Blow-up in exterior domains: existence and star-shapedness

The authors consider explosive solutions of the elliptic equations div(g(|∇u|)∇u)=f(u)k(|∇u|) defined in a domain Ω⊂RN, N≥2, satisfying the boundary condition limx→∞u(x)=+∞. Under the following main hypotheses: f∈C1((t0,+∞),R+), f(t+0)=0, f(+∞)=+∞, f′≥0, f′(t+0)<∞; g∈C2([0,+∞),R+), G(t):=(g(t)t)′>0 for all t≥0; k∈C1([0,∞),R+), they prove existence in smooth bounded domains, as well as in exterior domains under natural assumptions, and that certain solutions in an exterior star-shaped domain have star-shaped level sets. They also show the necessity of a certain condition in order that an explosive solution exists.