R~N上临界增长p-Kirchhoff型方程的非平凡解的存在性
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摘要
本文主要研究以下具临界增长的非线性p-Kirchhoff型方程的非平凡解的存在性:{-(a+b∫_(R~N)|▽u|~p)?_pu=|u|~(p*-2)u+μf (x)|u|~(q-2)u, x∈R~N,(0.1) u∈D~(1,p)(R~N),其中a≥0,b>0,1平凡解.
We study the existence of nontrivial solutions of the p-Kirchhoff type equation {-(a+b∫_(R~N)|▽u|~p)?_pu=|u|~(p*-2)u+μf (x)|u|~(q-2)u, x∈R~N,(0.1) u∈D~(1,p)(R~N),with critical nonlinearity, where a≥0,b>0,1
We study the existence of nontrivial solutions of the p-Kirchhoff type equation {-(a+b∫_(R~N)|▽u|~p)?_pu=|u|~(p*-2)u+μf (x)|u|~(q-2)u, x∈R~N,(0.1) u∈D~(1,p)(R~N),with critical nonlinearity, where a≥0,b>0,1
引文
1 Kirchhoff G.Mechanik.Leipzig:Teubner,1883
2 Lions J L.On some questions in boundary value problems of mathematical physics.In:Contemporary Development in Continuum Mechanics and Partial Differential Equations.Amsterdam-New York:North-Holland,1978,284-346
3 Arosio A,Panizzi S.On the well-posedness of the Kirchhoff string.Trans Amer Math Soc,1996,348:305-331
4 Alves C O,Corrêa F J S A.On existence of solutions for a class of problem involving a nonlinear operator.Comm Appl Nonlinear Anal,2001,8:43-56
5 Cavalcanti M M,Domingos Cavalcanti V N,Soriano J A.Global existence and uniform decay rates for the KirchhoffCarrier equation with nonlinear dissipation.Adv Difference Equ,2001,6:701-730
6 D’Ancona P,Spagnolo S.Global solvability for the degenerate Kirchhoff equation with real analytic data.Invent Math,1992,108:247-262
7 Guo H,Zhang Y M,Zhou H S.Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential.Commun Pure Appl Anal,2018,17:1875-1897
8 Brézis H,Nirenberg L.Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents.Comm Pure Appl Math,1983,36:437-477
9 Alves C O,Corrêa F J S A,Figueiredo G M.On a class of nonlocal elliptic problems with critical growth.Differ Equ Appl,2010,2:409-417
10 Figueiredo G M.Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument.J Math Anal Appl,2013,401:706-713
11 Fiscella A,Valdinoci E.A critical Kirchhoff type problem involving a nonlocal operator.Nonlinear Anal,2014,94:156-170
12 Naimen D.The critical problem of Kirchhoff type elliptic equations in dimension four.J Differential Equations,2014,257:1168-1193
13 Naimen D.Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent.NoDEANonlinear Differential Equations Appl,2014,21:885-914
14 Figueiredo G M,Santos Junior J R.Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth.Differential Integral Equations,2012,25:853-868
15 He X,Zou W.Ground states for nonlinear Kirchhoff equations with critical growth.Ann Mat Pura Appl(4),2014,193:473-500
16 Liu J,Liao J F,Tang C L.Positive solutions for Kirchhoff-type equations with critical exponent in RN.J Math Anal Appl,2015,429:1153-1172
17 Huang J C,Jiang Z M,Li Z Y,et al.Multiplicity of solutions for a p-Kirchhoff equation.Bound Value Probl,2017,41:1-16
18 Zhou C X,Song Y Q.Multiplicity of solutions for elliptic problems of p-Kirchhoff type with critical exponent.Bound Value Probl,2015,223:1-12
19 Li G.The existence of quasilinear elliptic equation with critical Sobolev exponents on unbounded domain.Acta Math Sci Ser B Engl Ed,1994,14:64-74
20 Li G,Ye H.Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3.J Differential Equations,2014,257:566-600
21 Sciunzi B.Classification of positive D1,p(RN)-solutions to the critical p-Laplace equation in RN.Adv Math,2016,291:12-23
22 Azzollini A.A note on the elliptic Kirchhoff equation in RNperturbed by a local nonlinearity.Commun Contemp Math,2015,17:1450039
23 Willem M.Minimax Theorems.Boston:Birkh?user,1996
24 Lions P L.The concentration-compactness principle in the calculus of variations:The limit cases,I and II.Rev Mat Iberoam,1985,1:145-201;45-121
25 J′ulio F,Corrêa S A,Figueiredo G M.On an elliptic equation of p-Kirchhoff type via variational methods.Bull Aust Math Soc,2006,74:263-277
26 Chabrowski J.Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents.Calc Var Partial Differential Equations,1995,3:493-512
2 Lions J L.On some questions in boundary value problems of mathematical physics.In:Contemporary Development in Continuum Mechanics and Partial Differential Equations.Amsterdam-New York:North-Holland,1978,284-346
3 Arosio A,Panizzi S.On the well-posedness of the Kirchhoff string.Trans Amer Math Soc,1996,348:305-331
4 Alves C O,Corrêa F J S A.On existence of solutions for a class of problem involving a nonlinear operator.Comm Appl Nonlinear Anal,2001,8:43-56
5 Cavalcanti M M,Domingos Cavalcanti V N,Soriano J A.Global existence and uniform decay rates for the KirchhoffCarrier equation with nonlinear dissipation.Adv Difference Equ,2001,6:701-730
6 D’Ancona P,Spagnolo S.Global solvability for the degenerate Kirchhoff equation with real analytic data.Invent Math,1992,108:247-262
7 Guo H,Zhang Y M,Zhou H S.Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential.Commun Pure Appl Anal,2018,17:1875-1897
8 Brézis H,Nirenberg L.Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents.Comm Pure Appl Math,1983,36:437-477
9 Alves C O,Corrêa F J S A,Figueiredo G M.On a class of nonlocal elliptic problems with critical growth.Differ Equ Appl,2010,2:409-417
10 Figueiredo G M.Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument.J Math Anal Appl,2013,401:706-713
11 Fiscella A,Valdinoci E.A critical Kirchhoff type problem involving a nonlocal operator.Nonlinear Anal,2014,94:156-170
12 Naimen D.The critical problem of Kirchhoff type elliptic equations in dimension four.J Differential Equations,2014,257:1168-1193
13 Naimen D.Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent.NoDEANonlinear Differential Equations Appl,2014,21:885-914
14 Figueiredo G M,Santos Junior J R.Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth.Differential Integral Equations,2012,25:853-868
15 He X,Zou W.Ground states for nonlinear Kirchhoff equations with critical growth.Ann Mat Pura Appl(4),2014,193:473-500
16 Liu J,Liao J F,Tang C L.Positive solutions for Kirchhoff-type equations with critical exponent in RN.J Math Anal Appl,2015,429:1153-1172
17 Huang J C,Jiang Z M,Li Z Y,et al.Multiplicity of solutions for a p-Kirchhoff equation.Bound Value Probl,2017,41:1-16
18 Zhou C X,Song Y Q.Multiplicity of solutions for elliptic problems of p-Kirchhoff type with critical exponent.Bound Value Probl,2015,223:1-12
19 Li G.The existence of quasilinear elliptic equation with critical Sobolev exponents on unbounded domain.Acta Math Sci Ser B Engl Ed,1994,14:64-74
20 Li G,Ye H.Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3.J Differential Equations,2014,257:566-600
21 Sciunzi B.Classification of positive D1,p(RN)-solutions to the critical p-Laplace equation in RN.Adv Math,2016,291:12-23
22 Azzollini A.A note on the elliptic Kirchhoff equation in RNperturbed by a local nonlinearity.Commun Contemp Math,2015,17:1450039
23 Willem M.Minimax Theorems.Boston:Birkh?user,1996
24 Lions P L.The concentration-compactness principle in the calculus of variations:The limit cases,I and II.Rev Mat Iberoam,1985,1:145-201;45-121
25 J′ulio F,Corrêa S A,Figueiredo G M.On an elliptic equation of p-Kirchhoff type via variational methods.Bull Aust Math Soc,2006,74:263-277
26 Chabrowski J.Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents.Calc Var Partial Differential Equations,1995,3:493-512