一类非线性p-Laplace方程解的存在性
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摘要
本文应用单调叠代方法证明一类非线性p-Laplace椭圆方程在有界区域上Dirichlet(?)问题解的存在性,再通过区域扩张的手段证明在无界区域上解的存在性。通过类似的过程,解决相应的抛物方程在无界区域上解的存在性。并在过程中用变分方法证明p-Laplace方程有界区域上解的存在性,以及带Neumann边值条件的抛物方程有界区域解的存在性。
This article concerns existence of solution of nonlinear PDEs on Rn with upper and lower solutions. We prove the existence of solution of a nonlinear elliptic p-Laplace equation on bounded domain, and then use domain-extention method to prove the existence on Rn. By a similar scheme, we get the result in the form of parabolic equation. A variational method is used in proving existence of solution of nonlinear el-liptic equation on bounded domain, and of nonlinear parabolic equation with Neumann boundary value.
This article concerns existence of solution of nonlinear PDEs on Rn with upper and lower solutions. We prove the existence of solution of a nonlinear elliptic p-Laplace equation on bounded domain, and then use domain-extention method to prove the existence on Rn. By a similar scheme, we get the result in the form of parabolic equation. A variational method is used in proving existence of solution of nonlinear el-liptic equation on bounded domain, and of nonlinear parabolic equation with Neumann boundary value.
引文
[1]D. Arcoya, D. Ruiz, The Ambrosetti-Prodi problem for the p-Laplacian operator, Comm. Par. Diff. Equ.31 (2006) 849-865.
[2]S. Bo, W. G. Ge, Existence and iteration of positive solutions for some p-Laplacian, Nonl. Anal.67(2007)1820-1830.
[3]C. Bandle, I. Stakgold, The formation of the dead core in parabolic reaction-diffusion prob-lems, Trans. Amer. Math. Soc.286 (1984) 307-317.
[4]A. N. Carvalho, C. B. Gentile, Comparison results for nonlinear parabolic equations with monotone principal part, J. Math. Anal. Appl.259 (2001) 319-337.
[5]H. J. Choe, A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle Problems, Arch. Rat. Mech. Anal.114 (1991) 383-394.
[6]L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Anna. Inst. Henri Poincare Nonl. Anal.15(1998)493-516.
[7]J. Deuel, P. Hess, Nonlinear parabolic boundary value problems with upper and lower solutions, Israel J. Math.29 (1978).
[8]董不明,关于P-Laplace方程解的梯度Holder连续性的一个注记,辽宁大学学报自然科学版,24(1997)15.
[9]范先令,王非之,无界区域上p-Laplace方程正解的存在性,西北民族学院学报(自然科学版),20(1999)219-223.
[10]S. Heikkil, V. Lakshmikantham, Monotone iterative techniques for discontinuous nonlinear differential equations, Bull. Amer. Math. Soc.18 (1988) 65-67.
[11]J. Nieto, An abstract monotone iterative technique, Nonl. Anal. Theo. Meth. Appl.28 (1997)1923-1933.
[12]G. S. Ladde, V. Lakshmikantham, A. S. Vatsala, Monotone Iterative Techniques For Nonlin-ear Differential Equation, Pitman Publishing Company, Boston London Melbourne, (1985)
[13]J. L. Lions著,郭柏灵,汪礼礽译,非线性边值问题的一些解法,中山大学出版社,(1992)
[14]W. M. Ni, On the elliptic equation Δu+K(x)un+2/n-2= 0, its generalizations,and applications in geometry, Indiana Univ Math. J.31 (1982) 493-529.
[15]D. X. Ma, Z. J. Du, W. G. Ge, Existence and iteration of monotone position solutions for multipoint boundary value problem wit p-laplacian operator, Comp. Math. Appl.50 (2005) 729-739.
[16]P. Pucci, J. Serrin, The strong maximum principle revisited, J. Diff. Equ.196 (2004) 1-66.
[17]S. Pigola, M. Rigoli, A. G. Setti, Maximum principles and singular elliptic inequalities, J. Func. Anal.193 (2002) 224-260.
[18]D. H. Sattinger, Topics In Stability And Bifurcation Theory, Lecture Notes in Mathematics, Springer-Verlag, Berlin and New York,309 (1973).
[19]S. Heikkila, V. Lakshmikantham, Monotone Iterative Techniques For Discontinuous Non-linear Differential Equations, Marcel Dekker, Inc. New York, (1994).
[20]P. J. Torres, M. Zhang, A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nachr.251(2003) 101-107.
[21]R. N. Wang, T. J. Xiao, J. Liang, A comparison principle for nonlocal coupled systems of fully nonlinear parabolic equations, Appl. Math. Let.19 (2006) 1272-1277.
[22]Q. L. Yao, Q. M. Sheng, Existence Of positive radial solutions for some semilinear elliptic equations in annulus, Appl. Matb. Mech. Eng. Ed.23 (2002).
[23]Z. X. Zhang, J. Y. Wang, The upper and lower solution method for a class of nonlinear second order three-point boundary value problem, J. Comp. Appl. Math.147 (2002)41-52.
[24]周文书，一类拟线性退缩抛物方程的比较原理及其应用,吉林大学学报,40(2002)219-223.
[2]S. Bo, W. G. Ge, Existence and iteration of positive solutions for some p-Laplacian, Nonl. Anal.67(2007)1820-1830.
[3]C. Bandle, I. Stakgold, The formation of the dead core in parabolic reaction-diffusion prob-lems, Trans. Amer. Math. Soc.286 (1984) 307-317.
[4]A. N. Carvalho, C. B. Gentile, Comparison results for nonlinear parabolic equations with monotone principal part, J. Math. Anal. Appl.259 (2001) 319-337.
[5]H. J. Choe, A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle Problems, Arch. Rat. Mech. Anal.114 (1991) 383-394.
[6]L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Anna. Inst. Henri Poincare Nonl. Anal.15(1998)493-516.
[7]J. Deuel, P. Hess, Nonlinear parabolic boundary value problems with upper and lower solutions, Israel J. Math.29 (1978).
[8]董不明,关于P-Laplace方程解的梯度Holder连续性的一个注记,辽宁大学学报自然科学版,24(1997)15.
[9]范先令,王非之,无界区域上p-Laplace方程正解的存在性,西北民族学院学报(自然科学版),20(1999)219-223.
[10]S. Heikkil, V. Lakshmikantham, Monotone iterative techniques for discontinuous nonlinear differential equations, Bull. Amer. Math. Soc.18 (1988) 65-67.
[11]J. Nieto, An abstract monotone iterative technique, Nonl. Anal. Theo. Meth. Appl.28 (1997)1923-1933.
[12]G. S. Ladde, V. Lakshmikantham, A. S. Vatsala, Monotone Iterative Techniques For Nonlin-ear Differential Equation, Pitman Publishing Company, Boston London Melbourne, (1985)
[13]J. L. Lions著,郭柏灵,汪礼礽译,非线性边值问题的一些解法,中山大学出版社,(1992)
[14]W. M. Ni, On the elliptic equation Δu+K(x)un+2/n-2= 0, its generalizations,and applications in geometry, Indiana Univ Math. J.31 (1982) 493-529.
[15]D. X. Ma, Z. J. Du, W. G. Ge, Existence and iteration of monotone position solutions for multipoint boundary value problem wit p-laplacian operator, Comp. Math. Appl.50 (2005) 729-739.
[16]P. Pucci, J. Serrin, The strong maximum principle revisited, J. Diff. Equ.196 (2004) 1-66.
[17]S. Pigola, M. Rigoli, A. G. Setti, Maximum principles and singular elliptic inequalities, J. Func. Anal.193 (2002) 224-260.
[18]D. H. Sattinger, Topics In Stability And Bifurcation Theory, Lecture Notes in Mathematics, Springer-Verlag, Berlin and New York,309 (1973).
[19]S. Heikkila, V. Lakshmikantham, Monotone Iterative Techniques For Discontinuous Non-linear Differential Equations, Marcel Dekker, Inc. New York, (1994).
[20]P. J. Torres, M. Zhang, A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nachr.251(2003) 101-107.
[21]R. N. Wang, T. J. Xiao, J. Liang, A comparison principle for nonlocal coupled systems of fully nonlinear parabolic equations, Appl. Math. Let.19 (2006) 1272-1277.
[22]Q. L. Yao, Q. M. Sheng, Existence Of positive radial solutions for some semilinear elliptic equations in annulus, Appl. Matb. Mech. Eng. Ed.23 (2002).
[23]Z. X. Zhang, J. Y. Wang, The upper and lower solution method for a class of nonlinear second order three-point boundary value problem, J. Comp. Appl. Math.147 (2002)41-52.
[24]周文书，一类拟线性退缩抛物方程的比较原理及其应用,吉林大学学报,40(2002)219-223.