New competition phenomena in Dirichlet problems

Alexandru Kristály, Gheorghe MoroŞanu

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

We study the multiplicity of nonnegative solutions to the problem,. (Pλ)-Δu=λa(x)up+f(u)inΩ,u=0on ∂Ω, 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 H01- and L-norm estimates of solutions are also given.

Original languageEnglish
Pages (from-to)555-570
Number of pages16
JournalJournal des Mathematiques Pures et Appliquees
Volume94
Issue number6
DOIs
Publication statusPublished - Dec 1 2010

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Keywords

  • Critical and supercritical growth
  • Dirichlet problem
  • Indefinite potential
  • Oscillatory nonlinearity
  • Variational method

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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