带有对数项的临界Kirchhoff-Choquard方程解的存在性和多重性OA

The critical nonlocal Choquard equation can be used to describe the theory of the polaron at rest,and can also be used as a self-gravitational model to describe the motion of a single particle in the self-gravitational field.In recent years,the study of the existence,multiplicity and related properties of solutions of Choquard equations has become a hot issue in the field of nonlinear analysis.This paper studied the existence of solutions for a class of critical Kirchhoff-Choquard equations with logarithmic term.Firstly,the Hardy-Littlewood-Sobolev exponent was classified and an appropriate truncation function was introduced to overcome the lack of compactness of the embedding caused by the critical exponent.Secondly,it was difficult to verify the boundedness of the PS(Palais-Smale)sequence due to the uncertainty of the logarithmic term and the fact that the logarithmic term does not satisfy the Ambrosetti-Rabinowitz type condition.Therefore,the local PS condition was obtained by using the concentration compactness principle,and the existence theorem of infinitely many solutions of the problem was established by means of the symmetric mountain pass lemma.Finally,the existence of the local minimal solution with negative energy was considered on the appropriate sphere(a constrained region of function space),and then the additional conditions were applied to the parameters.It is proved that the local minimal solution is also the ground state solution.The results of this paper extend the research on Kirchhoff-Choquard equation or logarithmic equation in the relevant literature,and analyze the influence of the parameters in the Kirchhoff operator and the logarithmic perturbation term and Hardy-Littlewood-Sobolev embedding constants on the number of solutions.