Let \( V \) be a differentiable manifold of \( \mathbb{R}^n \) with or without boundary. Suppose that \( V \) and \( \partial V \) (when not empty) are smooth enough, \( \bar{V} = V\cup\partial V \) is compact, and \( \beta_0 \), \( \gamma_0:\bar{V}\to \mathbb{R} \), \( \varphi: \partial V\to \mathbb{R} \) are given functions such that
\begin{align*}
& -\infty\leq\beta_0(x) \lt \gamma_0(x) \leq +\infty
\quad\text{ for all } x\in \bar{V};\\
& \beta_0(x)\lt \varphi(x)\lt \gamma_0(x)
\quad\text{for all }x\in \varphi V.
\end{align*}
The authors consider the following Dirichlet problem (\( \pi_0 \))
\begin{align*}
& \operatorname{div} A(x,\partial u, u) = a(x,\partial u, u)
\quad\text{in } V,\\
& u|_{\partial V} = \varphi, \, \beta_0(x)\lt u(x) \lt \gamma_0(x)
\quad\text{in } \bar{V},
\end{align*}
where \( A= (A^1,\dots,A^n) \) and \( a \) are vectorial respectively scalar real-valued functions,
\[
\partial u = \Bigr(\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\Bigr).
\]
Suppose that \( \operatorname{div} A \) defines an elliptical differential operator for \( x\in V \), \( u(x)\in(\beta_0(x), \gamma_0(x)) \). The authors introduce the notion “over (sub)solution” as a real-valued function for which \( \operatorname{div} A(x,\partial u, u)\lt a(x,\partial u, u) \) in \( V \), \( u|_{\partial V}\lt \varphi \) (\( \operatorname{div} A > a \), \( u|_{\partial V} > \varphi \)), \( \beta_0(x)\lt u(x)\lt \gamma_0(x) \) in \( \bar{V} \). The second problem which is studied in this paper is the following: \( (\pi) \) Being given an oversolution \( \gamma \) and a subsolution \( \beta \), find a solution of \( \pi_0 \) for which \( \beta(x) \leq u(x) \leq\gamma(x) \). The authors give sufficient conditions for the existence and uniqueness of the solution to the problem \( \pi \) and conditions for the non-existence of a solution to the problem \( \pi_0 \). The conditions are given by some inequalities and the proofs use the maximum principle and Leray–Schauder’s fixed point theorem.
