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

AbstractWe study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation $$\begin{aligned} iu_t+\Delta u=\lambda |u|^p+\mu |\nabla u|^q+w(x),\quad t>0,\, x\in {\mathbb {R}}^N, \end{aligned}$$ i u t + Δ u = λ | u | p + μ | ∇ u | q + w ( x ) , t > 0 , x ∈ R N , where $$N\ge 1$$ N ≥ 1 , $$p,q>1$$ p , q > 1 , $$\lambda ,\mu \in {\mathbb {C}}$$ λ , μ ∈ C , $$\lambda \ne 0$$ λ ≠ 0 , and $$u(0,\cdot ), w\in L^1_{\mathrm{loc}}({\mathbb {R}}^N,{\mathbb {C}})$$ u ( 0 , · ) , w ∈ L loc 1 ( R N , C ) . We consider both the cases where $$\mu =0$$ μ = 0 and $$\mu \ne 0$$ μ ≠ 0 , respectively. We establish existence/nonexistence of global weak solutions. In ea…