Super poly-harmonic property of solutions for Navier boundary problems on a half space

Liouville type theorem for higher order Hénon equations on a half space

“…The proof of Theorem 1.7 is entirely similar to that of Theorem 6 in [6] (see also Theorem 1.13 in [19]). We only need to replace the Liouville theorems for non-critical order Lane-Emden equations in R n (see Lin [28] for fourth order and Wei and Xu [41] for general even order) by Liouville theorems for critical order equations in R n (see Bidaut-Véron and Giacomini [1] for n = 2 and Chen, Dai and Qin [4] for n ≥ 4, see also Theorem 1.1), and replace the Liouville theorems for non-critical order Lane-Emden equations on R n + (Theorem 5 in [6], or further, Theorem 1.10 in [19]) by Theorem 1.6 in the proof. Thus we omit the details of the proof.…”

“…As a consequence of Theorem 1.16, we derive the following Liouville theorem for the generalized critical order PDEs (1.18 The rest of this paper is organized as follows. In section 2, we prove the super poly-harmonic properties for nonnegative solutions to (1.1) (i.e., Theorem 1.2) via a variant of the method used in [6]. In section 3, we show the equivalence between PDE (1.1) and IE (1.3), namely, Theorem 1.3.…”

“…Now we list some essential properties of the above Green's functions, which have been proved in [3] and [6]. First,…”

“…In [6], the authors prove the following property of Green's functions with different orders on a half space:…”

Liouville type theorem for some nonlocal elliptic equations

“…After the work of [7], many nonexistence results for elliptic equations with general nonlinearities were obtained, we refer the readers to [ In this paper, we study the nonexistence result for some elliptic equation involving nonlocal nonlinearity and nonlocal boundary value condition. The equation is 4) where N − 2 ≤ α < N and f, g, F, G are some nonlinear functions. We note that both the nonlinear term and the boundary value condition are nonlocal, so it is different from the equations in the references mentioned above.…”