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A NOTE ON THE UNIQUENESS OF POSITIVE SOLUTIONS OF ROBIN PROBLEM*

Published online by Cambridge University Press:  01 September 2008

QIUYI DAI
Affiliation:
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, P.R. China e-mail: daiqiuyi@yahoo.com.cn
YUXIA FU
Affiliation:
Department of Applied Mathematics, Hunan University, Changsha, Hunan 410082, P.R. China
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Abstract

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This note is devoted to prove some uniqueness results of positive solutions of a Robin problem for semi-linear elliptic equations and systems.

Keywords

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Ni, W. M. and Nussbaum, R. D., Uniqueness and non-uniqueness for positive solutions of Δ u+f(r,u)=0, Commun. Pure Appl. Math. 38 (1985), 67108.CrossRefGoogle Scholar
2.Dai, Q. Y., Entire positive solutions for inhomogeneous semilinear elliptic systems, Glasgow Math. J. 47 (2005), 97114.Google Scholar
3.Dai, Q. Y., Fu, Y. X. and Gu, Y. G., Uniqueness of positive solutions of semilinear elliptic equations, Sci. China A 50 (2007), 11411156.Google Scholar
4.Gidas, B. and Spruck, J., A priori bounds for positive solutions of semilinear elliptic equations, Commun. Part. Differ. Equ. 6 (1981), 883901.CrossRefGoogle Scholar
5.Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order (Springer-Velag, Berlin, Germany, 2001).CrossRefGoogle Scholar
6.Damascelli, L., Grossi, M. and Pacella, F., Qualitative properties of positive solutions of semilinear equations in symmetric domains via maximum principle, Ann. Inst. Henri Poincare Analse non lineaire 16 (1999), 631652.Google Scholar
7.Dancer, E. N., The effect of domain shape on the number of positive solutions of certain weakly nonlinear equations, J. Differ. Equ. 74 (1988), 120156.CrossRefGoogle Scholar
8.Dancer, E. N., On the influence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann. 285 (1989), 647669.CrossRefGoogle Scholar
9.Wang, X. J., Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differ. Equ. 93 (1991), 283310.Google Scholar
10.Zou, H., On the effect of domain geometry on uniqueness of positive solutions of Δ u+u p=0, Ann. Scuola Norm. Super. Pisa 21 (1994), 343356.Google Scholar
11.Zou, H., A priori estimates for semilinear elliptic system without variational structure and their application, Math. Ann. 323 (2002), 713735.CrossRefGoogle Scholar
12.de Figueiredo, D. and Felmer, P., A Liouville type theorem for systems, Ann. Scuola Norm. Super. Pisa 21 (1994), 387397.Google Scholar
13.Wei, J. C. and Zhang, L. Q., On the effect of the domain shape on the existence of large solutions of some superlinear problems, Preprint (2006).Google Scholar
14.Berestycki, H., Nirenberg, L. and Varadhan, S. N. S., The principle eigenvalues and maximum principle for second order elliptic operators in general domains, Commun. Pure Appl. Math. 47 (1994), 4792.Google Scholar
15.Zhang, L. Q., Uniqueness of positive solutions of Δ u+u p=0 in a convex domain in R 2, Preprint (1992).Google Scholar
16.Zhang, L. Q., Uniqueness of positive solutions of Δ u+u+u p=0 in a finite ball, Commun. Part. Differ. Equ. 47 (1992), 11411164.Google Scholar
17.Lin, C. S., Uniqueness of solutions minimizing the functional ∫Ω|∇ u|2/(∫Ω|u|p+1)2/(p+1) in R2, Manuscript Math. 84 (1) (1994), 1319.CrossRefGoogle Scholar