Semilinear elliptic equations with Hardy potential and subcritical source term

Original · EN

Let Ω be a smooth bounded domain in Rⁿ and δ(x)=dist(x,∂ Ω). Assume μ>0, ν is a nonnegative finite measure on ∂ Ω and g ∈ C(Ω× R+). We study positive solutions of (P) -Δu - μδ² u = g(x,u) in Ω, tr*(u)=ν. Here tr*(u) denotes the normalized boundary trace of u which was recently introduced by M. Marcus and P. T. Nguyen. We focus on the case 0<μ< Cₕ(Ω) (the Hardy constant for Ω) and provide some qualitative properties of solutions of (P). When g(x,u)=uq with q>1, we prove that there is a critical value q* (depending only on N, μ) for (P) in the sense that if 1<q<q* then (P) admits a solution under a smallness assumption on ν, but if q ≥ q* this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where g is subcritical. We also investigate the case where the g is linear or sublinear and give some existence results for (P).