The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus

Li, Yongxiang; Wang, Yanyan

doi:10.3390/axioms13060383

This is an early access version, the complete PDF, HTML, and XML versions will be available soon.

Open AccessArticle

The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus

by

Yongxiang Li

^*

and

Yanyan Wang

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

^*

Author to whom correspondence should be addressed.

Axioms 2024, 13(6), 383; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13060383

Submission received: 24 April 2024 / Revised: 29 May 2024 / Accepted: 1 June 2024 / Published: 4 June 2024

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)

Download Versions Notes

Abstract

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation

▵^{2} u = f (| x |, u, | \nabla u |, ▵ u)

in an annular domain

Ω = {x \in R^{N} : r_{1} < | x | < r_{2}}

(

N \geq 2

) with the boundary conditions

{u |}_{\partial Ω} = 0

and

{▵ u |}_{\partial Ω} = 0

, where

f : [r_{1}, r_{2}] \times R \times R^{+} \times R \to R

is continuous. Under certain inequality conditions on f involving the principal eigenvalue

λ_{1}

of the Laplace operator

- ▵

with boundary condition

{u |}_{\partial Ω} = 0

, an existence result and a uniqueness result are obtained. The inequality conditions allow for

f (r, ξ, ζ, η)

to be a superlinear growth on

ξ, ζ, η

as

| (ξ, ζ, η) | \to \infty

. Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates.

Keywords: biharmonic elliptic equation; radial solution; annular domain; Leray–Schauder fixed point theorem; principal eigenvalue

Share and Cite

MDPI and ACS Style

Li, Y.; Wang, Y. The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus. Axioms 2024, 13, 383. https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13060383

AMA Style

Li Y, Wang Y. The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus. Axioms. 2024; 13(6):383. https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13060383

Chicago/Turabian Style

Li, Yongxiang, and Yanyan Wang. 2024. "The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus" Axioms 13, no. 6: 383. https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13060383

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus

Abstract

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI