## Abstract

In this paper, we consider the nonhomogeneous semilinear elliptic equation (*)λ -△u + u = λK(x)u^{p} + h(x) in ℝ ^{N},u ε H^{1} in (ℝ^{n}) where λ ≥ 0, 1 < p < (N + 2)/(N - 2), if N ≥ 3, 1 < p < ∞, if N = 2, h(x) ε H^{-1}(ℝ^{N}), 0 ≢ h(x) ≥ 0 in ℝ^{N}, K(x) is a positive, bounded and continuous function on ℝ^{N}. We prove that if K(x) ≥ K_{∞} > 0 in ℝ^{N}, and lim K(x) = K∞, |x|-∞ then there exists a positive constant λ* such that (*)λ has at least two solutions if λ π (0, λ*) and no solution if λ > λ*. Furthermore, (*)λ has a unique solution for λ = λ* provided that h(x) satisfies some suitable conditions. We also obtain some further properties and bifurcation results of the solutions of (1.1)λ at λ = λ*. Copyright Clearance Centre, Inc.

Original language | English |
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Pages (from-to) | 349-370 |

Number of pages | 22 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 72 |

Issue number | 3 |

DOIs | |

State | Published - 12 2005 |