## Differential and Integral Equations

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

Yasuhiro Fujita

#### 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 $\nabla w\cdot \nabla u$.

#### Article information

Source
Differential Integral Equations, Volume 16, Number 8 (2003), 969-979.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356060578

Mathematical Reviews number (MathSciNet)
MR1989596

Zentralblatt MATH identifier
1030.35059

Subjects
Primary: 35J60: Nonlinear elliptic equations

#### Citation

Fujita, Yasuhiro. An existence result for nonlinear elliptic equations on $\Bbb R^d$ without sign condition. Differential Integral Equations 16 (2003), no. 8, 969--979. https://projecteuclid.org/euclid.die/1356060578