Existence Of Solutions For Quasilinear Equations With A Kirchhoff Term

In this thesis,we consider the solution of the following Kirchhoff type problem with a critical Sobolev exponent:where BR?RN is a ball,a,b,λ>0,1<p<N,p≠2,p*=Np/N-p,p≤q<p*.In chapter 1,we summarize the relevant background and progress on(Qa,b,λ),and we briefly explain the structure of this thesis.In chapter 2,we introduce some necessary definitions and lemmas.In chapter 3,we obtain solvability of the(Qa,b,λ)when p=q by applying the variational method,the theory of critical point and a scaling technique.Firstly,we establish a special equivalence between(Qa,b,λ)and(Qα),so we can avoid the difficulties caused by the Kirchhoff term.Secondly,we give some necessary results in proving main theorems.Thirdly we show that the solution of(Qα)is the only radially solution.Finally,we obtain solutions of(Qa,b,λ).In chapter 4,we obtain solvability of the(Qa,b,λ)when p<q by applying the variational method,critical point theory and a scaling technique.However,the equation(Qa,b,λ)and the equivalence are different from those in chapter 3,so we need a more complex calculation.We combine results which we found in the literature about the solution of(Qα),and we prove that(Qα)has only one radially solution.Finally,we establish the existence theory of solutions for the(Qa,b,λ).In chapter 5,we summarize and prospect the whole thesis.