Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

Abstract

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem $\left\{\begin{array}{cc}-\mathrm{div}\left(M\left(x\right)\nabla u\right)+{\left|u\right|}^{p-1}u=-\mathrm{div}\left(uE\left(x\right)\right)+f\left(x\right),\hfill & x\in \Omega ,\hfill \\ u\left(x\right)=0,\hfill & x\in \partial \Omega ,\hfill \end{array}\right.$ where $\Omega \subset {ℝ}^N\left(N>2\right)$ is a bounded smooth domain with $0\in \Omega$ , $f$ belongs to the Lebesgue space $L^m\left(\Omega \right)$ with $m\ge 1,p>0.$ The main innovation point of this paper is the combined effects of the convection terms and lower-order terms in elliptic equations.