Variational Methods: Applications To Nonlinear Differential And Difference Equations

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.