Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity

We consider the following semirelativistic nonlinear Schrödinger equation (SNLS): { i ∂ t u ± ( m 2 − Δ ) 1 / 2 u = λ | u | p , a m p ; ( t , x ) ∈ [ 0 , T ) × R d , u ( 0 , x ) = u 0 ( x ) , a m p ; x ∈ R d , \begin{equation} \left \{ \begin {array}{ll} i\partial _t u \pm (m^2-\Delta )^{1/2} u = \lambda |u|^{p}, & (t,x)\in [0,T)\times \mathbb {R}^d, \\ u(0,x)=u_0(x), & x \in \mathbb {R}^d, \end{array} \right . \notag \end{equation} where m ≥ 0 m\geq 0 , λ ∈ C ∖ { 0 } \lambda \in \mathbb {C} \setminus \{ 0\} , d ∈ N d\in \mathbb {N} , T > 0 T>0 , and ∂ t = ∂ / ∂ t \partial _t=\partial /\partial t . Here ( m 2 − Δ ) 1 / 2 := F − 1 ( m 2 + | ξ | 2 ) 1 / 2 F (m^2-\Delta )^{1/2}:=\mathcal {F}^{-1} (m^2+|\xi |^2 )^{1/2} \mathcal {F} , where F \mathcal {F} denotes the Fourier transform. Fujiwara and Ozawa proved the nonexistence of global weak solutions to SNLS for some initial data in the case of d = 1 d=1 , m = 0 m=0 , and 1 > p ≤ 2 1>p\leq 2 by a test function method. In this paper, we extend their result to a more general setting: for example, m ≥ 0 m\geq 0 , d ∈ N d\in \mathbb {N} , or p > 1 p>1 . Moreover, we obtain the upper estimates of weak solutions to SNLS. The key to the proof is to choose an appropriate test function.