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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.

thanks for your comments.

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