Bulletin

Processing math: 100%

Volume 37 • Number 4 • 2014

• On a Class of Degenerate Nonlocal Problems with Sign-Changing Nonlinearities
Nguyen Thanh Chung and Hoang Quoc Toan

Abstract.
Using variational techniques, we study the nonexistence and multiplicity of solutions for the degenerate nonlocal problem

$\begin{equation*} \begin{cases} \begin{array}{rlll} - M\left(\int_\Omega |x|^{-ap}|\nabla u|^pdx\right)\operatorname{div}\left(|x |^{-ap}|\nabla u|^{p-2}\nabla u\right) &= \lambda |x|^{-p(a+1)+c} f(x,u) & \text{ in } \Omega,\\ u &= 0 & \text{ on } \partial\Omega, \end{array} \end{cases} \end{equation*}$ where

$\Omega \subset \mathbb{R}^N$ (

$N \geq 3$ ) is a smooth bounded domain,

$0 \in \Omega$ ,

$0 \leq a < \frac{N-p}{p}$ ,

$1 < p < N$ ,

$c > 0$ ,

$M: \mathbb{R}^+\to \mathbb{R}^+$ is a continuous function that may be degenerate at zero,

$f:\Omega \times \mathbb{R}\to \mathbb{R}$ is a sign-changing Carath\'eodory function and

$\lambda$ is a parameter.

2010 Mathematics Subject Classification: 35D35, 35J35, 35J40, 35J62

Full text: PDF