| Popis: |
We investigate the rigidity of global minimizersu≥0u\ge 0of the Alt-Phillips functional involving negative power potentials∫Ω(∣∇u∣2+u−γχ{u>0})dx,γ∈(0,2),\mathop{\int }\limits_{\Omega }(| \nabla u{| }^{2}+{u}^{-\gamma }{\chi }_{\left\{u\gt 0\right\}}){\rm{d}}x,\hspace{1.0em}\gamma \in \left(0,2),when the exponentγ\gammais close to the extremes of the admissible values. In particular, we show that global minimizers inRn{{\mathbb{R}}}^{n}are one-dimensional ifγ\gammais close to 2 andn≤7n\le 7, or ifγ\gammais close to 0 andn≤4n\le 4. |
