Structure of Nonnegative Nontrivial and Positive Solutions of Singularly Perturbed <em>p</em>-Laplace Equations

ZHANG Zheng-ce; LI Kai-tai

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ZHANG Zheng-ce, LI Kai-tai. Structure of Nonnegative Nontrivial and Positive Solutions of Singularly Perturbed p-Laplace Equations[J]. Applied Mathematics and Mechanics, 2004, 25(8): 847-854.

Citation:

ZHANG Zheng-ce, LI Kai-tai. Structure of Nonnegative Nontrivial and Positive Solutions of Singularly Perturbed p-Laplace Equations[J]. Applied Mathematics and Mechanics, 2004, 25(8): 847-854.

Citation:

ZHANG Zheng-ce, LI Kai-tai. Structure of Nonnegative Nontrivial and Positive Solutions of Singularly Perturbed p-Laplace Equations[J]. Applied Mathematics and Mechanics, 2004, 25(8): 847-854.

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Structure of Nonnegative Nontrivial and Positive Solutions of Singularly Perturbed p-Laplace Equations

College of Sciences, Xi'an Jiaotong University, Xi'an 710049, P. R. China

Received Date: 2002-11-05
Rev Recd Date: 2004-03-08
Publish Date: 2004-08-15

Abstract

Abstract

Structure of nonnegative nontrivial and positive solutions was precisely studied for some singularly perturbed p-Laplace equations. By virtue of sub- and supersolution method, it is shown that there are many nonnegative nontrivial spike-layer solutions and positive intermediate spike-layer solutions. Moreover, the upper and lower bound on the measure of each spike-layer were estimated when the parameter is sufficiently small.
- p-Laplace equation,
- nonnegative nontrivial solution,
- positive solution,
- spike-layer solution,
- sub- and supersolution

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References(18)

References

[1]	Diaz J I.Nonlinear Partial Differential Equation and Free Boundaries—Ⅰ：Elliptic Equations[M].London：Pitman,1985.
[2]	杨作东,陆启韶.一类非牛顿渗流系统爆破界的估计[J]. 应用数学和力学,2001,22(3):287—294.
[3]	白占兵.一类四阶p-Laplace 方程正解的存在性及多解性[J].应用数学和力学, 2001,22(12):1324—1328.
[4]	Guo Z M, Webb J R L, Large and small solutions of a class of quasilinear elliptic eigenvalue problems[J].J Differential Equations,2002,180(1):1—50.
[5]	ZHANG Zheng-ce, LI Kai-tai.Spike-layered solutions of singularly perturbed quasilinear Dirichlet problems[J].J Math Anal Appl,2003,283(2):667—680. doi: 10.1016/S0022-247X(03)00333-0
[6]	Dancer E N, Wei J C.On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem[J].Proc Roy Soc Edinburgh Sect A,1997,127(4):691—701. doi: 10.1017/S0308210500023775
[7]	Ni W M, Takagi I, Wei J C.On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problems: Intermediate solutions[J].Duke Math J,1998,94(3):597—618. doi: 10.1215/S0012-7094-98-09424-8
[8]	Dancer E N, Wei J C.On the location of spikes of solutions with two sharp layers for a singularly perturbed semilinear Dirichlet problem[J].J Differential Equations,1999,157(1):82—101. doi: 10.1006/jdeq.1998.3619
[9]	Vzquez J L.A strong maximum principle for some quasilinear elliptic equations[J].Appl Math Optim,1984,12(3):191—202. doi: 10.1007/BF01449041
[10]	Diaz J I, Herrero M A.Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems[J].Proc Roy Soc Edinburgh Sect A,1981,89(2):249—258. doi: 10.1017/S0308210500020266
[11]	Gidas B, Ni W M,Nirenberg L.Symmetry and related properties via the maximum principle[J].Comm Math Phys,1979,68(3):209—243. doi: 10.1007/BF01221125
[12]	Brock F.Continuous rearrangement and symmetry of solutions of elliptic problems[J].Proc Indian Acad Sci Math Sci,2000,110(2):157—204. doi: 10.1007/BF02829490
[13]	Brock F.Radial symmetry for nonnegative solutions of semilinear elliptic equations involving the p-Laplacian[A，J]. In:Amann H,Bandle C,Chipot M,et al Eds.Progress in Partial Differential Equations[C].Vol 1.Pont--Mousson 1997,1—12；Pitman Res Notes Math Ser,Harlow-New York:Longman, 1998,383:46—58.
[14]	?tani M, Teshima T. On the first eigenvalue of some quasilinear elliptic equations[J]. Proc Japan Acad Ser A,1988,64(1):8—10. doi: 10.3792/pjaa.64.8
[15]	Guo Z M.Structure of nontravial nonnegative solutions to singularly perturbed semilinear Dirichlet problems[J].Proc Roy Soc Edinburgh Sect A,2003,133(2):363—392. doi: 10.1017/S0308210500002432
[16]	Caada A, Drbek P, Gamez J L. Existence of positive solutions for some problems with nonlinear diffusion[J].Trans Amer Math Soc,1997,349(10):4231—4249. doi: 10.1090/S0002-9947-97-01947-8
[17]	Guo Z M.Uniqueness and flat core of positive solutions for quasilinear elliptic eigenvalue problems in general smooth domains[J].Math Nachr,2002,243(1):43—74. doi: 10.1002/1522-2616(200209)243:1<43::AID-MANA43>3.0.CO;2-U
[18]	Guo Z M. Structure of large positive solutions of some semilinear elliptic problems where the nonlinearity changes sign[J].Top Methods Nonlinear Anal,2001,18(1):107—128.