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.