A global multiplicity result for N-Laplacian with critical nonlinearity of concave-convex type

Let Ω⊂RNΩ⊂RN, N⩾2N⩾2, be a bounded domain. We consider the following quasilinear problem depending on a real parameter λ>0λ>0:(Pλ){−ΔNu=λf(u)u>0}in Ω,u=0on ∂Ω, where f(t)f(t) is a nonlinearity that grows like etN/N−1etN/N−1 as t→∞t→∞ and behaves like tαtα, for some α∈(0,N−1)α∈(0,N−1), as t→0+t→0+. More precisely, we require f   to satisfy assumptions (A1)–(A5) in Section 1. With these assumptions we show the existence of Λ>0Λ>0 such that (Pλ)(Pλ) admits at least two solutions for all λ∈(0,Λ)λ∈(0,Λ), one solution for λ=Λλ=Λ and no solution for all λ>Λλ>Λ. We also study the problem (Pλ)(Pλ) posed on the ball B1(0)⊂RNB1(0)⊂RN and show that the assumptions (A1)–(A5) are sharp for obtaining global multiplicity. We use a combination of monotonicity and variational methods to show multiplicity on general domains and asymptotic analysis of ODEs for the case of the ball.