Study On The Solutions And Properties Of Elliptic Equations With Nonlocal Term

In recent years,elliptic equations with nonlocal terms have attracted extensive attention,especially Kirchhoff equation and Schrodinger-Poisson system.Among them,Kirchhoff equation was first proposed by Kirchhoff in the process of studying the classical D’Alembert wave equation of free vibration of retractable rope and non Newtonian mechanics are widely used in other fields;Schrodinger-Poisson system is coupled from Schrodinger equation and Poisson equation in physics,which describes the interaction between particles and electromagnetic field generated by motion in quantum mechanics(non relativity).Based on the strong physical background of the above related equations,in this paper,the existence,multiplicity and correlation of sign-changing solutions for Elliptic Equations with nonlocal terms are studied by using the variational method.Firstly,the 4-sublinear growth nonlinear Schrodinger-Kirchhoff-Poisson system on a bounded domain is considered.By making appropriate assumptions about the potential function Q(x),the existence and multiplicity of sign-changing solutions of the system are proved by using the descent flow invariant set method when 2<p<4 in the nonlinear term Q(x)|u|p-2u and the parameter λ before thenonlocal term are sufficient small.In particular,if λ=1,2<p<12/5,the existence of infinitely many sign-changing solutions for this system is established.Compared with the researches about Schrodinger-Poisson system on the whole space R3,this conclusion shows that Schrodinger-Poisson system has different dynamic behavior on R3 and bounded domain Ω.Secondly,the cubic nonlinear Schrodinger-Poisson system on a bounded domain Ω is studied.Under the premise of λ<λ1,through a new construction technique,it is proved that the system has a ground state sign-changing solution if and only if the parameter before the nonlinear term satisfy μ>0.This result generalizes the result of khoutir(J.math.Phys.,62(2021))[38].Thirdly,the Schrodinger-Poisson system with indefinite potentials is explored.Under suitable assumptions,using the constrained variational method and deformation lemma,it is proved that the system has a ground state sign-changing solution when 4<p<6 in the nonlinear term |u|p-2u.Furthermore,the nodal domains and energy doubling characteristics of the ground state sign-change solution are established.Due to lim|x|→∞a(x)=a∞=0 is allowed,this new situation can not be handled by Batista and Furtado[14]with the help of limit equation,so this result can be regarded as a supplement and expansion in a certain sense.Finally,the combined nonlinear kirchhoff equation is considered,in which the nonlinear term is |u|p-1u+|u|q-1u.By introducing a special sign-changing Nehari-Pohozaev constraint,and then exploring the minimization problem on this constraint,it is proved that for any 1<p≤q<5,the equation has at least one sign-changing solution.Note that p=q is allowed,which actually includes the case that the nonlinear term is pure power type,namely,|u|q-1u,1<q<5.In particular,due to the introduction of this constraint,it is easier to deal with the case of 1<q≤3.