Variational study of nonlinear diffential equations

用变分方法研究了一类非线性微分方程的存在性问题。我们主要讨论了非线性椭圆型问题和哈密顿系统。研究了无界域上非线性标量场方程正解的存在性和多重性。特别地，我们关注依赖于空间变量x的方程，我们研究了非齐次性（依赖于空间变量x）对标量场方程解集的影响。我们发现了一种非常微妙的依赖关系——非常小的非均匀性会引起解集的大变化——并且我们找到了一个非线性标量场方程的例子，该方程在非常小但不是零的pert扰动后有4个正解。我们还研究了非线性椭圆型问题的奇异摄动问题。得到2个结果：(a)在一维情况下，引入了一个新的有限维约简，并成功地证明了非齐次Allen-Cahn型方程具有一簇内层或边界层解的存在性。我们还成功地证明了非线性薛定谔方程具有一簇尖峰解的存在性。(b)给出了一类广泛的非线性椭圆型方程正解的山口刻划。作为一个应用，我们证明了一类广泛的非线性椭圆型方程（包括渐近线性方程）的峰值解的存在性。对于哈密顿系统，我们处理具有二体奇点的奇异哈密顿系统。当势V(q)具有3个以上的强力型奇点时，我们得到了一系列与符号动力系统有关的非常复杂的解。我们还处理了奇异点集合S不是点的情况它的体积是正的。我们考虑了V(q) ~ - 1/ dist (q， S)^α的情况，得到了所有正α > 0的非碰撞解的存在性，即即使弱力情况0 < α < 2也是如此。少

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