Heisenberg群上带不确定权的p-次Laplacian特征值问题.pdf

第40卷第6期 数 学 进 展 Vo1．40．No．6 2011年12月 ADVANCESIN MATHEMATICS Dec．，2011 EigenvalueProblem softheP-·sub——Laplacian W ith IndefiniteW eighton theHeisenbergGroup DOU Jingbo ． HAN Yazhou ． Schoo1ofStatistics，Xi’anUniversityofFinanceandEconomies，Xi’an，Shaan~，710100，P· R Chin口：2．DepartmentofMathematics CollegeofScience，ChinaJiliangUniversity，Hangzhou， Zh~iang310018．P．R．ChinQ) A bstract：Inthispaperwediscusstheexistenceofsolutionsforeigenvalueproblemsofp- sub—Laplacianwithindefiniteweight0ntheHeisenberggroup，andprovethatthefirsteigenvalue associatedtoanonnegativeeigenfunction issimple，isolated andunique． Keywords：Heisenberggroup；p-sub-L·aplacian；Palais—Smalecondition；simplicity；isola- tion MR(2000)SubjectClassification：35P30；35J35；35J60|CLCnumber：O175．25 Documentcode：A ArticleID：1000．．0917(2011)06--0687-10 0Introduction TheeigenvalueproblemsoftheLaplacianandp-LaplacianintheEuclideanspacehavebeen wide1vstudied[1—61Thesimplicity andisolationconcerningthefirsteigenvaluehavealsobeen ． consideredbymanymathematicians． Recently，eigenvalueproblemsforthesubellipticoperatorshavebeen given attentionby manyscholars．Theauthorsin7【】discussedtheeigenvalueproblemsontheboundeddomainofr thesubelliptieoDeratorconstructedbyH5rmander’Svectorfields．InthecaseoftheHeisenberg groupⅡ ．existenceresultsofthenonlinearsub—Laplacianequationhavebeeninvestigatedbyes— tablishingthetheoryofLeray—SchauderdegreeontheHeisenberggroup[引．Eigenvalueestimates ofrthesub—Laplacianhavebeenstudiedin[9]．ByconsideringanL Hardytypeinequalityon thehomogeneousgrouP，theexistenceoftheindefiniteeigenvalueproblem forthesub—Laplacian andthesimplicityofthefirsteigenvaluewer