Influence of nonlinear perturbed terms on the oscillation of elliptic equations

Authors:
Naoto Yamaoka and Jitsuro Sugie

Journal:
Proc. Amer. Math. Soc. **132** (2004), 2281-2290

MSC (2000):
Primary 35B05, 35B20; Secondary 34C10, 35J60

DOI:
https://doi.org/10.1090/S0002-9939-04-07489-1

Published electronically:
March 25, 2004

MathSciNet review:
2052404

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Our concern is to solve the nonlinear perturbation problem for the semilinear elliptic equation in an exterior domain of with . The lower limit of the nonlinear perturbed term is given for all nontrivial solutions to be oscillatory. The tools for obtaining our theorems are the so-called ``supersolution-subsolution'' method and some results concerning the oscillation and nonoscillation of solutions of the ordinary differential equation associated with the elliptic equation. A simple example is given to illustrate the main results.

**1.**C. Clark and C. A. Swanson,*Comparison theorems for elliptic differential equations*, Proc. Amer. Math. Soc.,**16**(1965), 886-890. MR**31:4983****2.**I. M. Glazman,*Direct methods of qualitative spectral analysis of singular differential operators*, Israel Program for Scientific Translations, Jerusalem, 1965; Daniel Davey and Co., New York, 1966. MR**32:8210****3.**V. B. Headley,*Some oscillation properties of selfadjoint elliptic equations*, Proc. Amer. Math. Soc.,**25**(1970), 824-829. MR**41:3961****4.**V. B. Headley and C. A. Swanson,*Oscillation criteria for elliptic equations*, Pacific J. Math.,**27**(1968), 501-506. MR**38:4797****5.**K. Kreith and C. C. Travis,*Oscillation criteria for selfadjoint elliptic equations*, Pacific J. Math.,**41**(1972), 743-753. MR**47:7189****6.**M. Naito, Y. Naito and H. Usami,*Oscillation theory for semilinear elliptic equations with arbitrary nonlinearities*, Funkcial. Ekvac.,**40**(1997), 41-55. MR**98i:35051****7.**E. S. Noussair and C. A. Swanson,*Positive solutions of quasilinear elliptic equations in exterior domains*, J. Math. Anal. Appl.,**75**(1980), 121-133. MR**81j:35007****8.**J. Sugie,*Oscillation criteria of Kneser-Hille type for second order differential equations with nonlinear perturbed terms*, to appear in Rocky Mountain J. Math.**9.**C. A. Swanson,*Comparison and Oscillation Theory of Linear Differential Equations*, Mathematics in Science and Engineering, vol. 48, Academic Press, New York and London, 1968. MR**57:3515**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35B05,
35B20,
34C10,
35J60

Retrieve articles in all journals with MSC (2000): 35B05, 35B20, 34C10, 35J60

Additional Information

**Naoto Yamaoka**

Affiliation:
Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan

Email:
yamaoka@math.shimane-u.ac.jp

**Jitsuro Sugie**

Affiliation:
Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan

Email:
jsugie@math.shimane-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-04-07489-1

Keywords:
Oscillation,
nonlinear perturbation,
elliptic equation,
exterior domain,
supersolution-subsolution method

Received by editor(s):
March 17, 2003

Published electronically:
March 25, 2004

Additional Notes:
The second author was supported in part by Grant-in-Aid for Scientific Research 11304008

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2004
American Mathematical Society