On the existence threshold for positive solutions of p-Laplacian equations with a concave–convex nonlinearity

Abstract

We study the following boundary value problem with a concave–convex nonlinearity: [Formula: see text] Here Ω ⊂ ℝnis a bounded domain and 1 < q < p < r < p*. It is well known that there exists a number Λq, r> 0 such that the problem admits at least two positive solutions for 0 < Λ < Λq, r, at least one positive solution for Λ = Λq, r, and no positive solution for Λ > Λq, r. We show that [Formula: see text] where λ1(p) is the first eigenvalue of the p-Laplacian. It is worth noticing that λ1(p) is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case q = p.