Critical Robin Problems on Manifolds with Noncompact Boundary

We work on connected, noncompact n-dimensional Riemannian manifolds (M, g), n ≥ 3, that are complete with respect to dg and have noncompact boundary and bounded geometry. We study the existence of positive solutions in C∞(M) ∩ L2^*(M) ∩ L∞(M) to the Yamabe problem: $$ \{{aligned} -Δ_g u+n-2/4(n-1) R_g u & =C_0 un+2/n-2 in M, \\ ∂ u/∂ ν+n-2/2(n-1) H_g u & =C_1 un/n-2 on ∂ M, {aligned}. $$ where $C_0=YC(M), C_1=YC(M) C d̄2/n-2$, the parameter $C ≥ 0$ is fixed, and $$ YC(M):=inf _{u ∈ C_c∞(M) \{0\}} {E_g(u)}{(\|u\|_{L2^*(M)}2^*+C\|u\|_{L^{2#}(∂ M)}2^*)2 / 2^*} . $$ Here $E_g$ is the associated quadratic energy. We assume $YC(M)0$, where $YC, ∞$ is the corresponding variational level at infinity. We approach this problem using nonnegative solutions $u_j ∈ W∂1,2(B_g(O, j)), j ∈ N, j ≥ j_0$, for which we prove that $$ j ≥ j_0\|u_j\|_{L∞(B_g(O, j-1))}<∞ . $$ For $C>0$, we also prove that, along a subsequence, $\|u_j\|_{L^2{ }(B_g(O, j) ∩ ∂ M)} → d̄$, with $0<d̄ ≤ C-1 / 2^*$. For $C=0$, we set $d̄=0$.