Concentration with a single sign-changing layer at the higher critical exponents

Abstract

AbstractWe exhibit a new concentration phenomenon for the supercritical problem-\Delta v=\lambda v+|v|^{p-2}v\quad\text{in }\Omega,\qquad v=0\quad\text{on }% \partial\Omega,as {p\rightarrow 2_{N,m}^{\ast}} from below, where {2_{N,m}^{\ast}:=\frac{2(N-m)}{N-m-2}}, {1\leq m\leq N-3}, is the so-called {(m+1)}-th critical exponent. We assume that Ω is of the form\Omega:=\bigl{\{}(x_{1},x_{2})\in\mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1}:(|x_% {1}|,x_{2})\in\Theta\bigr{\}},where Θ is a bounded smooth domain in {\mathbb{R}^{N-m}} such that {\overline{\Theta}\subset(0,\infty)\times\mathbb{R}^{N-m-1}}. Under some symmetry assumptions, we show that there exists {\lambda_{\ast}\geq 0} such that for each {\lambda\in(-\infty,\lambda_{\ast})\cup\{0\}}, there exist a sequence {p_{k}\in(2,2_{N,m}^{\ast})} with {p_{k}\rightarrow 2_{N,m}^{\ast}} and a sequence of solutions {v_{k}} which concentrate and blow up along an m-dimensional sphere of minimal radius contained in {\partial\Omega}, developing a single sign-changing layer as {p_{k}\rightarrow 2_{N,m}^{\ast}}. In contrast with previous results, the asymptotic profile of this layer on each space perpendicular to the blow-up sphere is not a sum of positive and negative bubbles, but a rescaling of a sign-changing solution to the critical problem-\Delta u=|u|^{{4}/({N-m-2})}u,\quad u\in D^{1,2}(\mathbb{R}^{N-m}).Moreover, {\lambda_{\ast}>0} if {m\geq 2}.