## Differential and Integral Equations

### Signed solutions for a semilinear elliptic problem

Pavol Quittner

#### Abstract

We show existence of signed solutions with positive energy of the problem $\Delta u+u_+^p-u_-^q=0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $<q<1<$, $p<(N+2)/(N-2)$ if $N>2$ and the domain $\Omega\subset\mathbb{R}^N$ is bounded and "sufficiently large.'' Our proof is based on the study of the dynamical system associated with the corresponding parabolic problem and it can be easily extended to more general problems. In particular, it does not rely on the uniqueness of the negative solution in contrast to the variational proof in [2] where the authors obtained signed solutions with negative energy.

#### Article information

Source
Differential Integral Equations, Volume 11, Number 4 (1998), 551-559.

Dates
First available in Project Euclid: 30 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367341033

Mathematical Reviews number (MathSciNet)
MR1666269

Zentralblatt MATH identifier
1131.35339

#### Citation

Quittner, Pavol. Signed solutions for a semilinear elliptic problem. Differential Integral Equations 11 (1998), no. 4, 551--559. https://projecteuclid.org/euclid.die/1367341033