Ground State Solutions to a Coupled Nonlinear Logarithmic Hartree System

Abstract

In this paper, we study the following coupled nonlinear logarithmic Hartree system

$ \left\{\begin{array}{ll}-\Delta u+\lambda_{1} u=\mu_{1}\left(-\frac{1}{2 \pi} \ln |x| * u^{2}\right) u+\beta\left(-\frac{1}{2 \pi} \ln |x| * v^{2}\right) u, & x \in \mathbb{R}^{2}, \\-\Delta v+\lambda_{2} v=\mu_{2}\left(-\frac{1}{2 \pi} \ln |x| * v^{2}\right) v+\beta\left(-\frac{1}{2 \pi} \ln |x| * u^{2}\right) v, & x \in \mathbb{R}^{2}.\end{array}\right.$

where β,µ_i,λ_i (i = 1,2) are positive constants, ∗ denotes the convolution in $\mathbb{R}^{2}$. By considering the constraint minimum problem on the Nehari manifold, we prove the existence of ground state solutions for β >0 large enough. Moreover, we also show that every positive solution is radially symmetric and decays exponentially.

Keywords

Hartree system / Logarithmic convolution potential / ground state solution / radial symmetry