Work

Existence results for nonlinear fourth-order elliptic boundary value problems

Year: 2025

Type: article

Abstract: This paper is concerned with the existence of a solution of the nonlinear fourth-order elliptic boundary value problem $$ \left \{ \textstyle\begin{array}{l} {\Delta }^{2} u = f(x,\,u,\,\Delta u),\qquad x\in \Omega , \\ u=\Delta u=0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$ where Ω is a bounded smooth domain in $\mathbb{R}^{N}$, $f: \overline{\Omega }\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a continuous function. We firstly present an existence result under a growth condition of $f(x,\,\xi ,\,\eta )$ on ξ and η, then use this result and a truncating function technique to establish an upper and lower solution theorem for the existence without monotonicity hypothesis of f. Using the upper and lower solution theorem we obtain an existence result of positive solution.

Source: Journal of Inequalities and Applications

DOI: 10.1186/s13660-025-03408-1

Authors: Yongxiang Li, Yanyan Wang

Institution: Northwest Normal University

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,190

Funder: National Natural Science Foundation of China