Multiple positive solutions for a superlinear elliptic problem in R2R2 with a sublinear Neumann boundary condition

Let Ω⊂R2Ω⊂R2 be a bounded domain with C2C2 boundary. Let u∈H1(Ω)u∈H1(Ω) be a weak solution of the following problem: equation(Pλ){−Δu+u=p(u)euαu>0}in Ω,∂u∂ν=λψuqon ∂Ω, where α∈(0,2],λ>0,q∈[0,1)α∈(0,2],λ>0,q∈[0,1) and ψ≥0ψ≥0, a Hölder continuous function on Ω¯. Here p(u)p(u) is a polynomial perturbation of euα as u→∞u→∞. Using variational methods we show that there exists 0<Λ<∞0<Λ<∞ such that (Pλ)(Pλ) admits at least two solutions if λ∈(0,Λ)λ∈(0,Λ), no solution if λ>Λλ>Λ and at least one solution when λ=Λλ=Λ.