Bifurcation properties of semilinear elliptic equations in Rn


We study the bifurcation properties of the semilinear equation Δu + λf(x)(u+h(u))=0, x ∈ Rn, where h: R → R is a bounded Hölder continuous function satisfying and f: Rn → R is a positive asymptotically radial function satisfying ∫Rn f(x) dx < ∞. We show the existence of a connected branch of positive solutions, u satisfying |x|n-2u(x)→ c > 0 as |x| → ∞. Such a branch bifurcates from infinity at λ1, where λ1 is the principal eigenvalue of the linear equation Δu + λ f(x)u = 0, and from the trivial solutions at Lambda;0 = λ1/(1 +a). We use the Leray-Schauder degree theory applied to the corresponding operator equations, and a global bifurcation result of Rabinowitz. © 1994, Khayyam Publishing.

Publication Title

Differential and Integral Equations

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