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 i u t + Δ u = λ | u | p + μ | ∇ u | q + w ( x ) , t > 0 , x ∈ R N , where N ≥ 1 , p , q > 1 , λ , μ ∈ C , λ ≠ 0 , and u ( 0 , · ) , w ∈ L loc 1 ( R N , 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.