Multiplicity of Solutions For a Convex-concave Problem With a Nonlinear Boundary Condition

Abstract

Abstract We study the existence of multiple positive solutions for a convex-concave problem, denoted by (Pλμ), with a nonlinear boundary condition involving two critical exponents and two positive parameters λ and μ. We obtain a continuous strictly decreasing function f such that K1 ≡ {(f(μ), μ) : μ ∈ [0, ∞)} divides [0, ∞) × [0, ∞) \ {(0, 0)} in two connected sets K0 and K2 such that problem (Pλμ) has at least two solutions for (λ, μ) ∈ K2, at least one solution for (λ, μ) ∈ K1 and no solution for (λ, μ) ∈ K0.