Infinitely many sign-changing solutions of the Neumann problem in some FLRW spacetimes

Abstract

We are concerned with the Neumann problem in some FLRW spacetimes(P){div(∇uf(u)f(u)2−|∇u|2)+f′(u)f(u)2−|∇u|2(N+|∇u|2f(u)2)=λNg(|x|,u)inB(R),|∇u|<f(u)inB(R),∂u∂ν=0on∂B(R), where ∂u∂ν denotes the outward normal derivative of u, B(R) is the Euclidean ball RN, N≥1, centered at 0 with radius R, f∈C2(I) is a positive function, I⊆R is an open interval containing 0, g:[0,R]×R→R is continuous, λ>0 is a parameter. We show that (P) has infinitely many radially symmetric sign-changing solutions under some appropriate conditions. The proof of our main result is based upon bifurcation techniques.