Liouville type results for semilinear biharmonic problems in exterior domains

Nonexistence of nontrivial nonnegative classical solutions is obtained for the problems: 0.1Δ2u=upinRN\B¯,u=Δu=0on∂B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u=u^p \;\;\; &{}\text{ in } {\mathbb {R}}^N \backslash {\overline{B}},\\ u=\Delta u=0 \;\;\; &{}\text{ on } \partial B \end{array} \right. \end{aligned}$$\end{document}with 1<p≤N+4N-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p\le \frac{N+4}{N-4}$$\end{document}, and 0.2Δ2u=upinRN\B¯,u=∂u∂ν=0on∂B,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u=u^p \;\;\; &{}\text{ in } {\mathbb {R}}^N \backslash {\overline{B}},\\ u=\frac{\partial u}{\partial \nu }=0 \;\;\; &{}\text{ on } \partial B, \end{array} \right. \end{aligned}$$\end{document}where 1<p<N+4N-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\frac{N+4}{N-4}$$\end{document}, B⊂RN(N≥5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B \subset {\mathbb {R}}^N \; (N \ge 5)$$\end{document} is the unit ball, ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} is the unit outward normal vector of ∂B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial B$$\end{document} relative to B. The interesting features in our proof are that neither asymptotic behavior of u at infinity nor symmetric property of u are required. Moreover, when p=N+4N-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=\frac{N+4}{N-4}$$\end{document}, we can also obtain nonexistence of nontrivial nonnegative classical radial solutions of (0.2). Nonexistence of nontrivial nonnegative classical solutions without symmetry property of (0.2) with p=N+4N-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=\frac{N+4}{N-4}$$\end{document} is still open. It is well known that problems (0.1) and (0.2) admit a unique positive radial solution u∈C4(RN\B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \in C^4 (\mathbb {R}^N \backslash B)$$\end{document} for p>N+4N-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>\frac{N+4}{N-4}$$\end{document} respectively.