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New competition phenomena in Dirichlet problems
Title / Series / Name
Journal de Mathematiques Pures et Appliquees
Publication Volume
94
Publication Issue
Pages
Editors
Keywords
URI
https://hdl.handle.net/20.500.14018/10293
Abstract
We study the multiplicity of nonnegative solutions to the problem, (Pλ) where Ω is a smooth bounded domain in RN, f:[0,∞)→R oscillates near the origin or at infinity, and p>0, λ∈R. While oscillatory right-hand sides usually produce infinitely many distinct solutions, an additional term involving up may alter the situation radically. Via a direct variational argument we fully describe this phenomenon, showing that the number of distinct non-trivial solutions to problem (Pλ) is strongly influenced by up and depends on λ whenever one of the following two cases holds:
•p⩽1 and f oscillates near the origin;
•p⩾1 and f oscillates at infinity (p may be critical or even supercritical).
The coefficient a∈L∞(Ω) is allowed to change its sign, while its size is relevant only for the threshold value p=1 when the behaviour of f(s)/s plays a crucial role in both cases. Various - and L∞-norm estimates of solutions are also given.
Topic
Publisher
Place of Publication
Type
Journal article
Date
2010