论文标题
Riemannian歧管上的一些问题的存在结果
Existence results for some problems on Riemannian manifolds
论文作者
论文摘要
通过使用各种技术，我们为Yamabe类型方程提供了新的存在结果，并在紧凑型$ D $数（$ d \ geq 3 $）的Riemannian歧管上设置了亚临界扰动，而没有边界。作为我们主要定理的直接结果，我们证明至少存在一种解决方案，以解决以下单一的Yamabe型问题$$ \ left \ left \ lbrace \ begin {array} {ll}-Δ_gw +α（σ） + f（w）\ right），\ quadσ\ in \ Mathcal {m}＆\\＆\\ w \ in H^2_α（\ Mathcal {M}），\ quad W> 0 \ \ \ \ \ \ \ \ \ \ \ mbox {in} $Δ_g$表示$（\ nathcal {m}，g）$，$α，k：\ nathcal {m} \ to \ athbb {r} $是正（本质上是）有界函数，$ r \ in（1,1）$ f：[0,1）$ f：[0,.0，$ f：tyty for：是亚临界连续功能。通过立体投影将自己限制在单位球体$ {\ mathbb {s}}^d $上，我们还解决了欧几里得案例中的一些参数化的emden-fowler方程。
By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main theorems, we prove the existence of at least one solution to the following singular Yamabe-type problem $$ \left\lbrace \begin{array}{ll} -Δ_g w + α(σ)w = μK(σ) w^\frac{d+2}{d-2} +λ\left( w^{r-1} + f(w)\right), \quad σ\in\mathcal{M} &\\ &\\ w\in H^2_α(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ where, as usual, $Δ_g$ denotes the Laplace-Beltrami operator on $(\mathcal{M},g)$, $α, K:\mathcal{M}\to\mathbb{R}$ are positive (essentially) bounded functions, $r\in(0,1)$, and $f:[0,+\infty)\to[0,+\infty)$ is a subcritical continuous function. Restricting ourselves to the unit sphere ${\mathbb{S}}^d$ via the stereographic projection, we also solve some parametrized Emden-Fowler equations in the Euclidean case.
