On a Hardy–Sobolev-type inequality and applications

Abstract

In this paper, we prove a new Friedrich-type inequality. As an application, we derive some existence and non-existence results to the quasilinear elliptic problem with Robin boundary condition $$\left\{\begin{array}{cc}-\text{div}(|\nabla u{|}^{N-2}\nabla u)+h(x)|u{|}^{q-2}u=\lambda k(x)|u{|}^{p-2}u\hfill & \text{in }\mathrm{\Omega },\hfill \\ |\nabla u{|}^{N-2}(\nabla u\cdot \nu )+|u{|}^{N-2}u=0\hfill & \text{on }\partial \mathrm{\Omega },\hfill \end{array}\right.$$ where $\mathrm{\Omega }\subset {ℝ}^N$ is an exterior domain such that $0\notin \overline{\mathrm{\Omega }}$.