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This paper studies Laplace's equation $-\Delta\,u=0$ in an exterior region$U\varsubsetneq{\mathbb R}^N$, when $N\geq3$, subject to the nonlinear boundarycondition $\frac{\partialu}{\partial\nu}=\lambda{\left\vert{u}\right\vert}^{q-2}u+\mu{\left\vert{u}\right\vert}^{p-2}u$on $\partial U$ with $1<q<2<p<2^*$.  In the function space $\mathscr{H}\left(U\right)$, 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$k\to\infty$; on the other hand, when $\lambda\in\mathbb R$ and $\mu>0$arbitrary, then there exists a sequence $\left\{\tilde{u}_k\right\}$ ofsolutions with positive and unbounded energy.  Also, associated with the $p$-Laplacian equation $-\Delta_p\,u=0$, theexterior $p$-harmonic Steklov eigenvalue problems are described.

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