Multiple solutions for the fractional \(p\)-Laplacian equation with Hardy-Sobolev exponents
The authors study the fractional \(p\)-Laplacian problem with Hardy-Sobolev exponents \[ \left\{ \begin{array}{ll} -\Delta_p^s u = \lambda |h|^{r-2} u + \frac{|u|^{p_{s, \alpha^{-2}}^*} u}{|x|^\alpha} & \text{in }\Omega, \\ u=0 & \text{on }\partial\Omega , \end{array} \right. \] where \(\Omega \subseteq \mathbb{R}^N\) be a bounded domain containing the origin, \(p\in (1,\infty)\), \(s\in (0,1)\), \(1<r<p\), \(0\leq \alpha < sp < N\), and \(\lambda > 0\) is a parameter, \(p^*_{s, \alpha} = ((N - \alpha)p)/(N- ps)\) is the fractional Hardy-Sobolev exponent. The authors prove that there is a \(\lambda_0 > 0\) such that for any \(\lambda \in (0, \lambda_0)\), the above problem possesses infinitely many solutions. They use variational methods, more specifically, the Nehari manifold and Ljusternik-Schnirelmann theory.
scientific article; zbMATH DE number 7393770
scientific article