Nonexistence of signed solutions for a semilinear elliptic problem (Q1891567)

This paper proves that no solution of \[ \Delta u + u^p_- -u^q_- =0 \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial \Omega \] which changes sign in \(\Omega\) exists, provided \(\Omega\) is a ``small enough'' domain. The proof by contradiction uses a priori estimates and the maximum principle. For ``sufficiently large'' domains, the existence of solutions had already been established.