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

Sugie, Jitsuro; Yamaoka, Naoto

doi:10.1090/S0002-9939-02-06681-9

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.

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