## 3 Results  
In the course of applying the algorithm, we obtained propositions under which Problems (1) and (3) have a solution with a large gradient in the vicinity of the curve \(\gamma\). Proposition 1: _There is an isolated solution \(u=\varphi^{(\mp)}(v,\mathbf{x})\) to each of the equations \(f^{(\mp)}(u,v,\mathbf{x},0)=0\) on \(I_{v}\times\overline{D}^{(\mp)}\), and the inequalities hold:_  
\[\varphi^{(-)}(v,\mathbf{x})<\varphi^{(+)}(v,\mathbf{x}),\ \ v\in I_{v},\ \ \mathbf{x}\in\gamma;\] \[f_{u}^{(\mp)}(\varphi^{(\mp)}(v,\mathbf{x}),v,\mathbf{x},0)>0,\ \ (v,\mathbf{x})\in I _{v}\times\overline{D}^{(\mp)}.\]  
Furthermore, we denote \(h^{(\mp)}(v,\mathbf{x})=g^{(\mp)}(\varphi^{(\mp)}(v,\mathbf{x}),v,\mathbf{x},0)\). Proposition 2: _There is an isolated solution \(v=\psi^{(\mp)}(\mathbf{x})\) to each of the equations \(h^{(\mp)}(v,\mathbf{x})=0\) on the set \(\overline{D}^{(\mp)}\), and the inequalities hold:_  
\[\psi^{(-)}(\mathbf{x})<\psi^{(+)}(\mathbf{x}),\ \ \mathbf{x}\in\gamma\,;\ \ \ h^{(\mp)}_{v}(\psi^{(\mp)}(\mathbf{x}),\mathbf{x})>0,\ \ \mathbf{x}\in\overline{D}^{(\mp)}.\]  
In this paper, we study a stationary solution to Problem (3) that is close to the functions \((\varphi^{(-)},\psi^{(-)})\) in the domain \(D^{(-)}\) and to the functions \((\varphi^{(+)},\psi^{(+)})\) in \(D^{(+)}\) which changes sharply from the values of \((\varphi^{(-)},\psi^{(-)})\) up to \((\varphi^{(+)},\psi^{(+)})\) in a small vicinity of the curve \(\gamma\). In what follows, we will refer to this vicinity as the internal transition layer. The next proposition is called the \"quasi-monotonicity conditions\". Proposition 3: _Let one of four pairs of inequalities hold_  
\[f_{v}^{(\mp)}(u,v,\mathbf{x},\varepsilon)<0,\ \ g_{u}^{(\mp)}(u,v,\mathbf{x}, \varepsilon)<0\ \ \text{(NN type)};\] \[f_{v}^{(\mp)}(u,v,\mathbf{x},\varepsilon)>0,\ \ g_{u}^{(\mp)}(u,v,\mathbf{x}, \varepsilon)>0\ \ \text{(PP type)};\] \[f_{v}^{(\mp)}(u,v,\mathbf{x},\varepsilon)<0,\ \ g_{u}^{(\mp)}(u,v,\mathbf{x}, \varepsilon)>0\ \ \text{(NP type)};\] \[f_{v}^{(\mp)}(u,v,\mathbf{x},\varepsilon)>0,\ \ g_{u}^{(\mp)}(u,v,\mathbf{x}, \varepsilon)<0\ \ \text{(PN type)}\]  
_for all \((u,v)\in I_{u}\times I_{v},\varepsilon\in(0,\varepsilon_{0}]\) with \(\mathbf{x}\in\overline{D}^{(\mp)}\)._  
When constructing asymptotic approximations for solutions to boundary value problems with internal transitional and boundary layers, an important stage is the study of the so-called adjoint equations [17]. In this paper, the properties of the adjoint problems determine the existence of functions that describe the internal transition layer. To construct these functions, it is expedient to use local coordinates in a small vicinity \(\Omega(\gamma)\) of the curve \(\gamma\) in the same way as in [19]. We denote with \(\mathbf{n}_{\gamma}\) the normal to the curve \(\gamma\) directed inside the domain \(D^{(+)}\). We introduce local coordinates \((\mathbf{y},r)\) in \(\Omega(\gamma)\) as follows:  
\[\mathbf{x}=\mathbf{y}+r\mathbf{n}_{\gamma}(\mathbf{y}),\ \ (\mathbf{x}\in\Omega(\gamma),\mathbf{y}\in \gamma). \tag{4}\]The \(r\) coordinate is considered positive if \(\mathbf{x}\in D^{(+)}\) and negative if \(\mathbf{x}\in D^{(-)}\). As noted in the introduction, the transition layer has two spatial scales.