New competition phenomena in Dirichlet problems

TitleNew competition phenomena in Dirichlet problems
Publication TypeJournal Article
AuthorsKristály, Alexandru, and Gheorghe Morosanu
Journal titleJournal de Mathematiques Pures et Appliquees

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.

Publisher link
Department of Mathematics and its Applications
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