Four positive solutions for the semilinear elliptic equation: - Δu + u = a(x)up + f(x) in ℝN

Shinji Adachi, Kazunaga Tanaka

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Abstract

We consider the existence of positive solutions of the following semilinear elliptic problem in ℝN: (Formula Presented) where 1 < p < N + 2/N - 2 (N ≥ 3), 1 < p < ∞ (N = 1, 2), a(x) ∈ C(ℝN), f(x) ∈ H-1 (ℝN) and f(x) ≥ 0. Under the conditions: 1° a(x) ∈ (0, 1) for all x ∈ ℝN, 2° a(x) → 1 as |x| → ∞, 3° there exist δ > 0 and C > 0 such that a(x) - 1 ≥ -Ce-(2+δ)|x| for all x ∈ ℝN, 4° a(x) ≢ 1, we show that (*) has at least four positive solutions for sufficiently small ||f||H-1(ℝN) but f ≢ 0.

Original languageEnglish
Pages (from-to)63-95
Number of pages33
JournalCalculus of Variations and Partial Differential Equations
Volume11
Issue number1
DOIs
Publication statusPublished - 2000 Aug

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ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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