Existence Solutions For The Chern-Simons-Schr?dinger Equation With Concave-convex Nonlinearities

In this paper,we mainly study the existence of solutions for Chern-Simons-Schr?dinger system by using the variational methods.Firstly,we study the existence of sign-changing solutions for the Chern-SimonsSchr?dinger equation with concave-convex nonlinearities:(0.0.3)where ω,λ>0 and K ∈ Lp/(2-p)(R2,R+),R+:=(0,∞),1<p<2,q>6.Using constrained minimization arguments and the quantitative deformation lemma,we prove that there exists a constant λ*>0 such that for any λ<λ*,equation(0.0.3)has a sign-changing solution uλ with positive energy.Secondly,we consider the existence of positive solutions for the Chern-Simons-Schr?dinger system with concave-convex nonlinearities:(0.0.4)where λ,μ>0,1<q<2,4<p<6 and V ∈ C(R2,R)is a steep potential well.By using Ekeland variational principle,we prove the above problem have a positive solution uλ+ with negative energy.