On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations

Abstract

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation $$\left(\sqrt{-c^2\Delta +m^2{c}^4}-m{c}^2\right)u+\mu u=|u{|}^{p-1}u\text{in}{\mathbb{R}}^n(n\ge 2)$$ involving an $H^{1/2}$-critical/supercritical power-type nonlinearity, i.e., $p\ge \frac{n+1}{n-1}$. We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is $H^{1/2}$-critical/supercritical but it is $H^1$-subcritical. On the other hand, we also show that there is no nontrivial bounded solution either $(i)$ if the nonlinearity is $H^{1/2}$-critical/supercritical in the ultra-relativistic regime or $(ii)$ if the nonlinearity is $H^1$-critical/supercritical in all cases.

References in corpus (2)

• Solitary Wave Solutions for the Nonlinear Dirac Equations
• Optimal convergence rate of nonrelativistic limit for the nonlinear pseudo-relativistic equations

Cited by in corpus (2)

• On a generalized pseudorelativistic Schrödinger equation with supercritical growth
• Uniqueness and symmetry of ground states for higher-order equations