SEMILINEAR PROBLEMS PERTURBED THROUGH THE BOUNDARY CONDITION∗

Abstract. We analyze the semilinear diffusion equation ∆u = a(x)up in a smooth bounded domain Ω subjected to the boundary condition ∂u/∂ν = λu, where ν is the outward unit normal to ∂Ω, λ is a real parameter. The coefficient a(x) is a nonnegative weight function, which could even vanish in a whole smooth subdomain Ω0 of Ω. We consider both cases p> 1 and 0 < p < 1, and give a detailed description of the existence, uniqueness or multiplicity and asymptotic behavior of nonnegative solutions. As an additional special feature, the adherence of the portions Ω ∩ ∂Ω0, ∂Ω ∩ ∂Ω0 of the boundary of Ω0 are allowed to meet each other in a smooth manifold. Key words. Bifurcation, Steklov problem, boundary blow-up, perturbation of domains. AMS subject classifications. 35J25, 35B40. 1. Introduction. I