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Remarks on the Range Properties of Certain Semilinear Problems of Landesman–Lazer Type

✍ Scribed by Pavel Drábek; Petr Girg; Francisco Roca

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
90 KB
Volume
257
Category
Article
ISSN
0022-247X

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✦ Synopsis

where g is a continuous function having finite limits at plus and minus infinity:

Ž . Ž . g yϱg qϱ . Imposing g yϱg sg qϱ for any s g ‫ޒ‬ we formulate a necessary condition. Our main result provides a characteristic of the set of w x Ž . functions f g C 0, T , such that 1 has a solution. However, to prove this we need that g is odd, increasing, and satisfies a certain asymptotic condition at infinity, in addition. The main tools used are Lyapunov᎐Schmidt reduction and certain asymptotical methods.

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It is proved that the singular semilinear elliptic equation y⌬u s p x g u , Ž . n Ž . 1 ŽŽ . Ž .. lim g s s qϱ, and g g C 0, ϱ , 0, ϱ which is s ª 0 Ž . 2q␣ Ž n . strictly decreasing in 0, ϱ , has a unique positive C R solution that decays to l o c ϱ Ž . Ž . Ž . zero near ϱ provided H t t dt -ϱ, w