涉及p(x)-Laplacian的一些变分问题
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摘要
在这篇博士学位论文中,我们分别研究了下面三个具非标准增长条件的变分问题解的存在性:这里位势F(x,t)=∫_0~t(x,s)ds在原点的任意邻域(或+∞)有适当的振荡行为,g是扰动项.这里ε>0是一个参数,Ω(?)R~N是光滑有界区域,f∈C((?)×R)且函数f(x,t)关于t是奇的,g∈C((?)×R).和这里Ω(?)R~N是有界区域,带有光滑边界(?)Ω,η是边界(?)Ω的外单位法向量,λ是一个正数,p_i,q∈C((?))和q(x)>1,对于所有的i∈{1,2…,N},2≤p_i(x) 我们主要运用变分法和变指数空间的理论作为工具研究了这几个问题.
对于第一个问题,其工作空间是W~(1,p(x))(R~N).由于在区域R~N上,嵌入不是紧的,因此(P.S.)条件不满足.为了克服紧性的缺失,一些方法已经被应用,例如径向法,集中紧性原理.对于这个问题我们主要应用加权函数法去克服紧性的缺失.另外我们还运用B.Ricceri的一般变分原理去研究该问题多重解的存在性.
对于第二个方程,函数f(x,t)关于t是奇的,次临界增长.我们知道如果ε=0,在适当的条件下方程有无穷多解.但是非线性项有扰动后,即ε>0时,方程还有无穷多解吗?退一步,方程还会有多重解存在吗?我们给出了当函数g∈C((?)×R),对于任意的j∈N,当ε>0充分小时,方程至少有j个不同的弱解.
对于最后一个方程,我们考虑了Neumann边界条件下一个各向异性拟线性方程在不同情况下本征值的存在性.对于各向异性算子,它比通常的p(x)-Laplacian算子更为复杂,因为不同的空间方向,具有不同的作用函数.同时我们给出了各向异性算子的具体性质以及一个有关各向异性广义Sobolev空间的嵌入定理.
In this doctoral thesis we study the existence of solutions of the following three variational problems with nonstandard growth conditions:where the potential F(x, t)=∫_0~tf (x,s) ds has a suitable oscillating behavior in any neighborhood of the origin(respectively +∞).whereε> 0 is a parameter,Ω(?) R~N is a smooth bounded domian,f∈C((?)×R)and f (x,t) is odd with respect to t,g∈C((?)×R).andwhereΩ(>)R~N is a bounded domain with smooth boundary (?)Ω,ηis the outside unit normal to (?)Ω,λis a positive number,p_i,q∈C ((?)) and q(x) > 1, 2≤p_i(x) < N, for all i∈{1,2,…,N}.
We study these variational problems respectively by the variational method and the theory of variable exponent spaces.
For the first problem, its working space is W~(1,p(x))(R~N).Because of R~N,the embedding isn't compact, so (P.S.) condition don't satisfied. For overcoming this difficulty, some methods have been applied, for example, radical method and compensate-compactness principle. For this problem we apply weight function method to overcome the difficulty. Moreover we consider the existence of multiple solutions of this equation by a general variational principle due to B. Ricceri.
For the second problem, the function f (x, t) is odd with respect to t and subcritical growth. We know that this equation possesses the infinitely many solutions under suitable conditions ifε=0.But when adding a perturbed term, that isε> 0, does the equation still possess the infinitely many solutions? or does the equation still possess multiple solutions? We find that if function g∈C ((?)×R),for any j∈N and ifε> 0 small enough, then the equation possesses at least j distinct weak solutions.
For the last problem, we consider the existence of a eigenvalue of an anisotropic quasilinear elliptic equation with Neumann boundary in different cases. For an anisotropic operator, it is more complex than for the usual p(x)-Laplacian operator, because it has different role in different space directions. Meanwhile we give some concrete properties of anisotropic operator and a relative embedding theorem involving anisotropic generalized Sobolev space.
对于第一个问题,其工作空间是W~(1,p(x))(R~N).由于在区域R~N上,嵌入不是紧的,因此(P.S.)条件不满足.为了克服紧性的缺失,一些方法已经被应用,例如径向法,集中紧性原理.对于这个问题我们主要应用加权函数法去克服紧性的缺失.另外我们还运用B.Ricceri的一般变分原理去研究该问题多重解的存在性.
对于第二个方程,函数f(x,t)关于t是奇的,次临界增长.我们知道如果ε=0,在适当的条件下方程有无穷多解.但是非线性项有扰动后,即ε>0时,方程还有无穷多解吗?退一步,方程还会有多重解存在吗?我们给出了当函数g∈C((?)×R),对于任意的j∈N,当ε>0充分小时,方程至少有j个不同的弱解.
对于最后一个方程,我们考虑了Neumann边界条件下一个各向异性拟线性方程在不同情况下本征值的存在性.对于各向异性算子,它比通常的p(x)-Laplacian算子更为复杂,因为不同的空间方向,具有不同的作用函数.同时我们给出了各向异性算子的具体性质以及一个有关各向异性广义Sobolev空间的嵌入定理.
In this doctoral thesis we study the existence of solutions of the following three variational problems with nonstandard growth conditions:where the potential F(x, t)=∫_0~tf (x,s) ds has a suitable oscillating behavior in any neighborhood of the origin(respectively +∞).whereε> 0 is a parameter,Ω(?) R~N is a smooth bounded domian,f∈C((?)×R)and f (x,t) is odd with respect to t,g∈C((?)×R).andwhereΩ(>)R~N is a bounded domain with smooth boundary (?)Ω,ηis the outside unit normal to (?)Ω,λis a positive number,p_i,q∈C ((?)) and q(x) > 1, 2≤p_i(x) < N, for all i∈{1,2,…,N}.
We study these variational problems respectively by the variational method and the theory of variable exponent spaces.
For the first problem, its working space is W~(1,p(x))(R~N).Because of R~N,the embedding isn't compact, so (P.S.) condition don't satisfied. For overcoming this difficulty, some methods have been applied, for example, radical method and compensate-compactness principle. For this problem we apply weight function method to overcome the difficulty. Moreover we consider the existence of multiple solutions of this equation by a general variational principle due to B. Ricceri.
For the second problem, the function f (x, t) is odd with respect to t and subcritical growth. We know that this equation possesses the infinitely many solutions under suitable conditions ifε=0.But when adding a perturbed term, that isε> 0, does the equation still possess the infinitely many solutions? or does the equation still possess multiple solutions? We find that if function g∈C ((?)×R),for any j∈N and ifε> 0 small enough, then the equation possesses at least j distinct weak solutions.
For the last problem, we consider the existence of a eigenvalue of an anisotropic quasilinear elliptic equation with Neumann boundary in different cases. For an anisotropic operator, it is more complex than for the usual p(x)-Laplacian operator, because it has different role in different space directions. Meanwhile we give some concrete properties of anisotropic operator and a relative embedding theorem involving anisotropic generalized Sobolev space.
引文
[1] E. Acerbi, G. Mingione, Regularity results for a class of functional with nonstandard growth,Arch. Ration. Mech. Anal. 156 (2001) 121-140.
[2] CO.Alves, M.A.S. Souto, Existence of solutions for a class of problems in R~N involving the p(x)-Laplacian, Progress in Nonlinear Differential Equations and Their Applications.66(2005)17-32.
[3] G. Anello,G.Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities,J.Differenrial Equations. 234 (2007) 80-90.
[4] S. Antontsev, S. Shmarev, Elliptic equations and systems with nonstandard growth conditions:Existence, uniqueness and localization properties of solutions, Nonlinear Anal. 65 (2006) 728-761.
[5] Y.M. Chen, S. Levine and M. Rao: Variable exponent, linear growth functionals in image restoration,SIAM J.Appl.Math. 66(4)(2006) 1383-1406.
[6] L. Diening, P. H(a|¨)st(o|¨),A.Nekvinda, Open problems in variable exponent Lebesgue and Soblev spaces,FSDONA04 Proceedings (Drabek and Rakosnik(eds.),Milovy, Czech Republic, 2004)38-58.
[7] M. Degiovanni, S. Lancelotti, Perturbations of even nonsmooth functionals, Differential and Integral Equations, 8 (1995) 981-992.
[8] D.E. Edmunds, J. Rakosnik, Sobolev embeddings with variable exponent, Studia Math.143(3)(2000) 267-293.
[9] X.L. Fan, A constrained minimization problem involving the p(x)-Laplacian in R~N,Nonlinear Anal. 69 (2008) 3661-3670.
[10] X.L. Fan, Eigenvalues of the p(x)-Laplacian Neumann problems, Nonlinear Anal. 67 (2007)2982-2992.
[11] X.L. Fan, Global C~(1,α) regularity for variable exponent elliptic equations in divergence form,J.Differenrial Equations. 235 (2007) 397-417.
[12] X.L. Fan, C. Ji, Existence of infinitely many solutions for a Neumann problems involving the p(x)-Laplacian, J. Math. Anal. Appl. 334 (2007) 248-260.
[13] X.L. Fan, X.Y. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in R~N,Nonlinear Anal. 59 (2004) 173-188.
[14] X.L. Fan, W.M. Liu, An exact inequality involving Luxemburg norm and Conjugate-Orlicz norm in L~(p(x))(Ω),Chinese Annals of Mathematics, Ser. A, 27 (2)(2006) 177-188.(Chinese) The English translation is published in: Chinese Journal of Contemporary Mathematics,27(2X2006)147-158.
[15] X.L. Fan, J.S. Shen, D.Zhao, Sobolev embedding theorems for spaces W~(k,p(x)(Ω)),J.Math.Anal. Appl.262 (2001) 249-260.
[16] X.L. Fan, D. Zhao, On the Spaces L~(p(x))(Ω) and W~(m,p(x))(Ω),J.Math. Anal. Appl. 263 (2001)424-446.
[17] X.L. Fan, D. Zhao, Regularity of minimizers of variational integrals with continuous p(x)-growth conditions, Chinese J. Contemp.Math.17(1996)327-336.
[18] X.L. Fan, D. Zhao, The quasi-minimizer of integral functionals with m(x) growth conditions,Nonlinear Anal. 39 (2000).
[19] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003) 1843-1852.
[20] X.L. Fan, Q. H. Zhao, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math.Anal. Appl. 302 (2005) 306-317.
[21] X.L. Fan, Y.Z. Zhao, D. Zhao, Compact imbedding theorems with symmetry of Strauss-Lions type for the space W~(1,p(x))(Ω),J.Math.Anal. Appl. 255 (2001) 333-348.
[22] P. Harjulehto, P.H(a|¨)st(o|¨), An overview of variable exponent Lebesgue and Sobolev spaces. In:D. Herron, Editor, Future Trends in Geometric Function Theory, RNC Workshop, Rep. Univ.Jyv(a|¨)skyl(a|¨),Dep. Math. Stat. vol. 92 (2003) 85-94.
[23] O.Kov(?)(?)ik,J.R(?)kosn(?)k,On spaces L~(p(x))(Ω) and W~(k,p(x))(Ω),Czechoslovak Math. J. 41(1991)592-618.
[24] S J. Li, Z.L. Liu, Perturbations from Symmetric Elliptic Boundary Value Problems, J. Differential Equations. 185 (2002) 271-280.
[25] PL. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part 1,Ann.Inst.H.Poincar(?) Anal,non lin(?)aire 1 (1984) 109-145.
[26] PL. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part 2,Ann.Inst.H.Poincar(?) Anal,non lin(?)aire 1 (1984) 223-283.
[27] P. Marcellini, Regularity and existence of solutions of elliptic equations with (p,q)-growth conditions, J. Differential Equations. 90 (1991) 1-30.
[28]M.Mih(?)ilescu,G.Morosanu,Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions,Applicable Analysis.(2009) 1-15.
[29]P.H.Rabinowitz,“Minimax Methods in Critical Point Theory with Applications to Differential Equations,”CBMS,No.65,Amer.Math.Soc.Providence,RI,1985.
[30]B.Ricceri,A general variational principle and some of its applications,J.Comput.Appl.Math.113(2000)401-410.
[31]M.R(?)(?)i(?)ka,Electrorheological Fluids:Modeling and Mathematical Theory,Lecture Notes in Math,Vol.1748,Springer-Verlag,Berlin,2000.
[32]S.Samko,Denseness of C_0~∞(R~N) in the generalized Sobolev spaces W~(m,p(x))(R~N),Dokl.Ross.Akad.Nauk 369(4)(1999) 142-160.
[33]S.Samko,On a progress in the theory of Lebesgue spaces with variable exponent:maximal and singular operators,Integral Transforms and Special Functions.16(2005) 461-482.
[34]M.Struwe,Variational Methods,2nd ed.,Springer-Verlag,Berlin,1996.
[35]M.Willem,Minimax theorems,Progress in Nonlinear Differential Equations and their Applications,Vol.24,Birkhauser,1996.
[36]V.V.Zhikov:Averaging of functional of the calculus of variations and elasticity theory,Math.USSR Izvestiya,29(1987) 33-36.
[37]V.V.Zhikov,On some variational problems,Russian J.Math.Phys.5(1997) 105-116.
[2] CO.Alves, M.A.S. Souto, Existence of solutions for a class of problems in R~N involving the p(x)-Laplacian, Progress in Nonlinear Differential Equations and Their Applications.66(2005)17-32.
[3] G. Anello,G.Cordaro, Perturbation from Dirichlet problem involving oscillating nonlinearities,J.Differenrial Equations. 234 (2007) 80-90.
[4] S. Antontsev, S. Shmarev, Elliptic equations and systems with nonstandard growth conditions:Existence, uniqueness and localization properties of solutions, Nonlinear Anal. 65 (2006) 728-761.
[5] Y.M. Chen, S. Levine and M. Rao: Variable exponent, linear growth functionals in image restoration,SIAM J.Appl.Math. 66(4)(2006) 1383-1406.
[6] L. Diening, P. H(a|¨)st(o|¨),A.Nekvinda, Open problems in variable exponent Lebesgue and Soblev spaces,FSDONA04 Proceedings (Drabek and Rakosnik(eds.),Milovy, Czech Republic, 2004)38-58.
[7] M. Degiovanni, S. Lancelotti, Perturbations of even nonsmooth functionals, Differential and Integral Equations, 8 (1995) 981-992.
[8] D.E. Edmunds, J. Rakosnik, Sobolev embeddings with variable exponent, Studia Math.143(3)(2000) 267-293.
[9] X.L. Fan, A constrained minimization problem involving the p(x)-Laplacian in R~N,Nonlinear Anal. 69 (2008) 3661-3670.
[10] X.L. Fan, Eigenvalues of the p(x)-Laplacian Neumann problems, Nonlinear Anal. 67 (2007)2982-2992.
[11] X.L. Fan, Global C~(1,α) regularity for variable exponent elliptic equations in divergence form,J.Differenrial Equations. 235 (2007) 397-417.
[12] X.L. Fan, C. Ji, Existence of infinitely many solutions for a Neumann problems involving the p(x)-Laplacian, J. Math. Anal. Appl. 334 (2007) 248-260.
[13] X.L. Fan, X.Y. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in R~N,Nonlinear Anal. 59 (2004) 173-188.
[14] X.L. Fan, W.M. Liu, An exact inequality involving Luxemburg norm and Conjugate-Orlicz norm in L~(p(x))(Ω),Chinese Annals of Mathematics, Ser. A, 27 (2)(2006) 177-188.(Chinese) The English translation is published in: Chinese Journal of Contemporary Mathematics,27(2X2006)147-158.
[15] X.L. Fan, J.S. Shen, D.Zhao, Sobolev embedding theorems for spaces W~(k,p(x)(Ω)),J.Math.Anal. Appl.262 (2001) 249-260.
[16] X.L. Fan, D. Zhao, On the Spaces L~(p(x))(Ω) and W~(m,p(x))(Ω),J.Math. Anal. Appl. 263 (2001)424-446.
[17] X.L. Fan, D. Zhao, Regularity of minimizers of variational integrals with continuous p(x)-growth conditions, Chinese J. Contemp.Math.17(1996)327-336.
[18] X.L. Fan, D. Zhao, The quasi-minimizer of integral functionals with m(x) growth conditions,Nonlinear Anal. 39 (2000).
[19] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003) 1843-1852.
[20] X.L. Fan, Q. H. Zhao, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math.Anal. Appl. 302 (2005) 306-317.
[21] X.L. Fan, Y.Z. Zhao, D. Zhao, Compact imbedding theorems with symmetry of Strauss-Lions type for the space W~(1,p(x))(Ω),J.Math.Anal. Appl. 255 (2001) 333-348.
[22] P. Harjulehto, P.H(a|¨)st(o|¨), An overview of variable exponent Lebesgue and Sobolev spaces. In:D. Herron, Editor, Future Trends in Geometric Function Theory, RNC Workshop, Rep. Univ.Jyv(a|¨)skyl(a|¨),Dep. Math. Stat. vol. 92 (2003) 85-94.
[23] O.Kov(?)(?)ik,J.R(?)kosn(?)k,On spaces L~(p(x))(Ω) and W~(k,p(x))(Ω),Czechoslovak Math. J. 41(1991)592-618.
[24] S J. Li, Z.L. Liu, Perturbations from Symmetric Elliptic Boundary Value Problems, J. Differential Equations. 185 (2002) 271-280.
[25] PL. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part 1,Ann.Inst.H.Poincar(?) Anal,non lin(?)aire 1 (1984) 109-145.
[26] PL. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part 2,Ann.Inst.H.Poincar(?) Anal,non lin(?)aire 1 (1984) 223-283.
[27] P. Marcellini, Regularity and existence of solutions of elliptic equations with (p,q)-growth conditions, J. Differential Equations. 90 (1991) 1-30.
[28]M.Mih(?)ilescu,G.Morosanu,Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions,Applicable Analysis.(2009) 1-15.
[29]P.H.Rabinowitz,“Minimax Methods in Critical Point Theory with Applications to Differential Equations,”CBMS,No.65,Amer.Math.Soc.Providence,RI,1985.
[30]B.Ricceri,A general variational principle and some of its applications,J.Comput.Appl.Math.113(2000)401-410.
[31]M.R(?)(?)i(?)ka,Electrorheological Fluids:Modeling and Mathematical Theory,Lecture Notes in Math,Vol.1748,Springer-Verlag,Berlin,2000.
[32]S.Samko,Denseness of C_0~∞(R~N) in the generalized Sobolev spaces W~(m,p(x))(R~N),Dokl.Ross.Akad.Nauk 369(4)(1999) 142-160.
[33]S.Samko,On a progress in the theory of Lebesgue spaces with variable exponent:maximal and singular operators,Integral Transforms and Special Functions.16(2005) 461-482.
[34]M.Struwe,Variational Methods,2nd ed.,Springer-Verlag,Berlin,1996.
[35]M.Willem,Minimax theorems,Progress in Nonlinear Differential Equations and their Applications,Vol.24,Birkhauser,1996.
[36]V.V.Zhikov:Averaging of functional of the calculus of variations and elasticity theory,Math.USSR Izvestiya,29(1987) 33-36.
[37]V.V.Zhikov,On some variational problems,Russian J.Math.Phys.5(1997) 105-116.
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