|Table of Contents|

The symmetry of solutions for a class of Baouendi-Grushin equations(PDF)


Research Field:
Publishing date:


The symmetry of solutions for a class of Baouendi-Grushin equations
 QIAN Hongli HUANG Xiaotao
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
 Baouendi-Grushin equation existence symmetry constrained minimization method Schwarz rearrangement1 引言及主要结论
O 175.5
To study the symmetry for solutions of a class of Baouendi-Grushin equations,first the symmetry is converted into a constrained minimization problem. Then,by Sobolev embedding theorem and some priori estimates, it is proved that the solutions after Schwarz rearrangement are also the solutions of its Lagrangian minimization functional.Thus the existence and symmetry results of the solutions of Baouendi-Grushin equations.


[1] MOSER J.On Harnack’s theorem for elliptic differential equations[J].Communications on Pure and Applied Mathematics,1961,14(3):577-591.
[2] TRUDINGER N S.On harnack type inequalities and their application to quasilinear elliptic equations[J].Communications on Pure and Applied Mathematics,1967,20(4):721-747.
[3] CAFFARELLI L A.Interior a priori estimates for solutions of fully non-linear equations[J].The Annals of Mathematics,1989,130(1):189.
[4] DIBENEDETTO E.C1+α local regularity of weak solutions of degenerate elliptic equations[J].Nonlinear Analysis:Theory,Methods & Applications,1983,7(8):827-850.
[5] HÖRMANDER L.Hypoelliptic second order differential equations[J].Acta Mathematica,1967,119:147-171.
[6] 韩彦武,钮鹏程.广义Baouendi-Grushin方程的非线性Liouville型定理[J].宁夏大学学报(自然科学版),2003,24(3):229-234. HAN Y W,NIU P C.Liouville type theorems to semilinear generalized baouendi-grushin equations[J].Journal of Ningxia University(Natural Science Edition),2003,24(3):229-234.(in Chinese)
[7] JERISON D,LEE J M.The Yamabe problem on CR manifolds[J].Journal of Differential Geometry,1987,25(2):167-197.
[8] FRANCHI B.Weighted Sobolev-Poincar? inequalities and pointwise estimates for a class of degenerate elliptic equations[J].Transactions of the American Mathematical Society,1991,327(1):125-158.
[9] WANG L H.Hölder estimates for subelliptic operators[J].Journal of Functional Analysis,2003,199(1):228-242.
[10] MONTI R,MORBIDELLI D.Kelvin transform for Grushin operators and critical semilinear equations[J].Duke Mathematical Journal,2006,131(1):167-202.
[11] ALEXANDROV A D.A characteristic property of spheres[J].Annali Di Matematica Pura Ed Applicata,Series 4,1962,58(1):303-315.
[12] SERRIN J.A symmetry problem in potential theory[J].Archive for Rational Mechanics and Analysis,1971,43(4):304-318.
[13] GIDAS B,NI W M,NIRENBERG L.Symmetry and related properties via the maximum principle[J].Communications in Mathematical Physics,1979,68(3):209-243.
[14] CAFFARELLI L A,GIDAS B,SPRUCK J.Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[J].Communications on Pure and Applied Mathematics,1989,42(3):271-297.
[15] BERESTYCKI H,LIONS P L.Nonlinear scalar field equations,II existence of infinitely many solutions[J].Archive for Rational Mechanics and Analysis,1983,82(4):347-375.
[16] DIPIERRO S,PALATUCCI G,VALDINOCI E.Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian[J].Le Matematiche,2012(68):201-216.
[17] XIE L L,HUANG X T,WANG L H.Radial symmetry for positive solutions of fractional p-Laplacian equations via constrained minimization method[J].Applied Mathematics and Computation,2018,337:54-62.
[18] SONG Q Z,WANG L H,LI D S.Hölder estimates for a class of degenerate elliptic equations[J].Acta Mathematica Scientia,2013,33(4):1202-1218.
[19] FRANCHI B,LANCONELLI E.An embedding theorem for Sobolev spaces related to non-smooth vector fieldsand harnack inequality[J].Communications in Partial Differential Equations,1984,9(13):1237-1264.
[20] KAWOHL B.Rearrangements and convexity of level sets in PDE[M].Heidelberg:Springer,1985.


Last Update: 2019-10-07