Infinitely many small-energy solutions of a nonlinear boundary value problem with Dirichlet-to-Neumann operator

In this study, we investigate the following nonlinear boundary value problem$\{\begin{array}{cc}-\Delta u+u=|u{|}^{p-2}u,& \text{in}\Omega \\ \frac{\partial u}{\partial \nu }=Du,& \text{on}\partial \Omega ,\end{array}$ where $2<p<+\infty$ if $N=2$ and $2<p\le 2^{⁎}$ if $N\ge 3$, Ω is a bounded domain in $R^N$ ($N\ge 2$) with $C^{\infty }$ smooth boundary, $u:\Omega \to R$, Δu is the Laplacian operator, ν is the unit outer normal vector at ∂Ω, and $D$ is the Dirichlet-to-Neumann operator. We prove that this equation has a sequence of solutions $\left\{u_m\right\}$ that satisfies ${\lim }_{m\to \infty }{\Vert u_m\Vert }_{L^p\left(\Omega \right)}=0$ and ${\lim \inf }_{m\to \infty }{\Vert u_m\Vert }_{L^{\infty }\left(\Omega \right)}\ge 1$. To prove this, a new critical point theorem without the usual Palais-Smale condition is used.