【作者】 王楠；

【导师】 刘轼波；

【作者基本信息】 厦门大学 ， 基础数学， 2008， 硕士

【摘要】 本文利用变分方法研究全空间上一类拟线性椭圆方程的无穷多解的存在性以及二阶差分方程的多重解的存在性.首先,我们应用喷泉定理研究了拟线性椭圆方程的无穷多解的存在性.其中Δ_pu=div(|（?）u|^p-2（?）u),1<p<∞.这里我们考虑超线性的情形:但与通常的结果不同,我们的f（x,u）不满足Ambrosetti-Rabinowit的超线性条件.其次,我们应用三临界点定理和Clark定理研究如下的非线性差分方程多重m-周期解的存在性.这里m≥2为一固定的整数,δ>0,Δx_n=x_n+1-x_n,Δ²x_n=Δ（Δx_n）,{p_n}是个m-周期非负实序列,f是Z×R上的连续函数,且关于第二变元是m-周期的.我们约定（-1）^δ=-1.更多还原

【Abstract】 In this thesis, the existence of infinitely many solutions for quasilinear elliptic equation on R^N, and multiple periodic solutions for nonlinear difference equations are obtained by variational methods. First, we use the Fountain theorem to obtain infinitely many solutions for quasilinear elliptic equationwhere△_pu = （?）, 1 < p <∞. Here we consider the superlinear case: （?）. Unlike the usual results, our f（x,u） does not satisfy the superlinear condition of Ambrosetti-Rabinowitz.Next, using the three critical points theorem and the clark theorem, we consider the existence of multiple periodic solutions for nonlinear difference equations of the formwhere m≥2 is a fixed integer,δ> 0,△is the forward difference operator defined by△x_n = x_n+1 - x_n, {p_n} is a nonnegative m-periodic real sequence; f is a continuous function on Z×R, which is m-periodic on the second variable. Throughout this thesis, the convention （-1）^δ= -1 is made.更多还原