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Smoothing the domain out for positive solutions

Published online by Cambridge University Press:  17 April 2009

Yaping Liu
Yaping Liu
Affiliation:
Department of Mathematics, Pittsburg State University, Pittsburg KS 66762-7500, United States of America
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Abstract

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For a given nonlinear partial differential equation defined on a bounded domain with irregular boundary, the available analytical tools are very limited in relation to the study of positive solutions. In this paper wer first use weak convergence methods to show that for an elliptic equation of a certain type, classical positive solutions on nearby smooth domains approach a generalised positive solution on the given domain. The idea is then applied to sublinear elliptic problems to obtain existence and uniqueness results.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 61 , Issue 3 , June 2000 , pp. 405 - 413
Copyright
Copyright © Australian Mathematical Society 2000

References

REFERENCES

[1]Amann, H., ‘Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709.CrossRefGoogle Scholar
[2]Appell, J., ‘The superposition operator in function spaces – a survey, Exposition Math. 6 (1988), 207–270.Google Scholar
[3]Babuška, I. and Výborný, R., ‘Continuous dependence of eigenvalues on the domain, Czechoslovak Math. J. 15 (1965), 169–178.CrossRefGoogle Scholar
[4]Berestycki, H. and Lions, P. L., ‘Some applications of the method of super and subsolutions’, in Bifurcation and Nonlinear Eigenvalue Problems, Lecture Notes in Mathematics 782 (Springer-Verlag, Berlin, Heidelberg, New York, 1980), pp. 16–41.CrossRefGoogle Scholar
[5]Berestycki, H., Nirenberg, L. and Varadhan, S.R.S., ‘The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), 47–92.CrossRefGoogle Scholar
[6]Blat, J. and Brown, K.J., ‘Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 21–34.CrossRefGoogle Scholar
[7]Brown, K.J., ‘Spatially inhomogeneous steady-state solutions for systems of equations describing interacting populations, J. Math. Anal. Appl. 95 (1983), 251–264.CrossRefGoogle Scholar
[8]Dancer, E.N., ‘The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations 74 (1988), 120–156.CrossRefGoogle Scholar
[9]Dancer, E.N., ‘The effect of domain shape on the number of positive solutions of certain nonlinear equations, II, J. Differential Equations 87 (1990), 316–339.CrossRefGoogle Scholar
[10]Dancer, E.N. and Daners, D., ‘Domain perturbation for elliptic equations subject to Robin boundary conditions, J. Differential Equations 138 (1997), 86–132.CrossRefGoogle Scholar
[11]Dancer, E.N. and Schmitt, K., ‘On positive solutions of semilinear elliptic equations, Proc. Amer. Math. Soc. 101 (1987), 445–452.CrossRefGoogle Scholar
[12]Daners, D., ‘Domain perturbation for linear and nonlinear parabolic equations, J. Differential Equations 129 (1996), 358–402.CrossRefGoogle Scholar
[13]Gilbarg, D. and Trudinger, N.S., Elliptic partial differential equations of second order, (2nd edition) (Springer-Verlag, Berlin, Heidelberg, New York, 1983).Google Scholar
[14]Heikkilä, S., ‘On an elliptic boundary value problem with discontinuous nonlinearity, App. Anal. 37 (1990), 183–189.CrossRefGoogle Scholar
[15]Hernández, J., ‘Maximum principles and decoupling for positive solutions of reaction-diffusion systems’, in Reaction-diffusion Equations (Brown, K. J. and Lacey, A. A., Editors), Oxford Science Publications, 1988 (Oxford University Press, New York, 1990), pp. 199–224.CrossRefGoogle Scholar
[16]Jahnke, E., Emde, F. and Lösch, F., Tables of higher functions, (6th edition) (McGraw-Hill, New York, 1960).Google Scholar
[17]Kuttler, J.R. and Sigillito, V.G., ‘Eigenvalues of the Laplacian in two dimensions, SIAM Rev. 26 (1984), 163–193.CrossRefGoogle Scholar
[18]Leung, A., Systems of nonlinear partial differential equations: Applications to biology and engineering, Mathematics and its Applications (Kluwer Academic Publishers, Dordrecht, 1989).CrossRefGoogle Scholar
[19]Li, L., ‘Coexistence theorems of steady-states for predator-prey interacting systems, Trans. Amer. Math. Soc. 305 (1988), 143–166.CrossRefGoogle Scholar
[20]Li, L. and Ghoreishi, A., ‘On positive solutions of general nonlinear elliptic symbiotic interacting systems, Appl. Anal. 14 (1991), 281–295.CrossRefGoogle Scholar
[21]Li, L. and Liu, Y., ‘Spectral and nonlinear effects in certain elliptic systems of three variables, SIAM J. Math. Anal. 24 (1993), 480–498.CrossRefGoogle Scholar
[22]Lions, P.L., ‘On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441–467.CrossRefGoogle Scholar
[23]Rauch, J. and Taylor, M., ‘Potential and scattering theory on wildly perturbed domains, J. Func. Anal. 18 (1975), 27–59.CrossRefGoogle Scholar
[24]Skrypnik, I.V., Methods for analysis of nonlinear elliptic boundary value problems, Translations of Mathematical Monographs 139 (Amer. Math. Soc., Providence, RI, 1994).CrossRefGoogle Scholar
[25]Stummel, F., ‘Perturbation theory for Sobolev spaces, Proc. Roy. Soc. Edinburgh, Sect. A 73 (1974), 5–49.CrossRefGoogle Scholar
[26]Stummel, F., Perturbation of domains in elliptic boundary value problems, Lecture Notes in Mathematics 503 (Springer-Verlag, Berlin, Heidelberg, New York, 1975), pp. 110–135.Google Scholar
[27]Sweers, G., ‘Semilinear elliptic problems on domains with corners, Comm. Partial Differential Equations 14 (1989), 1229–1247.CrossRefGoogle Scholar