| Abstract: |
| We investigate the asymmetry of positive solutions for the H\`{e}non equation in a reflectionally symmetric or a point symmetric domain $\Omega$, which is unbounded but it is getting narrower near the infinity.
We call $u(x)$ a least energy solution if it is a minimizer of the Rayleigh quotient associated with the H\`{e}non equation. We shall prove that no least energy solution is reflectionally symmetric and even.
Moreover, we prove the existence of a positive solution which has the exact symmetry of reflection. |
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