eigenvalue problem:

$$\label{Eq_2-the main principal eigenvalue}
\left\{\begin{array}{ll}
d\phi_{xx}+b(x)\phi+\lambda\phi=0,&x\in(0,h_{0}),\vspace{7 pt}\\
\phi_{x}(0)=\phi(h_{0})=0.
\end{array}\right.$$

Problem admits a positive principal eigenvalue $\lambda_{1}$ determined by $$\label{by-variational-method}
\lambda_{1}=\inf_{\phi\in W^{1,2}((0,h_{0}))}\left\{\int^{h_{0}}_{0}[d\phi^{2}_{x}-b(x)\phi^{2}]dx,~\phi_{x}(0)=\phi(h_{0})=0,\int^{h_{0}}_{0}\phi^{2}dx=1\right\}.$$ We state two hypotheses that we refer to when needed. We use a generic symbol $B(x)$ in the statement of the hypotheses. The function $B(x)$ will be replaced accordingly (by $b,$ $b_1$ or $b_2$) in the rest of this Section. $$\tag{\bf $B_{2}$}\label{B3}  B(x)\in C^{1}\left([0,+\infty)\right)\cap L^{\infty}\left([0,+\infty)\right) \text{ and }B(x)\text{ is positive somewhere in}~(0,h_{0}).$$ $$\tag{\bf $B_{3}$}\label{B4}
\left\| \begin{array}{l}
 B(x)\in C^{1}\left([0,+\infty)\right)\text{ and } \underline{b}<B(x)<\bar{b} \text{ for all } x\in[0,+\infty), \\
 \text{where } \underline{b} \text{ and }\bar{b} \text{ are two positive constants.}
\end{array}\right.$$

In order to compare the principal eigenvalues $\lambda_{1}$ associated with different parameters, we denote the principal eigenvalue $\lambda_{1}$ by $\lambda_{1}(d,h_{0})$. When we fix $h_{0}$ and study the property of $\lambda_{1}$ as $d$ varies, we write $\lambda_{1}=\lambda_{1}(d)$. Similarly, we write $\lambda_{1}=\lambda_{1}(h_{0})$ when $d$ is fixed while $h_{0}$ varies.

We gather the following known results about the dependance of $\lambda_1$ on $d$ and $h$.

\[the case about d\] Suppose that $b(x)$ satisfies , where $B(x)$ is replaced by $b(x)$. Then, $\lambda_{1}=\lambda_{1}(d)$ has the following properties:

1.  $\lambda_{1}(d)$ is increasing with respect to $d$.\

2.  $\lambda_{1}(d)\rightarrow+\infty$ as $d\rightarrow+\infty$ and $\lambda_{1}(d)\rightarrow -\displaystyle{\max_{x\in[0,l]}b(x)<0}$ 