Boundary-layers for a Neumann problem at higher critical exponents

We consider the Neumann problem $$(P)\qquad - \Delta v + v= v^{q-1} \ \text{in }\ \mathcal{D}, \ v > 0 \ \text{in } \ \mathcal{D},\ \partial_\nu v = 0 \ \text{on } \partial\mathcal{D} ,$$ where $\mathcal{D}$ is an open bounded domain in $\mathbb{R}^N,$ $\nu$ is the unit inner normal at the boundary and $q>2.$ For any integer, $1\le h\le N-3,$ we show that, in some suitable domains $\mathcal D,$ problem $(P)$ has a solution which blows-up along a $h-$dimensional minimal submanifold of the boundary $\partial\mathcal D$ as $q$ approaches from either below or above the higher critical Sobolev exponent ${2(N-h)\over N-h-2}.$Comment: 13 page

Concentration of Solutions for a Singularly Perturbed Neumann Problem in non smooth domains

We consider the equation $-\epsilon^{2}\Delta u + u = u^ {p}$ in a bounded domain $\Omega\subset\R^{3}$ with edges. We impose Neumann boundary conditions, assuming $1<p<5$, and prove concentration of solutions at suitable points of $\partial\Omega$ on the edges.Comment: 24 pages. Second Version, minor changes. To appear in Annales de l'Institut Henri Poincar\'e - Analyse non lin\'eair

Higher-Order Energy Expansions and Spike Locations

We consider the following singularly perturbed semilinear elliptic problem: (I)\left\{ \begin{array}{l} \epsilon^{2} \Delta u - u + f(u)=0 \ \ \mbox{in} \ \Omega, \\ u>0 \ \ \mbox{in} \ \ \Omega \ \ \mbox{and} \ \frac{\partial u}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega, \end{array} \right. where \Om is a bounded domain in R^N with smooth boundary \partial \Om, \ep>0 is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional J_\ep defined by J_\ep [u]:= \int_\Om \left(\frac{\ep^2}{2} |\nabla u|^2 + \frac{1}{2} u^2- F(u)\right) dx \ \ \ \ \ \mbox{for} \ u \in H^1 (\Om), where F(u)=\int_0^u f(s)ds. Ni and Takagi proved that for a single boundary spike solution u_\ep, the following asymptotic expansion holds: J_\ep [u_\ep] =\ep^{N} \Bigg[ \frac{1}{2} I[w] -c_1 \ep H(P_\ep) + o(\ep)\Bigg], where c_1>0 is a generic constant, P_\ep is the unique local maximum point of u_\ep and H(P_\ep) is the boundary mean curvature function at P_\ep \in \partial \Om. In this paper, we obtain a higher-order expansion of J_\ep [u_\ep]: J_\ep [u_\ep] =\ep^{N} \Bigg[ \frac{1}{2} I[w] -c_1 \ep H(P_\ep) + \ep^2 [c_2 (H(P_\ep))^2 + c_3 R (P_\ep)]+ o(\ep^2)\Bigg] where c_2, c_3 are generic constants and R(P_\ep) is the Ricci scalar curvature at P_\ep. In particular c_3 >0. Some applications of this expansion are given

A Higher-Order Energy Expansion to Two-Dimensional Singularly Neumann Problems

Of concern is the following singularly perturbed semilinear elliptic problem \begin{equation*} \left\{ \begin{array}{c} \mbox{${\epsilon}^2\Delta u -u+u^p =0$ in $\Omega$}\\ \mbox{$u>0$ in $\Omega$ and $\frac{\partial u}{\partial \nu}=0$ on $\partial \Omega$}, \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in ${\mathbf{R}}^N$ with smooth boundary $\partial \Omega$, $\epsilon>0$ is a small constant and $1< p<\left(\frac{N+2}{N-2}\right)_+$. Associated with the above problem is the energy functional $J_{\epsilon}$ defined by \begin{equation*} J_{\epsilon}[u]:=\int_{\Omega}\left(\frac{\epsilon^2}{2}{|\nabla u|}^2 +\frac{1}{2}u^2 -F(u)\right)dx \end{equation*} for $u\in H^1(\Omega)$, where $F(u)=\int_{0}^{u}s^p ds$. Ni and Takagi (\cite{nt1}, \cite{nt2}) proved that for a single boundary spike solution $u_{\epsilon}$, the following asymptotic expansion holds: \begin{equation*} (1) \ \ \ \ \ \ \ \ J_{\epsilon}[u_{\epsilon}]=\epsilon^{N} \left[\frac{1}{2}I[w]-c_1 \epsilon H(P_{\epsilon})+o(\epsilon)\right], \end{equation*} where $I[w]$ is the energy of the ground state, $c_1 >0$ is a generic constant, $P_{\epsilon}$ is the unique local maximum point of $u_{\epsilon}$ and $H(P_{\epsilon})$ is the boundary mean curvature function at $P_{\epsilon}\in \partial \Omega$. Later, Wei and Winter (\cite{ww3}, \cite{ww4}) improved the result and obtained a higher-order expansion of $J_{\epsilon}[u_{\epsilon}]$: \begin{equation*} (2) \ \ \ \ \ \ J_{\epsilon}[u_{\epsilon}]=\epsilon^{N} \left[\frac{1}{2}I[\omega]-c_{1} \epsilon H(P_{\epsilon})+\epsilon^2 [c_2(H(P_\epsilon))^2 +c_{3} R(P_\epsilon)]+o(\epsilon^2)\right], \end{equation*} where $c_2$ and $c_3>0$ are generic constants and $R(P_\epsilon)$ is the scalar curvature at $P_\epsilon$. However, if $N=2$, the scalar curvature is always zero. The expansion (2) is no longer sufficient to distinguish spike locations with same mean curvature. In this paper, we consider this case and assume that $2 \leq p <+\infty$. Without loss of generality, we may assume that the boundary near P\in\partial\Om is represented by the graph $\{ x_2 = \rho_{P} (x_1) \}$. Then we have the following higher order expansion of $J_\epsilon[u_\epsilon]:$ \begin{equation*} (3) \ \ \ \ \ J_\epsilon [u_\epsilon] =\epsilon^N \left[\frac{1}{2}I[w]-c_1 \epsilon H({P_\epsilon})+c_2 \epsilon^2(H({P_\epsilon}))^2 ] +\epsilon^3 [P(H({P_\epsilon}))+c_3S({P_\epsilon})]+o(\epsilon^3)\right], \end{equation*} where H(P_\ep)= \rho_{P_\ep}^{''} (0) is the curvature, $P(t)=A_1 t+A_2 t^2+A_3 t^3$ is a polynomial, $c_1$, $c_2$, $c_3$ and $A_1$, $A_2$,$A_3$ are generic real constants and S(P_\epsilon)= \rho_{P_\ep}^{(4)} (0). In particular $c_3<0$. Some applications of this expansion are given

Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball

In [40], it was shown that the following singularly perturbed Dirichlet problem \ep^2 \Delta u - u+ |u|^{p-1} u=0, \ \mbox{in} \ \Om,\] \[ u=0 \ \mbox{on} \ \partial \Om has a nodal solution u_\ep which has the least energy among all nodal solutions. Moreover, it is shown that u_\ep has exactly one local maximum point P_1^\ep with a positive value and one local minimum point P_2^\ep with a negative value and, as \ep \to 0, \varphi (P_1^\ep, P_2^\ep) \to \max_{ (P_1, P_2) \in \Om \times \Om } \varphi (P_1, P_2), where \varphi (P_1, P_2)= \min (\frac{|P_1-P_2}{2}, d(P_1, \partial \Om), d(P_2, \partial \Om)). The following question naturally arises: where is the {\bf nodal surface} \{ u_\ep (x)=0 \}? In this paper, we give an answer in the case of the unit ball \Om=B_1 (0). In particular, we show that for \epsilon sufficiently small, P_1^\ep, P_2^\ep and the origin must lie on a line. Without loss of generality, we may assume that this line is the x_1-axis. Then u_\ep must be even in x_j, j=2, ..., N, and odd in x_1. As a consequence, we show that \{ u_\ep (x)=0 \} = \{ x \in B_1 (0) | x_1=0 \}. Our proof is divided into two steps: first, by using the method of moving planes, we show that P_1^\ep, P_2^\ep and the origin must lie on the x_1-axis and u_\ep must be even in x_j, j=2, ..., N. Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of u_\ep (which implies the odd symmetry of u_\ep in x_1). Similar results are also proved for the problem with Neumann boundary conditions

Boundary clustered layer positive solutions for an elliptic Neumann problem with large exponent

Let $\mathcal{D}$ be a smooth bounded domain in $\mathbb{R}^N$ with $N\geq3$, we study the existence and profile of positive solutions for the following elliptic Neumann problem $$\begin{cases}-\Delta \upsilon+\upsilon=\upsilon^p,\quad \upsilon>0 \quad\textrm{in}\ \mathcal{D},\\[1mm] \frac{\partial \upsilon}{\partial\nu}=0\qquad\textrm{on}\ \partial\mathcal{D}, \end{cases}$$ where $p>1$ is a large exponent and $\nu$ denotes the outer unit normal vector to the boundary $\partial\mathcal{D}$. For suitable domains $\mathcal{D}$, by a constructive way we prove that, for any integers $l$, $m$ with $0\leq l\leq m$ and $m\geq1$, if $p$ is large enough, such a problem has a family of positive solutions with $l$ interior layers and $m-l$ boundary layers which concentrate along $m$ distinct $(N-2)$-dimensional minimal submanifolds of $\partial\mathcal{D}$, or collapse to the same $(N-2)$-dimensional minimal submanifold of $\partial\mathcal{D}$ as $p\rightarrow+\infty$