## Topological Methods in Nonlinear Analysis

### Singularly perturbed Neumann problems with potentials

Alessio Pomponio

#### Abstract

The main purpose of this paper is to study the existence of single-peaked solutions of the Neumann problem $$\begin{cases} -\varepsilon^2 {\rm div} (J(x)\nabla u)+V(x)u=u^p & \text{in }\Omega, \\ \dfrac{\partial u}{\partial \nu}=0 & \text{on }\partial\Omega, \end{cases}$$ where $\Omega$ is a smooth bounded domain of $\mathbb R^N$, $N\ge 3$, $1< p< (N+2)/(N-2)$ and $J$ and $V$ are positive bounded scalar value potentials. We will show that, for the existence of concentrating solutions, one has to check if at least one between $J$ and $V$ is not constant on $\partial \Omega$. In this case the concentration point is determined by $J$ and $V$ only. In the other case the concentration point is determined by an interplay among the derivatives of $J$ and $V$ calculated on $\partial \Omega$ and the mean curvature $H$ of $\partial \Omega$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 23, Number 2 (2004), 301-322.

Dates
First available in Project Euclid: 31 May 2016

https://projecteuclid.org/euclid.tmna/1464731411

Mathematical Reviews number (MathSciNet)
MR2078194

Zentralblatt MATH identifier
1129.35404

#### Citation

Pomponio, Alessio. Singularly perturbed Neumann problems with potentials. Topol. Methods Nonlinear Anal. 23 (2004), no. 2, 301--322. https://projecteuclid.org/euclid.tmna/1464731411

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