On a class of Hardy-Sobolev critical quasilinear elliptic systems on compact Riemannian manifolds

In this work, we investigate the existence and regularity of solutions to the critical system {-Delta(p,g)u + a(x)|u|(p-2)u + b(x) [(p - 1)|u|(p-2) + |v|(p-2)]/p v = alpha/p*(s) f(x)u|u|(alpha-2)|v|(beta)/d(g)(x, x(0))(s) in M, -Delta(p,g)v + b(x) [(p - 1)|v|(p-2) + |u|(p-2)]/p u + c(x)|v|(p-2)v = beta/p*(s) f(x)v|v|(beta-2)|u|(alpha)/d(g)(x, x(0))(s) in M, where (M, g) is a smooth closed Riemannian manifold of dimension n >= 2, d(g) is the Riemannian distance, Delta(p,g) = div(g)(|del(g)u|(p-2)del(g)u) is the p-Laplace-Beltrami operator on (M, g), p is an element of(1, n), a, b, c is an element of C-0,C-rho(M) for some rho is an element of(0, 1), with b equivalent to 0 when 1 < p < 2, x(0) is an element of M, s is an element of[0, p), f is a smooth function in M with f(x0) = max(M) f > 0, and alpha > 1, beta > 1are two real numbers such that alpha + beta = p* (s), where p* (s) = p(n - s)/(n - p) denotes the critical Hardy-Sobolev exponent. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.