The Existence Of Solutions For Quasilinear Schr?dinger-Poisson System With Supercritical Terms

Nonlinear partial differential equations,an important branch of modern mathematics,has always been widely concerned and applied in the natural sciences,physics and engineer-ing.Up to now,the existence and uniqueness,the multiplicity and the stability of solutions for the partial differential equation have already had certain research results,also includ-ing the existence and asymptotic properties of global solutions for initial value problems of partial differential equations.As the most basic and also an all-important embranchment in the Nonlinear partial differential equation,the existence and multiplicity of solutions for Schr?dinger equations which involving in this article has always been a problem of great interest to the mathematicians.Thus,in this paper,by using the Mountain path theorem,Moser iteration,truncation technique and variable substitution method,we consider the exis-tence of solutions for the following Schr?dinger-Poisson system with critical and supercritical terms:where K ∈ R,λ>0,V ∈ C（R³,R）,f∈C（R,R）The thesis consists of two sections,in Chapter 1,we study the existence of solutions for（0.2）with compactness condition.The case of κ>0 and κ<0 are discussed respectively.We assume that the nonlinear term f and the potential function V satisfy the following conditions:（V₀）V∈（R³,R）and there exists V₀>0,such that 0<V₀≤V_X and lim_|X|→∞V_X=∞for all X∈R³.（f₁）f∈C（R,R）and|f（t）|≤C(1+|t|^q-1);（f₂）lim（t→0）f（t）/t=0;（f₃）0<qF（t）:=q（?）≤f（t）t,t∈R?{0},where κ<0,8<q<12 and κ>0,4<q<6.Under the above conditions,firstly,we choose the variable substitution method to overcome the undefined problem about the energy functional of system（0.2）in H1（R³）.Then,the nonlinear term after the change of variables is truncated and we prove that the natural energy functional of the truncated equation satisfies the mountain pass structure Cerami condition.And we prove the natural energy functional owns a nontrivial solution uκ.Finally,we estimate the solution vκ by using Moser iterative method in L∞（R³）and obtain the existence of nontrivial solutions for the system（0.2）.The main conclusions are as follows:Theorem 1.1.1 Suppose that V=1,（f₁）-（f₃）are hold,p>12 and 8<q<12.Then there exists λk>0 such that the system（0.2）possesses a nontrivial solution u_k,λ for all k^<0 when λ∈（0,λk）.Theorem 1.1.2 Suppose that（V₀）,（f₁）-（f₃）are hold,p>6 and 4<q<6.Then there exist k1>0 and λ₁>0 such that the system（0.2）possesses a nontrivial solution u_k which satisfies ||u_k||∞≤（?） for all κ∈（0,κ1）and λ∈（0,λ₁）.In Chapter 2,we consider the problem of the existence of nontrivial solutions for the system（0.2）without compactness condition.Assuming that the potential function V and the nonlinear term f satisfy the following conditions:（V₀）The poteutial function V∈C（R³,R）,and there exist V₀,V_∞>0,such that 0<V₀≤V（x）≤V_∞:=lim_|x|→∞V（x）.（f₀）f∈C¹（R,R）,lim（t→0）f（t）/t=0 and f（t）t≥0,t∈R;（f₁）There exist 8<q<12,C>0,such that∈R,f（t）≤C(1+|t|^q-1);（f₂）f（t）/t⁷ is increasing in（0,+∞） and decreasing in （-∞,0）;（f₃）lim_|t|→∞f（t）/t⁷=∞.In this chapter,we consider the existence of nontrivial solutions for the system（0.2）withκ<0.Firstly,we adopt the change of variables to solve the difficult which we can not find a suitable space to define the energy functional due to the existence of the quasilinear item u△u².Then,the nonlinear term is truncated and the existence of nontrivial solutions for the truncated quasilinear Schr?dinger-Poisson system has been proved by using approach method and the concentration-compactness principle.Finally,we obtain the existence of nontrivial solutions for the system（0.2）by using Moser iterative method.In this chapter,the main conclusion reads as follows:Theorem 2.1.1 Suppose that p>12,（V₀）and（f₀）-（f₃）are hold.Then,for any κ<0 there exists λκ>0 such that the system（0.2）has a nontrivial solution when λ ∈（0,λ_κ）.