## Differential and Integral Equations

- Differential Integral Equations
- Volume 8, Number 5 (1995), 1259-1263.

### Nonexistence of signed solutions for a semilinear elliptic problem

#### Abstract

Let $ \Omega$ be a bounded, smooth domain in ${\Bbb R}^N$, $N\ge 2$. We consider the
elliptic boundary value problem $$ \begin{align} \Delta u + u_+^p - u_-^q & =0
\quad\hbox{ in } \ \Omega , \\ u & =0 \quad\hbox{ on } \ \partial \Omega , \end{align}
$$ where $1< p < (N+2)/(N-2)$, $0<q<1$, $u_+ = \max\{ u,0\}$, $u_- = -\min\{
u,0\}$. We prove that for certain *small* domains $\Omega$, no solution to this
problem which changes sign in $\Omega$ exists. This answers affirmatively a conjecture
raised in [3] where existence of at least one signed solution is established under a
largeness condition for the domain.

#### Article information

**Source**

Differential Integral Equations, Volume 8, Number 5 (1995), 1259-1263.

**Dates**

First available in Project Euclid: 20 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1369056054

**Mathematical Reviews number (MathSciNet)**

MR1325556

**Zentralblatt MATH identifier**

0830.35042

**Subjects**

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations

Secondary: 35B99: None of the above, but in this section

#### Citation

del Pino, Manuel A. Nonexistence of signed solutions for a semilinear elliptic problem. Differential Integral Equations 8 (1995), no. 5, 1259--1263. https://projecteuclid.org/euclid.die/1369056054