On singular solutions of Lane-Emden equation on the Heisenberg group

Abstract

By applying the gluing method, we construct infinitely many axial symmetric singular positive solutions to the Lane-Emden equation: ΔHu+up=0,inHn\{0}{\Delta }_{{\mathbb{H}}}u+{u}^{p}=0\left,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{H}}}^{n}\backslash \left\{0\right\} on the Heisenberg group Hn{{\mathbb{H}}}^{n}, where n>1,Q⁄(Q−4)<p<pJL(Q−2)n\gt 1,Q/\left(Q-4)\lt p\lt {p}_{JL}\left(Q-2), and Q=2n+2Q=2n+2 is the homogeneous dimension of Hn{{\mathbb{H}}}^{n}.

Keywords

• singular solutions
• heisenberg group
• lane-emden equation
• supercritical exponent
• gluing method
• primary: 35a21, 35b09
• secondary: 35j60