Research On The Existence Of Solutions For Some Classes Of Quasilinear Elliptic Equations

In this thesis,the existence of solutions of three kinds of quasilinear elliptic equations is studied by using variational method.This thesis consists of the following four chapters:The first chapter introduces the research background of quasilinear Schrodinger equation and Kirchhoff type problem,the research status at home and abroad in recent years,and the main research content of this thesis.In the second chapter,we explored the existence of the quasilinear Schrodinger equation in RN of the form where N≥3,V:RN→R and h:R→R are functions.Using variational method and mountain pass lemma,we prove that the quasilinear Schrodinger equation has a positive solution when the function V and the nonlinear term h satisfy some appropriate conditions respectively.In the third chapter,the following fractional Kirchhoff-type problem where a,b>0 are constants,s∈(3/4,1),V:R3→R is a continuous function,and f:R3 × R→R is a continuous function,is considered.It is demonstrated that the fractional Kirchhoff-type equation(0-2)has a radial sign-changing solution ub and a radial solution ub when f does not satisfy the subcritical growth condition and the usual Nehari-type monotonicity condition.The main tools are the constraint variational method and some analysis techniques.In the fourth chapter,we consider the following Kirchhoff type problem with Hardy potential:where a,b>0 are constants,μ<1/4,1/|x|2 is called the Hardy potential and g:R→R is a continuous function that satisfies the Berestycki-Lion type condition.Using variational methods,we establish two existence results for problem(0-3)under different conditions for g.Furthermore,if μ<0,we prove that the mountain pass level value in H1(R3)can not be achieved.