|
On fourth-order elliptic boundary value problems
Author(s):
C.
V.
Pao
Journal:
Proc. Amer. Math. Soc.
128
(2000),
1023-1030.
MSC (1991):
Primary 35J40, 35J65;
Secondary 34B15
Posted:
August 3, 1999
MathSciNet review:
1676365
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
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:
- 1.
- A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl. 116 (1986), 415-426. MR 87g:34017
- 2.
- R. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differential Integral Eqs. 2 (1989), 91-110. MR 89k:34038
- 3.
- A. Cabada, The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, J. Math. Anal. Appl. 185 (1994), 302-320. MR 95h:34033
- 4.
- C. Cosner and P. W. Schaefer, A comparison principle for a class of fourth order elliptic operators, J. Math. Anal. Appl. 128 (1987), 488-494. MR 88k:35012
- 5.
- C. De Coster, C. Fabry and F. Munyamarere, Nonresonance conditions for fourth-order nonlinear boundary value problems, Internat. J. Math. Meth. Sci. 17 (1994), 725-740. MR 95g:34032
- 6.
- M. A. Del Pino and R. F. Manasevich, Existence for a fourth-order nonlinear boundary problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc. 112 (1991), 81-86. MR 91h:34027
- 7.
- C. P. Gupta, Existence and uniqueness theorem for the bending of an elastic beam equation, Applicable Anal. 26 (1988), 289-304. MR 89m:34027
- 8.
- A. C. Lazer and P. J. McKenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl. 181 (1994), 648-655. MR 95b:34033
- 9.
- R. Y. Ma, J. H. Zhang and S. M. Fu, The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl. 215 (1997), 415-422. MR 98i:34037
- 10.
- A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear problem, Nonlinear Analysis 31 (1998), 895-908. MR 99d:35059
- 11.
- -, Nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Analysis 34 (1998), 509-523. CMP 98:15
- 12.
- C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. MR 94c:35002
- 13.
- -, On nonlinear reaction diffusion-diffusion systems, J. Math. Anal. Appl. 87 (1982), 165-198. MR 83i:35094
- 14.
- M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice Hall, Englewood Cliffs, N.J., 1967. MR 86f:35034; MR 36:2935
- 15.
- K. Schmitt, Boundary value problems for quasilinear second order elliptic equations, Nonlinear Analysis 2 (1978), 263-309. MR 80b:35064
- 16.
- J. Schroder, Fourth-order two-point boundary value problems; estimate by two side bounds, Nonlinear Analysis 8 (1984), 107-114. MR 85g:65092
- 17.
- G. Tarantello, A note on a semilinear elliptic problem, Differential Integral Equations 5 (1992), 561-565. MR 93h:35073
- 18.
- L. Y. Tsai, Existence of solutions of nonlinear elliptic systems, Bull. Inst. Math. Acad. Sinica 8 (1980), 111-127. MR 81e:35052
- 19.
- R. A. Usmani, A uniqueness theorem for a boundary value problem, Proc. Amer. Math. Soc. 77 (1979), 320-335. MR 80j:34018
- 20.
- Y. Yang, Fourth-order two-point boundary value problem, Proc. Amer. Math. Soc. 104 (1988), 175-180. MR 89g:34021
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (1991):
35J40, 35J65,
34B15
Retrieve articles in all Journals with
MSC (1991):
35J40, 35J65,
34B15
Additional Information:
C.
V.
Pao
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205
Email:
cvpao@eos.ncsu.edu
DOI:
10.1090/S0002-9939-99-05430-1
PII:
S 0002-9939(99)05430-1
Keywords:
Fourth-order elliptic equation,
two-point boundary problem,
existence-uniqueness,
method of upper and lower solutions
Received by editor(s):
May 15, 1998
Posted:
August 3, 1999
Communicated by:
Hal L. Smith
Copyright of article:
Copyright
2000,
American Mathematical Society
|
