A].

For fixed $\lambda_i$, let (S1), (S2), and (S3) be the following standing assumptions.

[**(S1)**]{} [*For $1\leq i\leq 3$, $\widehat h_i(\cdot;\lambda_i)$ is $C^N$, there is some $x_0\in {\rm{Int}} D_i$ in the case $i=1, 2$ and $x_0\in D_3$ in the case of $i=3$ satisfying that $\widehat h_i(x_0;\lambda_i)=\widehat h_{i, \max}$, and the partial derivatives of $\widehat h_i(x;\lambda_i)$ up to order $N-1$ at $x_0$ are zero.* ]{}

[**(S2)**]{} [ *$| \lambda_i|( \widehat m_{i,\max}-\widehat m_{i, \min})<\inf_{x\in \bar D_i}\int_{D_i}\kappa(y-x)dy$ in the case of $i= 2$, and $|\lambda_i|(\widehat m_{i,\max}-\widehat m_{i, \min})<1$ in the case of $i=1, 3$.* ]{}

[**(S3)**]{} [*$\ds\int_{D_i}\frac{1}{\widehat h_{i,\max}-\widehat h_i(x;\lambda_i)}dx=\infty$ for $i=1, 2, 3$.*]{}

Note that, if $|\lambda_i|\ll 1$, then the condition (S2) is automatically satisfied for $i=1, 2, 3$. Note also that (S1) implies (S3), and when $i=1$ or $3$, (S3) holds if and only if $\ds\int_{D_i}\frac{1}{\widehat m_{i,\max}-\widehat m_i(x)}dx=\infty$.

\[pv-existence\]

-   For given $\lambda_i\in{\ensuremath{\mathbb{R}}}$, if (S1) or (S2) or (S3) is satisfied, then $\mu_i^n(\lambda_i)$ is the principal eigenvalue of $\mathcal{L}_i+\lambda_i m_i$.

-   For given $\lambda_i\in{\ensuremath{\mathbb{R}}}$, if $\mu_i^n(\lambda_i)$ is not the principal eigenvalue of $\mathcal{L}_i+\lambda_i m_i$, then $$\mu_i^n(\lambda_i)=\widehat h_{i,\max}.$$

\(1) The result that (S1) and (S2) are sufficient conditions for $\mu_i^n(\lambda_i)$ to be the principal eigenvalue follows from the argument in Theorem B in 