Measure boundary value problems for semilinear elliptic equations with critical Hardy potentials

Abstract

Let Omega subset of R-N be a bounded C-2 domain and L-kappa = -Delta - kappa/d(2) where d = dist(., partial derivative Omega) and 0 < K <= 1/4. Let alpha(+/-) = 1 +/- root 1 - 4K, lambda(kappa) the first eigenvalue of L-kappa with corresponding positive eigenfunction phi(kappa). If g is a continuous nondecreasing function satisfying integral(infinity)(1) (g(s) + 2_2f vertical bar g(-s)vertical bar)s(-2) (2N-2+alpha/2N-4+alpha+) ds < infinity, then for any Radon measures nu is an element of m(phi kappa) (Omega) and mu is an element of m (partial derivative Omega) there exists a unique weak solution to problem P nu,mu: L(kappa)u + g(u) = nu in Omega, u = mu on partial derivative Omega. If g(r) = vertical bar r vertical bar(q-1) u (q > 1), we prove that, in the supercritical range of q, a necessary and sufficient condition for solving P-0,P-mu with mu > 0 is that mu is absolutely continuous with W respect to the capacity associated with the space B2- (2+alpha+/2q)',(q)'(RN-1). We also characterize the boundary removable sets in terms of this capacity. In the subcritical range of q we classify the isolated singularities of positive solutions.