## 3 Results
#### 3.2.1 Construction of the Upper and Lower Solutions
 \tag{28}\]  
where  
\[\overline{U}_{n}^{(-)}=U_{n}^{(-)}(\mathbf{x},\varepsilon)+\varepsilon^{n} \Big{(}\alpha^{(-)}(\mathbf{x})+q^{(-)}u(\tau,\mathbf{y})+m^{(-)}u(\sigma, \mathbf{y})\Big{)}+\varepsilon^{n+2}C_{u}\mathbf{e}^{-k_{u}\sigma_{\Gamma}},\]  
\[\mathbf{x}\in\overline{D}^{(-)},\ \ \mathbf{y}\in\gamma,\ \ \tau\leq 0,\ \ \sigma\leq 0,\ \ \sigma_{\Gamma}\geq 0,\]  
\[\overline{U}_{n}^{(+)}=U_{n}^{(+)}(\mathbf{x},\varepsilon)+\varepsilon^{n} \Big{(}\alpha^{(+)}(\mathbf{x})+q^{(+)}u(\tau,\mathbf{y})+m^{(+)}u(\sigma, \mathbf{y})\Big{)}+\varepsilon^{n+2}\overline{\Omega}_{u}(\mathbf{y}),\]  
\[\mathbf{x}\in\overline{D}^{(+)},\ \ \mathbf{y}\in\gamma,\ \ \tau\geq 0,\ \ \sigma\geq 0;\]  
\[\underline{U}_{\underline{n}}^{(-)}=U_{n}^{(-)}(\mathbf{x}, \varepsilon)-\varepsilon^{n}\Big{(}\alpha^{(-)}(\mathbf{x})+q^{(-)}u(\tau, \mathbf{y})+m^{(-)}u(\sigma,\mathbf{y})\Big{)}-\varepsilon^{n+2}C_{u} \mathbf{e}^{-k_{u}\sigma_{\Gamma}},\]  
\[\mathbf{x}\in\overline{D}^{(-)},\ \ \mathbf{y}\in\gamma,\ \ \tau\leq 0,\ \ \sigma\leq 0,\ \ \sigma_{\Gamma}\geq 0,\]  
\[\underline{U}_{\underline{n}}^{(+)}=U_{n}^{(+)}(\mathbf{x}, \varepsilon)-\varepsilon^{n}\Big{(}\alpha^{(+)}(\mathbf{x})+q^{(+)}u(\tau, \mathbf{y})+m^{(+)}u(\sigma,\mathbf{y})\Big{)}+\varepsilon^{n+2}\underline{ \Omega}_{u}(\mathbf{y}),\]  
\[\mathbf{x}\in\overline{D}^{(+)},\ \ \mathbf{y}\in\gamma,\ \ \tau\geq 0,\ \ \sigma\geq 0; \tag{29}\]  
\[\overline{V}_{n}^{(-)}=V_{n}^{(-)}(\mathbf{x},\varepsilon)+ \varepsilon^{n}\Big{(}\beta^{(-)}(\mathbf{x})+q^{(-)}v(\tau,\mathbf{y})\Big{)}+ \varepsilon^{n+2}m^{(+)}v(\sigma,\mathbf{y})+\] \[+\varepsilon^{n+1}C_{v}\mathbf{e}^{-k_{u}\tau_{\Gamma}},\ \ \mathbf{x}\in\overline{D}^{(-)},\ \ \mathbf{y}\in\gamma,\ \ \tau\leq 0,\ \ \sigma\leq 0,\ \ \tau_{\Gamma}\geq 0,\] \[\mathbf{x}\in\overline{D}^{(-)},\ \ \mathbf{y}\in\gamma,\ \ \tau\leq 0,\ \ \sigma\leq 0,\ \ \sigma_{\Gamma}\geq 0,\]0 \[\mathbf{x}\in\overline{D}^{(-)},\ \ \mathbf{y}\in\gamma,\ \ \tau\leq 0,\ \ \sigma\leq 0,\ \ \sigma_{\Gamma}\geq 0,\]1 \[\mathbf{x}\in\overline{D}^{(-)},\ \ \mathbf{y}\in\gamma,\ \ \tau\leq 0,\ \ \sigma\leq 0,\ \ \sigma_{\Gamma}\geq 0,\]2 \[\mathbf{x}\in\overline{D}^{(-)},\ \ \mathbf{y}\in\gamma,\ \ \tau\leq 0,\ \ \sigma\leq 0,\ \ \sigma_{\Gamma}\geq 0,\]3 \[\mathbf{x}\in\overline{D}^{(-)},\ \ \mathbf{y}\in\gamma,\ \ \tau\leq 0,\ \ \sigma\leq 0,\ \ \sigma_{\Gamma}\geq 0,\]4 \[\mathbf{x}\in\overline{D}^{(-)},\ \ \mathbf{y}\in\gamma,\ \ \tau\leq 0,\ \ \sigma\leq 0,\ \ \sigma_{\Gamma}\geq 0,\]5  
The terms \(\varepsilon^{n+2}C_{u}\mathbf{e}^{-k_{u}\sigma_{\Gamma}}\) and \(\varepsilon^{n+1}C_{v}\mathbf{e}^{-k_{v}\tau_{\Gamma}}\) were added to ensure the fulfillment of the inequalities in Condition (A3). For this, we choose the constants \(C_{u}\), \(C_{v}\), \(k_{u}\) and \(k_{v}\) to be sufficiently large positives (see [23]), and \(\varepsilon^{n+2}\overline{\Omega}_{u}(\mathbf{y})\), \(\varepsilon^{n+2}\underline{\Omega}_{u}(\mathbf{y})\) are included to achieve continuity of the functions \(\overline{U}_{\underline{n}}(\mathbf{x},\varepsilon)\), \(\underline{U}_{\underline{n}}(\mathbf{x},\varepsilon)\) on \(\gamma\). They eliminate small residuals in the condition of continuous matching introduced by the boundary layer functions. The terms \(\varepsilon^{n+1}\overline{\Omega}_{v}(\mathbf{y})\) and \(\varepsilon^{n+1}\underline{\Omega}_{v}(\mathbf{y})\) eliminate jumps of order no lower than \(O(\varepsilon^{n+1})\) for the functions \(\overline{V}_{n}(\mathbf{x},\varepsilon)\), \(\underline{V}_{\underline{n}}(\mathbf{x},\varepsilon)\) on the curve \(\gamma\). We include the functions \(\alpha^{(\mp)}(\mathbf{x})\) and \(\beta^{(\mp)}(\mathbf{x})\) in the upper and lower solutions to fulfill Condition (A2) far from the internal transition layer.