Multiple solutions for a Neumann system involving subquadratic nonlinearities

In this paper, we consider the model semilinear Neumann system equation(NλNλ){−Δu+a(x)u=λc(x)Fu(u,v)inΩ,−Δv+b(x)v=λc(x)Fv(u,v)inΩ,∂u∂ν=∂v∂ν=0on∂Ω, where Ω⊂RNΩ⊂RN is a smooth open bounded domain, νν denotes the outward unit normal to ∂Ω∂Ω, λ≥0λ≥0 is a parameter, a,b,c∈L+∞(Ω)∖{0}, and F∈C1(R2,R)∖{0}F∈C1(R2,R)∖{0} is a nonnegative function which is subquadratic at infinity. Two nearby numbers are determined in explicit forms, λ¯ and λ¯ with 0<λ¯≤λ¯, such that for every 0≤λ<λ¯, system (Nλ)(Nλ) has only the trivial pair of solution, while for every λ>λ¯, system (Nλ)(Nλ) has at least two distinct nonzero pairs of solutions.