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 < p < 2, f ðuÞ ¼ jujq-2u with q > 22-pp and W ¼ fðr cos y; r sin yÞ: jyj < a; jr - 1j < sg with 0 < a < p and 0 < s < 1, then for all q > 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.
Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains
Donato Passaseo
2020-01-01
Abstract
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 < p < 2, f ðuÞ ¼ jujq-2u with q > 22-pp and W ¼ fðr cos y; r sin yÞ: jyj < a; jr - 1j < sg with 0 < a < p and 0 < s < 1, then for all q > 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
