A Liouville-type theorem for very weak solutions of nonlinear partial differential equations

Let us consider the variational equation in $\mathbb{R}^n$ $$div\bigl(a(x)F'(\vert \nabla_u\vert)\frac{\nabla_u}{\vert\nabla_u\vert}\bigl)= 0$$ where $0 < \lambda 0\leq a(x)\leq\Lambda_0 < \infty$ and $F$ is a convex increasing function verifying suitable conditions. We prove that the \textit{very weak solutions} of such equation, whose gradient belongs to a suitable Orlicz space, must be constant almost everywhere. The result applies, in particular, to the case in which $F$ is the power $F(t) = t^p (p > 1)$, i.e. to the variational equation in $\mathbb{R}^n$ $$div\bigl(a(x)\vert\nabla_u\vert^{p-2}\nabla_u\bigl) = 0$$.