On a Class of Degenerate Nonlocal Problems with Sign-Changing Nonlinearities

Nguyen Thanh Chung ; Hoang Quoc Toan

Bulletin of the Malaysian Mathematical Society, Tome 37 (2014) no. 4

Résumé

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.

Détail

Classification : 35D35, 35J35, 35J40, 35J62