Existence Of Ground State Solution And Positive Solution For Elliptic Equations With Exponential Critical Growth

We have studied the existence of ground state solution and positive solution for the following N-Laplacian equation with critical exponent in this dissertation.In chapter 2,we list the existence of ground state solution for the following N-Laplacian equation with critical exponent-?Nu+|u|N-2u=f(u)x?RN where u>0,u?W1,N(RN),N?2.?Nu=div(|?u|N-2?u),the nonlinear term f(u)has critical exponential growth.By using mountain pass theorem,the existence of the week solutions of the equation is established and prove the existence of ground state solution.In chapter 3,we study the existence of positive solution for elliptic equation as follows-?Nu+(?V(x))+Z(x))|u|N-2u=f(u)x ?RN where N?2,u?W1,N(RN),?Nu=div(|?u|N-2?u),the nonlinear term f(u)has critical exponential growth.By using Trudinger-Moser inequality and mountain pass theorem,prove the existence of positive solution.