Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains

The nonexistence of nontrivial solution is discussed for the following quasilinear elliptic equations: (equation presented) for the case where Ω is an exterior domain such that Ω = R^N \Ω₀, with Ω₀ bounded and starshaped. It should be noted that because of the degeneracy of the equation at ∇u = 0, the nontrivial solutions of (E) are not twice differentiable. Therefore there arises the necessity of discussing the nonexistence of solutions in a framework of weak solutions. When Ω is bounded, the second author introduced a "Pohozaev-type inequality" valid for a class of weak solutions, which is effective enough for discussing the nonexistence. We here give an exterior-domain version of Pohozaev-type inequality, whence some results on the nonexistence of solutions are derived. These suggest that concerning the existence and nonexistence of nontrivial solutions, there may exist an interesting duality between the interior problems and the exterior problems for starshaped domains.