Multiplicity of Positive Solutions of Nonlinear Elliptic Equation with Gradient Term

In this paper, we consider the following nonlinear elliptic equation with gradient term: $$\{\begin{array}{c}{\displaystyle -\mathrm{\Delta }u-\frac{1}{2}(x\cdot \mathrm{\nabla }u)+(\lambda a(x)+b(x))u=\beta u^q+u^{2^{\ast }-1},\text{hfill}0<u\in H_K^1({\mathbb{R}}^N),\text{hfill}}\end{array}$$ where $\lambda ,\beta \in (0,\mathrm{\infty }),q\in (1,2^{\ast }-1),2^{\ast }=2N\mathrm{/}(N-2),N\ge 3,a(x),b(x):{\mathbb{R}}^N\to \mathbb{R}$ are continuous functions, and $a(x)$ is nonnegative on ${\mathbb{R}}^N$. When $\lambda$ is large enough, we prove the existence and multiplicity of positive solutions to the equation.