A nonlinear elliptic system with a transport term and singular data

Abstract

We consider the nonlinear elliptic system $\begin{array}{r}\{\begin{array}{ll}u\in W_0^{\frac{N}{N-1}}\left(\mathrm{\Omega }\right):& -\text{div}\left(M\right(x\left)\mathrm{\nabla u}\right)+u=-\text{div}\left(uM\right(x\left)\mathrm{\nabla \psi }\right)+f\left(x\right),\\ {\displaystyle \psi \in W_0^{1,2}\left(\mathrm{\Omega }\right):}& -\text{div}\left(M\right(x\left)\mathrm{\nabla \psi }\right)+\psi =R\left(u\right)+E\left(x\right)\mathrm{\nabla \psi },\end{array}\end{array}$where Ω is a bounded, open subset of ${\mathbb{R}}^N$, $N\ge 3$; $M\left(x\right)$ is a coercive, symmetric matrix with $L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ coefficients; $f\left(x\right)$ and $E\left(x\right)$ belong to some Lebesgue space, and $R\left(s\right)$ is a continuous function such that $\begin{array}{r}0\le R\left(s\right)\le |s{|}^{\theta },\text{for}\theta <\frac{2}{N}.\end{array}$Using a duality technique, we prove existence of at least a weak solution $(u,\psi )$. Moreover, if N=3 or N=4, we prove under stronger assumptions on $f\left(x\right)$ and $E\left(x\right)$ that the solution u belongs to $W_0^{1,2}\left(\mathrm{\Omega }\right)$.

Keywords:

• Elliptic systems
• finite energy solutions
• weak data
• regularity of solutions
• chemotaxis

1991 Mathematics Subject Classifications:

• 35J47
• 35A01
• 35B09

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The third author has been supported by Ministerio de Ciencia e Innovación (Spain) [grant number PID2022-141114NB-I00].