【作者】 黄欣；

【导师】 蒲志林；

【作者基本信息】 四川师范大学 ， 基础数学， 2008， 硕士

【摘要】 本文研究三类在无穷远处具渐近线性性质的非线性偏微分方程解的存在性.首先,研究带非负势的渐近线性椭圆方程的Dirichlet问题解的存在性.通过定义一个约束变分问题,使得在没有(AR)条件的情况下,利用一种改进了的山路引理,证明了方程(0-1)的正解存在性问题.其次,研究带无界势的渐近线性椭圆方程解的存在性.通过使用插值估计,得到新的约束变分问题解的存在性,进而通过变形的广义山路引理得到方程(0-2)非平凡弱解的存在性.最后,研究K - P方程解的存在性.其中f(x,y,u)关于u满足在无穷远处的渐近线性性质.通过定义一个约束变分问题,使得在没有(AR)条件的情况下,利用一种改进了的山路引理,证明了方程(0-3)行波解的存在性.更多还原

【Abstract】 This paper is concerned with the existence of the solutions forthree types of nonlinear partial di?erential equations with asymptotically linearproperty at infinity.At first we study the existence of the solutions of Dirichlet problems forthe asymptotically linear elliptic equations with nonnegative potentialBy defining a constraint variational problem and using the improved mountainpass theorem, the existence of positive solutions of the Dirichlet problems for(0-1) is obtained without (AR) condition.Next we investigate the existence of the solutions for the asymptoticallylinear elliptic equations with unbounded potentialBy using the interpolation estimate, the existence of the solutions for a new con-strained variational problem is obtained. Furthermore, by using the deformedmountain pass theorem, the existence of nontrivial solutions for (0-2)is gotten.At last, we study the existence of the travelling wave solutions for K-PEquationwhere f(x,y,w) is asymptotically linear with respect to w at infinity. Bydefining a constraint variational problem and using an improved mountain pass theorem, the existence of the travelling wave solutions for (0-3) is obtainedwithout (AR) condition.更多还原