Laplace's equation with concave and convex boundary nonlinearities on an exterior region

This paper studies Laplace’s equation $-\mathrm{\Delta }u=0$ in an exterior region $U⊊{\mathbb{R}}^N$, when $N\ge 3$, subject to the nonlinear boundary condition $\frac{\partial u}{\partial\nu}=\lambda{\left\vert{u}\right\vert}^{q-2}u+\mu{\left\vert{u}\right\vert}^{p-2}u$on$\partial U$with$1<q<2<p<2^*$. In the function space $\mathcal{H}\left(U\right)$, one observes when $\lambda >0$ and $\mu \in \mathbb{R}$ arbitrary, then there exists a sequence $\left\{u_k\right\}$ of solutions with negative energy converging to $0$ as $k\to \mathrm{\infty }$; on the other hand, when $\lambda \in \mathbb{R}$ and $\mu >0$ arbitrary, then there exists a sequence $\left\{{\tilde{u}}_k\right\}$ of solutions with positive and unbounded energy. Also, associated with the $p$-Laplacian equation $-{\mathrm{\Delta }}_{p}u=0$, the exterior $p$-harmonic Steklov eigenvalue problems are described.