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Dany NABAB : Multiplicity of solutions for a class of strongly non-linear PDEs



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The study of elliptic PDEs of type [ ] is an area of research that has seen a sharp increase in activity in recent years, by virtue of the ability of these problems to model complex scientific phenomena, particularly when pi : Ω → R+ and fi : Ω × R × R → R are strongly nonlinear functions, which paradoxically makes them very difficult to tackle from a mathematical point of view.  In this respect, to date there is little work in the literature that addresses the questions of the existence and multiplicity of solutions for this type of problem, given that most analytical methods based on algebraic or variational considerations are not applicable if the assumptions on pi and fi are too general. At this stage, only methods based on topological considerations,such as the fixed point method and the topological degree method, are best suited to meet this challenge.

The aim of this seminar is to present a new application of Amann's fixed point index, which is an extension of Leray-Schauder's topological degree, in order to prove that if a fixed point problem of the form [] where E is a Banach space, admits at least an odd number N of non-trivial solutions, then under certain conditions on the functional F, it necessarily admits at least one additional solu- tion which is also non-trivial.  This result will subsequently be used to prove the existence of at least two non-trivial solutions for the Hammerstein equation associated with the initial problem, the existence of the first non-trivial solution already being guaranteed following the results obtained in [1].

 

 

[1]    A. Moussaoui, D. Nabab, J. Vélin, Singular quasilinear convective systems involving variable exponents, Opuscula Mathematica  44 (2024),  pp. 105-134.
[2]    H. Amann,Lectures on some fixed point theorem, Conselho nacional de pesquisas, Instituto de Matematica Pura e  Aplicada, 1975.
[3]    H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM review, 18(4),  620-709,  1976.

 

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