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Bifurcation of positive entire solutions for a semilinear elliptic equation

Published online by Cambridge University Press:  17 April 2009

Tsing-San Hsu  and
Huei-Li Lin
Tsing-San Hsu
Affiliation:
Center for General Education Chang Gung University, Kwei-San, Tao-Yuan 333, Taiwan, Republic of China
Huei-Li Lin
Affiliation:
Center for General Education Chang Gung University, Kwei-San, Tao-Yuan 333, Taiwan, Republic of China
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In this paper, we consider the nonhomogeneous semilinear elliptic equation

,

where λ ≥ 0, 1 < p < (N + 2)/(N − 2), if N ≥ 3, 1 < p < ∞, if N = 2, h(x) ∈ H−l(ℝN), 0 ≢ h(x) ≥ 0 in ℝN, K(x) is a positive, bounded and continuous function on ℝN. We prove that if K(x) ≥ K∞ > 0 in ℝN, and lim∣x∣⃗∞K(x) = K∞, then there exists a positive constant λ✶ such that (✶)λ has at least two solutions if λ ∈ (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✶.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 3 , December 2005 , pp. 349 - 370
Copyright
Copyright © Australian Mathematical Society 2005

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