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Paper   IPM / M / 17138
School of Mathematics
  Title:   A nonlinear elliptic problem involving the gradient on a half space
  Author(s):  Asadollah Aghajani (Joint with C. Craig and L. Shiu Hong)
  Status:   Published
  Journal: Discrete Contin. Dyn. Syst.
  Vol.:  43
  Year:  2023
  Pages:   378-391
  Supported by:  IPM
We consider perturbations of the diffusive Hamilton-Jacobi equation \begin{equation*} %\label{non_pert} \left\{ \begin{array}{lcl} \hfill -\Delta u &=& (1+g(x))| \nabla u|^p\qquad \mbox{ in } \IR^N_+, \\ \hfill u &=& 0 \hfill \mbox{ on } \partial \IR^N_+, \end{array}\right. \end{equation*} for $ p>1$. We prove the existence of a classical solution provided $ p \in (\frac{4}{3},2)$ and $g$ is bounded with uniform radial decay to zero.

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