Bifurcation properties of semilinear elliptic equations in $\bf R^n$
Allan L. Edelson, Adolfo J. Rumbos · Differential and Integral Equations · 1994
We study the bifurcation properties of the semilinear equation ~u+Af(x)(u+h(u))=O, xERn, where h : R -+ R is a bounded Holder continuous function satisfying lim h(~) =a> 0 E-+O+ ~ ' and f : Rn -+ R is a positive asymptotically radial function satisfying JRn f(x) dx 0 as JxJ -+ oo.Such a branch bifurcates from infinity at AI, where AI is the principal eigenvalue of the linear equation ~u + Aj(x)u = 0, and from the trivial solutions at Ao = AI/ (1 +a).We use the Leray-Schauder degree theory applied to the corresponding operator equations, and a global bifurcation result of Rabinowitz.