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

TL;DR

This paper investigates the existence threshold for positive solutions of p-Laplacian equations with a concave-convex nonlinearity, showing that the critical parameter converges to the first eigenvalue of the p-Laplacian as the nonlinearity approaches the linear case.

Contribution

It establishes that the critical parameter for solution existence converges to the first eigenvalue of the p-Laplacian as the nonlinearity exponent approaches p.

Findings

The threshold parameter mbda_{q,r} converges to mbda_1(p) as q approaches p.

Existence of multiple positive solutions for mbda < mbda_{q,r}.

No positive solutions for mbda > mbda_{q,r}.

Abstract

We study the following boundary value problem with a concave-convex nonlinearity: \begin{equation*} \left\{ \begin{array}{r c l l} -\Delta_p u & = & \Lambda\,u^{q-1}+ u^{r-1} & \textrm{in }\Omega, \\ u & = & 0 & \textrm{on }\partial\Omega. \end{array}\right. \end{equation*} Here $\Omega \subset \mathbb{R}^n$ is a bounded domain and $1<q<p<r<p^*$. 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 \[ \lim_{q \to p} \Lambda_{q,r} = \lambda_1(p), \] 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 $q=p$.