Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 13, pp. 1-17.

Title: Existence of solutions to fractional p-Laplacian problems with Robin boundary conditions

Authors: Junhui Xie (Hubei Univ. of Education, Wuhan, China)
         Pengfei Li (Fuzhou Univ., Fujian, China)
  
Abstract:
This article  studies the existence of solutions for the 
fractional p-Laplacian problem
$$\displaylines{
(-\Delta)_p^su=\lambda|u|^{q-2}u+ \frac{|u|^{r-2}u}{|x|^\alpha},
\quad \hbox{in } \Omega,\cr
{N}_{s,p}u(x)+\beta(x)|u|^{p-2}u=0, 
\quad\hbox{in }\mathbb{R}^n\backslash\Omega,
}$$
where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^n$ containing 
$0$ with smooth boundary, $(-\Delta)_p^s$ denotes the fractional p-Laplace 
operator and $\lambda>0$, $1<q<p<r<p_\alpha^*$, $p_\alpha^*$ is the 
fractional critical Hardy-Sobolev exponent for $0\leq \alpha<ps<n$ and 
$0<s<1$. By using fibering maps and Nehari manifold, we obtain the 
existence of solution for Hardy-Sobolev 
subcritical and critical cases.


Submitted September 28, 2024. Published February 18, 2025.
Math Subject Classifications: 35R11, 35S15, 35A15, 47G20.
Key Words: Fractional p-Laplacian; Nehari manifold; Robin boundary; Hardy-Sobolev exponent.