On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations

초록

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation 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. Copyright © Royal Society of Edinburgh 2019.

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