Construction of Solutions for a Nonlinear Elliptic Problem on Riemannian Manifolds with Boundary

Abstract

Abstract Let ( M , g ) ${(M,g)}$ be a smooth compact n-dimensional Riemannian manifold ( n ≥ 2 ${n\geq 2}$ ) with smooth ( n - 1 ) ${(n-1)}$ -dimensional boundary ∂ ⁡ M ${\partial M}$ . We prove that the stable critical points of the mean curvature of the boundary generates H 1 ⁢ ( M ) ${H^{1}(M)}$ solutions for the following singularly perturbed elliptic problem with Neumann boundary conditions: - ε 2 ⁢ Δ g ⁢ u + u = u p - 1 ⁢ in ⁢ M , u > 0 ⁢ in ⁢ M , ∂ ⁡ u ∂ ⁡ ν = 0 ⁢ on ⁢ ∂ ⁡ M , $-\varepsilon^{2}\Delta_{g}u+u=u^{p-1}\text{ in }M,\quad u>0\text{ in }M,\quad% \frac{\partial u}{\partial\nu}=0\text{ on }\partial M,$ when ε is small enough. Here p is subcritical.