Signed solutions for a semilinear elliptic problem

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.