50 research outputs found
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 is a bounded domain and . It is well known that
there exists a number such that the problem admits at least
two positive solutions for , at least one positive
solution for , and no positive solution for . We show that
where is the first eigenvalue of the p-laplacian. It is worth
noticing that is the threshold for existence/nonexistence of
positive solutions to the above problem in the limit case
A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities
THE NEUMANN PROBLEM FOR THE â-LAPLACIAN AND THE MONGE-KANTOROVICH MASS TRANSFER PROBLEM
Abstract. We consider the natural Neumann boundary condition for the â-Laplacian. We study the limit as p â â of solutions of ââpup = 0 in a domain ⊠with |Dup | pâ2 âup/âÎœ = g on ââŠ. 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 ââŠ. 1. Introduction. In this paper we study the natural Neumann boundary conditions that appear when one considers the â-Laplacian in a smooth bounded domain as limit of the Neumann problem for the p-Laplacian as p â â. This problem is related to the Monge-Kantorovich mass tranfer problem when the involved measures are supporte