Some Studies On The Existence Of Positive Solutions For A Class Of Quasilinear Elliptic Equations

We have studied the existence of positive solutions of a class of quasilinear elliptic equations in this dissertation.In the second chapter, we list some results about the semilinear elliptic equation-△u=k(u),(1) and under some assumptions on the nonlinear term k(v), equation (1) has ground state solution.In the third chapter, we consider the quasilinear elliptic equation as follows-△u-△(u2)u+V(x)u=λh(u),(2) where x∈RN, N≥3. Under some assumptions on the potential function V(x) and the nonlinear term h(s), by change of variables, the quasilinear problem (2) was transformed to a semilinear problem (1). Moreover, we can proof that the variational functional corresponding to problem (1) satisfy (PS) condition and Mountain-pass geometry. Consequently, we obtain that for almost every λ∈[1/2,1], equation (2) has positive solution. In the forth chapter, we discuss the existence of positive solution for the equation-△u△A(u2)u+V(x)u=h(u).(3) Set λ=1, equation (2) become into equation (3), but the potential function V(x) and the nonlinear term h(s) have to satisfy more restriction. According to the Pohozaev type identity and the decomposition of the sequence, we obtain a result about the existence of positive solution for equation (2) by approximation method.