一类p-Laplacian方程解的存在性与对称性研究

Abstract

Abstract:

The purpose of this paper is to investigate a spacial p-Laplace equation involving Baouendi-Grushin operators. Some existence and symmetry results for positive solutions are obtained by rearrangement of its corresponding constrained minimization. These results are in accordance with those for the classical p-Laplace equations and the Baouendi-Grushin type sub-Laplacian equations.

Key words: p-Laplace equations, Baouendi-Grushin operators, Existence and symmetry, rearrangement, Constrained minimization method

CLC Number:

O175.5

Hongli Qian,Xiaotao Huang. A Symmetry Result for a Class of p-Laplace Involving Baouendi-Grushin Operators via Constrained Minimization Method[J].Acta mathematica scientia,Series A, 2020, 40(3): 725-734.

Trendmd

References 24

1	Ladyzhenskaya A , Ural'tseva N N . Linear and Quasilinear Elliptic Equations. New York: Academic Press, 1968
2	Giorgi E , E D . Sulla differenziabilitae l'analiticitá delle estremali degli integrali multipli regolari. Accad Sci Torino Cl Sci Fis Mat Nat, 1957, 3, 25- 43
3	Serrin J . Local behavior of solutions of quasi-linear equations. Acta Mathematica, 1964, 111 (1): 247- 302
4	Trudinger N S . On Harnack type inequalities and their application to quasilinear elliptic equations. Communications on Pure and Applied Mathematics, 1967, 20 (4): 721- 747 doi: 10.1002/cpa.3160200406
5	Moser J . On Harnack's theorem for elliptic differential equations. Communications on Pure and Applied Mathematics, 1961, 14 (3): 577- 591 doi: 10.1002/cpa.3160140329
6	Auscher P , Qafsaoui M . Equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form. Journal of Functional Analysis, 2000, 177 (2): 310- 364 doi: 10.1006/jfan.2000.3643
7	Byun S S , Wang L . Elliptic equations with BMO coefficients in Reifenberg domains. Communications on Pure and Applied Mathematics, 2004, 57 (10): 1283- 1310 doi: 10.1002/cpa.20037
8	Kombe I . Nonlinear degenerate parabolic equations for Baouendi-Grushin operators. Mathematische Nachrichten, 2006, 279 (7): 756- 773 doi: 10.1002/mana.200310391
9	Jerison D , Lee J M . The Yamabe problem on CR manifolds. Journal of Differential Geometry, 1987, 25 (2): 167- 197 doi: 10.4310/jdg/1214440849
10	Monti R , Morbidelli D . Kelvin transform for Grushin operators and critical semilinear equations. Duke Mathematical Journal, 2006, 131 (1): 167- 202 doi: 10.1215/S0012-7094-05-13115-5
11	Ferrari F , Valdinoci E . Geometric PDEs in the Grushin plane:weighted inequalities and flatness of level sets. International Mathematics Research Notices, 2009, 2009 (22): 4232- 4270
12	Birindelli I , Ferrari F , Valdinoci E . Semilinear PDEs in the Heisenberg group:The role of the right invariant vector fields. Nonlinear Anal, 2010, 72 (2): 987- 997 doi: 10.1016/j.na.2009.07.039
13	Birindelli I , Valdinoci E . On the Allen-Cahn equation in the Grushin plane:A monotone entire solution that is not one-dimensional. Discrete Continuous Dynamical Systems, 2012, 29 (3): 823- 838 doi: 10.3934/dcds.2011.29.823
14	Franchi B . Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations. Transactions of the American Mathematical Society, 1991, 327 (1): 125- 158 doi: 10.2307/2001837
15	Wang L . Hölder estimates for subelliptic operators. Journal of Functional Analysis, 2003, 199 (1): 228- 242 doi: 10.1016/S0022-1236(03)00093-4
16	Song Q , Lu Y , Shen J , et al. Regularity of a class of degenerate elliptic equations. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 2011, 10 (3): 645- 667
17	Huang X , Ma F , Wang L . L_q regularity for p-Laplace type Baouendi-Grushin equations. Nonlinear Analysis:Theory, Methods Applications, 2015, 113, 137- 146 doi: 10.1016/j.na.2014.10.001
18	Lascialfari F , Pardo D . Compact embedding of a degenerate Sobolev space and existence of entire solutions to a semilinear equation for a Grushin-type operator. Rend Seminario Mat Univ Padova, 2002, 107, 139- 152
19	Berestycki H , Lions P L . Nonlinear scalar field equations, I:Existence of a ground state. Archive for Rational Mechanics and Analysis, 1983, 82 (4): 313- 345 doi: 10.1007/BF00250555
20	Dipierro S , Palatucci G , Valdinoci E . Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche Catania, 2013, 63, 201- 216
21	Xie L , Huang X , Wang L . Radial symmetry for positive solutions of fractional p-Laplacian equations via constrained minimization method. Applied Mathematics and Computation, 2018, 337, 54- 62 doi: 10.1016/j.amc.2018.05.028
22	Franchi B , Lanconelli E . An embedding theorem for sobolev spaces related to non-smooth vector fieldsand harnack inequality. Communications in Partial Differential Equations, 1984, 9 (13): 1237- 1264 doi: 10.1080/03605308408820362
23	Bandle C . Isoperimetric inequalities and applications. Monographs and Studies in Mathematics, 1980, 2, 453- 488
24	Kawohl B . Rearrangements and Convexity of Level Sets in PDE. Berlin: Springer-Verlag, 1985

[1]	REN Yan-Bo. Some New Isomorphic Characterizations of q-convex Banach Spaces [J]. Acta mathematica scientia,Series A, 2012, 32(6): 1086-1091.
[2]	WANG Hui-Jian, SUN Dao-Chun. B-valued Bi-random Dirichlet Series [J]. Acta mathematica scientia,Series A, 2010, 30(6): 1604-1611.
[3]	ZHANG Chuan-Zhou, ZHANG Xue-Ying. (C, α)-summability of Character System of p}-series Field in Kaczmarz Rearrangement [J]. Acta mathematica scientia,Series A, 2009, 29(6): 1623-1633.

A Symmetry Result for a Class of p-Laplace Involving Baouendi-Grushin Operators via Constrained Minimization Method

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 24

Related Articles 3

Metrics

Comments

Recommended 10