Study of nonlinear differential equations via variational methods

利用变分方法研究了一类非线性微分方程解的存在性和多解性。1.研究了非线性标量场方程：-Δu+V（x）u=f（u）在R^N中解的存在性和多解性。通常在这样的问题中，需要非线性f（u）的全局条件（例如全局Ambrosetti-Rabinowitz条件）来确保解的存在性。在本研究中，我们试图在没有这样的整体假设的情况下得到一个存在性结果，并且我们发现，如果我们要求势V（x）的衰减足够快，这是可能的。2.我们还研究了奇异摄动问题：-Δu+λ^2a（x）u=| u| ^<p-1>u在R^N中，其中a（x）[大于或等于]0。当λ→∞时，作为极限问题，Dirichlet边值问题-Δu=| u| ^<p-1>u，u|在Ω &lt;${x ∈R^N;a（x）=0}中出现_&lt;&lt;$Ω&gt;=0。我们假设Ω是由几个有界连通分支Ω 1，[三键]，Ω κ组成的，对于Ω i中Dirichlet问题的给定解u i（x），我们试图在R^N中找到一个解u λ（x），其极限是Ω i中的u i（x）（连通问题）.我们成功地找到了一个解决方案，加入山口解决方案没有非简并条件。3.对于一维Allen-Cahn方程和Schrodinger方程，我们研究了奇异摄动下一族解的特征.更确切地说，我们考虑一个家庭的解决方案，越来越多的层或尖峰。我们用“极限能量函数”或“包络函数”给出了此类族的特征。相反，可添加的模式，我们通过变分方法构建相应的家庭的解决方案。

We study the existence and multiplicity of solutions of nonlinear differential equations via variational methods. In particular, we study singular perturbation problems.1.We study the existence and multiplicity of solutions of nonlinear scalar field equations : -Δu+V(x)u=f(u) in R^N. Usually in such a problem global conditions on nonlinearity f(u)(ex.global Ambrosetti-Rabinowitz condition) are required to ensure the existence of solutions. In this study we tried to obtain an existence result without such global assumptions and we find that it is possible if we require sufficiently fast decay of the potential V(x).2.We also study singular perturbation problem : -Δu+λ^2a(x)u=|u|^<p-1>u in R^N, where a(x)【greater than or equal】0. As a limit problem as λ→∞, a Dirichlet boundary value problem -Δu=|u|^<p-1>u, u|_<∂Ω>=0 in Ω≡{x ∈R^N;a(x)=0} appears. We assume Ω consists of several bounded connected components Ω_1,【triple bond】, Ω_κ and for given solutions u_i(x) of the Dirichlet problem in Ω_i, we try to find a solution u_λ(x) in R^N whose limit is u_i(x) in Ω_i (connecting problem). We succeed to find a solution joining Mountain Pass solutions without non-degeneracy conditions. Also we show that there are infinitely many sign-changing solutions that are connectable with Mountain Pass solutions.3.For 1-dimensional Allen-Cahn equations and Schrodinger equaitons, we study the characterization of a family of solutions in the setting of singular perturbation. More precisely, we consider a family of solutios with increasing number of layers or spikes. We give a characterization of such a family using "limit enery function" or "envelop function". Conversely for addmissible patterns we construct corresponding families of solutions via variational methods.