Sign-changing solutions for supercritical elliptic problems in domains with small holes

Abstract

Let \(\mathcal {D}\) be a bounded, smooth domain in \(\mathbb {R}^N\) , N ≥ 3, \(P\in \mathcal {D}\) . We consider the boundary value problem in \(\Omega = \mathcal {D} \setminus B_\delta(P)\) , $$\begin{aligned}\Delta u + |u|^{p-1} u = 0\, \quad in\, \Omega,\\u = 0\quad on\, \partial\Omega,\end{aligned}$$ with p supercritical, namely \(p > \frac{N+2}{N-2}\) . Given any positive integer m, we find a sequence \(p_1 0 sufficiently small, this problem has a sign-changing solution which has exactly m + 1 nodal domains.