Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary

S. Bandyopadhyay; M. Chhetri; B. B. Delgado; N. Mavinga; R. Pardo; S. Bandyopadhyay; M. Chhetri; B. B. Delgado; N. Mavinga; R. Pardo

doi:10.3934/era.2022107

Electronic Research Archive

2022, Volume 30, Issue 6: 2121-2137. doi: 10.3934/era.2022107

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Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary

1.
UNC Greensboro, Greensboro, NC 27402, USA
2.
Universidad Autónoma de Aguascalientes, 20131 Aguascalientes, Mexico
3.
Swarthmore College, Swarthmore, PA 19081, USA
4.
Universidad Complutense de Madrid, 28040 Madrid, Spain

Received: 14 October 2021 Revised: 29 December 2021 Accepted: 16 January 2022 Published: 14 April 2022

This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.
- elliptic problem,
- nonlinear boundary conditions,
- maximal and minimal weak solution,
- pseudomonotone operator,
- Kato's inequality
Citation: S. Bandyopadhyay, M. Chhetri, B. B. Delgado, N. Mavinga, R. Pardo. Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary[J]. Electronic Research Archive, 2022, 30(6): 2121-2137. doi: 10.3934/era.2022107

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Abstract

This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.

References

[1]	M. Cuesta Leon, Existence results for quasilinear problems via ordered sub- and supersolutions, Ann. Fac. Sci. Toulouse Math., 6 (1997), 591–608. http://www.numdam.org/item?id=AFST_1997_6_6_4_591_0
[2]	J. Schoenenberger-Deuel, P. Hess, A criterion for the existence of solutions of non-linear elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 74 (1976), 49–54. https://doi-org.proxy.swarthmore.edu/10.1017/s030821050001653x doi: 10.1017/s030821050001653x
[3]	J. M. Arrieta, R. Pardo, A. Rodríguez-Bernal, Infinite resonant solutions and turning points in a problem with unbounded bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2885–2896. https://doi.org/10.1142/S021812741002743X doi: 10.1142/S021812741002743X
[4]	J. M. Arrieta, R. Pardo, A. Rodríguez-Bernal, Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 225–252.
[5]	J. M. Arrieta, R. Pardo, A. Rodríguez-Bernal, Equilibria and global dynamics of a problem with bifurcation from infinity, J. Differ. Equ., 246 (2009), 2055–2080. https://doi.org/10.1016/j.jde.2008.09.002 doi: 10.1016/j.jde.2008.09.002
[6]	P. Liu, J. Shi, Bifurcation of positive solutions to scalar reaction-diffusion equations with nonlinear boundary condition, J. Differ. Equ., 264 (2018), 425–454. https://doi.org/10.1016/j.jde.2017.09.014 doi: 10.1016/j.jde.2017.09.014
[7]	N. Mavinga, Generalized eigenproblem and nonlinear elliptic equations with nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 137–153. https://doi.org/10.1017/S0308210510000065 doi: 10.1017/S0308210510000065
[8]	N. Mavinga, R. Pardo, Bifurcation from infinity for reaction-diffusion equations under nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 649–671. https://doi.org/10.1017/S0308210516000251 doi: 10.1017/S0308210516000251
[9]	C. Morales-Rodrigo, A. Suárez, Some elliptic problems with nonlinear boundary conditions, in Spectral theory and nonlinear analysis with applications to spatial ecology, World Sci. Publ., Hackensack, NJ, 2005,175–199. https://doi.org/10.1142/9789812701589_0009
[10]	J. M. Arrieta, A. N. Carvalho, A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differ. Equ., 156 (1999), 376–406. https://doi.org/10.1006/jdeq.1998.3612 doi: 10.1006/jdeq.1998.3612
[11]	R. S. Cantrell, C. Cosner, On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains, J. Differ. Equ., 231 (2006), 768–804. https://doi.org/10.1016/j.jde.2006.08.018 doi: 10.1016/j.jde.2006.08.018
[12]	A. A. Lacey, J. R. Ockendon, J. Sabina, Multidimensional reaction diffusion equations with nonlinear boundary conditions, SIAM J. Appl. Math., 58 (1998), 1622–1647. https://doi.org/10.1137/S0036139996308121 doi: 10.1137/S0036139996308121
[13]	C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.
[14]	K. Akô, On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan, 13 (1961), 45–62, https://doi.org/10.2969/jmsj/01310045 doi: 10.2969/jmsj/01310045
[15]	H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J., 21 (1971), 125–146, https://doi.org/10.1512/iumj.1971.21.21012 doi: 10.1512/iumj.1971.21.21012
[16]	D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971), 979–1000. https://doi.org/10.1512/iumj.1972.21.21079 doi: 10.1512/iumj.1972.21.21079
[17]	H. Amann, M. G. Crandall, On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J., 27 (1978), 779–790. https://doi.org/10.1512/iumj.1978.27.27050 doi: 10.1512/iumj.1978.27.27050
[18]	E. N. Dancer, G. Sweers, On the existence of a maximal weak solution for a semilinear elliptic equation, Differ. Integral Equ., 2 (1989), 533–540.
[19]	D. Motreanu, A. Sciammetta, E. Tornatore, A sub-supersolution approach for {N}eumann boundary value problems with gradient dependence, Nonlinear Anal. Real World Appl., 54 (2020), 103096. https://doi.org/10.1016/j.nonrwa.2020.103096 doi: 10.1016/j.nonrwa.2020.103096
[20]	H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, in New developments in differential equations (Proc. 2nd Scheveningen Conf., Scheveningen, 1975), 1976, 43–63. North–Holland Math. Studies, Vol. 21.
[21]	R. A. Adams, J. J. F. Fournier, Sobolev spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.
[22]	S. Carl, V. K. Le, D. Motreanu, Nonsmooth variational problems and their inequalities, Springer Monographs in Mathematics, Springer, New York, 2007. https://doi.org/10.1007/978-0-387-46252-3 Comparison principles and applications.
[23]	A. Kufner, O. John, S. Fučík, Function spaces, Noordhoff International Publishing, Leyden; Academia, Prague, 1977, Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis.
[24]	P. Hess, On the solvability of nonlinear elliptic boundary value problems, Indiana Univ. Math. J., 25 (1976), 461–466. https://doi.org/10.1512/iumj.1976.25.25036 doi: 10.1512/iumj.1976.25.25036
[25]	J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.
[26]	H. Brezis, A. C. Ponce, Kato's inequality up to the boundary, Commun. Contemp. Math., 10 (2008), 1217–1241. https://doi.org/10.1142/S0219199708003241 doi: 10.1142/S0219199708003241
[27]	H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.

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