论文标题
带有内部和边界非线性反应的椭圆形替代品
A Fredholm Alternative for Elliptic with Interior and Boundary Nonlinear Reactions
论文作者
论文摘要
在本文中，我们研究了以下广义非线性两参数问题的解决方案\ begin {equation*} a（u，v）\; = \; λ\，b（u，m） +μ\，m（u，v） + \ varepsilon \，f（u，v），对于三重$（a，b，m）的连续，对称性双线性形式的三重$（a，b，m）的真实可分开的hilbert space $ v $ v $和非线性$ f $。这个问题是大量差分运算符的非线性问题的自然抽象，在微分方程中具有非线性的各种椭圆形PDE和/或边界条件是特殊的子类。首先，开发了相关线性两参数特征值问题的弗雷德霍尔姆替代方案，然后将其用于构建弗雷德霍尔姆替代方案的非线性版本。最后，使用Steklov-Robin Fredholm方程来说明抽象结果。
In this paper we study the existence of solutions to the following generalized nonlinear two-parameter problem \begin{equation*} a(u, v) \; =\; λ\, b(u, m) + μ\, m(u, v) + \varepsilon\, F(u, v), \end{equation*} for a triple $(a, b, m)$ of continuous, symmetric bilinear forms on a real separable Hilbert space $V$ and nonlinear form $F$. This problem is a natural abstraction of nonlinear problems that occur for a large class of differential operators, various elliptic pde's with nonlinearities in either the differential equation and/or the boundary conditions being a special subclass. First, a Fredholm alternative for the associated linear two-parameter eigenvalue problem is developed, and then this is used to construct a nonlinear version of the Fredholm alternative. Lastly, the Steklov-Robin Fredholm equation is used to exemplify the abstract results.
