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Positivity of solutions of elliptic equations with nonlocal terms*

Published online by Cambridge University Press:  14 November 2011

W. Allegretto  and
A. Barabanova
W. Allegretto
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, CanadaT6G 2G1
A. Barabanova
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, CanadaT6G 2G1
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Extract

In this paper we study a nonlocal problem for a second-order partial differential equation which depends on a parameter n. We prove the existence of critical values 0 < and 0 > such that for all ≦ɳ≦ and for all non-negative right-hand sides, our problem has nonnegative solutions. We obtain a formula for ɳ0, the maximal possible value of , and find the exact value of ɳ for spherical ɳ. We also study the corresponding eigenvalue problem. At the end of the paper, we consider the application of our results to the nonlinear system describing the distribution of temperature and potential in a microsensor.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 3 , 1996 , pp. 643 - 663
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Allegretto, W., Shen, Bing, Haswell, P., Lai, Zhongsheng and Robinson, A. M.. Numerical modelling and optimization of micromachined thermal conductivity pressure sensor. IEEE Transactions on Computer Aided Design 13 (1994), 1247–56.CrossRefGoogle Scholar
2Allegretto, W. and Xie, H.. Ca(Ω) solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem. SIAM J. Math. Anal. 22 (1991), 1491–9.Google Scholar
3Allegretto, W. and Xie, H.. A nonlocal thermistor problem. European J. Appl. Math. 6 (1995), 83–94.CrossRefGoogle Scholar
4Allegretto, W., Xie, H. and Yang, Shixin. Existence and uniqueness of solutions to the electrochemistry system. Applicable Analysis 59 (1995), 27–48.CrossRefGoogle Scholar
5Bobisud, L. E., Calvert, J. E. and Royalty, W. D.. A steady-state melting problem in a moving medium. Appl. Anal. 52 (1994), 177–91.Google Scholar
6Catchpole, E. A.. A Cauchy problem for an ordinary integro-differential equation. Proc. Roy. Soc. Edinburgh Sect. A 72 (1972/1973), 39–55.CrossRefGoogle Scholar
7Chipot, M. and Rodrigues, J.-F.. On a class of nonlocal nonlinear elliptic problems. Math. Modelling Numer. Anal. 26 (1992), 447–68.CrossRefGoogle Scholar
8Cimatti, G.. A bound for temperature in the thermistor problem. IMA J. Appl. Math. 40 (1988), 15–22.CrossRefGoogle Scholar
9Cimatti, G. and Prodi, G.. Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor. Ann. Math. Pura Appl. (4) 151–152 (1988), 227–36.CrossRefGoogle Scholar
10Fiedler, B. and Poláčik, P.. Complicated dynamics of a scalar reaction–diffusion equation with a nonlocal term. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 167–92.CrossRefGoogle Scholar
11Freitas, P.. A nonlocal Sturm–Louiville eigenvalue problem. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 169–88.CrossRefGoogle Scholar
12D. Gilbarg and Trudinger, N. S.. Elliptic Partial Differential Equations of the Second Order (Berlin: Springer, 1983).Google Scholar
13Hu, Bei and Yin, Hong-Ming. Semilinear parabolic equations with prescribed energy (Preprint).Google Scholar
14Krasnosel'skii, M. A.. Positive Solutions of Operator Equations (Amsterdam: Noordhoff, 1964).Google Scholar
15Luckhaus, S. and Zheng, Songum. A nonlinear boundary value problem involving a nonlocal term. Nonlinear Anal. 22 (1994), 129–35.CrossRefGoogle Scholar
16Mikhailov, V. P.. Partial Differential Equations (Moscow: Mir, 1978).Google Scholar
17Stampacchia, G.. Équations Elliptiques du Second Ordre à Coefficients Discontinus (Montreal: Les Presses de l'Universite de Montréal, 1966).Google Scholar
18Troianiello, G. M.. Elliptic Differential Equations and Obstacle Problems (New York: Plenum Press, 1987).CrossRefGoogle Scholar
19Tsai, L.-Y.. On bounded entire solutions of integro–differential equations. Chinese J. Math. 16 (1988), 265–88.Google Scholar
20Tsai, L.-Y. and Wu, S.-L.. On nonexistence results for some integro–differential equations of elliptic type. Chinese J. Math. 21 (1993), 349–85.Google Scholar
21Tsai, L.-Y. and Wu, S. T.. Existence of solutions for elliptic integro–differential systems. Tamkang J. Math. 25(1994), 61–70.CrossRefGoogle Scholar