###### Abstract  
We are concerned with positive solutions of two types of nonlinear elliptic boundary value problems (BVPs). We present conditions for existence, uniqueness and multiple positive solutions of a first type of elliptic BVPs. For a second type of elliptic BVPs, we obtain conditions for existence and uniqueness of positive global solutions. We employ mathematical tools including strictly upper (SU) and strictly lower (SL) solutions, iterative sequence method and Amann theorem. We present our research findings in new original theorems. Finally, we summarize and indicate areas of future study and possible applications of the research work. positive (global) solution; (strict) upper and lower solutions; multiplicity of positive solutions; elliptic BVPs +
Footnote †: journal: Applied Sciences  
_Article_  
New Positive Solutions of Nonlinear Elliptic PDEs Mustafa Inc 1\({}^{\text{\textcircled{R}}}\), [PERSON] 2\({}^{\text{\textcircled{R}}}\), [PERSON] 3\({}^{\text{\textcircled{R}}}\), [PERSON] 2, [PERSON] 4,5,*\({}^{\text{\textcircled{R}}}\), [PERSON] 6,7\({}^{\text{\textcircled{R}}}\) and [PERSON] 8  
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**Theorem 4**. : _([31]) Assume that \(E\) is a Banach space. Assume also that \(K\subset E\) is a normal solid cone. Suppose that there are \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), \(\beta_{2}\in E\) by \(\beta_{1}<\alpha_{1}<\beta_{2}<\alpha_{2}.\) The operator \(\mathcal{A}:\left[\beta_{1},\alpha_{2}\right]\to E\) satisfying_  
\[\beta_{1}\leq\mathcal{A}\beta_{1},\,\mathcal{A}\alpha_{1}\leq\alpha_{1},\, \beta_{2}<\mathcal{A}\beta_{2},\,\mathcal{A}\alpha_{2}\leq\alpha_{2}.\]  
_has at least three fixed points, \(u_{1}\), \(u_{2}\), \(u_{3}\) such that_  
\[\beta_{1}<u_{1}<\alpha_{1},\,\,\beta_{2}<u_{2}\leq\alpha_{2},\,\,\beta_{2} \
leq u_{3}\
leq\alpha_{1}.\]  
**Theorem 5**. : _Assume that \(\left(H_{1}\right)\) holds. Suppose that \(\beta_{1},\,\,\beta_{2}\) are \(LS\), and \(\alpha_{1}\), \(\alpha_{2}\) are \(US\) of \(\left(1\right)\) such that \(\beta_{2}\), \(\alpha_{1}\) are strict with \(\beta_{1}<\alpha_{1}<\beta_{2}<\alpha_{2}\). In this case, \(\left(1\right)\) has at least three solutions \(u_{1}\), \(u_{2}\), \(u_{3}\) such that_  
\[\beta_{1}\leq A\beta_{1},\,\mathcal{A}\alpha_{1}\leq\alpha_{1},\,\beta_{2}< \mathcal{A}\beta_{2},\,\mathcal{A}\alpha_{2}\leq\alpha_{2}.\]  
Proof. : We shall show that \(\mathcal{A}\) is strongly increasing operator. Equivalently saying, or all \(u_{1}\), \(u_{2}\in\left[\beta_{2},\alpha_{2}\right]\) with \(u_{1}<u_{2}\)\(u_{1}\left(y\right)\leq u_{2}\left(y\right)\) and \(u_{1}\left(y\right)\
eq u_{2}\left(y\right)\). In view of \(\left(H_{1}\right)\), we have  
\[c\left(y,u_{1}\right)-c\left(y,u_{2}\right)\geq 0\text{ for all }y\in\Omega.\]  
As a result that there exists a neighborhood \(\Omega^{\prime}\subset\Omega\) such that \(u_{1}\left(y\right)\leq u_{2}\left(y\right)\) for \(y\in\Omega^{\prime}\) since \(u_{1}\left(y\right)\
eq u_{2}\left(y\right)\). Hence, by \(\left(H_{1}\right)\), we have for all \(y\in\Omega\)  
\[c\left(y,u_{1}\right)-c\left(y,u_{2}\right)\geq 0,\,y\in\Omega^{\prime}.\] (8)  
By \(\left(\ref{eq:2}\right)\), we have  
\[-\left(\mathcal{A}-\sigma\right)\left(\mathcal{Q}u_{2}-\mathcal{Q}u_{1}\right) =c\left(y,u_{1}\right)-c\left(y,u_{2}\right)+\sigma\left(u_{2}-u_{1}\right)\]  
\[\geq-\sigma\left(u_{2}-u_{1}\right)+\sigma\left(u_{2}-u_{1}\right)\geq 0 \text{ for all }y\in\Omega.\]  
Therefore, \(\mathcal{Q}u_{2}<\mathcal{Q}u_{1}\) in \(\Omega\) by strong maximum principle, and we conclude that \(\mathcal{Q}\) is a strongly increasing operator. Now, we prove \(\beta_{1}\leq\mathcal{Q}\beta_{1}\). Consider the following problem:  
\[\left\{ \begin{array}{l}-\left(\mathcal{A}-\sigma\right)\left(\mathcal{Q} \beta_{1}-\beta_{1}\right)=c\left(y,\beta_{1}\right)+\sigma\beta_{1},\text{ in }\Omega,\\ \mathcal{N}\left(\mathcal{Q}\beta_{1}-\beta_{1}\right)=h,\text{ on }\partial\Omega.\end{array}