For a second order elliptic problem an upper solution above a lower solution implies existence of a solution in between. The authors address the question what happens if semisolutions for the semilinear boundary value problem \({\mathcal L} u=f(x,u)\), \({\mathcal B} u=0 \) have reversed order. Introducing an admissible box \([a,b]\times [c,d] \subset L^p \times L^p\) with the functions \(a,b,c,d\) bounding \(\liminf_{s\to \pm\infty}\) and \(\limsup_{s \to\pm \infty} f(x,s)/s\), their main result is that, if such a box exists and if there is a reversely ordered pair of semisolutions, a solution \(u\) exists with \(u\) in between for at least one point. A box is admissible if for \((q_+,q_-) \in[a,b] \times [c,d]\) the Fučík type eigenvalue problem \({\mathcal L} u=q_+ u^+- q_-u^-\), \({\mathcal B} u=0\) only has fixed sign solutions and if for \(q_+\), \(q_-\) in this box with \(q_+\) \(q_- \geq\lambda_1\) there exists a nondegenerate homotopy to some \(q^0_+ =q^0_-\). The paper extends earlier results of J. P. Gossez and P. Omari in [Commun. Partial Differ. Equations 19, No. 7-8, 1163-1184 (1994; Zbl 0814.35019)].