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UNIQUENESS FOR SINGULAR SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS II

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  21 July 2015

D. D. HAI  and
R. C. SMITH
D. D. HAI
Affiliation:
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA e-mail: dang@math.msstate.edu, smith@math.msstate.edu
R. C. SMITH
Affiliation:
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA e-mail: dang@math.msstate.edu, smith@math.msstate.edu
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Abstract

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We prove uniqueness of positive solutions for the boundary value problem

\begin{equation*} \left\{ \begin{array}{l} -\Delta u=\lambda f(u)\text{ in }\Omega , \\ \ \ \ \ \ \ \ u=0\text{ on }\partial \Omega , \end{array} \right. \end{equation*}
where Ω is a bounded domain in $\mathbb{R}$n with smooth boundary ∂ Ω, λ is a large positive parameter, f:(0,∞) → [0,∞) is nonincreasing for large t and is allowed to be singular at 0.

MSC classification

Primary: 35J75: Singular elliptic equations
Secondary: 35J92: Quasilinear elliptic equations with $p$-Laplacian
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 2 , May 2016 , pp. 461 - 469
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

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.An, L., On the local Holder continuity of the inverse of the p-Laplace operator, Proc. Amer. Math. Soc. 135 (2007), 3553–3560.Google Scholar
3.Crandall, M. G., Rabinowitz, P. H. and Tartar, L., On a Dirichlet problem with a singular nonlinearitiy, Comm. Partial Differ. Equ. 2 (1977), 193–222.CrossRefGoogle Scholar
4.Dancer, E. N., On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc. 53 (1986), 429–452.CrossRefGoogle Scholar
5.Hai, D. D., On a class of singular p-Laplacian boundary value problems, J. Math. Anal. Appl. 383 (2011), 619–626.CrossRefGoogle Scholar
6.Hai, D. D. and Smith, R. C., Uniqueness for singular semilinear elliptic boundary value problems, Glasgow Math. J. 55 (2013), 399–409.CrossRefGoogle Scholar
7.Hai, D. D. and Smith, R. C., On uniqueness for a class of nonlinear boundary value problems, Proc. Roy. Soc. Edinburgh 136 (2006), 779–784.Google Scholar
8.Lazer, A. C. and McKenna, P. J., On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc. 111 (1991), 721–730.Google Scholar
9.Lin, S. S., On the number of positive solutions for nonlinear elliptic equations when a parameter is large, Nonlinear Anal. 16 (1991), 283–297.Google Scholar
10.Sakaguchi, S., Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet Problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14 (1987), 403–421.Google Scholar
11.Schuchman, V., About uniqueness for nonlinear boundary value problems, Math. Ann. 267 (1984), 537–542.CrossRefGoogle Scholar
12.Wiegner, M., A uniqueness theorem for some nonlinear boundary value problems with a large parameter, Math. Ann. 270 (1985), 401–402.Google Scholar