Applications of phase plane analysis of a Liénard system to positive solutions of Schrödinger equations
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- by Jitsuro Sugie and Naoto Yamaoka PDF
- Proc. Amer. Math. Soc. 131 (2003), 501-509 Request permission
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
- T. A. Burton, On the equation $x^{\prime \prime }+f(x)h(x^{\prime } )x^{\prime } + g(x)=e(t)$, Ann. Mat. Pura Appl. (4) 85 (1970), 277–285. MR 262595, DOI 10.1007/BF02413538
- Adrian Constantin, Positive solutions of Schrödinger equations in two-dimensional exterior domains, Monatsh. Math. 123 (1997), no. 2, 121–126. MR 1430499, DOI 10.1007/BF01305966
- John R. Graef, On the generalized Liénard equation with negative damping, J. Differential Equations 12 (1972), 34–62. MR 328200, DOI 10.1016/0022-0396(72)90004-6
- E. S. Noussair and C. A. Swanson, Positive solutions of semilinear Schrödinger equations in exterior domains, Indiana Univ. Math. J. 28 (1979), no. 6, 993–1003. MR 551163, DOI 10.1512/iumj.1979.28.28072
- E. S. Noussair and C. A. Swanson, Positive solutions of quasilinear elliptic equations in exterior domains, J. Math. Anal. Appl. 75 (1980), no. 1, 121–133. MR 576278, DOI 10.1016/0022-247X(80)90310-8
- C. A. Swanson, Bounded positive solutions of semilinear Schrödinger equations, SIAM J. Math. Anal. 13 (1982), no. 1, 40–47. MR 641539, DOI 10.1137/0513003
- C. A. Swanson, Criteria for oscillatory sublinear Schrödinger equations, Pacific J. Math. 104 (1983), no. 2, 483–493. MR 684305
- Jitsuro Sugie, Da-Li Chen, and Hideaki Matsunaga, On global asymptotic stability of systems of Liénard type, J. Math. Anal. Appl. 219 (1998), no. 1, 140–164. MR 1607102, DOI 10.1006/jmaa.1997.5773
- 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).
- J. Sugie, N. Yamaoka and Y. Obata, Nonoscillation theorems for a nonlinear self-adjoint differential equation, Nonlinear Anal., 47 (2001), 4433–4444.
Additional Information
- Jitsuro Sugie
- Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
- MR Author ID: 168705
- Email: jsugie@math.shimane-u.ac.jp
- Naoto Yamaoka
- Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
- MR Author ID: 688560
- Email: yamaoka@math.shimane-u.ac.jp
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1933341