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.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.