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

We study the large-time behavior of solutions for the inhomogeneous nonlinear Schrodinger equation iu(t) + Delta u = lambda vertical bar u vertical bar(p) + mu vertical bar Delta u vertical bar(q) + omega(x), t > 0, x is an element of R-N, where N >= 1, p,q > 1, lambda,mu is an element of C, lambda not equal 0, and u(0, .),omega is an element of L-loc(1) (R-N,C). We consider both the cases where mu = 0 and mu not equal 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 mu not equal 0, we show that the nonlinearity vertical bar del u vertical bar(q) induces an interesting phenomenon of discontinuity of the Fujita critical exponent.