EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH

doi:10.1016/S0252-9602(17)30077-2

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (5): 1348-1360.doi: 10.1016/S0252-9602(17)30077-2

EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH

章国庆¹, 张卫国¹, 刘三阳²

1. College of Sciences, University of Shanghai for Science and Technology, Shanghai 200093, China;
2. College of Mathematics and Statistics, Xidian University, Xi'an 710071, China

收稿日期:2016-02-29 修回日期:2017-04-28 出版日期:2017-10-25 发布日期:2017-10-25
作者简介:Guoqing ZHANG,E-mail:shzhangguoqing@126.com;Weiguo ZHANG,E-mail:zwgzwm@126.com;Sanyang LIU,E-mail:liusanyang@126.com
基金资助:
Supported by Shanghai Natural Science Foundation (15ZR1429500) and NNSF of China (11471215).

EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH

Guoqing ZHANG¹, Weiguo ZHANG¹, Sanyang LIU²

1. College of Sciences, University of Shanghai for Science and Technology, Shanghai 200093, China;
2. College of Mathematics and Statistics, Xidian University, Xi'an 710071, China

Received:2016-02-29 Revised:2017-04-28 Online:2017-10-25 Published:2017-10-25
Supported by:
Supported by Shanghai Natural Science Foundation (15ZR1429500) and NNSF of China (11471215).

摘要/Abstract

摘要：

In this paper,we consider a class of N-Laplacian equations involving critical growth

where Ω is a bounded domain with smooth boundary in RN (N > 2),f (x,u) is of critical growth.Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti,we prove the existence of a nontrivial solution for any λ > λ₁,λ_l=λ_l(l=2,3,…),and λ_l is the eigenvalues of the operator (-△_N,W₀^1,N(Ω)), which is defined by the Z₂-cohomological index.

关键词: nonstandard linking theorem, N-Laplacian equation, critical growth

Abstract:

In this paper,we consider a class of N-Laplacian equations involving critical growth

where Ω is a bounded domain with smooth boundary in RN (N > 2),f (x,u) is of critical growth.Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti,we prove the existence of a nontrivial solution for any λ > λ₁,λ_l=λ_l(l=2,3,…),and λ_l is the eigenvalues of the operator (-△_N,W₀^1,N(Ω)), which is defined by the Z₂-cohomological index.

Key words: nonstandard linking theorem, N-Laplacian equation, critical growth

章国庆, 张卫国, 刘三阳. EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH[J]. 数学物理学报(英文版), 2017, 37(5): 1348-1360.

Guoqing ZHANG, Weiguo ZHANG, Sanyang LIU. EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH[J]. Acta mathematica scientia,Series B, 2017, 37(5): 1348-1360.

参考文献

[1] Adimurthi. Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian. Ann Scuola Norm Sup Pisa, 1990, 17:393-413
[2] Adimurthi, Sandeep K. A singular Moser-Trudinger embedding and its applications. Nonl Differ Equ Appl, 2007, 13:583-603
[3] Adimurthi, Struwe M. Global compactness properties of semilinear elliptic equations with critical exponential growth. J Funct Anal, 2000, 175:125-167
[4] Degiovanni M, Lancelotti S. Linking over cones and nontrival solutions for p-Laplace equations with psuperlinear nonlinearity. Ann Inst H Pioncaré Anal Non Linéaire, 2007, 24:907-919
[5] Degiovanni M, Lancelotti S. Linking solutions for p-Laplace equations with nonlinearity at critical growth. J Funct Anal, 2009, 256:3643-3659
[6] Degiovanni M, Magrone P. Linking solutions for quasilinear equations at critical growth involving the "1-Laplace" operator. Calc Var Partial Diff Equ, 2009, 36:591-609
[7] de Souza M, do Ó J M. On a singular and nonhomogeneous N-Laplacian equation involving critical growth. J Math Anal Appl, 2011, 380:241-263
[8] Di Bendetto E. C^1,α local regularity of weak solutions of degenerate elliptic equtions. Nonlinear Anal, 1983, 7:486-490
[9] do Ó J M. Semilinear Dirichlet problem for the N-Laplacian in R^N with nonlinearities in the critical growth range. Diff Integral Equ, 1996, 9:967-979
[10] Drábek P, Robinson S B. Resonance problems for the p-Laplacian. J Funct Anal, 1999, 169:189-200
[11] Fadell E R, Rabinowitz P H. Generalized cohomological index theoreies for Lie group actions with an application to bifurcation question for Hamiltonian systems. Invent Math, 1978, 45:139-174
[12] Guedda M, Veron L. Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal, 1989, 13:879-902
[13] Heinonen J. Lectures on Analysis on Metric Spaces. New York:Springer-Verlag, 2001
[14] Lindqvist P. On the equation div(|▽u|^p-2|▽u|)+λ|u|^p-2u=0. Proc Amer Math Soc, 1990, 109:157-164
[15] Lions P L. The concentration-compactness principle in the Calculus of Variations, The limit case, Part I. Rev Mat Iberoam, 1985, 1:145-201
[16] Moser J. A sharp form of an inequality by N. Trudinger. Indiana Univ Math, 1971, 20:1077-1092
[17] Panda P. On semilinear Neumann problems with critical growth for the N-Laplacian. Nonlinear Anal, 1996, 26:1347-1366
[18] Perera K, Agarwal R P, O'Regan D. Morse Theoretic Aspects of p-Laplacian Type Operators. Providence, RI:Amer Math Soc, 2010
[19] Perera K, Szulkin A. p-Laplacian problems where the nonlinearity acrosses an eigenvalue. Discrete Contin Dyn Syst, 2005, 13:743-753
[20] Trudinger N S. On imbedding onto Orlicz spaces and some applications. J Math Mech, 1967, 17:473-484
[21] Wang Z, Zhou H. Solutions for a nonhomogeneous elliptic problem involving critical Sobolev-Hardy exponent in R^N. Acta Math Sci, 2006, 26B(3):525-536
[22] Zhang G, Shao J, Liu S. Linking Solutions for N-Laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Comm Pure Appl Anal, 2011, 10:571-581

EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH

EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH

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