Existence And Concentration Of Ground State Solutions To Two Classes Of Elliptic Equations

Firstly,we consider the critical Kirchhoff-Schrodinger equation with general nonlinear-ities and study the existence of semiclassical ground state solutions of Nehari-Pohozaev type to(SKε)where f(u)～|u|q-2u for q ∈(2,4],which is seldom studied.With some decay assumption on V,we establish an existing result which improves some exiting works which only handle q ∈(4,6).With some monotonicity condition on V,we also get a ground state solution vε and analysis its concentrating behaviour around global minimum xε of Ⅴas ε→0.Furthermore,for any sequence εn→0,vεn(εnx+xεn)converges in H1(R3)to a ground state solution of the following problem where 0<V0=minx∈R3 V(x).Secondly,we discuss the following Schrodinger-Possion system where K ∈C(R,(0,∞))and p ∈(3,∞).By using some new variational framework and some new tricks,we prove that the above system has a ground state solution possessing the least energy in the axially symmetric functions space.