Resumen de Liouville type theorems for poly-harmonic Dirichlet problems of Hénon-Hardy type equations on a half space or a ball

Wei Dai

• In this paper, we are concerned with the poly-harmonic Dirichlet problems for Hénon-Hardy type equations \begin{aligned} (-\Delta )^{m}u(x)=f(x,u(x)) \,\,\,\,\,\,\,\,\,\,\,\, \text {in} \,\,\, {\mathbb {R}}^{n}_{+} \,\,\, \text {or} \,\,\, B_{R}(0) \end{aligned} with n\ge 2, m\ge 1 and R>0. We prove Liouville theorems for nonnegative solutions to the above poly-harmonic Dirichlet problems and equivalent integral equations in {\mathbb {R}}^{n}_{+} and B_{R}(0) under general assumptions on f. A typical case is the Hénon-Hardy type nonlinearity f(x,u)=|x|^{a}u^{p} with a\in (-2m,+\infty ) and p>0. Our results extend the Liouville results on poly-harmonic Dirichlet problems in Reichel and Weth (Math. Z. 261:805–827, 2009), Fang and Chen (Adv. Math. 229:2835–2867, 2012), Pucci and Serrin (Indiana Univ. Math. J. 35:681–703, 1986, J. Math. Pures Appl. 69:55–83, 1990) from f=u^{p} to general f(x, u).