# Elliptic problem on half space; infinite boundary values; Liouville theorem

In a the study of a boundary value problem the following problem is arising:

$-\Delta v(x)= e^{v(x)}$ in $R^N_+$
$v= - \infty$ $\qquad$ on $\partial R^N_+$ $\qquad$ $v \le 0$ in $R^N_+$.

Here $R_+^N$ is the upper half space $\{ x \in R^N: x_N >0\}$.

The question I am interested in is whether one can prove the non-existence of a classical solution to the above problem. I attempted using a change of variables of the form $u(x)= e^{t v(x)}$ where $t>0$. Then one arrives at an equation involving an advection term and at least one has the more standard boundary values of $u(x)=0$ on the boundary. The advection term is $\frac{ | \nabla u(x)|^2}{u(x)}$ and so there are some regularity issues on the boundary. In any case I would appreciated any comments on how to prove a non-existence result (or maybe this is not true). I have some added assumptions on $v$ which may help.