Normalized solutions for Schrödinger-Poisson-Slater equations with unbounded potentials

We study the following minimization problem(P)mp(ρ)=inf⁡{Ep(u):u∈Sρ}, where Ep(u) is the Schrödinger-Poisson-Slater functional with unbounded potentialEp(u)=12∫R3(|∇u|2+V(x)|u|2)dx+14∫R3∫R3|u(x)|2|u(y)|2|x−y|dxdy−1p+2∫R3|u|p+2dx, and the constraint Sρ is given bySρ={u:∫R3|u|2dx=ρ and ∫R3|∇u|2+V(x)|u|2dx<∞}. We prove that, when 00; when p=p⁎, (P) is attained if and only if ρ≤ρp⁎ with ρp⁎>0 given by (1.11). Moreover, for each fixed ρ>ρp⁎, we show the concentration behavior of minimizers as the exponent p tends from below to the critical exponent p⁎.