# On multiple solutions to a family of nonlinear elliptic systems in divergence form coupled with an incompressibility constraint

Taheri, Ali and Vahidifar, Vahideh (2022) On multiple solutions to a family of nonlinear elliptic systems in divergence form coupled with an incompressibility constraint. Nonlinear Analysis Theory Methods and Applications, 221. a112889. ISSN 0362-546X

## Abstract

The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint:
\begin{align*}
\left\{
\begin{array}{ll}
\dive\{\A(|x|,|u|^2,|\nabla u|^2) \nabla u\} + \B(|x|,|u|^2,|\nabla u|^2) u
= \dive \{ \mcP(x) [{\rm cof}\,\nabla u] \} \quad &\text{ in} \ \Omega , \\
\text{det}\, \nabla u = 1 \ &\text{ in} \ \Omega , \\
u =\varphi \ &\text{ on} \ \partial \Omega,
\end{array}
\right.
\end{align*}
where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) is a bounded domain, $u=(u_1, \dots, u_n)$ is a vector-map and $\varphi$ is a prescribed boundary condition. Moreover $\mathscr{P}$ is a hydrostatic pressure associated with the constraint $\det \nabla u \equiv 1$ and $\A = \A(|x|,|u|^2,|\nabla u|^2)$, $\B = \B(|x|,|u|^2,|\nabla u|^2)$ are sufficiently regular scalar-valued functions satisfying suitable growths at infinity. The system arises in diverse areas, e.g., in continuum mechanics and nonlinear elasticity, as well as geometric function theory to name a few and a clear understanding of the form and structure of the solutions set is of great significance. The geometric type of solutions constructed here draws upon intimate links with the Lie group ${\bf SO}(n)$, its Lie exponential and the multi-dimensional curl operator acting on certain vector fields. Most notably a discriminant type quantity $\Delta=\Delta(\A,\B)$, prompting from the PDE, will be shown to have a decisive role on the structure and multiplicity of these solutions.

Item Type: Article Nonlinear elliptic systems, Incompressible maps, Maximal tori, Determinant constraint, Multiple solutions, Curl Operator School of Mathematical and Physical Sciences > Mathematics Analysis and Partial Differential Equations Research Group Mx Elements Account Mx Elements Account 15 Mar 2022 09:20 31 Mar 2022 07:00 http://sro.sussex.ac.uk/id/eprint/104861