Lehto-type growth conditions and a Picard theorem for the gradient of entire solutions to the Euler -- Lagrange equations of variational integrals with linear growth in the plane including a small extension of Moser's Bernstein result

Abstract

We mainly consider entire solutions $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ of the Euler -- Lagrange equation for a variational integral with density $g(|\nabla u|)$ of linear growth. Assuming a Lehto-type condition at infinity, we show that either $\nabla u$ is a constant vector or the image of $\nabla u$ is the whole plane with the possible exception of at most one point. This result is extended to certain energy densities of nearly linear growth. Moreover, we slightly generalize the Bernstein result of Moser.