Multiplicity of Positive Solutions of Nonlinear Elliptic Equation with Gradient Term

Abstract

In this paper, we consider the following nonlinear elliptic equation with gradient term: \[ \left\{ \begin{gathered} - Δu - \frac{1}{2}(x \cdot \nabla u) + (λa(x)+b(x))u = βu^q +u^{2^*-1}, \hfill 0<u \in {H_K^{1}(\mathbb{R}^N)}, \hfill \\ \end{gathered} \right . \] where $\lambda ,\beta \in (0,\infty ),q\in (1,2^{\ast }-1),2^{\ast }=2N/(N-2),N\ge 3,a(x),b(x):{\mathbb{R}}^N\to \mathbb{R}$ 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.