We obtain the existence of a 1-parameter family of nontrivial exterior domains \({\Omega }\subset \mathbb {R}^{N}\) which support a positive solution of \(\begin{aligned} \Delta u=0\,\, \text{ in }\,\,{\Omega }, \,\, u=u_0,\,\,\partial _\nu u=\gamma H+C_0\,\,\text{ on }\,\,\partial {\Omega },\,\, \lim _{r\rightarrow +\infty }u=0, \end{aligned}\) if \(N\ge 4\) and \(\begin{aligned} \Delta u=0\,\, \text {in}\,\,{\Omega }, \,\, u=u_0,\,\,\partial _\nu u=\gamma K+C_0\,\,\text {on}\,\,\partial {\Omega },\,\, \lim _{r\rightarrow +\infty }u=+\infty , \end{aligned}\) if \(N=2,\) where H is the mean curvature of \(\partial {\Omega },\) K is the curvature of \(\partial {\Omega }\) , \(\gamma \) is a parameter, \(u_0\) and \(C_0\) are constants. These domains bifurcates from the complement of a ball.
