Of interest in this note is the simplicity of the first eigenvalue of the following problem
where and satisfy
with . A simple variational argument shows that exists and can be characterized by
Further arguments show that is achieved by some positive pair in the sense that for any test pair , there holds
By the simplicity of we mean the eigenspace associated with is of dimension one in the sense that whenever we must have and for some constants .
To prove the simplicity of , we first recall the following Picone identity for -Laplacian
with the equality occurs whenever is constant. Here by the density, we only require that and . Upon multiplying both sides by and integrating by parts, we arrive at
In a similar way, for any with , we also obtain
Suppose that is also an eigenpair associated with , we first obtain from the definition of the following
The above estimates show that and are constants as claimed.