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

We study the following boundary value problem with a concave–convex nonlinearity: $$\left\{\begin{array}{@{}l@{\quad}l@{}} -\Delta_{p} u = \Lambda u^{q-1} + u^{r-1} & \textrm{in } \Omega,\\[3pt] u = 0 & \textrm{on } \partial \Omega.\end{array}\right.$$ Here Ω ⊂ ℝn is a bounded domain and 1 0 such that the problem admits at least two positive solutions for 0 Λq, r. We show that $$\lim_{q \to p} \Lambda_{q,r} = \lambda_{1}(p),$$ 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.