The Existence Of Solutions For A Class Of Quasilinear Elliptic Equations

In this paper, we study a class of quasilinear elliptic equations of the form where △u:= div(|▽u|p-2▽u) is the p-Laplace operator. Under some suitable hy-potheses on the potential V(x) and nonlinearity g(x,u) which satisfy super-p growth and asymptotical-p growth conditions, we study the existence and multiplicity of non-trivial solutions to the equations (1).This paper will be divided into the following four chapters:Chapter 1:We introduce the background, the present research situation and the main work of this paper.Chapter 2:We stated some theorem and preliminary assumptions which will be used in the sequel.Chapter 3:Under the hypotheses of super-p growth on g(x,u) and sign-changing potential V(x) conditions, we will prove the existence and multiplicity of the solutions to the quasilinear elliptic equations (1) by using the change of variables, Ekeland’s variational principle and symmetric mountain pass theorem.Chapter 4:Under the hypotheses of the asymptotical-p growth on g(x,u) condi-tions, we will establish the existence of the nontrivial solutions to the quasilinear elliptic equations (1) through applying the change of variables and mountain pass theorem.