MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHR&Ouml;DINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN RN

Huifang JIA; Gongbao LI

doi:10.1016/S0252-9602(18)30756-2

2018 , Vol. 38 >Issue 2: 391 - 418

DOI: https://doi.org/10.1016/S0252-9602(18)30756-2

Articles

MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHRÖDINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN R^N

Huifang JIA ,
Gongbao LI

Expand

Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received date: 2017-07-17

Revised date: 2017-12-18

Online published: 2018-04-25

Supported by

This work was supported by Natural Science Foundation of China (11371159 and 11771166), Hubei Key Laboratory of Mathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University #IRT_17R46.

Fold

Abstract

In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of Schrödinger-Kirchhoff type
-ε^pM(ε^p-N ∫_R^N|▽u|^p)△_pu + V (x)|u|^p-2u=f(u)
in R^N, where △_p is the p-Laplacian operator, 1< p < N, M:R⁺→ R⁺ and V:R^N → R⁺ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and LyusternikSchnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.

Key words： Schrödinger-Kirchhoff type equation; variational methods; multiple positive solutions; concentrating solution; penalization method

Cite this article

Huifang JIA , Gongbao LI . MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHRÖDINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN R^N[J]. Acta mathematica scientia, Series B, 2018 , 38(2) : 391 -418 . DOI: 10.1016/S0252-9602(18)30756-2

References

[1] Alves C O, Figueiredo G M. Multiplicity and concentration of positive solutions for a class of quasilinear problems via penalization methods. Adv Nonlinear Stud, 2011, 11(2):265-294
[2] Benci V, Cerami G. Multiple positive solutions for some elliptic problems via the Morse theory and the domain topology. Calcular of Variations, 1993, 2(1):29-48
[3] Damascelli L. Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonity results. Annales Institut Henri Poincare Nonlinear Analysis, 1998, 15(4):493-516
[4] del Pino M, Felmer P L. Local mountain pass for semilinear elliptic problems in unbounded domains. Calc Var Partial Differ Equ, 1996, 4:121-137
[5] Figueiredo G M, Santos J R. Multiplicity and concentration behavior of positive solutions for a SchrödingerKirchhoff type problem via penalization method. Esaim Control Optimisation and Calculus of Variations, 2014, 20:389-415
[6] Figueiredo G M, Ikoma N, Júnior J R S. Existence and concentration result for the Kirchhoff type equation with general nonlinearities. Arch Rational Mech Anal, 2014, 213(3):931-979
[7] He X, Zou W. Existence and concentration behavior of positive solutions for a kirchhoff equation in R³. J Differential Equations, 2012, 252(2):1813-1834
[8] He Y, Li G. Standing waves for a class of Kirchhoff type problems in R³ involving critical Sobolev exponents. Calc Var Partial Differential Equations, 2015, 54(3):3067-3106
[9] He Y, Li G, Peng S. Concentrating bounded states for Kirchhoff type problems in RN involving critical Sobolev exponents. Advanced Nonlinear Studies, 2014, 14(2):483-510
[10] Li G, Ye H. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R³. J Differ Equ, 2014, 257(2):566-600
[11] Li G, Ye H. Existence of positive solutions for nonlinear Kirchhoff type equations in R³ with critical Sobolev exponent. Math Meth Appl Sci, 2015, 37(16):2570-2584
[12] Lions P L. The concentration-compactness principle in the calculus of variations. The locally compact case, part Ⅱ. Ann Inst H Poincare Anal Non Linéeaire, 1984, 1:223-283
[13] Moser J. A new proof of de Giorgiś theorem concerning the regularity problem for elliptic differential equations. Commun Pure Appl Math, 1960, 13:457-460
[14] Szulkin A, Weth T. The method of Nehari manifold. Handbook of Nonconvex Analysis and Applications. Boston:International Press, 2010:597-632
[15] Wang J, Tian L, Xu J, et al. Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J Differential Equations, 2012, 253(7):2314-2351
[16] Wang X. On concentration of positive bound states of nonlinear Schrödinger equations. Commun Math Phys, 1993, 53:229-244
[17] Willem M. Minimax Theorems. Boston:Birkhäusee, 1996

Options

Abstract

Outlines

模态框（Modal）标题

Abstract

Cite this article

References