关于椭圆型半变分不等式问题解的存在性及多解性问题的研究
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摘要
这篇硕士论文集中了作者在攻读硕士学位期间的主要研究成果.
在第二章我们首先考虑关于以下p-Laplacian型(p-Laplacian type)方程非平凡解及多解的存在性
对于带有p-Laplacian算子的椭圆拟线性半边分不等式问题,为应用非光滑的山路引理证明解的存在性,在证明方程所对应的能量泛函满足非光滑的PS条件时,需利用Sobolev空间的一致凸性,但是对于具有更一般形式的算子的p-Laplacian型方程,不具备上述性质,在文中为克服这一困难,本人对位势泛函做了一致凸的假设,从而证明了解的存在性,并应用推广的Ricceri定理,证明了方程三个解的存在性.
在第三章我们研究如下p(x)-Laplacian椭圆方程多重正解的存在性问题.难点在于p(x)-Laplace算子比p-Laplace算子具有更复杂的性质.本文应用了一定的技巧来进行不等式的放缩,克服了变指数p(x)带来的困难,从而在非光滑临界点理论的基础上,证明了此问题多重正解的存在性.
在第四章,我们考虑一个无界域上具有非光滑位势泛函的Schrodinger方程,即
其中V>0,是连续的周期泛函,j(x,u)是关于变量u的局部Lipschitz连续泛函.由于该方程的解就是其所对应的能量泛函的临界点,通常都要在一些紧型条件(如PS条件、C-条件)的基础上来证明其所对应的能量泛函临界点的存在性,但当我们在无界域上考虑该椭圆方程解的存在性时,Sobolev空间H_0~(1,2)(R~N)到L~(2*)(R~N)的嵌入非紧,从而导致所对应的能量泛函失去紧性,本章在非光滑临界点理论的基础上,应用周期逼近的方法证明该问题非平凡解的存在性.
This dissertation collects the main results obtained by the author during the period when she has applied for the M.D.
The contents are the following:
In chapter two, the existence and multiplicity results for the following equation of p-Laplacian type are obtained.For the elliptic quasilinear hemivariational inequality involving the p-Laplacian operator,in order to use the mountain pass theorem proving the existence result, the authors usually need to use the uniform convexity of the Sobolev space to prove the energy function satisfies the PS condition. But for the p-Laplacian type equation mentioned above, this method is no use. To overcome this difficulty, the potential function is assumed to be convex, then I prove the existence result and by using the extension of the Ricceri theorem, the multiplicity result for the problem is obtained.
Chapter three studies the multiplicity of positive solutions for the following Dirich-
let problem involving p(x)-Laplacian operator.The main difficulty is that the p(x)-Laplacian operator possesses more complicated nonlinearities than the p-Laplacian. To overcome this difficulty, our approach is variational based on critical point theory, and the multiplicity of positive solutions is proved by using some special techniques.
Chapter four deals with the existence result for the problem about the Schrodinger equation with nonsmooth potential in unbounded domain, that is:where V > 0, it is a continuous periodic potential function, the function j(x,u) is locally Lipschitz in u. Since the solutions of this problem are the critical points of the associated energy function. One generally needs some compactness such as PScondition or C-condition to prove the existence of critical points of the energy func-tion, but when we study the elliptic equation in R~N, the compactness condition does not always hold since the imbedding of the Sobolev space H_0~(1,2)(Ω)into L~(2*)(Ω) is not compact. In this chapter, based on the nonsmooth critical point theory, and by using the approximation technique with periodic function, the existence of nontrivial solution is obtained.
在第二章我们首先考虑关于以下p-Laplacian型(p-Laplacian type)方程非平凡解及多解的存在性
对于带有p-Laplacian算子的椭圆拟线性半边分不等式问题,为应用非光滑的山路引理证明解的存在性,在证明方程所对应的能量泛函满足非光滑的PS条件时,需利用Sobolev空间的一致凸性,但是对于具有更一般形式的算子的p-Laplacian型方程,不具备上述性质,在文中为克服这一困难,本人对位势泛函做了一致凸的假设,从而证明了解的存在性,并应用推广的Ricceri定理,证明了方程三个解的存在性.
在第三章我们研究如下p(x)-Laplacian椭圆方程多重正解的存在性问题.难点在于p(x)-Laplace算子比p-Laplace算子具有更复杂的性质.本文应用了一定的技巧来进行不等式的放缩,克服了变指数p(x)带来的困难,从而在非光滑临界点理论的基础上,证明了此问题多重正解的存在性.
在第四章,我们考虑一个无界域上具有非光滑位势泛函的Schrodinger方程,即
其中V>0,是连续的周期泛函,j(x,u)是关于变量u的局部Lipschitz连续泛函.由于该方程的解就是其所对应的能量泛函的临界点,通常都要在一些紧型条件(如PS条件、C-条件)的基础上来证明其所对应的能量泛函临界点的存在性,但当我们在无界域上考虑该椭圆方程解的存在性时,Sobolev空间H_0~(1,2)(R~N)到L~(2*)(R~N)的嵌入非紧,从而导致所对应的能量泛函失去紧性,本章在非光滑临界点理论的基础上,应用周期逼近的方法证明该问题非平凡解的存在性.
This dissertation collects the main results obtained by the author during the period when she has applied for the M.D.
The contents are the following:
In chapter two, the existence and multiplicity results for the following equation of p-Laplacian type are obtained.For the elliptic quasilinear hemivariational inequality involving the p-Laplacian operator,in order to use the mountain pass theorem proving the existence result, the authors usually need to use the uniform convexity of the Sobolev space to prove the energy function satisfies the PS condition. But for the p-Laplacian type equation mentioned above, this method is no use. To overcome this difficulty, the potential function is assumed to be convex, then I prove the existence result and by using the extension of the Ricceri theorem, the multiplicity result for the problem is obtained.
Chapter three studies the multiplicity of positive solutions for the following Dirich-
let problem involving p(x)-Laplacian operator.The main difficulty is that the p(x)-Laplacian operator possesses more complicated nonlinearities than the p-Laplacian. To overcome this difficulty, our approach is variational based on critical point theory, and the multiplicity of positive solutions is proved by using some special techniques.
Chapter four deals with the existence result for the problem about the Schrodinger equation with nonsmooth potential in unbounded domain, that is:where V > 0, it is a continuous periodic potential function, the function j(x,u) is locally Lipschitz in u. Since the solutions of this problem are the critical points of the associated energy function. One generally needs some compactness such as PScondition or C-condition to prove the existence of critical points of the energy func-tion, but when we study the elliptic equation in R~N, the compactness condition does not always hold since the imbedding of the Sobolev space H_0~(1,2)(Ω)into L~(2*)(Ω) is not compact. In this chapter, based on the nonsmooth critical point theory, and by using the approximation technique with periodic function, the existence of nontrivial solution is obtained.
引文
[1] Z. Naniewicz, P.D. Panagiotopoulos. Mathematical theory of hemivariational inequalities and applications[M]. New York: Marcel-Dekker, 1995.
[2] P.D.Panagiotopoulos. Hemivarivational inequalities: Applications in mechanics and engineering, Springer-Verlag[M]. New York: Berlin, 1993.
[3] K.C. Chang. Variational methods for nondifferentiable functionals and their applications to partial differential equations[J]. J. Math. Anal. Appl., 1981(80):102— 129.
[4] F. H.Clarke. Optimization and Nonsmooth Analysis[M]. New York: Wiley, 1983.
[5] G.Bonanno. Some remarks on a three critical points theorem[J]. Nonlinear Analysis, 2003(54):651-665.
[6] S.A. Marano. On some elliptic hemivariational and variational-hemivariational inequalities[J]. Nonlinear Analysis, 2005(62):757-774.
[7] S.Hu, N.S. Papageorgiou. Multiple positive solutions for nonlinear eigenvalue problems with the p-laplacian[J]. Nonlinear Analysis, 2008(69):4286-4300.
[8] P.De Napoli, M.C.Mariani. Mountain pass solutions to equations of p-Laplacian type, Nonlinear Analysis[J]. Nonlinear Analysis, 2003(54): 1205-1219.
[9] D. M.Duc, N. T. Vu. Nonuniformly elliptic equations of p-Laplacian type[J]. Nonlinear Analysis, 1971 (61): 1483-1495.
[10] A.Kristaly, H.Lisei, C.Varga. Multiple solutions for p-Laplacian type equations[J]. Nonlinear Analysis, 2008(68): 1375-1381.
[11] S.A.Marano, D.Motreanu. On a three critical points theorem for non- differentiable functions and applications to nonlinear boundary value problems [J]. Nonlinear Analysis, 2002(48):37-52.
[12] B.Ricceri. On a three critical points theorem [J]. Arch Math(Basel), 2000(75):220-226.
[13] Minbo Yang, Zifei Shen. Multiplicity results for quasilinear elliptic systems with neumann boundary condition[J]. 应用泛函分析学报, 2005(7): 146-150.
[14] Z.Denkowski, S.Migorski, N.S.Papageorgiou. An introduction to nonlinear analysis: Theory[M]. New York: Kluwer/Plenum, 2003.
[15] C.Maya, R.Shivaji. Multiple positive solutions of a class of semilinear elliptic boundary value problems[J]. Nonlinear Analalysis, 1999(38):497-504.
[16] K.Perera. Multiple positive solutions of a class of quasilinear elliptic boundary value problems[J]. Electron J. Differencial Equations, 2003(7):1 - 5.
[17] L.Diening. Theoretical and numerical results for electrorheological fluids[D].Ph.D. thesis, University of Freiburg, New York, 2002.
[18] X. L.Fan, D. Zhao. On the spaces L~p(Ω) and W~(m,p(x))(Ω)[J]. J. Math. Anal. Appl., 2001(262):749-760.
[19] X. L.Fan, Q. H.Zhang. Existence of solutions for p(x)-Laplacian Dirichlet problem[J]. J. Math. Anal. Appl., 2003(52): 1843-1852.
[20] Q.Zhang. A strong maximum principle for differential equation with nonstandard p(x)-growth conditions[J]. J. Math. Anal. Appl., 2005(312):24-32.
[21] A.A.Pankov, K. Plfuger. On a semilinear Schrodinger equation with periodic potential[J]. Nonlinear Analysis, 1998(33):593-609.
[22] Y.H. Ding, Lee Cheng. Multiple solutions of Schrodinger equation with indefinite linear part and super or asymptotically linear terms[J]. J. Differential Equations, 2006(222): 137-163.
[23] T. Bartsch, Y.H.Ding. On a nonlinear Schrodinger equation with periodic potential[J]. Math. Ann., 1999(7): 15-37.
[24] M.Willem, Wenming Zou. On a Schrodinger equation with periodic potential and spectrum point zero[J]. Indiana Univ. Math. J., 2003(52): 109-132.
[25] N. Ackermann. On a periodic Schrodinger equation with nonlocal superlinear part[J]. Math Z., 2004(248):423-443.
[26] J.Chabrowski, A.Szulkin. On a semilinear Schrodinger equation with critical Sobolev exponent[J]. Pro, Amer. Math, sco., 2001(130):85-93.
[27] E.H.Papageorgiou, N.S. Papageorgiou. Nonlinear second-order periodic systems with nonsmooth potential[J]. Indian Acad, Sci (Math sci), 2004(114):269-298.
[28] L. Gasinski, N.S.Papageorgiou. A multiple result for nonlinear second-order periodic equations with nonsmooth potential[J]. Bull Belgian Math Soc, 2002(9): 1-14.
[29] M. Filippakis, L. Gasinski, N.S.Papageorgiou. Periodic problems with asymmetric nonlinearities and nonsmooth potentials [J]. Nonlinear Analysis, 2004(58):683-702.
[30] D.Motreanu, P.D.Panagiotopoulos. Minimax theorem and qualitative properties of the solutions of the solutions of hemivariational inequalities[M]. London: Kluwer Academic publishers, 1999.
[31] E.H.Papageorgiou, N.S. Papageorgiou. Existence of solutions and of multiple solutions for nonlinear nonsmooth periodic systems [J]. Czechoslovak math J., 2004, 54(129):347-371.
[32] C.L.Tang. Existence and multiplicity of periodic solutions for nonautonomous second order systems[J]. Nonlinear Anal., 1998(32):299-304.
[33] P.H.Rabinowitz. A note on a semilinear elliptic equation on R~n[J]. Nonlinear Anal., 1991:307-318.
[34] S.Hu, N.S. Papageorgiou. Handbook of multivalued analysis[M], vol. I:Theory, Kluwer, Dordrecht. The Netherlands, 1997.
[2] P.D.Panagiotopoulos. Hemivarivational inequalities: Applications in mechanics and engineering, Springer-Verlag[M]. New York: Berlin, 1993.
[3] K.C. Chang. Variational methods for nondifferentiable functionals and their applications to partial differential equations[J]. J. Math. Anal. Appl., 1981(80):102— 129.
[4] F. H.Clarke. Optimization and Nonsmooth Analysis[M]. New York: Wiley, 1983.
[5] G.Bonanno. Some remarks on a three critical points theorem[J]. Nonlinear Analysis, 2003(54):651-665.
[6] S.A. Marano. On some elliptic hemivariational and variational-hemivariational inequalities[J]. Nonlinear Analysis, 2005(62):757-774.
[7] S.Hu, N.S. Papageorgiou. Multiple positive solutions for nonlinear eigenvalue problems with the p-laplacian[J]. Nonlinear Analysis, 2008(69):4286-4300.
[8] P.De Napoli, M.C.Mariani. Mountain pass solutions to equations of p-Laplacian type, Nonlinear Analysis[J]. Nonlinear Analysis, 2003(54): 1205-1219.
[9] D. M.Duc, N. T. Vu. Nonuniformly elliptic equations of p-Laplacian type[J]. Nonlinear Analysis, 1971 (61): 1483-1495.
[10] A.Kristaly, H.Lisei, C.Varga. Multiple solutions for p-Laplacian type equations[J]. Nonlinear Analysis, 2008(68): 1375-1381.
[11] S.A.Marano, D.Motreanu. On a three critical points theorem for non- differentiable functions and applications to nonlinear boundary value problems [J]. Nonlinear Analysis, 2002(48):37-52.
[12] B.Ricceri. On a three critical points theorem [J]. Arch Math(Basel), 2000(75):220-226.
[13] Minbo Yang, Zifei Shen. Multiplicity results for quasilinear elliptic systems with neumann boundary condition[J]. 应用泛函分析学报, 2005(7): 146-150.
[14] Z.Denkowski, S.Migorski, N.S.Papageorgiou. An introduction to nonlinear analysis: Theory[M]. New York: Kluwer/Plenum, 2003.
[15] C.Maya, R.Shivaji. Multiple positive solutions of a class of semilinear elliptic boundary value problems[J]. Nonlinear Analalysis, 1999(38):497-504.
[16] K.Perera. Multiple positive solutions of a class of quasilinear elliptic boundary value problems[J]. Electron J. Differencial Equations, 2003(7):1 - 5.
[17] L.Diening. Theoretical and numerical results for electrorheological fluids[D].Ph.D. thesis, University of Freiburg, New York, 2002.
[18] X. L.Fan, D. Zhao. On the spaces L~p(Ω) and W~(m,p(x))(Ω)[J]. J. Math. Anal. Appl., 2001(262):749-760.
[19] X. L.Fan, Q. H.Zhang. Existence of solutions for p(x)-Laplacian Dirichlet problem[J]. J. Math. Anal. Appl., 2003(52): 1843-1852.
[20] Q.Zhang. A strong maximum principle for differential equation with nonstandard p(x)-growth conditions[J]. J. Math. Anal. Appl., 2005(312):24-32.
[21] A.A.Pankov, K. Plfuger. On a semilinear Schrodinger equation with periodic potential[J]. Nonlinear Analysis, 1998(33):593-609.
[22] Y.H. Ding, Lee Cheng. Multiple solutions of Schrodinger equation with indefinite linear part and super or asymptotically linear terms[J]. J. Differential Equations, 2006(222): 137-163.
[23] T. Bartsch, Y.H.Ding. On a nonlinear Schrodinger equation with periodic potential[J]. Math. Ann., 1999(7): 15-37.
[24] M.Willem, Wenming Zou. On a Schrodinger equation with periodic potential and spectrum point zero[J]. Indiana Univ. Math. J., 2003(52): 109-132.
[25] N. Ackermann. On a periodic Schrodinger equation with nonlocal superlinear part[J]. Math Z., 2004(248):423-443.
[26] J.Chabrowski, A.Szulkin. On a semilinear Schrodinger equation with critical Sobolev exponent[J]. Pro, Amer. Math, sco., 2001(130):85-93.
[27] E.H.Papageorgiou, N.S. Papageorgiou. Nonlinear second-order periodic systems with nonsmooth potential[J]. Indian Acad, Sci (Math sci), 2004(114):269-298.
[28] L. Gasinski, N.S.Papageorgiou. A multiple result for nonlinear second-order periodic equations with nonsmooth potential[J]. Bull Belgian Math Soc, 2002(9): 1-14.
[29] M. Filippakis, L. Gasinski, N.S.Papageorgiou. Periodic problems with asymmetric nonlinearities and nonsmooth potentials [J]. Nonlinear Analysis, 2004(58):683-702.
[30] D.Motreanu, P.D.Panagiotopoulos. Minimax theorem and qualitative properties of the solutions of the solutions of hemivariational inequalities[M]. London: Kluwer Academic publishers, 1999.
[31] E.H.Papageorgiou, N.S. Papageorgiou. Existence of solutions and of multiple solutions for nonlinear nonsmooth periodic systems [J]. Czechoslovak math J., 2004, 54(129):347-371.
[32] C.L.Tang. Existence and multiplicity of periodic solutions for nonautonomous second order systems[J]. Nonlinear Anal., 1998(32):299-304.
[33] P.H.Rabinowitz. A note on a semilinear elliptic equation on R~n[J]. Nonlinear Anal., 1991:307-318.
[34] S.Hu, N.S. Papageorgiou. Handbook of multivalued analysis[M], vol. I:Theory, Kluwer, Dordrecht. The Netherlands, 1997.