On a singularly perturbed semi-linear problem with Robin boundary conditions

This paper is concerned with a semi-linear elliptic problem Robin boundary condition: style='text-indent:20px;'> \begin{document}$ \begin{equation} \left\{\begin{array}{lll} \varepsilon \Delta w-\lambda w^{1+\chi} = 0, &\text{in} \ \Omega\\ \nabla w \cdot \vec{n}+\gamma & \text{on} \partial \Omega \end{array}\right. \end{equation} ~~~~~~~~~~~~~~~~~~~~(*)$\end{document} style='text-indent:20px;'>where \begin{document}$ \subset {\mathbb R}^N (N\geq 1) $\end{document} bounded domain smooth boundary, id="M2">\begin{document}$ \vec{n} denotes the unit outward normal vector of id="M3">\begin{document}$ and id="M4">\begin{document}$ \gamma \in R}/\{0\} $\end{document}. id="M5">\begin{document}$ id="M6">\begin{document}$ \lambda are positive constants. The (*) derived from well-known singular Keller-Segel system. When id="M7">\begin{document}$ \gamma>0 $\end{document}, we show there only trivial solution id="M8">\begin{document}$ 0 id="M9">\begin{document}$ \gamma<0 id="M10">\begin{document}$ B_R(0) ball, that has non-constant which converges to zero uniformly as id="M11">\begin{document}$ tends zero. main idea this transform nonlocal Dirichelt by Cole-Hopf type transformation then use shooting method obtain existence transformed Dirichlet problem. With results for (*), get stationary solutions original