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On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on R-N

Abstract : Using the 'monotonicity trick: introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais-Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the form [GRAPHICS] We assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s(-1) --> a is an element of]0, infinity] as s --> + infinity; and (ii) f(x, s)s(-1) is non decreasing as a function of s greater than or equal to 0, a.e. x is an element of R-N.
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https://hal-upec-upem.archives-ouvertes.fr/hal-00693898
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  • HAL Id : hal-00693898, version 1

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L Jeanjean. On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on R-N. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Cambridge University Press (CUP), 1999, 129 (?), pp.787--809. ⟨hal-00693898⟩

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