Normalized solutions for nonlinear fractional Kirchhoff type systems

Abstract

In this paper, we consider the existence of positive solutions with prescribed normalizations for strongly coupled fractional Kirchhoff type systems in the whole space $\mathbb{R}^N$ $(N=2,3)$. Under constant vanishing potentials and attractive interspecies interactions, two cases are studied: one is $L^2$-subcritical and the other is $L^2$-supercritical. In the first case, we prove the existence of a positive solution by the constrained minimizing methods. In the second case, by using a minimax procedure, we prove the existence of a mountain pass type solution under high perturbations of the coupling parameter, which is also a ground state solution. Moreover, we study the $L^2$-critical case under certain type of trapping potentials. In this case, we are concerned with not only attractive but also repulsive interspecies interactions, and prove the existence of a positive solution by introducing some auxiliary minimization problems. These conclusions extend some known ones in previous papers.