| Abstract: |
| We consider the following equation
\[ \Delta u - u + |x|^\alpha u^p = 0 \textrm{ in } \Omega, \ \frac{\partial u}{\partial n} = 0 \textrm{ on }
\partial \Omega.\]
Here $\Omega$ is a domain in the unit ball $B(0,1)$ in $\mathbf{R}^N$ with $\partial \Omega \cap \partial B(0,1) \ne \emptyset.$ We are concerned on the least energy solutions of the problem
for $p \in (1, (N+2)/(N-2)]$ and large $\alpha > 0.$ The asymptotic shape of the solutions for large
$\alpha > 0$ strongly depends on the range of $p \in (1,N/(N-2)], p \in (N/(N-2),(N+2)/(N+2)), p = (N+2)/(N-2)$ and a geometric shape of $\partial \Omega$ near $\partial B(0,1).$ |
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