# Semi-linear boundary problems of composition type in $L_p$-related spaces

## TL;DR

This paper investigates semi-linear elliptic boundary value problems with nonlinear composition perturbations in various $L_p$-related function spaces, establishing existence results using topological methods and advanced regularity theory.

## Contribution

It extends existence results for semi-linear elliptic problems to broad $L_p$-Sobolev, Besov, and Triebel--Lizorkin spaces, employing novel regularity techniques.

## Key findings

- Existence of solutions in $L_p$-Sobolev spaces.
- Extension to Besov and Triebel--Lizorkin spaces.
- Application of Leray--Schauder and Landesman--Lazer conditions.

## Abstract

The class of problems treated here are elliptic partial differential equations with a homogeneous boundary condition and a non-linear perturbation obtained by composition with a fixed smooth function. The existence of solutions is obtained from the Leray--Schauder theorem or under a Landesman--Lazer condition on the data. Existence is carried over to a wide range of $L_p$-Sobolev spaces, using a non-trivial procedure to obtain a general regularity result. In fact the results are obtained in the general scales of Besov and Triebel--Lizorkin spaces.