Existence Of Solutions For A Class Of Quasilinear Elliptic Equations

In this dissertation,the existence and asymptotic properties of positive solutions for quasilinear elliptic equation were considered by introducing the change of variables and critical point theory.This type of equation derives from the superfluid film equation in plasma physics and so on.It obtained the attention of many mathematical researchers because of its background.It is easy to know that the functional with respect to this equation can't be well defined in the usual Sobolev space since the equation includes a quasilinear term.So the critical point theory can't be used directly.In this dissertation,we will introduce an improved change of variables to study the existence and asymptotic behavior of solutions for a general quasilinear elliptic equation with superlinear condition.Moreover,the existence of nontrivial solutions was considered when the nonlinear term involves concave and convex nonlinearities.The main research works are as follows:In Chapter 3,we transform the quasilinear elliptic equation into semi-linear elliptic equation which are studied relatively easily by introducing an improved change of variables.This indicates that we can consider the new equation in Sobolev space,but the nonlinear term of the new equation is more complex.Firstly,based on the critical point theory in variational method,the existence of positive solutions for new equation is obtained.Then,the existence of positive solutions of the quasilinear elliptic equation is got by using some estimates and properties of changes.It extends the range of nonlinear term exponent related to references in this type of equation.Moreover,using the change of variables and constraint variation method,the existence and asymptotic behavior of ground state for quasilinear elliptic equation with constant potential is obtained.In Chapter 4,the existence of positive solution for a general quasilinear elliptic equation with concave convex nonlinearities is considered.By using the same change of variables as the previous chapter,the general quasilinear elliptic reduces to a semi-linear elliptic equation.In this time,the nonlinear term of new equation becomes more complex since the nonlinear term involves concave convex nonlinearities.Using mountain pass lemma and Lions compactness lemma,the existence of positive solutions for quasilinear elliptic equation with concave convex nonlinearities is obtained.In Chapter 5,we summarize the contents of this dissertation and make some expectation.Our main innovation is an improved change of variables.It can extend the range of the exponent of nonlinear term.But the asymptotic behavior of positive solutions with the non-constant potential has not been studied,and the range of the exponents of nonlinear term in quasilinear elliptic equations with concave convex nonlinear terms is likely to be further expanded.