Existence And Concentrating Behavior Of Solutions To Several Elliptic Equations And Systems

In this thesis,we mainly study the existence and concentration behavior of the ground state solutions for the kirchhoff type equations with steep potential well,the existence,concentration and multiplicity of solutions for the Schr???dinger-Kirchhoff type p-Laplacian equation and for the fractional Schr???dinger-Kirchhoff type problem,the existence,stability and quantitative properties of the standing waves for nonlinear schrodinger systems with partial confinement.The thesis consists of five chapters:In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some preliminary results and notations used in the whole thesis.In Chapter Two,we study the existence and concentrating behavior of solutions for the following Kirchhoff type equations with steep potential well???where a,b,?>0,V?C?R³,R?is a potential well,q?x?is a positive bounded function,f?s?is either asymptotically linear or asymptotically 3-linear in s at in-finity.Under some other suitable conditions on V,q and f,we prove the existence of solutions for problem?E₁?with the asymptotically linear nonlinearity via vari-ational methods.In particular,we consider the existence of solutions for problem?E₁?with sign-changing potential V.Furthermore,the existence of concentrating ground state solutions to problem?E₁?with the asymptotically 3-linear nonlinearity is also discussed.We mainly extend the results in Sun and Wu?J.Differ.Equations.2014?,which dealt with Kirchhoff type equations with nonnegative potential well,to Kirchhoff type equations with sign-changing potential well.The main results in this chapter have been published in?J.Math.Anal.Appl.,467,893-915?2018??.In Chapter three,study the multiplicity and concentration behavior of pos-itive solutions for the Schr???dinger-Kirchhoff type p-Laplacian equation???where ?_p is the p-Laplacian operator,N?3,1<p<N,M:R+?R+ and V:RN?R+ are continuous functions,? is a positive parameter,and f is a continuous function with subcritical growth.We assume that V satisfies the local condition introduced by del Pino and Felmer[28].By using the variational methods,penalization techniques and Lyusternik-Schnirelmann theory,we prove the existence,multiplicity and concentration of solutions for problem?E₂?.The main results in this chapter have been published in?Acta Math.Sci.Ser.B,38,391-418?2018??.In Chapter four,we study the existence,concentrating and multiplicity of ground state solutions for the following fractional Schr???dinger-Kirchhoff type prob-lem???where???is a nonlocal operator defined by???,? is a small positive parameter,s??3/4,1?,the operator?-??^s is the fractional Lapla-cian of order s,M,V,K and f are continuous functions.Under proper assumptions on M,V,K and f,we prove the existence and concentration phenomena of solutions of the problem?E₃?.With minimax theorems and the Ljusternik-Schnirelmann the-ory,we also obtain multiple solutions of problem?E₃?by employing the topology of the set where the potentials V?x?attains its minimum and K?x?attains its max-imum.The main results in this chapter have been published in?Ann.Acad.Sci.Fenn.Math.,43,991-1021?2018??.In Chapter five,we deal with the existence,qualitative and symmetry properties of normalized solutions for the following nonlinear Schr???dinger systems with partial confinement???where ?>0?i=1,2?,?>0,and the frequencies ?1,?2 are unknown and appear as Lagrange multipliers.We can get a normalized solution to problem?E4?by looking for a critical point of ???constrained on the sphere S?c₁?×S?c₂?,where ???.The functional I?u,v?is unbounded from below on S?c₁?×S?c₂?.By using the constrained minimization method on a suitable subset of S?c₁?x S?c₂?,we prove that for certain c₁,C2>0,I?u,v?has a local minimizer on S?c₁?×S?c₂?.In ad-dition,we study the stability of the corresponding standing waves for the related time-dependent Schr???dinger systems.We mainly extend the results in Bellazzini et al.?Commun.Math.Phys.2017?,which dealt with mass-supercritical nonlin-ear Schr???dinger equation with partial confinement,to cubic nonlinear Schr???dinger systems with partial confinement.