First and second critical exponents for an inhomogeneous Schrödinger equation with combined nonlinearities

We study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation iut+Δu=λ|u|p+μ|∇u|q+w(x),t>0,x∈RN,where N≥ 1 , p, q> 1 , λ, μ∈ C, λ≠ 0 , and u(0,·),w∈Lloc1(RN,C). We consider both the cases where μ= 0 and μ≠ 0 , respectively. We establish existence/nonexistence of global weak solutions. In each studied case, we compute the critical exponents in the sense of Fujita, and Lee and Ni. When μ≠ 0 , we show that the nonlinearity | ∇ u| q induces an interesting phenomenon of discontinuity of the Fujita critical exponent.