Optimal existence conditions for \(\varphi\)-Laplacian equations with upper and lower solutions in the reversed order (Q1586116)

Consider the scalar \(\varphi\)-Laplacian problem \[ -\varphi' (u'(t))' = f(t,u(t)),\quad u'(a) = 0 = u'(b),\tag \(*\) \] under the following conditions: (i) \(\varphi: \mathbb{R}\to \mathbb{R}\) is an increasing homeomorphism such that \(\varphi (0) = 0\), (ii) for every compact interval \([k_1,k_2]\) there exists \(K>0\) such that for all \(u,v \in [k_1,k_2]\) holds \((\varphi (u)-\varphi (v))(u-v) \geq K(u-v)^2\), and (iii) \(f: I \times \mathbb{R}\to \mathbb{R}\) is an \(L^1\)-Carathéodory function. The authors prove an anti-maximum comparision principle for the operator \(-(\varphi(u'(t))'-Mu(t))\) and use it to establish the existence of extremal solutions to \((*)\) via the monotone method in the presence of upper and lower solutions in the reversed order.