Yong-Li Tang

On the Classification of Standing Wave Solutions for the Schrödinger Equation

In this article, we consider the following semilinear elliptic equation where n ≥ 3, p > 1 and λ > 0. We provide the existence and uniqueness of the singular radial solution of the above equation for specific ranges of n and p. In addition, we also clarify the entire structure of radial solutions for various types according to their behaviors at the origin and infinity.

Existence, uniqueness and stability of positive solutions to sublinear elliptic systems

Reaction–diffusion systems are used to model many chemical and biological phenomena in the natural world [25, 26], and systems of coupled partial differential equations are also used in other physical models such as nonlinear Schrodinger systems in multi-component Bose–Einstein condensates and nonlinear optics [19, 23]. The steady-state solutions or standing-wave solutions of such systems of nonlinear partial differential equations satisfy a nonlinear elliptic system with more than one equation. Much effort has been devoted to the existence of solutions of such systems (see, for example, [4, 8, 10, 11, 13, 15–17, 24, 30, 33, 34]), but it is usually difficult to determine whether or not the solution is unique. We consider the positive solutions of a semilinear elliptic system of the form

Uniqueness of finite total curvatures and the structure of radial solutions for nonlinear elliptic equations

In this article, we are concerned with the semilinear elliptic equation where n > 2, p > 1, and K(lxl) > 0 in R n . The correspondence between the initial values of regularly positive radial solutions of the above equation and the associated finite total curvatures will be derived. In addition, we also conduct the zeros of radial solutions in terms of the initial data under specific conditions on K and p. Furthermore, based on the Pohozaev identity and openness for the regions of initial data corresponding to certain types of solutions, we obtain the whole structure of radial solutions depending on various situations.

Erratum to “On the Classification of Standing Wave Solutions for the Schrödinger Equation”

In [1], Lemma 3.2 guaranteed the monotonicity of regular solutions of (1.4) on some fixed interval near the origin in terms of initial values, so as to prove the existence of singular solutions of (1.1). To make the arguments more clear for readers, the estimate of g t on page 295, line 23 in the original proof of Lemma 3.2 needs to be modified. Here, we provide a revised statement and proof of this lemma, and add an extra remark in the following. Refer to [1] for all notations and labeled equations appearing below.