两类非线性椭圆问题解的多重性
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摘要
本文研究两类非线性椭圆问题的可解性和多解性.设Ω(?)N(N≥1)是光滑有界区域.首先讨论下面带有奇异项的拟线性椭圆问题的多解性,N0是常数,λ是非负的参数.我们运用Ricceri三解定理证明了问题(P1)存在三个不同的正解,主要的方法是上下解以及截断的技巧.
其次讨论了如下的非线性椭圆问题的可解性和多解性其中A：Ω×RN→R,A=A(χ，ξ)在Ω×RN关于ξ有连续的偏导数,α(χ,.)=(?)A(χ,·)=A'.当f在无穷远处是(p-1)次超线性时,我们应用山路引理得到两个解；当f在无穷远处是(p-1)次次线性时,由Ricceri三解定理得到三个不同的解.
In this paper we study the existence and multiplicity of solutions of two classes nonlinear elliptic problems. LetΩ(?) RN is a bounded open set with smooth boundary. The first is about a quasilinear elliptic problem with a singular term N≥1, N< p<+∞,γ> 0 is a constant, andλ> 0 is a parameter. Three weak solutions of the problem (P1) are obtained by a three critical points theorem of B. Ricceri. Our methods are mainly based on the super-sub solutions and the cutoff method. The second problem is the existence and multiplicity of solutions of the following nonlinear elliptic problem (?) LetA:Ωx RN→R, A= A(x,ξ) be a continuous function inΩ×RN with continuous derivative with respect toξ, a(x,·)=(?)A(x,·)= A'. Two weak solutions are obtained by mountian pass lemma provided the nonlinear term via a (p - 1)-superlinear growth at infinity. Three weak solutions are obtained by a three critical points theorem provided the nonlinear term via a (p - 1)-sublinear growth at infinity.
其次讨论了如下的非线性椭圆问题的可解性和多解性其中A：Ω×RN→R,A=A(χ，ξ)在Ω×RN关于ξ有连续的偏导数,α(χ,.)=(?)A(χ,·)=A'.当f在无穷远处是(p-1)次超线性时,我们应用山路引理得到两个解；当f在无穷远处是(p-1)次次线性时,由Ricceri三解定理得到三个不同的解.
In this paper we study the existence and multiplicity of solutions of two classes nonlinear elliptic problems. LetΩ(?) RN is a bounded open set with smooth boundary. The first is about a quasilinear elliptic problem with a singular term N≥1, N< p<+∞,γ> 0 is a constant, andλ> 0 is a parameter. Three weak solutions of the problem (P1) are obtained by a three critical points theorem of B. Ricceri. Our methods are mainly based on the super-sub solutions and the cutoff method. The second problem is the existence and multiplicity of solutions of the following nonlinear elliptic problem (?) LetA:Ωx RN→R, A= A(x,ξ) be a continuous function inΩ×RN with continuous derivative with respect toξ, a(x,·)=(?)A(x,·)= A'. Two weak solutions are obtained by mountian pass lemma provided the nonlinear term via a (p - 1)-superlinear growth at infinity. Three weak solutions are obtained by a three critical points theorem provided the nonlinear term via a (p - 1)-sublinear growth at infinity.
引文
[1]陈亚浙,吴兰成,二阶椭圆型方程与椭圆型方程组,科学出版社,北京,2006.
[2]江泽坚,孙善利,泛函分析(第2版),高等教育出版社,北京,2005.
[3]汪林,泛函分析中的反例,高等教育出版社,北京,1994.
[4]张恭庆,临界点理论及其应用,上海科技出版社,上海,1986.
[5]钟承奎,范先令,陈文源,非线性泛函分析引论,兰州大学出版社,兰州,1998.
[6]解晓霞,一类p(x)-Laplacian Dirichlet问题正解的多重性,兰州大学硕士学位论文2009.
[7]陈海鸿,一类具有双参数和对流项的拟线性奇异椭圆问题,兰州大学硕士学位论文2009.
[8]R.A. Adams, J.J.F. Fournier, Sobolev spaces, Second edition, Academic Press, New York,2003.
[9]A. Ambrosetti, J. Azorero, and I. Peral, Multiplicity results for some non-linear elliptic equations, J. Funct. Anal.137 (1996) 219-242
[10]A. Ambrosetti, H. Brezis, Combined effects of concave and convex nonlin-earities in some elliptic problems, J. Funct. Anal.122 (1994), no.2,519-543.
[11]A. Ambrosetti, A. Malchiodi, Nonlinear analysis and semilinear elliptic prob-lems. Cambridge Studies in Advanced Mathematics,104. Cambridge Uni-versity Press, Cambridge,2007.
[12]A. Ambrosetti, Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal.14 (1973),349-381.
[13]D. Averna, G. Bonanno, A mountain pass theorem for a suitable class of functions, Rocky Mountain J. Math.39 (2009),707-727.
[14]J. P. Garcia Azorero, I. Peral Alonso, Juan J. Manfredi, Sobolev versus Holder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math.2 (2000) 385-404.
[15]A. Bahri, P. L. Lions, Morse index of some min-max critical points. I. Ap-plication to multiplicity results. Comm. Pure Appl. Math.41 (1988),1027-1037.
[16]G. Bonanno, existence of three solutions for a two point boundary value problem, Appl. Math. Lett.13 (2000),53-57.
[17]G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal.54 (2003) 651-665.
[18]H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36 (1983), 437-477.
[19]H. Brezis, L. Nirenberg, H1 versus C1 local minimizers, C. R. Acad. Sci. Paris Ser. I Math.317 (1993),465-472.
[20]K. Demiling, Nonlinear functional analysis, Springer-Verlag,1985.
[21]D. M. Duc, N. T. Vu, Non uniformly elliptic equations of p-Laplacian type. Nonlinear Anal.61 (2005),1483-1495.
[22]P. Enflo, A counter example to the approximation problem in Banach spaces, Acta Math,130 (1973) 309-317
[23]L.C. Evans, Partial differential equations, American Mathematical Society, 1997.
[24]X. Fan, on the sub-supersolution method for p(x)-Laplccian equations, J. Math. Anal. Appl.330 (2007) 665-682.
[25]D. Gilbarg, Neil S. Trudinger, Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, 2001.
[26]Z. M. Guo, Z. T. Zhang, W1,p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl.286 (2003), 32-50.
[27]A. Kristdly, H. Lisei, C. Varga, Multiple solutions for p-Laplacian type equa-tions, Nonlinear Anal.68 (2008) 1375-1381.
[28]P.L. Lions, On the existence of positive solutions of semilinear elliptic equa-tions, SIAM Rev.24 (1982) 441-467
[29]P. De Napoli, M.C. Mariani, Mountain pass solutions to equations of p-Laplacian type, Nonlinear Anal.54 (2003) 1205-1219.
[30]K. Perera, E.A.B. Silva, Existnce and multiplicity of positive solutions for singular quasilinear problems, J. Math. Anal. Appl.323 (2006) 1238-1252.
[31]K. Perera, Z. Zhang, multiple positive solutions of singular p-Laplacian prob-lems by variational methods, Boundary Value Problems 3 (2005) 377-382.
[32]P. Pucci, J. Serrin, Extensions of the mountain pass theorem. J. Funct. Anal. 59 (1984),185-210.
[33]P. Pucci, J. Serrin, A mountain pass theorem, J. Differential Equations (1985),142-149.
[34]J. Saint Raymond, On a minimax theorem, Arch. Math. (Basel) 74 (2000), 432-437.
[35]B. Ricceri, On a three critical points theorem, Arch. Math.75 (2000) 220-226.
[36]B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue prob-lems, Nonlinear operator theory. Math. Comput. Modelling 32 (2000),1485-1494.
[37]B. Ricceri, Three solutions for a Neumann problem, Topol. Methods Non-linear Anal.20 (2002),275-281.
[38]B. Ricceri, A three critical points theorem revisited, Nonlinear Anal.70 (2009),3084-3089.
[39]B. Ricceri, A further three critical points theorem, Nonlinear Anal.71 (2009),4151-4157.
[40]J. Shi, M. Yao, On a singular nonlinear semilinear ellptic problem, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 1389-1401.
[41]Y. Sun, S. Wu, Y. Long, Combined effects of singular and superlinear nonlin-earities in some singular boundary value problems, J. Differential Equations 176 (2001) 511-531.
[42]J. L. Vazquez, A Strong Maximum Principle for Some Quasilinear Elliptic Equations, Appl. Math. Optim.12 (1984),191-202.
[43]Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincare Anal. Non Lineaire 8 (1991) 43-57.
[44]Z. Zhang, on a Dirichlet Problem with a Singular Nonlinearity, J. Math. Anal. Appl.194 (1995) 103-113.
[45]Z. Zhang, Critical points and positive solutions of singular elliptic boundary value problems, J. Math. Anal. Appl.302 (2005) 476-483.
[2]江泽坚,孙善利,泛函分析(第2版),高等教育出版社,北京,2005.
[3]汪林,泛函分析中的反例,高等教育出版社,北京,1994.
[4]张恭庆,临界点理论及其应用,上海科技出版社,上海,1986.
[5]钟承奎,范先令,陈文源,非线性泛函分析引论,兰州大学出版社,兰州,1998.
[6]解晓霞,一类p(x)-Laplacian Dirichlet问题正解的多重性,兰州大学硕士学位论文2009.
[7]陈海鸿,一类具有双参数和对流项的拟线性奇异椭圆问题,兰州大学硕士学位论文2009.
[8]R.A. Adams, J.J.F. Fournier, Sobolev spaces, Second edition, Academic Press, New York,2003.
[9]A. Ambrosetti, J. Azorero, and I. Peral, Multiplicity results for some non-linear elliptic equations, J. Funct. Anal.137 (1996) 219-242
[10]A. Ambrosetti, H. Brezis, Combined effects of concave and convex nonlin-earities in some elliptic problems, J. Funct. Anal.122 (1994), no.2,519-543.
[11]A. Ambrosetti, A. Malchiodi, Nonlinear analysis and semilinear elliptic prob-lems. Cambridge Studies in Advanced Mathematics,104. Cambridge Uni-versity Press, Cambridge,2007.
[12]A. Ambrosetti, Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal.14 (1973),349-381.
[13]D. Averna, G. Bonanno, A mountain pass theorem for a suitable class of functions, Rocky Mountain J. Math.39 (2009),707-727.
[14]J. P. Garcia Azorero, I. Peral Alonso, Juan J. Manfredi, Sobolev versus Holder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math.2 (2000) 385-404.
[15]A. Bahri, P. L. Lions, Morse index of some min-max critical points. I. Ap-plication to multiplicity results. Comm. Pure Appl. Math.41 (1988),1027-1037.
[16]G. Bonanno, existence of three solutions for a two point boundary value problem, Appl. Math. Lett.13 (2000),53-57.
[17]G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal.54 (2003) 651-665.
[18]H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36 (1983), 437-477.
[19]H. Brezis, L. Nirenberg, H1 versus C1 local minimizers, C. R. Acad. Sci. Paris Ser. I Math.317 (1993),465-472.
[20]K. Demiling, Nonlinear functional analysis, Springer-Verlag,1985.
[21]D. M. Duc, N. T. Vu, Non uniformly elliptic equations of p-Laplacian type. Nonlinear Anal.61 (2005),1483-1495.
[22]P. Enflo, A counter example to the approximation problem in Banach spaces, Acta Math,130 (1973) 309-317
[23]L.C. Evans, Partial differential equations, American Mathematical Society, 1997.
[24]X. Fan, on the sub-supersolution method for p(x)-Laplccian equations, J. Math. Anal. Appl.330 (2007) 665-682.
[25]D. Gilbarg, Neil S. Trudinger, Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, 2001.
[26]Z. M. Guo, Z. T. Zhang, W1,p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl.286 (2003), 32-50.
[27]A. Kristdly, H. Lisei, C. Varga, Multiple solutions for p-Laplacian type equa-tions, Nonlinear Anal.68 (2008) 1375-1381.
[28]P.L. Lions, On the existence of positive solutions of semilinear elliptic equa-tions, SIAM Rev.24 (1982) 441-467
[29]P. De Napoli, M.C. Mariani, Mountain pass solutions to equations of p-Laplacian type, Nonlinear Anal.54 (2003) 1205-1219.
[30]K. Perera, E.A.B. Silva, Existnce and multiplicity of positive solutions for singular quasilinear problems, J. Math. Anal. Appl.323 (2006) 1238-1252.
[31]K. Perera, Z. Zhang, multiple positive solutions of singular p-Laplacian prob-lems by variational methods, Boundary Value Problems 3 (2005) 377-382.
[32]P. Pucci, J. Serrin, Extensions of the mountain pass theorem. J. Funct. Anal. 59 (1984),185-210.
[33]P. Pucci, J. Serrin, A mountain pass theorem, J. Differential Equations (1985),142-149.
[34]J. Saint Raymond, On a minimax theorem, Arch. Math. (Basel) 74 (2000), 432-437.
[35]B. Ricceri, On a three critical points theorem, Arch. Math.75 (2000) 220-226.
[36]B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue prob-lems, Nonlinear operator theory. Math. Comput. Modelling 32 (2000),1485-1494.
[37]B. Ricceri, Three solutions for a Neumann problem, Topol. Methods Non-linear Anal.20 (2002),275-281.
[38]B. Ricceri, A three critical points theorem revisited, Nonlinear Anal.70 (2009),3084-3089.
[39]B. Ricceri, A further three critical points theorem, Nonlinear Anal.71 (2009),4151-4157.
[40]J. Shi, M. Yao, On a singular nonlinear semilinear ellptic problem, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 1389-1401.
[41]Y. Sun, S. Wu, Y. Long, Combined effects of singular and superlinear nonlin-earities in some singular boundary value problems, J. Differential Equations 176 (2001) 511-531.
[42]J. L. Vazquez, A Strong Maximum Principle for Some Quasilinear Elliptic Equations, Appl. Math. Optim.12 (1984),191-202.
[43]Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincare Anal. Non Lineaire 8 (1991) 43-57.
[44]Z. Zhang, on a Dirichlet Problem with a Singular Nonlinearity, J. Math. Anal. Appl.194 (1995) 103-113.
[45]Z. Zhang, Critical points and positive solutions of singular elliptic boundary value problems, J. Math. Anal. Appl.302 (2005) 476-483.