×

Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. (English) Zbl 0911.35009

Summary: We prove some weak and strong comparison theorems for solutions of differential inequalities involving a class of elliptic operators that includes the \(p\)-Laplacian operator. We then use these theorems together with the method of moving planes and the sliding method to get symmetry and monotonicity properties of solutions to quasilinear elliptic equations in bounded domains.

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J70 Degenerate elliptic equations
35B50 Maximum principles in context of PDEs
35J60 Nonlinear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Badiale, M.; Nabana, E., A note on radiality of solutions of \(p\)-laplacian equation, Applicable Anal., Vol. 52, 35-43 (1994) · Zbl 0841.35008
[2] Berestycki, H.; Nirenberg, L., On the method of moving planes and the sliding method, Bol. Soc. Brasileira de Mat. Nova Ser., Vol. 22, 1-37 (1991) · Zbl 0784.35025
[4] Di Benedetto, E., \(C^{1+α}\) local regularity of weak solutions of degenerate elliptic equations, Nonlin. Anal. T.M.A., Vol. 7, 8, 827-850 (1983) · Zbl 0539.35027
[5] Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., Vol. 68, 209-243 (1979) · Zbl 0425.35020
[6] Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order (1983), Springer · Zbl 0691.35001
[8] Guedda, M.; Veron, L., Quasilinear elliptic equations involving critical Sobolev exponents, Nonlin. Anal. T.M.A., Vol. 13, 8, 879-902 (1989) · Zbl 0714.35032
[9] Kesavan, S.; Pacella, F., Symmetry of positive solutions of a quasilinear elliptic equation via isoperimetric inequalities, Applicable Anal., Vol. 54, 27-37 (1994) · Zbl 0833.35040
[10] Tolksdorf, P., On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. in P.D.E., Vol. 8, 7, 773-817 (1983) · Zbl 0515.35024
[11] Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations, J. Diff. Eqns., Vol. 51, 126-150 (1984) · Zbl 0488.35017
[12] Trudinger, N. S., On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. on Pure and Applied Math., Vol. XX, 721-747 (1967) · Zbl 0153.42703
[13] Vazquez, J. L., A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., Vol. 12, 191-202 (1984) · Zbl 0561.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.