山路引理与渐近线性偏微分方程解的存在性
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摘要
本文研究三类在无穷远处具渐近线性性质的非线性偏微分方程解的存在性.
首先,研究带非负势的渐近线性椭圆方程的Dirichlet问题解的存在性.通过定义一个约束变分问题,使得在没有(AR)条件的情况下,利用一种改进了的山路引理,证明了方程(0-1)的正解存在性问题.
其次,研究带无界势的渐近线性椭圆方程解的存在性.通过使用插值估计,得到新的约束变分问题解的存在性,进而通过变形的广义山路引理得到方程(0-2)非平凡弱解的存在性.
最后,研究K - P方程解的存在性.其中f(x,y,u)关于u满足在无穷远处的渐近线性性质.通过定义一个约束变分问题,使得在没有(AR)条件的情况下,利用一种改进了的山路引理,证明了方程(0-3)行波解的存在性.
This paper is concerned with the existence of the solutions forthree types of nonlinear partial di?erential equations with asymptotically linearproperty at infinity.
At first we study the existence of the solutions of Dirichlet problems forthe asymptotically linear elliptic equations with nonnegative potentialBy defining a constraint variational problem and using the improved mountainpass theorem, the existence of positive solutions of the Dirichlet problems for(0-1) is obtained without (AR) condition.
Next we investigate the existence of the solutions for the asymptoticallylinear elliptic equations with unbounded potentialBy using the interpolation estimate, the existence of the solutions for a new con-strained variational problem is obtained. Furthermore, by using the deformedmountain pass theorem, the existence of nontrivial solutions for (0-2)is gotten.
At last, we study the existence of the travelling wave solutions for K-PEquationwhere f(x,y,w) is asymptotically linear with respect to w at infinity. Bydefining a constraint variational problem and using an improved mountain pass theorem, the existence of the travelling wave solutions for (0-3) is obtainedwithout (AR) condition.
首先,研究带非负势的渐近线性椭圆方程的Dirichlet问题解的存在性.通过定义一个约束变分问题,使得在没有(AR)条件的情况下,利用一种改进了的山路引理,证明了方程(0-1)的正解存在性问题.
其次,研究带无界势的渐近线性椭圆方程解的存在性.通过使用插值估计,得到新的约束变分问题解的存在性,进而通过变形的广义山路引理得到方程(0-2)非平凡弱解的存在性.
最后,研究K - P方程解的存在性.其中f(x,y,u)关于u满足在无穷远处的渐近线性性质.通过定义一个约束变分问题,使得在没有(AR)条件的情况下,利用一种改进了的山路引理,证明了方程(0-3)行波解的存在性.
This paper is concerned with the existence of the solutions forthree types of nonlinear partial di?erential equations with asymptotically linearproperty at infinity.
At first we study the existence of the solutions of Dirichlet problems forthe asymptotically linear elliptic equations with nonnegative potentialBy defining a constraint variational problem and using the improved mountainpass theorem, the existence of positive solutions of the Dirichlet problems for(0-1) is obtained without (AR) condition.
Next we investigate the existence of the solutions for the asymptoticallylinear elliptic equations with unbounded potentialBy using the interpolation estimate, the existence of the solutions for a new con-strained variational problem is obtained. Furthermore, by using the deformedmountain pass theorem, the existence of nontrivial solutions for (0-2)is gotten.
At last, we study the existence of the travelling wave solutions for K-PEquationwhere f(x,y,w) is asymptotically linear with respect to w at infinity. Bydefining a constraint variational problem and using an improved mountain pass theorem, the existence of the travelling wave solutions for (0-3) is obtainedwithout (AR) condition.
引文
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[8] Caldiroli P., Musina R., Existence of H-bubbles in a perturbative setting[J].Revista Matematica Iberoamericana, 2004, 20(2): 611-626.
[9] Wood J. C., Harmonic maps and harmonic morphisms[J]. Journal of Math-ematical Sciences, 1999, 94(2): 1263-1269.
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[14] 陈亚浙, 吴兰成, 二阶椭圆型方程与椭圆型方程组, 北京: 科学出版社,1997.
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[18] Costa D. G., Magalh?aes C. A., Variational elliptic problems which arenonquadrotic at infinity[J]. Nonl. Anal. TMA, 1994, 23: 1401-1412.
[19] Costa D. G., Magalh?aes C. A., Existence result for perturbations of thep-Laplacian, Nonlinear Analysis T.M.A.[J], 1995, 24(3): 409-418
[20] Jeanjean L., On the existence of bounded Palais-Smale sequencesand application to a Landesman-Lazer type problem set on RN[J].Proc. Royal. Soc. Edinburgh, Section A, 1999, 129A: 787-809.
[21] Schechter M., Superlinear elliptic boundary value problems[J]. ManuscriptaMath. 1995, 86: 253-265.
[22] Stuart C. A., Zhou H. S., A variational problems related to self-trappingof an electromagnetic field[J]. Math. Methods in Applied Sciences, 1996,19: 1397-1407.
[23] Stuart C. A., Zhou H. S., Applying the mountain pass theorem to asymp-totically linear elliptic equation on RN[J]. Comm. in PDE, 1999, 24: 1731-1758.
[24] Costa D. A., Miyagaki O. H., Nontrivial solutions for perturbations ofthe p-Laplacian on unbounded domains[J]. J. Math. Anal. Appl., 1995,193: 737-755.
[25] Zhou H. S., An application of a Mountain Pass Theorem[J]. Acta Math-ematica Sinica Sinica, English Series Jan., 2002, 18: 27-36.
[26] Gongbao Li, Huansong Zhou, Asymptotically linear Dirichlet problem forthe p-Laplacian, Nonlinear Analysis[J], 2001, 43: 1043-1055.
[27] 刘轼波, 李树杰, 一类超线性椭圆方程的无穷多解, 数学学报[J], 2003,46(4): 625-630.
[28] T.Bartsch, Infinitely many solutions of a symmetric Dirichlet problem,Nonlinear Analysis T.M.A.[J], 1993, 20:1205-1216.
[29] 戚桂杰, 山路引理的推广[J], 科学通报, 1986, 10: 724-727.
[30] 王广西, 许友军, 一类p-laplace方程正解的存在性, 数学理论与应用,2007, 27(3): 64-68.
[31] 章国庆, 刘三阳, 一类拟线性椭圆方程组非平凡解的存在性, 高校应用数学学报A辑, 2004, 19(1): 297-302.
[32] Xuan, B.-J., Nontrivial solitary waves of GKP equation in multi-dimensional spaces. Revista Colombiana de Matematicas, 2003, 37: 11-23.
[33] Xuan, B.-J. Nontrivial solitary waves of GKP equation in bounded do-main.Appliable Analysis, 2003, 82: 1039-1048.
[34] Bourgain, J., On the Cauchy problem for the Kadomtsev-Petviashvili equa-tion. Geom. and Funct. Anal. 1993, 4; 315-341.
[35] De Bouard, A., Saut, J. C., Sur les ondes solitarires des equations deKadomtsev-Petviashvili .C.R.Acad, Sciences Paris, 1995, 320: 315-328.
[36] De Bouard, A., Saut, J. C., Solitary waves of generalized Kadomtsev-Petviashvili equation. Ann. Inst. H. Poincare Anal. Non Lineaire, 1997,14; 211-236.
[37] Isaza, P., Mejia, J., Local and Global Cauchy problem for the Kadomtsev-Petviashvili equation in antisotropic Sobolev spaces with negative indeices.Comm. in P.D.E., 2001, 26:1027-1054.
[38] Willem, M., Minimax Theorems. Progress in Nonlinear Di?erential equa-tions and Their Applications, Birkhauser, 1996.
[39] Ablow, M. J., Segur, H., Wang, X. P., Wave collapse and instability ofsolitary waves of a generalized Kadomtsev-Petviashvili equation, Physica D,78, No.3-4.
[40] Willem, M., On the generalized Kadomtsev-Petviashvili equa-tion[J]Seminaire do Mathematique, 1995, 96: 213-222.
[41] He X. M., Zou W. M., Nontrivial Solitary Waves to the GeneralizedKadomtsev-Petviashvili Equations, Appl., Math. Comput. (2007)
[42] Struwe, M., Variational methocs:Applications to Nonlinear Partial Di?er-ential Equation and Hamiltonian Systems[M], Springer, New York, 1996.
[43] Ekeland, I., Convexity methods in Hamiltonian mechanics, Springer-Verlag, 1990.
[44] Schechter, M., A variation of the Mountain Pass lemma and applications,J. London Math. Soc., 1991, 44(2): 491-502.
[51] Evans, L., C., Partial Di?erential Equations, Graguate Stuolies in MAth.,AMS, Providence, Rhode Island, 1997, 19.
[46] Wang, Z. Q. Willem, M., A multiplicity result for the generalizedKadomtsev-Petviashivili Equation, Topolog. Meth. Nonl. Anal. 7(1996)261-270.
[47] 张恭庆, 临界点理论及其应用, 上海科学技术出版社(298号).
[48] Brezis, H., Nirenberg, L., Remarks on finding critical points,Commn. Pure Appl. Math. 44: 8/9, 939-963(1991)
[49] Mawhin, J., Willem, M., Critical point theory and Hamiltonian systems,Springer, 1989.
[50] Ljusternik, L., Schnirelmann, L., Methods topologiques dans les problemsvariationals, Hermann, Paris(1934)
[51] Evans L. C., Partial di?erential equations, Graguate Studies in Math.,AMS, Providence, Rhode Island, 1997, 19.
[2] Leslie F. M, Some constitutive equations for liquid crystals[J]. Archive forRational Mechanics and Analysis, 1968, 28(4): 265-283.
[3] Read N., Order parameter and Ginzburg-Landau theory for the fractionalquantum hall e?ect[J]. Phys Rev Lett, 1989, 62(1): 86-89.
[4] Nagai T., Lkeda T., Traveling waves in a chemotactic model[J]. Journal ofMathematical Biology, 1991, 30(2): 169-184.
[5] Weber E., Kirsner B.Reasons for Rank-Dependent utility evalua-tion[J].Journal of Risk and uncertainty, 1997, 14(1): 41-46.
[6] Yamabe H., On the deformations of Riemannian structures on compactmanifolds[J]. Osaka Math J, 1960, 12:21-37.
[7] Ambrosetti A., Malchiodi A., On the symmetric scalar curvature problemon Sn[J]. Journal of Di?erential Equations, 2001, 170(1): 228-245.
[8] Caldiroli P., Musina R., Existence of H-bubbles in a perturbative setting[J].Revista Matematica Iberoamericana, 2004, 20(2): 611-626.
[9] Wood J. C., Harmonic maps and harmonic morphisms[J]. Journal of Math-ematical Sciences, 1999, 94(2): 1263-1269.
[10] Diening L., Theoretical and Numerical Results for Electrorheological Flu-ids[D]. Berlin: University of Frieburg, 2002.
[11] Halsey T. C., Electrorheological ?uids[J]. Science, 1992, 258(5083): 761-766.
[12] Gilbarg D., Trudinger N. S., Elliptic equation of second order, Springer-Verlag. New York: Heidelberg, 1977.
[13] Ladyz?enskaja O. A., Ural’ceva H. H., Linear and quasilinear elliptic equa-tions, English Transl. New York: Academic Press, 1968.
[14] 陈亚浙, 吴兰成, 二阶椭圆型方程与椭圆型方程组, 北京: 科学出版社,1997.
[15] Ambrosetti A., Rabinowitz P. H., Dual variational methods in a criticalpoints theory and applications[J]. J. Funct. Anal., 1973, 14: 349-381.
[16] Rabinowitz P. H., Minimax methods in critical point theory with applica-tions to di?erential equations[M]. CBMS Reg. Conf. Ser. in Math. No. 65,Amer. Math. Soc., Providence, R. I., 1986.
[17] Rabinowitz P. H., Some critical point theorems and applica-tions to semilinear elliptic partial di?erential eatuations[J]. Ann. ScuolaNorm. Sup. Pisa CI. Sci. 1978, 5: 215-223.
[18] Costa D. G., Magalh?aes C. A., Variational elliptic problems which arenonquadrotic at infinity[J]. Nonl. Anal. TMA, 1994, 23: 1401-1412.
[19] Costa D. G., Magalh?aes C. A., Existence result for perturbations of thep-Laplacian, Nonlinear Analysis T.M.A.[J], 1995, 24(3): 409-418
[20] Jeanjean L., On the existence of bounded Palais-Smale sequencesand application to a Landesman-Lazer type problem set on RN[J].Proc. Royal. Soc. Edinburgh, Section A, 1999, 129A: 787-809.
[21] Schechter M., Superlinear elliptic boundary value problems[J]. ManuscriptaMath. 1995, 86: 253-265.
[22] Stuart C. A., Zhou H. S., A variational problems related to self-trappingof an electromagnetic field[J]. Math. Methods in Applied Sciences, 1996,19: 1397-1407.
[23] Stuart C. A., Zhou H. S., Applying the mountain pass theorem to asymp-totically linear elliptic equation on RN[J]. Comm. in PDE, 1999, 24: 1731-1758.
[24] Costa D. A., Miyagaki O. H., Nontrivial solutions for perturbations ofthe p-Laplacian on unbounded domains[J]. J. Math. Anal. Appl., 1995,193: 737-755.
[25] Zhou H. S., An application of a Mountain Pass Theorem[J]. Acta Math-ematica Sinica Sinica, English Series Jan., 2002, 18: 27-36.
[26] Gongbao Li, Huansong Zhou, Asymptotically linear Dirichlet problem forthe p-Laplacian, Nonlinear Analysis[J], 2001, 43: 1043-1055.
[27] 刘轼波, 李树杰, 一类超线性椭圆方程的无穷多解, 数学学报[J], 2003,46(4): 625-630.
[28] T.Bartsch, Infinitely many solutions of a symmetric Dirichlet problem,Nonlinear Analysis T.M.A.[J], 1993, 20:1205-1216.
[29] 戚桂杰, 山路引理的推广[J], 科学通报, 1986, 10: 724-727.
[30] 王广西, 许友军, 一类p-laplace方程正解的存在性, 数学理论与应用,2007, 27(3): 64-68.
[31] 章国庆, 刘三阳, 一类拟线性椭圆方程组非平凡解的存在性, 高校应用数学学报A辑, 2004, 19(1): 297-302.
[32] Xuan, B.-J., Nontrivial solitary waves of GKP equation in multi-dimensional spaces. Revista Colombiana de Matematicas, 2003, 37: 11-23.
[33] Xuan, B.-J. Nontrivial solitary waves of GKP equation in bounded do-main.Appliable Analysis, 2003, 82: 1039-1048.
[34] Bourgain, J., On the Cauchy problem for the Kadomtsev-Petviashvili equa-tion. Geom. and Funct. Anal. 1993, 4; 315-341.
[35] De Bouard, A., Saut, J. C., Sur les ondes solitarires des equations deKadomtsev-Petviashvili .C.R.Acad, Sciences Paris, 1995, 320: 315-328.
[36] De Bouard, A., Saut, J. C., Solitary waves of generalized Kadomtsev-Petviashvili equation. Ann. Inst. H. Poincare Anal. Non Lineaire, 1997,14; 211-236.
[37] Isaza, P., Mejia, J., Local and Global Cauchy problem for the Kadomtsev-Petviashvili equation in antisotropic Sobolev spaces with negative indeices.Comm. in P.D.E., 2001, 26:1027-1054.
[38] Willem, M., Minimax Theorems. Progress in Nonlinear Di?erential equa-tions and Their Applications, Birkhauser, 1996.
[39] Ablow, M. J., Segur, H., Wang, X. P., Wave collapse and instability ofsolitary waves of a generalized Kadomtsev-Petviashvili equation, Physica D,78, No.3-4.
[40] Willem, M., On the generalized Kadomtsev-Petviashvili equa-tion[J]Seminaire do Mathematique, 1995, 96: 213-222.
[41] He X. M., Zou W. M., Nontrivial Solitary Waves to the GeneralizedKadomtsev-Petviashvili Equations, Appl., Math. Comput. (2007)
[42] Struwe, M., Variational methocs:Applications to Nonlinear Partial Di?er-ential Equation and Hamiltonian Systems[M], Springer, New York, 1996.
[43] Ekeland, I., Convexity methods in Hamiltonian mechanics, Springer-Verlag, 1990.
[44] Schechter, M., A variation of the Mountain Pass lemma and applications,J. London Math. Soc., 1991, 44(2): 491-502.
[51] Evans, L., C., Partial Di?erential Equations, Graguate Stuolies in MAth.,AMS, Providence, Rhode Island, 1997, 19.
[46] Wang, Z. Q. Willem, M., A multiplicity result for the generalizedKadomtsev-Petviashivili Equation, Topolog. Meth. Nonl. Anal. 7(1996)261-270.
[47] 张恭庆, 临界点理论及其应用, 上海科学技术出版社(298号).
[48] Brezis, H., Nirenberg, L., Remarks on finding critical points,Commn. Pure Appl. Math. 44: 8/9, 939-963(1991)
[49] Mawhin, J., Willem, M., Critical point theory and Hamiltonian systems,Springer, 1989.
[50] Ljusternik, L., Schnirelmann, L., Methods topologiques dans les problemsvariationals, Hermann, Paris(1934)
[51] Evans L. C., Partial di?erential equations, Graguate Studies in Math.,AMS, Providence, Rhode Island, 1997, 19.