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: \[ \left\{ \begin{gathered} - \Delta u - \frac{1}{2}(x \cdot \nabla u) + (\lambda a(x)+b(x))u = \beta u^q +u^{2^*-1}, \hfill 0<u \in {H_K^{1}(\mathbb{R}^N)}, \hfill \\ \end{gathered} \right . \] where $\lambda, \beta \in (0,\infty), q \in (1,2^*-1), 2^* = 2N/(N-2), N\geq3, a(x), b(x): \mathbb{R}^N \to \mathbb{R}$ 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.