We study distributional solutions of semilinear biharmonic equations of the type Δ2u+f(u)=0 onℝN, {\Delta ^2}u + f(u) = 0\quad on\;{{\mathbb R} ^N}, where f is a continuous function satisfying f ( t ) t ≥ c | t | q +1 for all t ∈ ℝ with c > 0 and q > 1. By using a new approach mainly based on careful choice of suitable weighted test functions and a new version of Hardy- Rellich inequalities, we prove several Liouville theorems independently of the dimension N and on the sign of the solutions.

中文翻译：

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某些四阶椭圆问题和相关不等式的全解

我们研究 Δ2u+f(u)=0 onℝN, {\Delta ^2}u + f(u) = 0\quad on\;{{\mathbb R} ^N} 类型的半线性双调和方程的分布解，其中 f 是满足 f ( t ) t ≥ c | 的连续函数 吨 | q +1 for all t ∈ ℝ with c > 0 and q > 1. 通过使用主要基于仔细选择合适的加权测试函数和新版本的 Hardy-Rellich 不等式的新方法，我们证明了几个独立于维 N 和解的符号。