On critical and supercritical pseudo-relativistic nonlinear Schrodinger equations

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schr<spacing diaeresis>odinger equation - c2.+ m2c4 - mc2 u + mu u = |u|p-1u in Rn (n 2) involving an H1/2-critical/supercritical power-type nonlinearity, that is, p ((n + 1)/(n - 1)). We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is H1/2-critical/supercritical but it is H1-subcritical. On the other hand, we also show that there is no nontrivial bounded solution either (i) if the nonlinearity is H1/2-critical/supercritical in the ultra-relativistic regime or (ii) if the nonlinearity is H1-critical/supercritical in all cases.