Study of nonlinear differential equations via variational methods

用变分方法研究了非线性微分方程解的存在性和多重性。特别地，我们研究了奇异摄动问题。研究了非线性标量场方程-Δu+V(x)u=f(u) in R^N的解的存在性和多重性。通常在这类问题中，非线性f(u)（例如）的全局条件。全局Ambrosetti-Rabinowitz条件)来保证解的存在性。在这项研究中，我们试图在没有这种全局假设的情况下获得存在性结果，我们发现，如果我们要求电位V(x).2的衰变足够快，这是可能的。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)【大于等于】0。作为λ→∞的极限问题，一个Dirichlet边值问题-Δu=|u|^<p-1>u, u|_<∂Ω>=0在Ω≡{x∈R^N；会出现一个(x) = 0}。我们假设Ω由几个有界连通分量Ω_1，【三键】，Ω_κ组成，对于Ω_i中Dirichlet问题的已知解u_i(x)，我们试图在R^N中找到一个极限为Ω_i中u_i(x)的解（连接问题）。我们成功地找到了一个没有非简并条件的连接Mountain Pass解的解。并证明了有无穷多个变号解与Mountain Pass解可连通。对于一维Allen-Cahn方程和Schrodinger方程，研究了奇异摄动下一族解的性质。更准确地说，我们考虑一组具有越来越多的层或尖峰的解决方案。我们用“极限能函数”或“包络函数”给出了这种族的表征。相反，对于可容许模式，我们通过变分方法构造相应的解族。

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.