Combined effects of critical Hardy-Sobolev exponent and singular nonlinearities in nonlocal problems with variable weights

This article studies a fractional elliptic equation that involves a critical Hardy-Sobolev nonlinearity along with a singular term, (-ₚ) ˢ u = f (x) u^- g (x) |u|^{pₛ^* (t) -2u|x|ᵗ, u > 0 in, u = 0 RN, } where \ (\) is a bounded domain in \ (RN\) with a smooth boundary \ (\), and |90 \). The dimension \ (N\) satisfies \ (N > sp\), \ (s (0, 1) \), \ (> 0\), \ (0 < < 1\), and \ (p^*ₛ (t) = p (N-t) N-sp\) represent the critical Hardy-Sobolev exponent. The weight functions \ (f\) and \ (g\) are elements of \ (L^ () \) and satisfy specific positivity conditions, and \ ( (-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 Holder continuous. For more information and the latex file, see https: //ejde. math. txstate. edu/Volumes/2025/99/abstr. html