Nontrivial solutions for p-Laplacian systems

D. D. Hai, Haiyan Wang

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

Abstract

The paper deals with the existence and nonexistence of nontrivial nonnegative solutions for the sublinear quasilinear systemdiv (| ∇ ui |p - 2 ∇ ui) + λ fi (u1, ..., un) = 0 in  Ω, ui = 0 on  ∂ Ω, i = 1, ..., n, where p > 1, Ω is a bounded domain in RN (N ≥ 2) with smooth boundary, and fi, i = 1, ..., n, are continuous, nonnegative functions. Let u = (u1, ..., un), {norm of matrix} u {norm of matrix} = ∑i = 1n | ui |, we prove that the problem has a nontrivial nonnegative solution for small λ > 0 if one of lim{norm of matrix} u {norm of matrix} → 0 frac(fi (u), {norm of matrix} u {norm of matrix}p - 1) is infinity. If, in addition, all lim{norm of matrix} u {norm of matrix} → ∞ frac(fi (u), {norm of matrix} u {norm of matrix}p - 1) is zero, we show that the problem has a nontrivial nonnegative solution for all λ > 0. A nonexistence result is also obtained.

Original languageEnglish (US)
Pages (from-to)186-194
Number of pages9
JournalJournal of Mathematical Analysis and Applications
Volume330
Issue number1
DOIs
StatePublished - Jun 1 2007

Keywords

  • Elliptic system
  • Schauder Fixed-Point Theorem
  • p-Laplacian

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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