YAN Huimin,XIE Junhui.A priori estimate of positive solutions for a class of p-Laplace equations with Hardy term[J].Journal of Yanbian University,2024,(01):23-30.
一类带Hardy项的p-Laplace方程正解的先验估计
- Title:
- A priori estimate of positive solutions for a class of p-Laplace equations with Hardy term
- 文章编号:
- 1004-4353(2024)01-0023-08
- 关键词:
- p-Laplace方程; Hardy项; 先验估计; Doubling引理; Liouville定理
- 分类号:
- O175.25
- 文献标志码:
- A
- 摘要:
- 研究了一类带Hardy项椭圆型p-Laplace方程??pu=+h(x,u,?u)解的先验估计,其中3<<pN,|x|auqp?1<q<(N?a)(p?1)Np?,0<<a2.在假设h(x,u,?u)满足一定的条件下,首先运用Doubling引理证明了解的一个衰减估计,然后运用blowup技巧并结合Liouville定理证明了非负解的先验估计.
- Abstract:
- This paper is devoted to studying a priori estimate of a elliptic p-Laplacian equation ??pu= +h(x,u,?u)|x|a uq with Hardy term,where 3< <p N, p?1<q<(N?a)( p?1) N p? , 0< <a 2. Under suitable conditions of h ( x, u,?u ) , firstly,we prove the decay estimates of solutions by Doubling lemma,then combining blow up technique with Liouville theorem,we also prove a priori estimate for the positive solutions.
参考文献/References:
[1] AZIZIEH C,CLEMENT P. A priori estimates and continuation methods for positive solutions of p-Laplace equations[J]. Journal of Differential Equations,2002,179(1):213-245.
[2] VETOIS J. A priori estimates and application to the symmetry of solutions for critical p-Laplace equations[J]. Journal of Differential Equations,2016,260(1):149-161.
[3] SUN X Q,BAO J G. Pointwise a priori estimates for solutions to some p-Laplacian equations[J]. Acta Mathematica Sinica,English Series,2022,38(12):2150-2162.
[4] AGHAJANI A,MOTTAGHI S F. A priori estimates for semistable solutions of p-Laplace equations with general nonlinearity[J]. Asymptotic Analysis,2020,122(7):1-12.
[5] CHEN Z. A priori bounds and existence of non-negative solutions to a quasi-linear Schrodinger equation involving p-Laplacian[J]. Annals of Functional Analysis,2023,14(1):24.
[6] CHO K,CHOE H J. Nonlinear degenerate elliptic partial differential equations with critical growth conditions on the gradient[J]. Proceeding of the American Mathematical Society,1995,123(12):3789-3796.
[7] ITURRIAGA L,LORCA S,SANCHEZ J. Existence and multiplicity results for the p-Laplacian with a p-gradient term[J]. Nonlinear Differential Equations and Applications,2008,15(6):729-743.
[8] 李振杰,张正策. 拟线性椭圆型方程与方程组非负解的一个先验估计[J]. 应用数学学报,2014,37(2):379-384.
[9] LI J,YIN J,KE Y. Existence of positive solutions for the p-Laplacian with p-gradient term[J]. Journal of Mathematical Analysis and Applications,2011,383(1):147-158.
[10] LI P,XIE J. A priori bounds and existence of positive solutions to a p-Kirchhoff equations[J]. International Journal of Mathematics,2021,32(11):2150082.
[11] GHOUSSOUB N,YUAN C. Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents[J]. Transactions of the American Mathematical Society,2000,352(12):5703-5743.
[12] KANG D S. On the quasilinear elliptic problem with critical Sobolev-Hardy exponents and Hardy terms[J]. Nonlinear Analysis,2008,68(7):1973-1985.
[13] GIDAS B,SPRUCK J. A priori bounds for positive solutions of nonlinear elliptic equations[J]. Communications in Partial Differential Equations,1981,6(8):883-901.
[14] PHAN Q H,SOUPLET P. Liouville-type theorems and bounds of solutions of Hardy-Henon equations[J]. Journal of Differential Equations,2012,252(3):2544-2562.
[15] EVANS L C. Partial Differential Equations[M]. Providence,Rhode Island: American Mathematical Society,2010:618.
[16] AZORERO J G,ALONSO I P. Hardy inequalities and some critical elliptic and parabolic problems[J]. Journal of Differential Equations,1998,144(2):441-476.
[17] SERRIN J,ZOU H H. Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[J]. Acta Mathematica,2002,189(1):79-142.
[18] ZOU H H. A priori estimates and existence for quasi-linear elliptic equations[J]. Calculus of Variations and Partial Differential Equations,2008,33(4):417-437.
[19] BIRINDELLI I,DEMENGEL F. Some Liouville theorems for the p-Laplacian[C]// Jerome Busca. 2001-Luminy Conference on Quasilinear Elliptic and Parabolic Equations and Systems. Luminy France: Electronic Journal of Differential Equations,2002:35-46.
[20] POLACIK P,QUITTNER P,SOUPLET P. Singularity and decay estimates in superlinear problems via Liouville-type theorems,I:Elliptic equations and systems[J]. Duke Mathematical Journal,2007,139(3):555-579.
[21] XIAO Y B,KIM J K,HUANG N J. A generalization of Ascoli-Arzela theorem with an application[J]. Nonlinear Functional Analysis & Applications,2006,11(2):305-317.
[22] SAKAGUCHI S. Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems[J]. Annali Della Scuola Normale Superiore di Pisa,Classe di Scienze,1987,14(3):403-421.
备注/Memo
投稿日期:2023-09-22
基金项目:国家自然科学基金(11761030);湖北省自然科学基金(2022CFC016)第一作者:闫慧敏(1999— ),女,硕士研究生,研究方向为偏微分方程理论及其应用.
通信作者:谢君辉(1984— ),女,博士,副教授,研究方向为偏微分方程理论及其应用.