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A Landesman–Lazer-type condition for asymptotically linear second-order equations with a singularity

Published online by Cambridge University Press:  27 November 2012

Alessandro Fonda
Dipartimento di Matematica e Geoscienze, Università di Trieste, Piazzale Europa 1, 34127 Trieste, Italy (
Maurizio Garrione
SISSA – International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy (


We consider the T-periodic problem

where g: [0,T]×]0,+∞[→ℝ exhibits a singularity of a repulsive type at the origin, and an asymptotically linear behaviour at infinity. In particular, for large x, g(t, x) is controlled from both sides by two consecutive asymptotes of the T-periodic Fučik spectrum, with possible equality on one side. Using a suitable Landesman–Lazer-type condition, we prove the existence of a solution.

Research Article
Copyright © Royal Society of Edinburgh 2012

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