【作者】 章国庆；

【导师】 刘三阳；

【作者基本信息】 西安电子科技大学 ， 应用数学， 2005， 博士

【摘要】 数学、物理学、化学、生态学及经济学等学科产生的非线性微分差分问题,正日益引起人们的重视。目前,已有许多学者对非线性微分差分问题解的存在性与多重性利用不同的方法进行了深入和广泛的研究,这些方法主要有变分法、拓扑度法、单调迭代法与Kaplan-Yorke耦合系统法等。 本文主要利用变分法,分别研究了有界区域和全空间上拟线性椭圆型方程组、二维空间上半线性椭圆型方程(组)和二阶非线性差分方程(组)问题的解与多重解的存在性,具体内容如下: 1.建立了一个抽象的紧性定理,然后借此定理证明了对应于一类拟线性椭圆型方程组的泛函在比Boccardo和De.Figueiredo(2002)的条件更弱的条件下满足(C)条件,并利用山路引理证明了这类拟线性椭圆型方程组非平凡解的存在性,最后举出两个例子验证了文中所给条件的确比Boccardo和De.Figueiredo(2002)的条件弱。 2.讨论了在有界区域上一类含参数拟线性椭圆型方程组的Nehari流形,并利用Nehari流形的性质证明了该方程组存在非平凡解,且利用Picone恒等式,证明了解的不存在性,从而得到了其整体分支结论。 3.讨论了在全空间上拟线性椭圆型方程组的非平凡解的存在性。首先,利用上—下解方法考虑了一类含参数的拟线性椭圆型方程组,应用山路引理得到了该方程组存在非负上解,再利用Leray-Schauder不动点定理证明了其非平凡非负解的存在性;其次,研究了全空间上一类共振椭圆型方程组非平凡解的存在性,利用一种变形的山路引理,在一定条件下证明了该方程组非平凡解的存在性,所获结论改进了已有文献中相关结果。 4.在二维空间上考虑了半线性椭圆型方程(组)解的存在性。首先,讨论了R~2中一类带不定权且含临界位势的二阶椭圆型方程的特征值问题,并借此特征值问题的第一特征值性质,利用山路引理及Trudinger-Moser不等式,证明了R~2中一类带不定权且含临界位势的非线性椭圆型方程非平凡解的存在性;其次,利用广义环绕定理,Trudinger-Moser不等式及集中列紧原理,得到了R~2上一类具有强不定部分的半线性椭圆型方程组在非线性项分别为次临界增长和临界增长情形下非平凡解的存在性。 5.利用Ricceri建立的三临界点定理,证明了一类含参数二阶差分方程在某些新的条件下至少存在三个解;随后,利用临界点理论中的极小极大原理和环绕定理,讨论了一类二阶超线性差分方程组多重解的存在性。 6.给出了变分法在物理学中的两个应用。首先,讨论了非线性光学中的二次谐波产生的耦合方程组,利用变分法证明了耦合方程组非平凡解的存在性,然后进行了数值模拟,实验结果表明文中方法比经典的非线性光学中的方法有较大的改进,这对优化光倍频器件的设计将有所帮助;其次,将一类二阶半线性椭圆型方程组转化为一个变分问题,然后利用变分法和更多还原

【Abstract】 Many nonlinear differential and difference problems resulting from mathematics, physics, chemistry, biology and economics and so on have increasingly brought people ’s attention. Now, the existence and multiplicity of solution for nonlinear differential and difference problems have been studied extensively by various methods including variational methods, topology degree method, monotone iterative method and Kaplan-Yorke coupled system methods.The aim of this paper is to study the existence of solution and multiply results for quasilinear elliptic system in bounded domain and the whole space, semilinear elliptic equation in two dimension space, second order nonlinear difference equation by variational methods. The main results are listed in the following:1. An abstract compactness theorem is proved, and then, under weak Boccardo and De.Figueiredo (2002) condition, using above compactness theorem we prove that the functional corresponding to a class of quasilinear system of elliptic equations satisfies (C) condition. After that the existence of a nontrival solution for the elliptic system is proved by means of Mountain-Pass Lemma. At last, we present two examples to illustrate that our condition is weaker than Boccardo and De.Figueiredo (2002) condition really.2. The Nehari manifold for a class of quasilinear elliptic systems involving a pair of (p,q)-Laplacian operators and a parameter is studied. We prove the existence of a nonnegative nonsemitrivial solution for the systems by discussing property of the Nehari manifold, and so global bifurcation results are obtained. Thanks to Picone’s identity, we also prove a nonexistence result.3. The existence of weak solution for quasilinear elliptic system in the whole space is studied. Firstly, by using sub-solution and super-solution methods, a class of quasilinear elliptic systems involving a parameter in R~N is considered. The existence of a nonnegative super-solution is obtained by Mountain-Pass Lemma, and then, thanks to Leray-Schauder fixed point theorem, we prove the existence of its solution. Secondly, the problem of a class of resonant quasilinear elliptic systems in R~N is investigated, the existence of a nontrivial solution for the elliptic systems is obtained under some conditions by a variant of the Mountain Pass Lemma. Our results improve the related results in the literature.4. The semilinear elliptic problem in two dimension space is discussed. Firstly, the eigenvalue problem of a class of second order elliptic equation with critical potential and indefinite weights is considered. Then, using critical point theory, Trudinger-Moser inequality and the properties of the first eigenvalue, we prove the existence of a nontrivial solution for a class of nonlinear elliptic with critical potentialand indefinite weights in R~2. Secondly, we prove the existence of nontrivial solutions for a class of subcritical and critical elliptic systems with indefinite part in R~2 byusing a generalized linking theorem, Trudinger-Moser inequality and concentration-compactness principle.5. The existence of at least three weak solutions for discrete boundary value problem is established by using a three critical point theorem introduced by Ricceri. A complete novel assumption is presented. Then, we prove the existence of two nontrivial solutions for a class of second order superlinear difference systems with the help of Minimax principle and Linking theorem in critical point theory,.6. Two applications in physics are given. Firstly, a coupled differential system coming from second harmonic waves is discussed. The nontrivial solutions of the coupled differential system are proved by variational methods. The numercial simulation has been carried out, and the experimental results show that this method has been improved significantly comparing with the traditional one in nonlinear optics. Secondly, a second order system of semilinear elliptic equations, which comes from a mathematics model of the electric potential distribution of a media surrounded with the protein, is transformed into a variational problem, the existence of solution of it is proved by using variational principle and Trudinger—Moser inequality, and the relation between the solutions of variational problem and the solutions of its dual problem is obtained.更多还原