Existence Of Solutions For Schr(?)dinger Poisson Systems With Saturable Nonlinearity

In the second chapter,we are mainly concerned with the following problem of Schr(?)dinger Poisson system with saturable nonlinearity:(?)-Δu + V(x)u + λM(x)φu = K(x)Γu3/1+u2 x ∈ R3,(SP)-Δφ = M(x)u2 x ∈ R3.u3/1+u2 is generally called saturable nonlinearity.Where λ,Γ are positive parameter,V(x)、M(x)and K(x)are continuous potentials.The potential V(x)may vanish to 0 at infinity.but it may not be radically symmetric.Under some proper assumptions on V(x),M(x)and K(x),we prove that the functional corresponding of the problem(SP)has a nontrivial critical point in the weighted Sobolev space if λ and Γ are lie in a certain range.Furthermore,we prove that the nontrivial critical point is a bound state for the problem(SP).Finally,we can obtain the existence of ground state for the system(SP)via a constrained minimization technique and there is no any nontrivial positive solution for λ sufficiently large.In the third chapter,we consider the following Schr(?)dinger Poisson Slater equations in R3,-Δu + φu-Γu3/1+u2 = ωu.(SPS) Where Γ is a parameter,φ = ∫R3u2(y)/4π|x-y|dy.Note that the equation(SPS)is a special case of the system(SP)with V(x)= ω,M(x)= K(x)= 1 and λ = 1 when ω E R is a fixed and assigned parameter.But in this section,we are more interested in the solutions with prescribed L2-norm.That is u satisfying(SPS)with ‖ u ‖2= p.Therefore,it only needs to find the critical point u for I(u)on the constraint Bρ = {u∈H1(R3):‖u‖2=ρ}in the H1(R3),where I(u)= 1/2∫R3|(?)u |2 dx + 1/4 ∫R3φ|u|2 dx-ΓR3[u2-ln(1 + u2)]dx Obviously,if u is a critical point for I(u)on the Bρ,then u is a solution of the equation(SPS),where ω is characterized as a Lagrange multiplier.In fact,we focus on exploring the accessibility of minimizer for I(u)on Bρ in this chapter,we find that there no constrained critical points exist when Γ<Tρ,and Iρ<0 when Γ>Tρ.