An existence result for nonlinear elliptic equations on $\Bbb R^d$ without sign condition

Abstract

In this paper, we consider the nonlinear elliptic equation of the following type: $$ -\frac{1}{2} \Delta u(x) + \nabla w(x)\cdot \nabla u(x) + [\lambda + H(x,u(x))]u(x) = f(x),\qquad x \in {\mathbb R}^d, $$ where $\lambda$ is a given constant and $f$, $H$, and $w$ are given functions, respectively. The derivative $\nabla w$ of the function $w$ is unbounded on ${\mathbb R}^d$. Our purpose is to show the existence of a solution to this equation without sign conditions on $H$. Therefore, we can treat even the case that $H$ is unbounded below on ${\mathbb R}^d$. This is due to the existence of the term $\nabla w\cdot \nabla u$.