Qualitative analysis for the nonlinear fractional Hartree type system with nonlocal interaction

Jun Wang

doi:10.1515/anona-2021-0202

Abstract

Abstract In the present paperwe study the existence of nontrivial solutions of a class of static coupled nonlinear fractional Hartree type system. First, we use the direct moving plane methods to establish the maximum principle(Decay at infinity and Narrow region principle) and prove the symmetry and nonexistence of positive solution of this nonlocal system. Second, we make complete classification of positive solutions of the system in the critical case when some parameters are equal. Finally, we prove the existence of multiple nontrivial solutions in the critical case according to the different parameters ranges by using variational methods. To accomplish our results we establish the maximum principle for the fractional nonlocal system.

Highlights

Introduction and main resultsIn the present paper we study the coupled nonlinear fractional Hartree system in the following form ⎧ ⎛ ⎞ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨(−Δ) α 2 u = μ1 ⎝ ⎛RN|u(y)|p |x − y|N−γ dy⎠ |u|p−2 u +
In the present paper we study the existence of nontrivial solutions of a class of static coupled nonlinear fractional Hartree type system
We use the direct moving plane methods to establish the maximum principle(Decay at infinity and Narrow region principle) and prove the symmetry and nonexistence of positive solution of this nonlocal system

Summary

Introduction

If α = 2, the paper [41] proved the existence and some properties of solution of (1.7). For the fractional case 0 < α < 2, the paper [12] proved the regularity and classification of the solution of (1.7). Under some conditions for the potential function λi(x), i = 1, 2, the existence of a ground state solution of (1.10) for ε > 0 small and β > 0 sufficiently large was proved.

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Conclusion