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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
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Abstract: This paper deals with semilinear elliptic equations in an exterior domain of $\mathbb{R}^N$ with $N \ge 3$. 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.


References [Enhancements On Off] (What's this?)

  • 1. T. A. Burton, On the equation $x'' + f(x)h(x')x' + g(x) = e(t)$, 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.

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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

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