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 $$ \left\lbrace \begin{array}{ll} -\Delta_g w + \alpha(\sigma)w = \mu K(\sigma) w^\frac{d+2}{d-2} +\lambda \left( w^{r-1} + f(w)\right), \quad \sigma\in\mathcal{M} &\\ &\\ w\in H^2_\alpha(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ where, as usual, $\Delta_g$ denotes the Laplace-Beltrami operator on $(\mathcal{M},g)$, $\alpha, K:\mathcal{M}\to\mathbb{R}$ 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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黎曼流形上若干问题的存在性结果

通过使用变分技术，我们为 Yamabe 型方程提供了新的存在结果，其中亚临界扰动设置在无边界的紧凑 $d$ 维 ($d\geq 3$) 黎曼流形上。作为我们主要定理的直接结果，我们证明了以下奇异 Yamabe 类型问题的至少一个解的存在性 $$ \left\lbrace \begin{array}{ll} -\Delta_g w + \alpha(\sigma )w = \mu K(\sigma) w^\frac{d+2}{d-2} +\lambda \left( w^{r-1} + f(w)\right), \quad \sigma \in\mathcal{M} &\\ &\\ w\in H^2_\alpha(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \ end{array} \right.$$ 其中，像往常一样，$\Delta_g$ 表示 $(\mathcal{M},g)$, $\alpha, K:\mathcal{M}\to 上的 Laplace-Beltrami 算子\mathbb{R}$ 是正（本质上）有界函数，$r\in(0,1)$ 和 $f:[0,+\infty)\to[0, +\infty)$ 是亚临界连续函数。通过立体投影将自己限制在单位球面 ${\mathbb{S}}^d$ 中，我们还在欧几里得情况下求解了一些参数化的 Emden-Fowler 方程。