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

Abstract

Abstract By applying the gluing method, we construct infinitely many axial symmetric singular positive solutions to the Lane-Emden equation: Δ H u + u p = 0 , in H n \ { 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 H n {{\mathbb{H}}}^{n} , where n > 1 , Q ⁄ ( Q − 4 ) < p < p J L ( Q − 2 ) n\gt 1,Q/\left(Q-4)\lt p\lt {p}_{JL}\left(Q-2) , and Q = 2 n + 2 Q=2n+2 is the homogeneous dimension of H n {{\mathbb{H}}}^{n} .