Boundedness of solutions of Chern-Simons-Higgs systems involving the \(\Delta _{\lambda }\)-Laplacian

Abstract

The purpose of this paper is to study the boundedness of solutions of the Chern-Simons-Higgs equation

$$\begin{aligned} \partial _tw-\Delta _{\lambda } w = \left| w \right| ^2 \left( \beta ^2-\left| w \right| ^2\right) w-\frac{1}{2}\left( \beta ^2-\left| w \right| ^2 \right) ^2w \text{ in } \mathbb {R}\times \mathbb {R}^N \end{aligned}$$

and system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u -\Delta _\lambda u = u^2\left( 1-u^2-\gamma v^2\right) u-\frac{1}{2}\left( 1-u^2-\gamma v^2 \right) ^2u & \text { in } \mathbb {R}\times \mathbb {R}^N, \\ \partial _t v -\Delta _\lambda v = v^2\left( 1-v^2-\gamma u^2\right) v-\frac{1}{2}\left( 1-v^2-\gamma u^2 \right) ^2v & \text { in }\mathbb {R}\times \mathbb {R}^N,\\ \end{array}\right. } \end{aligned}$$

where \(\gamma >0\), \(\beta \) is a bounded continuous function and \(\Delta _{\lambda }\) is the strongly degenerate operator defined by

$$\begin{aligned} \Delta _{\lambda }:=\sum _{i=1}^N \partial _{x_i}\left( \lambda _i^2\partial _{x_i} \right) . \end{aligned}$$

Under some general hypotheses of \(\lambda _i\), we shall establish some boundedness properties of solutions of the equation and system above. Our result can be seen as an extension of that in [Li, Yayun; Lei, Yutian, Boundedness for solutions of equations of the Chern-Simons-Higgs type. Appl. Math. Lett.88(2019), 8-12.]. In addition, we provide a simple proof of the boundedness of solutions.