## 4 Weak Solutions for the First Problem  
Now, our main result shows the existence of a weak solution to problem \((1)\) with Dirichlet boundary condition (\(u=0\) i.e., \(h=0\) and \(\mathcal{N}u=u\)) under a unboundedness on \(c\). Before doing this, we introduce the following notion of weak solution to \((1)\). For that, we need Lemmas 5-7 and the assumptions \((H_{3})\) and \((H_{4})\) (see later). Note that the notion of weak solution to \((1)\) is essentially the same as in Definition 1, the only difference is that we now require that \(u\) belong to \(H_{0}^{1}\left(\Omega\right)\). **Definition 3**. : _Let \(u\in H_{0}^{1}\left(\Omega\right)\), \(u\) is said to be a weak solution of \(\left(\ref{eq:u}\right)\) if it satisfies_  
\[\int\limits_{\Omega}\
abla u\
abla\phi dx=\int\limits_{\Omega}c\left(u\right) \phi dx\quad\text{in}\quad\Omega.\]  
_A nonnegative function \(u,\ \overline{u}\in H_{0}^{1}\left(\Omega\right)\) is called a weak lower solution (WLS) and weak upper solution (WLS) of \(\left(\ref{eq:u}\right)\) if they satisfy_  
\[\int\limits_{\Omega}\
abla\underline{u}\
abla\phi dx\leq\int\limits_{\Omega}c \left(\underline{u}\right)\phi dx\quad\text{in}\quad\Omega,\]  
_and_  
\[\int\limits_{\Omega}\
abla\overline{u}\
abla\phi dx\geq\int\limits_{\Omega}c \left(\overline{u}\right)\phi dx\quad\text{in}\quad\Omega,\]  
_for all \(\phi\in H_{0}^{1}\left(\Omega\right)\)._  
**Lemma 5**. : _([26]) Let \(v\) solve \(\Delta v=c\) in \(\Omega\). If \(c\in C\left(\Omega\right)\), then \(v\in C^{1,\eta}\left(\Omega\right)\) for any \(\eta\in\left(0,1\right)\), thus in particular, \(v\) is continuous in \(\Omega\)._  
**Lemma 6**. : _([32]) For each \(c\in L^{2}\left(\Omega\right)\). Then, there exists a unique solution \(v\in H_{0}^{1}\left(\Omega\right)\) to problem \(\left(\ref{eq:u}\right)\)._  
**Lemma 7**. : _([33]) Assume that \(u\) and \(v\) are two non-negative functions such that_  
\[\left\{ \begin{array}{ll}-\Delta u\geq-\Delta v,\ \text{in}\ \ \Omega,\\ u=v=0,\qquad\text{in}\ \partial\Omega.\end{array} \right.\]  
_Then, \(u\geq v\), a.e., in \(\Omega\)._  
\(\left(H_{3}\right)\)_\(c\in C^{1}\left(\left(0,+\infty\right)\right)\) is increasing function such that_  
\[\lim\limits_{u\rightarrow+\infty}c\left(u\right)=+\infty.\]  
\(\left(H_{4}\right)\)_Moreover, \(c\in C^{1}\left(\left(0,+\infty\right)\right)\) satisfies_  
\[\lim\limits_{u\rightarrow+\infty}\frac{c\left(u\right)}{u}=0.\]  
**Theorem 10**. : _Let \(\left(H_{3}\right)\) and \(\left(H_{4}\right)\) hold. Then, problem \(\left(\ref{eq:u}\right)\) has a positive weak solution._  
Proof. : Let \(\sigma\) be the first eigenvalue of \(-\Delta\) with Dirichlet boundary conditions and \(\phi_{1}\) be the corresponding positive eigenfunction with \(\left\|\phi_{1}\right\|=1\). Let \(\delta>0\) be such that \(\left|\
abla\phi_{1}\right|^{2}-\sigma\phi_{1}^{2}>0\) on  
\[\overline{\Omega}_{\delta}=\left\{x\in\Omega\,:\,d\left(x,\partial\Omega \right)\leq\delta\right\}.\]  
Let us define  
\[\underline{u}=\frac{1}{2}\phi_{1}^{2}.\]  
We shall verify that \(\underline{u}\) is a weak LS of \(\left(\ref{eq:u}\right)\). Indeed, let \(\phi\in H_{0}^{1}\left(\Omega\right)\) with \(\phi>0\) in \(\Omega\). A simple calculation shows that  
\[\int\limits_{\Omega_{\delta}}\
abla\underline{u}\
abla\phi dx=\int\limits_{ \overline{\Omega}_{\delta}}\phi_{1}\
abla\phi_{1}\
abla\phi dx\]\[=\int\limits_{\overline{\Omega}_{\delta}}\
abla\phi_{1}\
abla\left( \phi_{1}\cdot\phi\right)dx-\int\limits_{\overline{\Omega}_{\delta}}\left|\
abla \phi\right|^{2}\phi dx\] \[=\int\limits_{\overline{\Omega}_{\delta}}\left(\sigma\phi_{1}^{2}- \left|\
abla\phi\right|^{2}\right)\phi dx.\]  
On \(\overline{\Omega}_{\delta}\), we have \(\left|\
abla\phi_{1}\right|^{2}-\sigma\phi_{1}^{2}>0\), then \(\sigma\phi_{1}^{2}-\left|\
abla\phi_{1}\right|^{2}<0\). Hence,  
\[\int\limits_{\overline{\Omega}_{\delta}}\
abla\underline{u}\
abla\phi dx<0.\]  
By \(\left(H_{3}\right)\), we get \(c\left(\underline{u}\right)>0\) (large enough).