【作者】 卢静；

【导师】 宋先发；

【作者基本信息】 天津大学 ， 应用数学， 2015， 硕士

【摘要】 在本文中,我们主要工作是解决下面的拟线性椭圆方程非平凡解的存在性.其中△pu=div((|▽u|)p-2▽u),1<p<N,p*=Np/(N-p)是Sobolev |临界指数,V,K,W:RN→R和g:RN×R→R是连续函数,并且h(x,u)=m1(W*(|u|)m2)(|u|)m1-2u+m2(W*(|u|)m1)(|u|)m2-2u.本文主要给出了二阶偏微分方程和变分法的一个应用实例,即利用变分法探讨一类带有临界指数的拟线性椭圆方程非平凡解的存在性.首先,我们简述了变分法的发展历史以及利用变分原理解决二阶偏微分方程问题的研究动态,并给出了本文的预备知识.随后,根据方程条件,我们给出了变分结构,将方程解的存在性问题转化为泛函存在临界点问题,利用山路定理的第一个推广形式找到一个(p.S.)。序列,并证明了有界性.因为研究区域是冗N,导致Sobolev嵌入不紧,我们引入了集中紧性原理并给出了临界值估计.然后,利用不等式的推导,Fatou’s引理及山路定理的第二个推广形式等证得所研究的带有临界指数的拟线性椭圆方程存在非平凡解.更多还原

【Abstract】 In this paper, we will consider the existence of nontrivial solution to the following quasilinear elliptic problem where △pu= div((|Vu|)p-2▽u),1< p< N, p*= Np/(N-p) is Sobolev critical exponent, V,K,W:RN→R and g:RN x R→ R are continuous functions, and h(x, u)= m1(W*(|u|)m2)(|u|)m1-2u+m2(W*(|u|)m1)(|u|)m2-2u.This paper gives an application of second order partial differential equations and calculus of variations, which investigates the existence of the nontrivial solu-tion to a quasilinear elliptic equations with Sobolev critical exponent by applying calculus of variations.At first, we outline the history of the calculus of variations’development and advances of solving second order partial differential equations by it and give some preliminary results.Subsequently, according to the equation conditions, we give the variational framework and transfer the problem which study the existence of the solution to a quasilinear elliptic equation into the problem which discusses the existence of the functional’s critical point. By the first version of the Mountain Pass Theorem, we obtain a (P.S.)c sequence, and we prove it is bounded. Because the domain is RN, the Sobolev embedding is not compact. So we introduce the Concentration-Compactness Principle and give the estimate of the critical value. Therefore, applying the inequalities, Fatou’s lemma, the second version of the Mountain Pass Theorem and so on, we prove the nontrivial existence of the equation we study.更多还原