Advances in Differential Equations

On the effect of critical points of distance function in superlinear elliptic problems

Massimo Grossi and Angela Pistoia

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study some perturbed semilinear problems with Dirichlet or Neumann boundary conditions, $$ \begin{cases} -{\varepsilon}^2\Delta u+u=u^p & \mbox{ in $\Omega$}\cr u>0 & \mbox{ in $\Omega$}\cr u=0\ \ \mbox{ or }\ \ {{\partial u}\over{\partial\nu}}=0 & \mbox{ in $\partial\Omega,$} \end{cases} $$ where $\Omega$ is a bounded, smooth domain of $\mathbb R^N$, $N\ge2$, ${\varepsilon}>0$, $1 <p <{{N+2}\over{N-2}}$ if $N\ge3$ or $p>1$ if $N=2$ and $\nu$ is the unit outward normal at the boundary of $\Omega$. We show that any ``suitable" critical point $x_0$ of the distance function generates a family of single interior spike solutions, whose local maximum point tends to $x_0$ as ${\varepsilon}$ tends to zero.

Article information

Adv. Differential Equations, Volume 5, Number 10-12 (2000), 1397-1420.

First available in Project Euclid: 27 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35B25: Singular perturbations 35J20: Variational methods for second-order elliptic equations


Grossi, Massimo; Pistoia, Angela. On the effect of critical points of distance function in superlinear elliptic problems. Adv. Differential Equations 5 (2000), no. 10-12, 1397--1420.

Export citation