广义Hardy型势对Hénon问题有限Morse指标解的影响：局部与非局部情形

中文摘要 AI 辅助

本文研究了半线性椭圆问题$$(-\nDelta)^s u+\lambda|x|^\alpha u=|x|^\beta |u|^{p-1}u \\; \mbox{in}\\; \mathcal{H}$$的稳定解的不存在性结果，其中$\mathcal{H}=\mathbb{R}^n$或$\mathcal{H}=\mathbb{R}^n_+=\{x=(x',x_n),\\, x'\in\mathbb{R}^{n-1},x_n>0\}$，在半空间情形下配备Dirichlet条件。我们假设$n\geq2s$，$p>1$，$\lambda>0$，$\alpha, \beta>-2s$且$0<s\leq1$。我们的目标是分析Hardy型幂势$\lambda|x|^\alpha u$在Hénon权重$|x|^\beta$存在时的影响，特别关注弱稳定解（包括可能无界或变号的解）的存在性与分类。我们建立了非平凡稳定解的不存在性结果，更一般地，对于在紧集外稳定的解，在无需额外假设的情况下覆盖了临界和超临界情形，并在对解的加权版本施加适当的$L^\infty$-小性条件下覆盖了次临界情形。我们的结果适用于局部情形$s=1$和非局部情形$0<s<1$，在全空间和半空间中均成立。证明基于能量估计与Pohozaev恒等式相结合。在超临界情形下，我们进一步建立了单调性公式，遵循Dávila等人（Trans. Amer. Math. Soc. 369, 6087-6104, 2017）的方法。

英文摘要

In this paper, we investigate nonexistence results for stable solutions of the semilinear elliptic problem $$(-Δ)^s u+λ|x|^αu=|x|^β|u|^{p-1}u \; \mbox{in}\; \mathcal{H}, $$ where $\mathcal{H}=\mathbb{R}^n$ or $\mathcal{H}=\mathbb{R}^n_+=\{x=(x',x_n),\, x'\in\mathbb{R}^{n-1},x_n>0\}$ , with Dirichlet conditions in the half-space case. We assume that $n\geq2s$, $p>1$, $λ>0,$ $α, β>-2s$ and $0<s\leq1$. Our aim is to analyze the influence of the Hardy-type power potential $λ|x|^αu$ in the presence of the Hénon weight $|x|^β$, with particular emphasis on the existence and classification of weak stable solutions, including solutions that may be unbounded or sign-changing. We establish nonexistence results for nontrivial stable solutions and, more generally, for solutions that are stable outside a compact set, covering critical and supercritical regimes without additional assumptions, and the subcritical regime under a suitable $L ^\infty$-smallness condition on a weighted version of the solution. Our results apply to both the local case $s = 1$ and the nonlocal case $0 < s < 1 $, in the whole space and in the half-space. The proofs are based on energy estimates combined with a Pohozaev identity. In the supercritical regime, we further establish a monotonicity formula, following the approach of Dávila et al. (Trans. Amer. Math. Soc. 369, 6087-6104, 2017).

发表机构

• King Faisal University（费萨尔国王大学）
• Institut Supérieur des Sciences Appliquées et de Technologie de Kairouan（凯鲁万应用科学与技术高等学院）

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