Monotonicity of the first eigenvalue and the global bifurcation diagram for the branch of interior peak solutions

Let B⊂RnB⊂Rn (n⩾1n⩾1) be a unit ball and D⊂RmD⊂Rm be a bounded domain. We study the global branch consisting of interior single-peak solutions of the elliptic Neumann problemΔu+λ(−u+up)=0in B,∂νu=0on ∂B and the monotonicity of the first eigenvalue along the branch for large λ, where1