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: \begin{equation*} \left\{ \begin{array}{r c l l} -Δ_p u & = & Λ\,u^{q-1}+ u^{r-1} & \textrm{in }Ω, \\ u & = & 0 & \textrm{on }\partialΩ. \end{array}\right. \end{equation*} Here $\Omega \subset {\mathbb{R}}^n$ is a bounded domain and $1<q<p<r<p^{\ast }$. It is well known that there exists a number ${\Lambda }_{q,r}>0$ such that the problem admits at least two positive solutions for $0<\Lambda <{\Lambda }_{q,r}$, at least one positive solution for $\Lambda ={\Lambda }_{q,r}$, and no positive solution for $\Lambda >{\Lambda }_{q,r}$. We show that \[ \lim_{q \to p} Λ_{q,r} = λ_1(p), \] where ${\lambda }_1(p)$ is the first eigenvalue of the p-laplacian. It is worth noticing that ${\lambda }_1(p)$ is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case $q=p$.