Existence results for semi-linear differential equations with nonlocal boundary value conditions - Bełdziński, Michał; Galewski, Marek; Kossowski, Igor (Topological Methods in Nonlinear Analysis, 2026)
The paper deals with a semi-linear elliptic equation with non-local boundary conditions. In order to tackle problem under consideration we employ the classical Schauder theorem and the theory of convergent matrices.Firstly, we examine the solvability of the auxiliary problem \begin{equation*} \begin{cases} -\Delta u(\boldsymbol{x}) = f(\boldsymbol{x},u(\boldsymbol{x})) & \text{for }\boldsymbol{x}\in \Omega, \\ \displaystyle u(\boldsymbol{x}) = \int_\Omega K(\boldsymbol{x},\boldsymbol{\xi}) h(\boldsymbol{\xi},u(\boldsymbol{\xi})) d\boldsymbol{\xi} & \text{for } \boldsymbol{x}\in \partial \Omega, \end{cases} \end{equation*}where the growth of $f$, $h$ and the kernel $K$ are limited and where $\Omega$ is open and bounded set with a Lipschitz bondary.Next, under the assumption of a sublinear growth for $f$ and $g$, and using the approximation methods, the solvability of the problem of the form \begin{equation*} \begin{cases} -\Delta u(\boldsymbol{x}) = f(\boldsymbol{x}, u(\boldsymbol{x})) & \text{for }\boldsymbol{x}\in \Omega, \\ u(\boldsymbol{x}) = \alpha(\boldsymbol{x}) g (u(\boldsymbol{p}_1),\ldots, u(\boldsymbol{p}_m))& \text{for }\boldsymbol{x}\in \partial \Omega, \end{cases} \end{equation*} is examined in the $H^2$-sense, where $\Omega$ is an open and bounded subset of $\mathbb{R}^2$ or $\mathbb{R}^3$ with a $C^2$ boundary and where $\boldsymbol{p}_1,\ldots, \boldsymbol{p}_m \in \Omega$. We focus mainly on having the nonlocal condition fixed at one point and then proceed to the multipoint case.