Critical Robin Problems on Manifolds with Noncompact Boundary

We work on connected, noncompact $n$-dimensional Riemannian manifolds $(\mathcal{M}, g), n \geq 3$, that are complete with respect to $d_g$ and have noncompact boundary and bounded geometry. We study the existence of positive solutions in $C^{\infty}(\mathcal{M}) \cap L^{2^*}(\mathcal{M}) \cap L^{\infty}(\mathcal{M})$ to the Yamabe problem: $$ \left\{\begin{aligned} -Δ_g u+\frac{n-2}{4(n-1)} R_g u & =\mathscr{C}_0 u^{\frac{n+2}{n-2}} \quad \text { in } \mathcal{M}, \\ \frac{\partial u}{\partial ν}+\frac{n-2}{2(n-1)} H_g u & =\mathscr{C}_1 u^{\frac{n}{n-2}} \quad \text { on } \partial \mathcal{M}, \end{aligned}\right. $$ where $\mathscr{C}_0=Y_{\mathscr{C}}(\mathcal{M}), \mathscr{C}_1=Y_{\mathscr{C}}(\mathcal{M}) \mathscr{C} \bar{d}^{\frac{2}{n-2}}$, the parameter $\mathscr{C} \geq 0$ is fixed, and $$ Y_{\mathscr{C}}(\mathcal{M}):=\inf _{u \in C_c^{\infty}(\mathcal{M}) \backslash\{0\}} \frac{E_g(u)}{\left(\|u\|_{L^{2^*}(\mathcal{M})}^{2^*}+\mathscr{C}\|u\|_{L^{2^{\#}}(\partial \mathcal{M})}^{2^*}\right)^{2 / 2^*}} . $$ Here $E_g$ is the associated quadratic energy. We assume $Y_{\mathscr{C}}(\mathcal{M})0$, where $Y_{\mathscr{C}, \infty}$ is the corresponding variational level at infinity. We approach this problem using nonnegative solutions $u_j \in W_{\partial}^{1,2}\left(B_g(\mathcal{O}, j)\right), j \in \mathbb{N}, j \geq j_0$, for which we prove that $$ \sup _{j \geq j_0}\left\|u_j\right\|_{L^{\infty}\left(B_g(\mathcal{O}, j-1)\right)}<\infty . $$ For $\mathscr{C}>0$, we also prove that, along a subsequence, $\left\|u_j\right\|_{L^2{ }_{\left(B_g(\mathcal{O}, j) \cap \partial \mathcal{M}\right)}} \rightarrow \bar{d}$, with $0<\bar{d} \leq \mathscr{C}^{-1 / 2^*}$. For $\mathscr{C}=0$, we set $\bar{d}=0$.