Existence Of Solutions To A Class Of Quasilinear Elliptic Equations

This thesis is concerned with the existence of nontrivial solutions for the quasi-linear elliptic equations -?_pu-?_p(|u|^2?)|u|^2?-2u+V(x)|u|^p-2u=|u|^q-2u,x?R^N,where ??1,1<p<N,p*=Np/N-p,?_p is the p-Laplace operator and the potential V(x)>0 is a continuous function.In this work we mainly focus on nontrivial solutions.When 2?p<q<p*,we establish the existence of nontrivial solutions by using Mountain-Pass lemma;when q>2?p*,by using a Pohozaev type variational identity,we prove that the equation has no nontrivial solutions.The first part of the present thesis is devoted to introduction,research back-ground and main results.In Chapter 2,we provide preliminary tools,which are important and essential for establishing our main results.We first give some important properties for the change of variables u=h(w)and then give some useful lemmas.In the end of this part,we present an auxiliary problem and some related results which are crucial in this work.In Chapter 3,the existence of nontrivial solutions to the above equation is established under certain conditions.Finally,in Chapter 4,we prove the non-existence of nontrivial solution to the above equation under certain conditions.To prove the non-existence of non-trivial solution,we first state a Pohozaev type variational identity,which is extremely use-ful and this identity enables us to prove that the equation has no nontrivial solution.