Existence of Two Boundary Blow-Up Solutions for Semilinear Elliptic Equations

In this paper we consider the boundary blow-up problemΔu=f(u) in Ω, u(x)→∞ as x→∂Ω,and its non-autonomous version in a bounded, convexC2-domainΩof N. We give growth conditions onfat ±∞ which imply the existence of two distinct blow-up solutions. The cases, (a)fhas a zero, and (b) min f>0, are fundamentally different. In case (a) we have a positive and a sign-changing blow-up solution. In case (b) we introduce a bifurcation parameterλinto the equationΔu=λf(u) and show that for 0<λ<λcritthere are blow-up solutions and forλ>λcritthere is no blow-up solution.