Applications of phase plane analysis of a Liénard system to positive solutions of Schrödinger equations

Authors:
Jitsuro Sugie and Naoto Yamaoka

Journal:
Proc. Amer. Math. Soc. **131** (2003), 501-509

MSC (2000):
Primary 35B05, 35J60; Secondary 34C10, 70K05

DOI:
https://doi.org/10.1090/S0002-9939-02-06681-9

Published electronically:
June 12, 2002

MathSciNet review:
1933341

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with semilinear elliptic equations in an exterior domain of with . Sufficient conditions are obtained for the equation to have a positive solution which decays at infinity. The main result is proved by means of a supersolution-subsolution method presented by Noussair and Swanson. By using phase plane analysis of a system of Liénard type, a suitable positive supersolution is found out. Asymptotic decay estimation on a solution of the Liénard system gains a positive subsolution. Examples are given to illustrate the main result.

**1.**T. A. Burton,*On the equation*, Ann. Mat. Pura Appl.,**85**(1970), 277-285. MR**41:7201****2.**A. Constantin,*Positive solutions of Schrödinger equations in two-dimensional exterior domains*, Monatsh. Math.,**123**(1997), 121-126. MR**97i:35026****3.**J. R. Graef,*On the generalized Liénard equation with negative damping*, J. Differential Equations,**12**(1972), 34-62. MR**48:6542****4.**E. S. Noussair and C. A. Swanson,*Positive solutions of semilinear Schrödinge equations in exterior domains*, Indiana Univ. Math. J.,**28**(1979), 993-1003. MR**81b:35031****5.**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****6.**C. A. Swanson,*Bounded positive solutions of semilinear Schrödinger equations*, SIAM J. Math. Anal.,**13**(1982), 40-47. MR**83c:35032****7.**C. A. Swanson,*Criteria for oscillatory sublinear Schrödinger equations*, Pacific J. Math.,**104**(1983), 483-493. MR**84c:35008****8.**J. Sugie, D.-L. Chen and H. Matsunaga,*On global asymptotic stability of systems of Liénard type*, J. Math. Anal. Appl.,**219**(1998), 140-164. MR**99c:34111****9.**J. Sugie, K. Kita and N. Yamaoka,*Oscillation constant of second order nonlinear self-adjoint differential equations*, to appear in Ann. Mat. Pura Appl. (4).**10.**J. Sugie, N. Yamaoka and Y. Obata,*Nonoscillation theorems for a nonlinear self-adjoint differential equation*, Nonlinear Anal.,**47**(2001), 4433-4444.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35B05,
35J60,
34C10,
70K05

Retrieve articles in all journals with MSC (2000): 35B05, 35J60, 34C10, 70K05

Additional Information

**Jitsuro Sugie**

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

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

**Naoto Yamaoka**

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

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

DOI:
https://doi.org/10.1090/S0002-9939-02-06681-9

Keywords:
Positive solution,
Schr\"{o}dinger equation,
exterior domain,
Li\'enard system

Received by editor(s):
September 19, 2001

Published electronically:
June 12, 2002

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

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2002
American Mathematical Society