log-concavity of eigenfunction and Brunn-Minkowski inequality of eigenvalue for weighted p-Laplace operator

In this paper, we investigate the log-concavity property of the first eigenfunction to the weighted $p$-Laplace operator in class of bounded, convex and smooth domain. Moreover, we prove a Brunn-Minkowski-type inequality for the first eigenvalue to the weighted $p$-Laplace operator in the class of $C^2$ convex bodies in ${\mathbb{R}}^n$