Uniqueness and asymptotic behavior of positive solution of quasilinear elliptic equations with Hardy potential

In this paper, we study the uniqueness and singular behavior at the origin of positive solution to -Delta(p)u = lambda u(p-1)/vertical bar x vertical bar p - vertical bar x vertical bar(sigma) f(u), x is an element of Omega\{0}, where N > p > 1, sigma > -p, 0 is an element of Omega and Omega subset of R-N( N >= 3) is a domain. For the nonlinear term f(u), we suppose that lim(u ->infinity) f(u)/u(q) = a > 0 and lim(u -> 0)+ f(u)/u(s) = b. When Omega = R-N, sigma >= 0 and sigma > N(s-p+1)- s/ p-1, we discuss the uniqueness of positive solution. When Omega is a bounded smooth domain, sigma >= 0 and sigma > N(q-p+1)-pq/p-1, we prove that each positive solution u(x) satisfies lim(vertical bar x vertical bar -> 0) u(x)vertical bar x vertical bar p+sigma/q-p+1 = (lambda/a + (p + sigma/q - p + 1)(p-1) [(1 + p + sigma)/q - p + 1) p - 1/a - N - 1 a])(1/) q-(p+1), and the equation with the boundary condition u = phi >= 0 on partial derivative Omega has a unique positive solution. When Omega = B-R (0) and sigma >= 0, the uniqueness of positive solution to the corresponding Dirichlet problem is also established. (c) 2020 Elsevier Ltd. All rights reserved.