Advances in Differential Equations

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

Massimo Grossi and Angela Pistoia

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Abstract

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

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

Dates
First available in Project Euclid: 27 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1356651227

Mathematical Reviews number (MathSciNet)
MR1785679

Zentralblatt MATH identifier
0989.35054

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

Citation

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. https://projecteuclid.org/euclid.ade/1356651227.


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