On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

Abstract

This paper deals with the following prescribed boundary mean curvature problem in ${\mathbb{B}^N}$

$$\left\{ {\matrix{{ - \Delta u = 0,\,u > 0,} \hfill & {y \in {\mathbb{B}^N},} \hfill \cr {{{\partial u} \over {\partial \nu }} + {{N - 2} \over 2}u = {{N - 2} \over 2}\tilde K(y){u^{{2^\sharp } - 1}},} \hfill & {y \in {\mathbb{S}^{N - 1}},} \hfill \cr } } \right.$$

where $\tilde K(y) = \tilde K(|{y^\prime }|,\tilde y)$ is a bounded nonnegative function with $y = ({y^\prime },\tilde y) \in {\mathbb{R}^2} \times {\mathbb{R}^{N - 3}},\,\,{2^\sharp } = {{2(N - 1)} \over {N - 2}}$. Combining the finite-dimensional reduction method and local Pohozaev type of identities, we prove that if N ≥ 5 and $\tilde K(r,\tilde y)$ has a stable critical point (r₀, $({r_0},{\tilde y_0})$) with r₀ > 0 and $\tilde K({r_0},{\tilde y_0}) > 0$, then the above problem has infinitely many solutions, whose energy can be made arbitrarily large. Here our result fill the gap that the above critical points may include the saddle points of $\tilde K(r,\tilde y)$.