### Abstract

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

Original language | English |
---|---|

Pages (from-to) | 63-95 |

Number of pages | 33 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 11 |

Issue number | 1 |

Publication status | Published - 2000 Aug |

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### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

**Four positive solutions for the semilinear elliptic equation : - Δu + u = a(x)u ^{p} + f(x) in ℝ^{N}.** / Adachi, Shinji; Tanaka, Kazunaga.

Research output: Contribution to journal › Article

^{p}+ f(x) in ℝ

^{N}',

*Calculus of Variations and Partial Differential Equations*, vol. 11, no. 1, pp. 63-95.

}

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.

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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 -