Differential and Integral Equations

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

Yasuhiro Fujita

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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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations


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

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