 35J62; Secondary: 35H10, 35R25.
Citation: Kazuaki Taira. The hypoelliptic Robin problem for quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1601-1618. doi: 10.3934/dcdss.2020091
References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Second edition, Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[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.  Google Scholar

[3]

H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295.  doi: 10.1007/BF01221152.  Google Scholar

[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.  Google Scholar

[5]

J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris, Sér. A., 265 (1967), 333-336.   Google Scholar

[6]

K. C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. Taira, The Yamabe problem and nonlinear boundary value problems, J. Differential Equations, 122 (1995), 316-372.  doi: 10.1006/jdeq.1995.1151.  Google Scholar

[18]

K. Taira, Boundary value problems for elliptic integro-differential operators, Math. Z., 222 (1996), 305-327.  doi: 10.1007/BF02621868.  Google Scholar

[19]

K. Taira, Existence and uniqueness theorems for semilinear elliptic boundary value problems, Adv. Differential Equations, 2 (1997), 509-534.   Google Scholar

[20]

K. Taira, Bifurcation theory for semilinear elliptic boundary value problems, Hiroshima Math. J., 28 (1998), 261-308.  doi: 10.32917/hmj/1206126761.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[24]

F. Tomi, Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 111 (1969), 350-366.  doi: 10.1007/BF01110746.  Google Scholar

[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.  Google Scholar

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.   Google Scholar
[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.  Google Scholar

[3]

H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150 (1976), 281-295.  doi: 10.1007/BF01221152.  Google Scholar

[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.  Google Scholar

[5]

J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris, Sér. A., 265 (1967), 333-336.   Google Scholar

[6]

K. C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

K. Taira, The Yamabe problem and nonlinear boundary value problems, J. Differential Equations, 122 (1995), 316-372.  doi: 10.1006/jdeq.1995.1151.  Google Scholar

[18]

K. Taira, Boundary value problems for elliptic integro-differential operators, Math. Z., 222 (1996), 305-327.  doi: 10.1007/BF02621868.  Google Scholar

[19]

K. Taira, Existence and uniqueness theorems for semilinear elliptic boundary value problems, Adv. Differential Equations, 2 (1997), 509-534.   Google Scholar

[20]

K. Taira, Bifurcation theory for semilinear elliptic boundary value problems, Hiroshima Math. J., 28 (1998), 261-308.  doi: 10.32917/hmj/1206126761.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[24]

F. Tomi, Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 111 (1969), 350-366.  doi: 10.1007/BF01110746.  Google Scholar

[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.  Google Scholar
Figure 1.  The unit outward normal $ \mathbf{n} $ and the conormal $ \boldsymbol\nu $ to $ \partial \Omega $
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Figure 2.  The open subset $ \Omega^{+} $ with boundary $ \partial \Omega^{+} $
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