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