EXISTENCE, NONEXISTENCE, AND DEGENERACY OF LIMIT SOLUTIONS TO p-LAPLACE PROBLEMS INVOLVING HARDY POTENTIALS AS p → 1 +

In this paper we analyze the asymptotic behavior as p -> 1(+) of solutions u(p) to {-Delta(p)u(p) = lambda /| x| (p)| u(p)| (p - 2)u(p)+f in Omega, up=0 on partial derivative Omega, where Omega is a bounded open subset of R-N with Lipschitz boundary, lambda is an element of R+, and f is a nonnegative datum in L (N, infinity) (Omega). Under sharp smallness assumptions on the data lambda and f, we estimate the family ( u(p) ) (p>1) uniformly in BV (Omega) and then we let p -> 1(+) in order to completely characterize its limit u. As a consequence of this limit procedure, we prove that u suitably solves the homogeneous Dirichlet problem { Delta (1)u=lambda/ | x| Sgn(u) +f in Omega, where, u=0 on partial derivative Omega, Delta (1)u = div( Du/ IDuI ) is the 1-Laplace operator. The main assumptions are further discussed through explicit examples in order to show their optimality.