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: \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 $q=p$

THE NEUMANN PROBLEM FOR THE ∞-LAPLACIAN AND THE MONGE-KANTOROVICH MASS TRANSFER PROBLEM

Abstract. We consider the natural Neumann boundary condition for the ∞-Laplacian. We study the limit as p → ∞ of solutions of −∆pup = 0 in a domain Ω with |Dup | p−2 ∂up/∂ν = g on ∂Ω. We obtain a natural minimization problem that is verified by a limit point of {up} and a limit problem that is satisfied in the viscosity sense. It turns out that the limit variational problem is related to the Monge-Kantorovich mass transfer problems when the measures are supported on ∂Ω. 1. Introduction. In this paper we study the natural Neumann boundary conditions that appear when one considers the ∞-Laplacian in a smooth bounded domain as limit of the Neumann problem for the p-Laplacian as p → ∞. This problem is related to the Monge-Kantorovich mass tranfer problem when the involved measures are supporte