Sarah Almutairi, Kamel Saoudi
Abstract:
This article studies a fractional elliptic equation that involves a critical
Hardy-Sobolev nonlinearity along with a singular term,
$$\displaylines{
(-\Delta_p)^s u = \lambda f(x) u^{-\gamma} \pm \frac{g(x)|u|^{p_s^*(t)-2}u}{|x|^t}
\quad\text{in }\Omega ,\cr
u > 0 \quad \text{in } \Omega, \cr
u = 0 \quad\text{in }\mathbb{R}^N\setminus\Omega,
}$$
where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with a smooth boundary
\(\partial \Omega\), and \(0 \in \Omega\). The dimension \(N\) satisfies \(N > sp\),
\(s \in (0,1)\), \(\lambda > 0\), \(0 < \gamma < 1\), and \(p^*_s(t) = \frac{p(N-t)}{N-sp}\)
represent the critical Hardy-Sobolev exponent.
The weight functions \(f\) and \(g\) are elements of \(L^\infty(\Omega)\) and satisfy specific
positivity conditions, and
\( (-\Delta_p)^s u \) is the fractional \(p\)-Laplacian operator.
We use the method of sub- and super-solutions combined with monotonicity arguments,
to establish the existence and nonexistence of solutions. Furthermore, we prove that
any weak solution is locally H\"older continuous.
Submitted May 12, 2025. Published October 20, 2025.
Math Subject Classifications: 34B15, 37C25, 35R20.
Key Words: Nonlocal operator; singular nonlinearity; existence and nonexistence of solutions;
sub- and supersolutions; monotonicity arguments.
DOI: 10.58997/ejde.2025.99
Show me the PDF file (409 KB), TEX file for this article.
| Sarah Almutairi College of sciences at Dammam Imam Abdulrahman Bin Faisal University 31441 Dammam, Kingdom of Saudi Arabia email: 2240500248@iau.edu.sa | |
 |
Kamel Saoudi College of sciences at Dammam Imam Abdulrahman Bin Faisal University 31441 Dammam, Kingdom of Saudi Arabia email: kmsaoudi@iau.edu.sa |
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