Multiple positive solutions for a superlinear elliptic problem in image with a sublinear Neumann boundary condition Original Research Article

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) View the MathML source{−Δu+u=p(u)euαu>0}in Ω,∂u∂ν=λψuqon ∂Ω, Turn MathJax on 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 View the MathML sourceΩ¯. Here p(u)p(u) is a polynomial perturbation of View the MathML sourceeuα 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.