On the planar weakly coupled nonlinear logarithmic Choquard systems

Abstract

In this paper, we study the following class of coupled nonlinear logarithmic Choquard equations$\{\begin{array}{ccc}-\Delta u+{\lambda }_{1}u& =(\log \frac{1}{|\cdot |}⁎u^2)u+(\log \frac{1}{|\cdot |}⁎v^2)u,& \text{in }{R}^2,\\ -\Delta v+{\lambda }_{2}v& =(\log \frac{1}{|\cdot |}⁎v^2)v+(\log \frac{1}{|\cdot |}⁎u^2)v,& \text{in }{R}^2.\end{array}$ We prove the existence of a nonnegative vector solution when ${\lambda }_1={\lambda }_2$. Moreover, we prove that if ${\lambda }_1\ne {\lambda }_2$, then the system admits only the semi-trivial solution. Our approach is based on minimization over Nehari manifold and a version of the Principle of Symmetric Criticality due to Palais.