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

Shinji Adachi, Kazunaga Tanaka

    Research output: Contribution to journalArticle

    76 Citations (Scopus)

    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
    Publication statusPublished - 2000 Aug

    Fingerprint

    Semilinear Elliptic Problem
    Semilinear Elliptic Equations
    Existence of Positive Solutions
    Positive Solution

    ASJC Scopus subject areas

    • Mathematics(all)
    • Analysis
    • Applied Mathematics

    Cite this

    @article{dedfc50792b74116958d07317d0893d8,
    title = "Four positive solutions for the semilinear elliptic equation: - Δu + u = a(x)up + f(x) in ℝN",
    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.",
    author = "Shinji Adachi and Kazunaga Tanaka",
    year = "2000",
    month = "8",
    language = "English",
    volume = "11",
    pages = "63--95",
    journal = "Calculus of Variations and Partial Differential Equations",
    issn = "0944-2669",
    publisher = "Springer New York",
    number = "1",

    }

    TY - JOUR

    T1 - Four positive solutions for the semilinear elliptic equation

    T2 - - Δu + u = a(x)up + f(x) in ℝN

    AU - Adachi, Shinji

    AU - Tanaka, Kazunaga

    PY - 2000/8

    Y1 - 2000/8

    N2 - 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.

    AB - 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.

    UR - http://www.scopus.com/inward/record.url?scp=0004435491&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0004435491&partnerID=8YFLogxK

    M3 - Article

    AN - SCOPUS:0004435491

    VL - 11

    SP - 63

    EP - 95

    JO - Calculus of Variations and Partial Differential Equations

    JF - Calculus of Variations and Partial Differential Equations

    SN - 0944-2669

    IS - 1

    ER -