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

Let , N 2, be a bounded domain. We consider the following quasilinear problem depending on a real parameter λ>0: where f(t) is a nonlinearity that grows like etN/N−1 as t→∞ and behaves like tα, for some α (0,N−1), as t→0+. More precisely, we require f to satisfy assumptions (A1)–(A5) in Section 1. With these assumptions we show the existence of Λ>0 such that (Pλ) admits at least two solutions for all λ (0,Λ), one solution for λ=Λ and no solution for all λ>Λ. We also study the problem (Pλ) posed on the ball 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.