In this paper, we consider the existence of positive solutions for a relativistic nonlinear Schrodinger equation with square-root nonlinearity. If N = 2, the existence and uniqueness of ground states for this equation was obtained by Colin [Adv. Differ. Equations 8, 1 (2003)]. In this paper, when N = 3, by modifying the original problem and proving some estimates, the existence of the positive solution uω(x) for this equation is obtained. In the meantime, uω(x) → 0 as ω → 0+. Moreover, using the same technique, the existence of positive solutions for more generalized nonlinear Schrodinger equations with square-root nonlinearity is also obtained.

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具有平方根非线性的相对论非线性薛定谔方程的正解

在本文中，我们考虑了具有平方根非线性的相对论非线性薛定谔方程的正解的存在性。如果 N = 2，则该方程的基态的存在性和唯一性由 Colin [Adv. 不同。方程 8, 1 (2003)]。在本文中，当N=3时，通过修改原问题并证明一些估计，得到该方程的正解uω(x)的存在性。同时，uω(x) → 0 为 ω → 0+。此外，使用相同的技术，还获得了具有平方根非线性的更广义非线性薛定谔方程的正解的存在性。