Variational study of nonlinear diffential equations

非线性微分方程的变分研究

• 批准号: 11640216
• 负责人: TANAKA Kazunaga
• 金额: $ 2.37万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640216/
• 关键词: variational problems; elliptic equatims; Hamiltonian systems; 非線型楕円型方程式

We study the existence problems for nonlinear differential equations via variational methods. We mainly dealt with nonlinear elliptic problems and Hamiltonian systems.1. We study the existence and multiplicity of positive solutions of nonlinear scalar field equations in unbounded domains. In particular, we are concerned with equations which depend on the space variable x and we investigate the effects of the inhomogeneity (dependence on the space variable x) on the set of solutions of the scalar field equation. We find a very delicate dependence ― very small inhomogeneity induces a big change in the set of soluitons ― and we find an example of nonlinear scalar field equation which has 4 positive solutions after very small but not zero pert perturbation.2. We also consider the singular perturbation problems for nonlinear elliptic problems. We get 2 results : (a) for 1-dimensional setting we introduce a new finite dimensional reduction and we succeed to prove the existence of solutions w … More ith a cluster of interior or boundary layers for inhomogeneous Allen-Cahn type equations. We also succeed to prove the existence of solutions with a cluster of spikes for nonlinear Schrodinger equations. (b) We give a mountain pass characterization of positive solutions for a wide class of nonlinear elliptic equations. As an application, we show the existence of a spike solution for a wide class of nonlinear elliptic equations including asymptotically linear equations.3. For Hamiltonian systems we deal with singular Hamiltonian systems with 2-body type singularities. In case the potential V(q) has more than 3 strong force type singularities we find a family of very complex solutions which are related with symbolic dynamical systems. We also deal with the case the set S of singularity is not a point and it has a positive volume. We consider the case where V(q) 〜 - 1/ dist (q, S)^α and we find the existence of non-collision solutions for all positive α > 0, that is, even for weak force case 0 < α < 2. Less

利用变分方法研究非线性微分方程解的存在性问题。主要研究了非线性椭圆问题和Hamilton系统.研究了无界区域上非线性标量场方程正解的存在性和多解性。特别是，我们关注的方程依赖于空间变量x和我们调查的不均匀性（依赖于空间变量x）的标量场方程的解决方案的集合上的效果。我们发现了一个非常微妙的依赖关系― ―非常小的不均匀性引起解集的大变化― ―并且我们发现了一个非线性标量场方程的例子，它在非常小但不为零的扰动后有4个正解.我们还考虑了非线性椭圆型问题的奇摄动问题。我们得到了两个结果：（a）对于1维情形，我们引入了一个新的有限维降阶，并成功地证明了解的存在性。 ...更多信息 它是非齐次Allen-Cahn型方程的一组内部或边界层。我们还成功地证明了非线性薛定谔方程的一簇尖峰解的存在性。(b)给出了一类非线性椭圆型方程正解的山路刻画。作为应用，我们证明了包括渐近线性方程在内的一类非线性椭圆型方程的尖峰解的存在性.对于哈密顿系统，我们处理奇异哈密顿系统的2-体类型的奇异性。当势函数V（q）具有3个以上的强力型奇点时，我们发现了一族与符号动力系统有关的非常复杂的解。我们还处理的情况下，集S的奇点不是一个点，它有一个积极的体积。我们考虑V（q）<$-1/dist（q，S）^α的情形，并且我们发现对于所有正的α> 0，也就是说，即使对于弱力情形0 <α <2，也存在非碰撞解.少

项目成果

S.Adachi, K.Tanaka: "Four positive solutions for the semilinear elliptic equation : -Δu+u=a(x)u^p+f(x) in R^N"Calculus of Variations and PDE. 11. 63-95 (2000)

S.Adachi、K.Tanaka：“半线性椭圆方程的四个正解：R^N 中的 -Δu+u=a(x)u^p+f(x)”变分和偏微分方程 11。63- 95 (2000)

S. Adachi and K. Tanaka: "Ttudinger type inequalities in R^N and their best exponents"Proc. Amer. Math. Soc.. 128. 2051-2057 (2000)

S. Adachi 和 K. Tanaka：“R^N 中的 Ttudinger 型不等式及其最佳指数”Proc。

T. Shibata: "Precise asymptotic formulas for semilinear eigenvalue problems"Ann. Inst. H. Poincare. 2. 713-732 (2001)

T. Shibata：“半线性特征值问题的精确渐近公式”Ann。

S. Adachi: "A positive solution of a nonhomogeneous elliptic equation in R^N with G-invariant nonlinearity"Comm. Partial Diff. Eq.. 27. 1-22 (2002)

S. Adachi：“具有 G 不变非线性的 R^N 中非齐次椭圆方程的正解”Comm。

L. Jeanjean and K. Tanaka: "A remark on least energy solutions in R^N"Proc. A ner. Math. Soc.. to appear.

L. Jeanjean 和 K. Tanaka：“关于 R^N 中最少能量解决方案的评论”Proc。