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Singular limit of quasilinear Neumann problems

Published online by Cambridge University Press:  14 November 2011

Xing-Bin Pan
Xing-Bin Pan
Affiliation:
Center for Mathematical Sciences, Zhejiang University, Hangzhou 310027, People's Republic of China
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Abstract

This paper is devoted to the study of the singular limit of the minimal solutions, as p → 1, of quasilinear Neumann problems involving p-Laplacian operators. It is established that the limit function is of bounded variation and is locally Höolder-continuous inside the domain.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 1 , 1995 , pp. 205 - 223
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1, Adimurthi, Pacella, F. and Yadava, S. L.. Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity. C.R Acad. Sci. Paris Ser. I Math. 314 (1992), 811–815.Google Scholar
2, Adimurthi and Yadava, S. L.. Critical exponent problem in R n (n≧4) with Neumann boundary condition. Proc. Indian Acad. Sci. Math. Sci. 10 (1990), 275–284.Google Scholar
3Budd, C., Knaap, M. C. and Peletier, L. A.. Asymptotic behaviour of solutions of elliptic equations with critical exponents and Neumann boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), 225–250.Google Scholar
4Comte, M. and Knaap, M. C.. Solutions of elliptic equations involving critical Sobolev exponents with Neumann boundary conditions. Manuscripta Math. 69 (1990), 43–70.Google Scholar
5DiBenedetto, E.. C 1+∝ local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (1983), 827–850.CrossRefGoogle Scholar
6Federer, H.. Geometric Measure Theory (Berlin: Springer, 1969).Google Scholar
7Fife, P. C.. Semilinear elliptic boundary value problems with small parameters. Arch.Rational Mech. Anal. 52 (1973), 205–232.CrossRefGoogle Scholar
8Fleming, W. and Rishel, R.. An integral formula for total gradient variation. Arch. Math.(Basel) 11 (1960), 218–222.Google Scholar
9Garcia-Azorero, J.P. and Peral-Alonso, I.. Existence and nonuniqueness for the p Laplacian: Nonlinear eigenvalues. Commun. Partial Differential Equations 12 (1987), 1389–1430.Google Scholar
10Giaquinta, M.. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems (Princeton: Princeton University Press, 1983).Google Scholar
11Giusti, E.. Minimal Surfaces and Functions of Bounded Variation (Boston:Birkhauser, 1984).Google Scholar
12Hardt, R. and Lin, F.-H.. Mappings minimizing the L p norm of the gradient. Comm. Pure Appl. Math. 40 (1987), 555–588.Google Scholar
13Lin, C.-S., Ni, W.-M. and Takagai, I.. Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations 72 (1988), 1–27.CrossRefGoogle Scholar
14Lions, P. L.. The concentration-compactness principle in the calculus of variations. The limit case. Rev. Mat. Iberoamericana 1 (1985), (1) 145–201; (2) 45–121.Google Scholar
15Lions, P. L., Pacella, F. and Tricarico, M.. Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions. Indiana Univ. Math. J. 37 (1988), 301–324.CrossRefGoogle Scholar
16Ni, W.-M., Pan, X.-B. and Takagi, I.. Singular behaviour of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents. Duke Math. J. 67 (1992), 1–20.Google Scholar
17Ni, W.-M. and Serrin, J.. Existence and nonexistence theorems for ground states for quasilinear partial differential equations. Atti. Accad. Naz. dei Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 77 (1986), 231–257.Google Scholar
18Ni, W.-M. and Serrin, J.. Nonexistence theorems for singular solutions of quasilinear partial differential equations. Comm. Pure Appl. Math. 39 (1986), 379–399.Google Scholar
19Ni, W.-M. and Takagi, I.. On the shape of least-energy solutions to a semilinear Neumannproblem. Comm. Pure Appl. Math. 44 (1991), 819–851.Google Scholar
20Ni, W.-M. and Takagi, I.. Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70 (1993), 247–281.Google Scholar
21Pan, X.-B.. Condensation of least-energy solutions of a semilinear Neumann problem. J. Partial Differential Equations (in press).Google Scholar
22Pan, X.-B.. Further study on the effect of boundary conditions. J. Differential Equations (to appear).Google Scholar
23Pucci, P. and Serrin, J.. Continuation and limit properties for solutions of strongly nonlinear second order differential equations (preprint).Google Scholar
24Schoen, R. and Uhlenbeck, K.. A regularity theory for harmonic maps. J. Differential Geometry 17 (1982), 307–335.CrossRefGoogle Scholar
25Simon, L. M.. Lectures on Geometric Measure Theory, Proc. CMA (Canberra:Australian National University, 1983).Google Scholar
26Talenti, G.. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353–372.Google Scholar
27Tolksdorf, P.. Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51 (1984), 126–150.Google Scholar
28Vázquez, J. L.. A strong maximum principle for some quasilinear elliptic equations.Appl. Math. Optim. 12 (1984), 191–202.Google Scholar
29Wang, X.-J.. Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Differential Equations 93 (1991), 283–310.Google Scholar
30Wang, X.-J.. Positive solutions of the Neumann problem of p-Laplacian equations (preprint).Google Scholar
31Wang, Z.-Q.. On the existence of multiple, single-peaked solutions of a semilinear Neumann problem. Arch. Rational Mech. Anal. 120 (1992), 375–399.Google Scholar