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Keywords:
weak comparison principle; quasilinear elliptic operator; $p$-Laplace operator
weak comparison principle; quasilinear elliptic operator; $p$-Laplace operator
Summary:
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 $a\colon \Omega \times \Bbb R^N \rightarrow \Bbb R^N$ 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.
References:
[1] Boccardo, L., Croce, G.: 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] Brezis, H.: 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] Chipot, M.: 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] Damascelli, L.: 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] D'Ambrosio, L., Farina, A., Mitidieri, E., Serrin, J.: 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] D'Ambrosio, L., Mitidieri, E.: 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] Mitrović, D., Žubrinić, D.: 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] Motreanu, D., Motreanu, V. V., Papageorgiou, N.: 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] Tolksdorf, P.: 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] Unai, A.: Sub- and super-solutions method for some quasilinear elliptic operators. Far East J. Math. Sci. (FJMS) 99 (2016), 851-867. DOI 10.17654/MS099060851 | MR 3842964 | Zbl 06627771
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