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

Abstract

We deal with nonlinear elliptic Dirichlet problems of the form $$ \mathrm {div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial\Omega $$ where $\Omega$ is a bounded domain in $\mathbb R^n$, $n\ge 2$, $p > 1$ 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 $\Omega$, arbitrarily close to domains with nontrivial topology, such that the problem does not have nontrivial solutions. For example, we prove that if $n=2$, $1 < p < 2$, $f(u)=|u|^{q-2}u$ with $q > {2p\over 2-p}$ and $\Omega={(\rho\cos\theta,\rho\sin\theta)\ :|\theta|<\alpha,\ |\rho -1| < s}$ with $0 < \alpha < \pi$ and $0 < s < 1$, then for all $q > {2p\over 2-p}$ there exists $\bar s > 0$ such that the problem has only the trivial solution $u\equiv 0$ for all $\alpha\in (0,\pi)$ and $s\in (0,\bar s)$.