arXiv 2008.051992020-08-12math.AP

黎曼流形上若干问题的存在性结果

Existence results for some problems on Riemannian manifolds

Giovanni Molica Bisci, Luca Vilasi, Dušan D. Repovš

发表信息：: Comm. Anal. Geom. 28:3 (2020), 677-706
DOI：: 10.4310/CAG.2020.v28.n3.a6

摘要：

利用变分方法，给出了一类具有次临界扰动的Yamabe型方程解的存在性新结果，该方程解位于紧致的无边界$d$维（$d\geq 3$）黎曼流形上.作为我们的主要定理的直接后果，我们证明了以下奇异Yamabe型问题$$ \left\lbrace \begin{array}{ll} -Δ_g w + α(σ)w = μK(σ) w^\frac{d+2}{d-2} +λ\left( w^{r-1} + f(w)\right), \quad σ\in\mathcal{M} &\\ &\\ w\in H^2_α(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$的至少一个解决方案的存在，像往常一样，$Δ_g$表示拉普拉斯-Beltrami算子$(\mathcal{M},g)$，$α, K:\mathcal{M}\to\mathbb{R}$是积极的（本质上）有界函数，$r\in(0,1)$，和$f:[0,+\infty)\to[0,+\infty)$是一个亚临界连续函数。通过赤平投影将我们的研究限制在单位球面${\mathbb{S}}^d$上，我们还求解了一些参数化的Euden-Fowler方程。

英文摘要：

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.

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