On fourth-order elliptic boundary value problems

Pao, C.

doi:10.1090/S0002-9939-99-05430-1

On fourth-order elliptic boundary value problems
HTML articles powered by AMS MathViewer

by C. V. Pao PDF

Proc. Amer. Math. Soc. 128 (2000), 1023-1030 Request permission

Abstract:

This paper is concerned with the existence and uniqueness of a solution for a class of fourth-order elliptic boundary value problems. The existence of a solution is proven by the method of upper and lower solutions without any monotone nondecreasing or nonincreasing property of the nonlinear function. Sufficient conditions for the uniqueness of a solution and some techniques for the construction of upper and lower solutions are given. All the existence and uniqueness results are directly applicable to fourth-order two-point boundary value problems.

References

A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl. 116 (1986), no. 2, 415–426. MR 842808, DOI 10.1016/S0022-247X(86)80006-3
Ravi P. Agarwal, On fourth order boundary value problems arising in beam analysis, Differential Integral Equations 2 (1989), no. 1, 91–110. MR 960017
Alberto Cabada, The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems, J. Math. Anal. Appl. 185 (1994), no. 2, 302–320. MR 1283059, DOI 10.1006/jmaa.1994.1250
Chris Cosner and Philip W. Schaefer, A comparison principle for a class of fourth-order elliptic operators, J. Math. Anal. Appl. 128 (1987), no. 2, 488–494. MR 917383, DOI 10.1016/0022-247X(87)90201-0
C. De Coster, C. Fabry, and F. Munyamarere, Nonresonance conditions for fourth order nonlinear boundary value problems, Internat. J. Math. Math. Sci. 17 (1994), no. 4, 725–740. MR 1298797, DOI 10.1155/S0161171294001031
Manuel A. del Pino and Raúl F. Manásevich, Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc. 112 (1991), no. 1, 81–86. MR 1043407, DOI 10.1090/S0002-9939-1991-1043407-9
Chaitan P. Gupta, Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. Anal. 26 (1988), no. 4, 289–304. MR 922976, DOI 10.1080/00036818808839715
A. C. Lazer and P. J. McKenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl. 181 (1994), no. 3, 648–655. MR 1264537, DOI 10.1006/jmaa.1994.1049
Ma Ruyun, Zhang Jihui, and Fu Shengmao, The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl. 215 (1997), no. 2, 415–422. MR 1490759, DOI 10.1006/jmaa.1997.5639
Anna Maria Micheletti and Angela Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal. 31 (1998), no. 7, 895–908. MR 1488379, DOI 10.1016/S0362-546X(97)00446-X
—, Nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Analysis 34 (1998), 509–523.
C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. MR 1212084
C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl. 87 (1982), no. 1, 165–198. MR 653613, DOI 10.1016/0022-247X(82)90160-3
Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825, DOI 10.1007/978-1-4612-5282-5
Klaus Schmitt, Boundary value problems for quasilinear second-order elliptic equations, Nonlinear Anal. 2 (1978), no. 3, 263–309. MR 512661, DOI 10.1016/0362-546X(78)90019-6
Johann Schröder, Fourth order two-point boundary value problems; estimates by two-sided bounds, Nonlinear Anal. 8 (1984), no. 2, 107–114. MR 734445, DOI 10.1016/0362-546X(84)90063-4
Gabriella Tarantello, A note on a semilinear elliptic problem, Differential Integral Equations 5 (1992), no. 3, 561–565. MR 1157488
Long Yi Tsai, Existence of solutions of nonlinear elliptic systems, Bull. Inst. Math. Acad. Sinica 8 (1980), no. 1, 111–127. MR 562733
Riaz A. Usmani, A uniqueness theorem for a boundary value problem, Proc. Amer. Math. Soc. 77 (1979), no. 3, 329–335. MR 545591, DOI 10.1090/S0002-9939-1979-0545591-4
Yi Song Yang, Fourth-order two-point boundary value problems, Proc. Amer. Math. Soc. 104 (1988), no. 1, 175–180. MR 958062, DOI 10.1090/S0002-9939-1988-0958062-3