Applications of Mathematics, Vol. 63, No. 4, pp. 483-498, 2018


A weak comparison principle for some quasilinear elliptic operators: it compares functions belonging to different spaces

Akihito Unai

Received April 20, 2018.   Published online July 17, 2018.

Abstract:  We shall prove a weak comparison principle for quasilinear elliptic operators $-{\rm div}(a(x,\nabla u))$ that includes the negative $p$-Laplace operator, where $a: \Omega\times\Bbb R^N \rightarrow\Bbb R^N$ satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.
Keywords:  weak comparison principle; quasilinear elliptic operator; $p$-Laplace operator
Classification MSC:  35B51, 35J62, 35J25


References:
[1] L. Boccardo, G. Croce: Elliptic Partial Differential Equations. Existence and Regularity of Distributional Solutions. De Gruyter Studies in Mathematics 55, De Gruyter, Berlin (2013). DOI 10.1515/9783110315424 | MR 3154599 | Zbl 1293.35001
[2] H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2011). DOI 10.1007/978-0-387-70914-7 | MR 2759829 | Zbl 1220.46002
[3] M. Chipot: Elliptic Equations: An Introductory Course. Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, Basel (2009). DOI 10.1007/978-3-7643-9982-5 | MR 2494977 | Zbl 1171.35003
[4] L. Damascelli: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15 (1998), 493-516. DOI 10.1016/S0294-1449(98)80032-2 | MR 1632933 | Zbl 0911.35009
[5] L. D'Ambrosio, A. Farina, E. Mitidieri, J. Serrin: Comparison principles, uniqueness and symmetry results of solutions of quasilinear elliptic equations and inequalities. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 90 (2013), 135-158. DOI 10.1016/j.na.2013.06.004 | MR 3073634 | Zbl 1287.35033
[6] L. D'Ambrosio, E. Mitidieri: A priori estimates and reduction principles for quasilinear elliptic problems and applications. Adv. Differ. Equ. 17 (2012), 935-1000. MR 2985680 | Zbl 1273.35138
[7] D. Mitrović, D. Žubrinić: Fundamentals of Applied Functional Analysis. Distributions - Sobolev Spaces - Nonlinear Elliptic Equations. Pitman Monographs and Surveys in Pure and Applied Mathematics 91, Longman, Harlow (1998). MR 1603811 | Zbl 0901.46001
[8] D. Motreanu, V. V. Motreanu, N. Papageorgiou: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014). DOI 10.1007/978-1-4614-9323-5 | MR 3136201 | Zbl 1292.47001
[9] P. Tolksdorf: On the Dirichlet problem for quasilinear equations in domains with conical boundary points. Commun. Partial Differ. Equations 8 (1983), 773-817. DOI 10.1080/03605308308820285 | MR 0700735 | Zbl 0515.35024
[10] A. Unai: Sub- and super-solutions method for some quasilinear elliptic operators. Far East J. Math. Sci. (FJMS) 99 (2016), 851-867. DOI 10.17654/MS099060851 | Zbl 06627771

Affiliations:   Akihito Unai, Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-0825, Japan, e-mail: unai@rs.kagu.tus.ac.jp


 
PDF available at: