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

 1 Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 Institute of Applied Mathematical Sciences and National Center for Theoretical Sciences (NCTS), National Taiwan University, Taipei 10617, Taiwan 3 Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

Received  September 2019 Published  January 2020

This paper is concerned with a semi-linear elliptic problem with Robin boundary condition:
 $$$\left\{\begin{array}{lll} \varepsilon \Delta w-\lambda w^{1+\chi} = 0, &\text{in} \ \Omega\\ \nabla w \cdot \vec{n}+\gamma w = 0, & \text{on} \ \partial \Omega \end{array}\right.$$ ~~~~~~~~~~~~~~~~~~~~(*)$
where
 $\Omega \subset {\mathbb R}^N (N\geq 1)$
is a bounded domain with smooth boundary,
 $\vec{n}$
denotes the unit outward normal vector of
 $\partial \Omega$
and
 $\gamma \in {\mathbb R}/\{0\}$
.
 $\varepsilon$
and
 $\lambda$
are positive constants. The problem (*) is derived from the well-known singular Keller-Segel system. When
 $\gamma>0$
, we show there is only trivial solution
 $w = 0$
. When
 $\gamma<0$
and
 $\Omega = B_R(0)$
is a ball, we show that problem (*) has a non-constant solution which converges to zero uniformly as
 $\varepsilon$
tends to zero. The main idea of this paper is to transform the Robin problem (*) to a nonlocal Dirichelt problem by a Cole-Hopf type transformation and then use the shooting method to obtain the existence of the transformed nonlocal Dirichlet problem. With the results for (*), we get the existence of non-constant stationary solutions to the original singular Keller-Segel system.
Citation: Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020083
References:

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