A note on a nonlinear elliptic problem with a nonlocal coefficient

In this paper, we investigate a nonlocal and nonlinear elliptic problem,equation(P){−a(∫Ω|∇u|2dx)Δu=λu+up in Ω,u=0 on ∂Ω, where N≤3N≤3, Ω⊂RNΩ⊂RN is a bounded domain with smooth boundary ∂Ω, a   is a nondegenerate continuous function, p>1p>1 and λ∈Rλ∈R. We show several effects of the nonlocal coefficient a   on the structure of the solution set of (P). We first introduce a scaling observation and describe the solution set by using that of an associated semilinear problem. This allows us to get unbounded continua of solutions (λ,u)(λ,u) of (P). A rich variety of new bifurcation and multiplicity results are observed. We also prove that the nonlocal coefficient can induce up to uncountably many solutions in a convenient way. Lastly, we give some remarks from the variational point of view.