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

We 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}.

Existence and Asymptotic Profile of Nodal Solutions to Supercritical Problems

We establish the existence of nodal solutions to the supercritical problemin a symmetric bounded smooth domain Ω of

Nonexistence Results of Sign-changing solutions for a Supercritical Problem of the Scalar Curvature Type

This paper is devoted to the study of the nonlinear elliptic problem with supercritical critical exponent (P

Beyond the Mountain Pass: Some Applications

We will survey three applications of the famous Mountain Pass Lemma by Antonio Ambrosetti and Paul H. Rabinowitz, [6]. Precisely, we will present some results on the following problems:A model of growth and roughening of surfaces, usually called Kardar-Parisi-Zhang model.A supercritical problem involving Hardy-Leray potential with the pole at the boundary.A fourth order nonlinear problem related to epitaxial growth.The common characteristic of all these problems is that we have some extra difficulties in order to apply the Mountain Pass Lemma.