Concentration for an elliptic equation with singular nonlinearity

AbstractWe are interested in nontrivial solutions of the equation:−Δu+χ[u>0]u−β=λup,u⩾0inΩ, with u=0 on ∂Ω, where Ω⊂RN, N⩾2, is a bounded domain with smooth boundary, 0<β<1, 1⩽p<N+2N−2 if N⩾3 (p⩾1 if N=2) and λ>0. If p>1 we prove existence of nontrivial solutions for every λ>0. As λ→+∞ we find that the least energy solutions concentrate around a point that maximizes the distance to the boundary. We also study the behavior as λ→0. When p=1 we have similar results, extending previous works for radial solutions in a ball