帶陡峭位勢的分?jǐn)?shù)階Schr?dinger-Poisson系統(tǒng)的基態(tài)變號解

摘 要：本文研究了帶陡峭位勢的分?jǐn)?shù)階Schr?dinger-Poisson系統(tǒng)的基態(tài)變號解的存在性，由于系統(tǒng)中的位勢是陡峭位勢，這使得系統(tǒng)的能量泛函緊性缺失。運用約束變分法將能量泛函限制在約束集Mλ中，證明能量泛函的下確界可以達到，采用形變引理，得到了系統(tǒng)有1個基態(tài)變號解，基態(tài)變號解有2個結(jié)點域，并且基態(tài)變號解的能量嚴(yán)格大于基態(tài)解能量的2倍。

關(guān)鍵詞：分?jǐn)?shù)階Schr?dinger-Poisson系統(tǒng)；約束變分法；基態(tài)變號解；陡峭位勢

中圖分類號：O177.91"" 文獻標(biāo)志碼：A""" 文章編號：1673-5072（2024）05-0488-07

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Ground State Sign-changing Solution

for Fractional Schr?dinger-Poisson System with Steep Potential Well

HUANG Xiao-qinga， LIAO Jia-fengab

（a.School of Mathematics amp; Information，b.College of Mathematics Education，China West Normal University，Nanchong Sichuan 637009，China）

Abstract：This paper studies the existence of ground state sign-changing solution for a fractional Schr?dinger-Poisson system with steep potential well.A lack of energy functional tightness is caused by the steep potential well in the system.The constraint variational method is employed to limit the energy functional to the constraint concentration Mλ，which proves that the lower boundary of the energy functional can be reached.With the aid of the deformation lemma，it is found that the system has a ground state sign-changing solution which has two nodal domains，and the energy of the ground state sign-changing solution is strictly more than twice the energy of the ground state solution.

Keywords：fractional Schr?dinger-Poisson system；constrained variation method；ground state sign-changing solution；steep potential well