Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains

We deal with nonlinear elliptic Dirichlet problems of the form divðjDujp-2DuÞ þ f ðuÞ ¼ 0 in W; u ¼ 0 on qW where W is a bounded domain in Rn, n b 2, p > 1 and f has supercritical growth from the viewpoint of Sobolev embedding. Our aim is to show that there exist bounded contractible non star-shaped domains W, arbitrarily close to domains with nontrivial topology, such that the problem does not have nontrivial solutions. For example, we prove that if n ¼ 2, 1 22-pp and W ¼ fðr cos y; r sin yÞ: jyj 22-pp there exists s > 0 such that the problem has only the trivial solution u C 0 for all a a ð0; pÞ and s a ð0; sÞ