Nodal solutions for Lane-Emden problems in almost-annular domains

In this paper, we prove an existence result to the problem -Δu=-u-p-1uu=0in Ω on∂Ωwhere Ω is a bounded domain in ℝNwhich is a perturbation of the annulus. Then there exists a sequence p1<p2 < ⋯ with lim pκ =κ → +∞+∞ such that for any real number p > 1 and p 6 ≠ pκ there exist at least one solution with m nodal zones. In doing so, we also investigate the radial nodal solution in an annulus: we provide an estimate of its Morse index and analyze the asymptotic behavior as p → 1.