### Singularly perturbed Neumann problems with potentials

DOI: http://dx.doi.org/10.12775/TMNA.2004.013

#### Abstract

The main purpose of this paper is to study the existence of single-peaked solutions of the

Neumann problem

$$

\cases

-\varepsilon^2 \text{\rm div} \left(J(x)\nabla u\right)+V(x)u=u^p

& \text{in }\Omega,

\\

\displaystyle \dfrac{\partial u}{\partial \nu}=0

& \text{on }\partial\Omega,

\endcases

$$

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 $\partial \Omega$.

Neumann problem

$$

\cases

-\varepsilon^2 \text{\rm div} \left(J(x)\nabla u\right)+V(x)u=u^p

& \text{in }\Omega,

\\

\displaystyle \dfrac{\partial u}{\partial \nu}=0

& \text{on }\partial\Omega,

\endcases

$$

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 $\partial \Omega$.

#### Keywords

Singularly perturbed Neumann problem; presence of potentials; concentrating solutions

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