The Study On Existence And Concentration Phenomenon Of Solutions To Elliptic Equations With Nonlocal Terms Via Variational Methods

In this paper,we first consider the following general nonlocal problem where ?(?)RN is a bounded domain with Lipschitz boundary ?? and LK is a nonlocal integrodifferential operator of fractional Laplacian type.Combining constraint variational method and quantitative deformation Lemma,we verify that the problem possesses one least energy sign-changing solution u0.Moreover,the energy of u0 is strictly larger than the ground state energy.Furthermore,we consider the above problem with concave and convex terms,that is,where 1<q<2<p<2s*:= 2N/N-2s,N>2s,s ?(0,1).We obtain the existence of infinitely many solutions via Bartsch's Fountain Theorem and its dual version.Lastly,we consider existence and concentration behavior of ground state solutions for the following fractional Kirchhoff type problem where M(t)= ?2sa + ?4s-3bt is a Kirchhoff function,0<s<1 and ?>0 is a small parameter,the potential V is a positive continuous function which has the global minimum,and f is supercubic but subcritical at infinity.The existence of a positive ground state solution was established via variational methods for small ?>0.We show that these ground state solutions converges in Hs(R3)to a ground state solution to the limit equation,and concentrate around the global minima of the potential V as ? ? 0+.Moreover,the decay estimate of solutions is also established.