Existence and Asymptotic Profile of Nodal Solutions to Supercritical Problems

Abstract

Abstract We establish the existence of nodal solutions to the supercritical problem - Δ ⁢ u = | u | p - 2 ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $-\Delta u=|u|^{p-2}u\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega,$ in a symmetric bounded smooth domain Ω of ℝ N ${\mathbb{R}^{N}}$ , N ≥ 3 ${N\geq 3}$ , for p > 2 N ∗ := 2 ⁢ N N - 2 ${p>2_{N}^{\ast}:=\frac{2N}{N-2}}$ , up to some range which depends on the symmetries, and we study their asymptotic behavior as p → 2 N ∗ ${p\rightarrow 2_{N}^{\ast}}$ . We exhibit solutions u p ${u_{p}}$ to this problem in symmetric domains with a shrinking hole, which concentrate at a single point as the hole shrinks and p approaches 2 N ∗ ${2_{N}^{\ast}}$ from above, and whose limit profile is a rescaling of a nonradial sign-changing solution to the limit problem - Δ ⁢ u = | u | 2 N ∗ - 2 ⁢ u , u ∈ D 1 , 2 ⁢ ( ℝ N ) . $-\Delta u=|u|^{2_{N}^{\ast}-2}u,\quad u\in D^{1,2}(\mathbb{R}^{N}).$