发表机构
Federal University of Pernambuco; Universidade Federal de Goiás（伯南布哥联邦大学; 戈亚斯联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究一类奇异椭圆问题，确定了存在非平凡非负弱解的尖锐阈值，并证明在阈值左侧邻域内至少存在两个不同解，且其能量符号与先前结果不同。
AI 中文摘要
我们考虑如下一类奇异椭圆问题：\\[-\Delta u = -\frac{u}{|u|^{\beta+1}}\chi_{\{|u|>0\}} + \lambda |u|^{p-1}u, \quad \text{在 }\Omega \text{ 内},\\] 边界条件为 \\(u=0\\) 于 \\(\partial\Omega\\)，其中 \\(\Omega\subset\mathbb R^N\\) 是有界光滑区域，\\(N\geq3\\)，\\(0<\beta,p<1\\)，且 \\(\lambda>0\\)。能量泛函在 \\(H_0^1(\Omega)\\) 中不是 \\(C^1\\) 类的，因此标准临界点理论不适用。我们证明：存在非平凡非负弱解当且仅当 \\(\lambda\\) 超过临界值 \\(\bar\lambda\\)，我们确定该临界值并将其严格定位在两个 Rayleigh 型参数之间：\\(0<\lambda_*<\bar\lambda<\lambda^*<+\infty\\)。这给出了一个尖锐的存在性阈值。我们还证明：对于 \\(\lambda^*\\) 左侧邻域内的每个 \\(\lambda\\)，至少存在两个不同的非平凡非负弱解。第一个是 Nehari 流形上的极小元，其能量在 \\(\lambda^*\\) 处变号。第二个具有严格正的能量。对于 \\(\lambda<\lambda^*\\) 且充分接近 \\(\lambda^*\\) 的情形，两个解的能量均为正。这与同一问题的先前多重性结果形成对比，先前结果中得到的两个解的能量符号相反。
英文摘要
We consider the following class of singular elliptic problems \[ \begin{cases} -Δu = -\dfrac{u}{|u|^{β+1}}χ_{\{|u|>0\}} +λ|u|^{p-1}u, & \text{in }Ω,\\[1mm] u=0, & \text{on }\partialΩ, \end{cases} \] where $Ω\subset\mathbb R^N$ is a bounded smooth domain, $N\geq3$, $0<β,p<1$, and $λ>0$. The energy functional is not of class $C^1$ in $H_0^1(Ω)$, and the standard critical point theory does not apply. We prove that a nontrivial nonnegative weak solution exists if and only if $λ$ exceeds a critical value $\barλ$, which we identify and locate strictly between two Rayleigh-type parameters $0<λ_*<\barλ<λ^*<+\infty$. This gives a sharp existence threshold. We also prove that for every $λ$ in a neighborhood on the left of $λ^*$, there exist at least two distinct nontrivial nonnegative weak solutions. The first is a minimizer on the Nehari manifold, with an energy changing sign at $λ^*$. The second has a strictly positive energy. For $λ<λ^*$ sufficiently close to $λ^*$, both solutions have positive energy. This contrasts with previous multiplicity results for the same problem, in which the two solutions obtained have energies of opposite signs.
Comments40 pages, 4 figures
