This paper is devoted to study the following equation \(-\Delta u+\lambda u= |u|^{p-1}u \;\;\text{in}\;\Omega \), with homogeneous Dirichlet or Neumann boundary conditions where \(p>1\), \(\lambda >0\), \(\Omega =\mathbb{R}^{n-k}\times \omega \), \(n\geq 2\), \(k\geq 1\), and \(\omega \) is a smoothly bounded domain of \(\mathbb{R}^{k}\). We prove Liouville-type theorems for \(C^{2}\) solutions which are stable or stable outside a compact set of \(\Omega \). We first provide an integral estimate using stability argument which combined with Pohozaev-type identity allow to obtain nonexistence results for \(p\in [p_{s}(n), p_{s}(n-k)]\), where \(p_{s}(n)\) is the classical Sobolev exponent in dimension \(n\). Also, we establish monotonicity formula to prove the nonexistence of nontrivial solution which is stable or stable outside a compact set of \(\Omega \) for all \(p>1\).

本文致力于用齐次Dirichlet或Neumann研究以下方程\（-\ Delta u + \ lambda u = | u | ^ {p-1} u \; \; \ text {in} \; \ Omega \）边界条件其中\（p> 1 \），\（\ lambda> 0 \），\（\ Omega = \ mathbb {R} ^ {nk} \ times \ omega \），\（n \ geq 2 \），\（k \ geq 1 \）和\（\ omega \）是\（\ mathbb {R} ^ {k} \）的平滑边界域。我们证明了\（C ^ {2} \）解的Liouville型定理在稳定的\（\ Omega \）集合外是稳定的或稳定的。我们首先使用稳定性参数提供一个积分估计，该参数与Pohozaev型身份相结合，可以获取\ [p \ in [p_ {s}（n），p_ {s}（nk）] \）的不存在结果，其中\（ p_ {s}（n）\）是维\（n \）中的经典Sobolev指数。另外，我们建立单调性公式来证明非平凡解的不存在性，该非平凡解对于所有\（p> 1 \）都在\（\ Omega \）的紧集内稳定或稳定。