非光滑型Ricceri变分原理在p(x)-Laplacian方程中的应用
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摘要
在这篇文章中,我们在有界域Ω上分别考虑了包含p(x)-Laplacian算子的Neumann型的微分包含问题和Dirichlet型的微分包含问题在对非线性项作适当假设后,我们分别在变指数Sobolev空间W~(1,(p(x)))(Ω)和W_0~(1,(p(x)))(Ω)中,利用非光滑型Ricceri变分原理得到了两类问题的无穷多解性.
In this paper, we consider differential inclusion problem in a bounded domainΩ, involving p(x)-Laplacian of Neumann-typeand Dirichlet-typeWith some suitable assumptions on nonlinearities, the existences of infinitely many solutions are obtained by using nonsmooth version Ricceri's variational principle in variable exponent Sobolev spaces W~(1,(p(x)))(Ω) and W_0~(1,(p(x)))(Ω), respectively.
In this paper, we consider differential inclusion problem in a bounded domainΩ, involving p(x)-Laplacian of Neumann-typeand Dirichlet-typeWith some suitable assumptions on nonlinearities, the existences of infinitely many solutions are obtained by using nonsmooth version Ricceri's variational principle in variable exponent Sobolev spaces W~(1,(p(x)))(Ω) and W_0~(1,(p(x)))(Ω), respectively.
引文
[1]S.N.Antontsev and S.I.Shmarev,A Model Porous Medium Equation with Variable Exponent of Nonlinearity:Existence,Uniqueness and Localization Properties of Solutions,Nonlinear Anal.60 (2005),515-545.
[2]S.N.Antontsev and J.F.Rodrigues,On Stationary Thermo-rheological Viscous Flows,Ann.Univ.Ferrara,Sez.7,Sci.Mat.52 (2006),19-36.
[3]E.Acerbi and G.Mingione:Regularity results for a class of functionals with non-standard growth,Arch.Ration.Mech.Anal,156(2001),121-140.
[4]G.Anello and G.Cordaro,Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian via a varitional priciple of Ricceri,Arch.Math.79 (2002)274-287.
[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]K.C.Chang,Variational methods for nondifferentiable functionals and their applications to partial differential equations,J.Math.Anal.Appl.80 (1981),102-129.
[7]F.Cammaroto,A.Chinni and B.Di Bella,Infinitely many solutions for the Dirichlet problem involving the p-Laplacian,Nonlinear Anal.61 (2005),41-49.
[8]F.H.Clarke (1983).Optimization and Nonsmooth Analysis.Wiley,New York.
[9]L.Diening,P.H(a|¨)st(o|¨) and A.Nekvinda,Open problems in variable exponent Lebesgue and Sobolev spaces,in:P.Drabek,J.Rakosnik,FSDONA04 Proceedings,Milovy,Czech Republic,2004,pp.38-58.
[10]X.L.Fan,Global C~(1,α) regularity for variable exponent elliptic equations in divergence form,J.Differential Equations 235 (2007) 397-417.
[11]X.L.Fan,On the sub-supersolution methods for p(x)-Laplacian equations,J.Math.Anal.Appl.330 (2007),665-682.
[12]X.L.Fan and C.Ji,Existence of infinitely many solutions for a Neumann problem involving the p(x)-Laplacian,J.Math.Anal.Appl.334 (2007),248-260.
[13]X.L.Fan and X.Y.Han,Existence and multiplicity of solutions for p(x)-LapIacian equations in R~N,Nonlinear Anal.59 (2004),173-188.
[14] X.L. Fan, J.S. Shen and D. Zhao, Sobolev embedding theorems for spaces W~(k,(p(x)))(Ω), J. Math. Anal. Appl. 262 (2001), 749-760.
[15] X.L. Fan and Q.H. Zhang, Existence of solutions for p(x) -Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003), 1843-1852.
[16] X.L. Fan, Q.H. Zhang and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306-317.
[17] X.L. Fan and D. Zhao, On the Spaces L~(p(x)) and W~(m,(p(x))) , J. Math. Anal. Appl. 263 (2001), 424-446.
[18] M. Galewski, A new variational for the p(x) -Laplacian equation, BULL. AUSTRAL. MATH. SOC. 72 (2005), 53-65.
[19] P. Harjulehto and P. Hasto, An overview of variable exponent Lebesgue and Sobolev spaces, Future Trends in Geometric Function Theory (D. Herron (ed.), RNC Work- shop), Jyv(?)skyl(?), 2003,85-93.
[20] O. Kov(?)(?)ik and J. R(?)kosn(?)k: On spaces L~(p(x)) and W~(k;(p(x))) ,Czech. Math. J., 41(116)(1991), 592-618.
[21] A. Krist(?)ly, Infinitely many solutions for a differential inclusion problem in R~N, J. Differential Equations 220 (2006) 511-530.
[22] N.C. Kourogenis and N.S. Papageorgiou, Nonsmooth crical point theory and nonlinear elliptic equation at resonance, KODAI Math. J. 23 (2000), 108-135.
[23] N.C. Kourogenis and N.S. Papageorgiou, Existence theorems for elliptic hemivariational inequalities involving the p-laplacian, Abstract and Applied Analysis, 7:5 (2002), 259-277.
[24] S. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian, J. Differential Equations 182(2002)108-120.
[25] J. Musielak: Orlicz spaces and modular spaces, Lecture Notes in Math., Vol. 1034, Springer- Verlag, Berlin, 1983.
[26] W. Orlicz: (?)ber konjugierte Exponentenfolgen, Studia Math., 3(1931), 200-211.
[27] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401-410.
[28] B. Ricceri, Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian, Bull. London Math. Soc. 33 (2001) 331-340.
[29]M.Ruzicka,Electrorheological Fluids:Modeling and Mathematical Theory,Springer-Verlag,Berlin,2000.
[30]S.Samko,On a progress in the theory of Lebesgue spaces with variable exponent Maximal and singular operators,Integral Transforms Spec.Funct.16 (2005),461-482.
[31]V.V.Zhikov (=V.V.Jikov),S.M.Kozlov,O.A.Oleinik,(Translated from the Russian by G.A.Yosifian),Homogenization of Differential Operators and Integral Functionals,Springer-erlag,Berlin,1994.
[32]V.V.Zhikov,On some variational problems,Russian J.Math.Phys.5 (1997),105-116.
[33]V.V.Zhikov,Averaging of functionals of the calculus of variations and elasticity theory,Math.USSR.Izv.9 (1987),33-66.
[2]S.N.Antontsev and J.F.Rodrigues,On Stationary Thermo-rheological Viscous Flows,Ann.Univ.Ferrara,Sez.7,Sci.Mat.52 (2006),19-36.
[3]E.Acerbi and G.Mingione:Regularity results for a class of functionals with non-standard growth,Arch.Ration.Mech.Anal,156(2001),121-140.
[4]G.Anello and G.Cordaro,Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian via a varitional priciple of Ricceri,Arch.Math.79 (2002)274-287.
[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]K.C.Chang,Variational methods for nondifferentiable functionals and their applications to partial differential equations,J.Math.Anal.Appl.80 (1981),102-129.
[7]F.Cammaroto,A.Chinni and B.Di Bella,Infinitely many solutions for the Dirichlet problem involving the p-Laplacian,Nonlinear Anal.61 (2005),41-49.
[8]F.H.Clarke (1983).Optimization and Nonsmooth Analysis.Wiley,New York.
[9]L.Diening,P.H(a|¨)st(o|¨) and A.Nekvinda,Open problems in variable exponent Lebesgue and Sobolev spaces,in:P.Drabek,J.Rakosnik,FSDONA04 Proceedings,Milovy,Czech Republic,2004,pp.38-58.
[10]X.L.Fan,Global C~(1,α) regularity for variable exponent elliptic equations in divergence form,J.Differential Equations 235 (2007) 397-417.
[11]X.L.Fan,On the sub-supersolution methods for p(x)-Laplacian equations,J.Math.Anal.Appl.330 (2007),665-682.
[12]X.L.Fan and C.Ji,Existence of infinitely many solutions for a Neumann problem involving the p(x)-Laplacian,J.Math.Anal.Appl.334 (2007),248-260.
[13]X.L.Fan and X.Y.Han,Existence and multiplicity of solutions for p(x)-LapIacian equations in R~N,Nonlinear Anal.59 (2004),173-188.
[14] X.L. Fan, J.S. Shen and D. Zhao, Sobolev embedding theorems for spaces W~(k,(p(x)))(Ω), J. Math. Anal. Appl. 262 (2001), 749-760.
[15] X.L. Fan and Q.H. Zhang, Existence of solutions for p(x) -Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003), 1843-1852.
[16] X.L. Fan, Q.H. Zhang and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306-317.
[17] X.L. Fan and D. Zhao, On the Spaces L~(p(x)) and W~(m,(p(x))) , J. Math. Anal. Appl. 263 (2001), 424-446.
[18] M. Galewski, A new variational for the p(x) -Laplacian equation, BULL. AUSTRAL. MATH. SOC. 72 (2005), 53-65.
[19] P. Harjulehto and P. Hasto, An overview of variable exponent Lebesgue and Sobolev spaces, Future Trends in Geometric Function Theory (D. Herron (ed.), RNC Work- shop), Jyv(?)skyl(?), 2003,85-93.
[20] O. Kov(?)(?)ik and J. R(?)kosn(?)k: On spaces L~(p(x)) and W~(k;(p(x))) ,Czech. Math. J., 41(116)(1991), 592-618.
[21] A. Krist(?)ly, Infinitely many solutions for a differential inclusion problem in R~N, J. Differential Equations 220 (2006) 511-530.
[22] N.C. Kourogenis and N.S. Papageorgiou, Nonsmooth crical point theory and nonlinear elliptic equation at resonance, KODAI Math. J. 23 (2000), 108-135.
[23] N.C. Kourogenis and N.S. Papageorgiou, Existence theorems for elliptic hemivariational inequalities involving the p-laplacian, Abstract and Applied Analysis, 7:5 (2002), 259-277.
[24] S. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian, J. Differential Equations 182(2002)108-120.
[25] J. Musielak: Orlicz spaces and modular spaces, Lecture Notes in Math., Vol. 1034, Springer- Verlag, Berlin, 1983.
[26] W. Orlicz: (?)ber konjugierte Exponentenfolgen, Studia Math., 3(1931), 200-211.
[27] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401-410.
[28] B. Ricceri, Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian, Bull. London Math. Soc. 33 (2001) 331-340.
[29]M.Ruzicka,Electrorheological Fluids:Modeling and Mathematical Theory,Springer-Verlag,Berlin,2000.
[30]S.Samko,On a progress in the theory of Lebesgue spaces with variable exponent Maximal and singular operators,Integral Transforms Spec.Funct.16 (2005),461-482.
[31]V.V.Zhikov (=V.V.Jikov),S.M.Kozlov,O.A.Oleinik,(Translated from the Russian by G.A.Yosifian),Homogenization of Differential Operators and Integral Functionals,Springer-erlag,Berlin,1994.
[32]V.V.Zhikov,On some variational problems,Russian J.Math.Phys.5 (1997),105-116.
[33]V.V.Zhikov,Averaging of functionals of the calculus of variations and elasticity theory,Math.USSR.Izv.9 (1987),33-66.