Singularly perturbed Neumann problems with potentials
Keywords
Singularly perturbed Neumann problem, presence of potentials, concentrating solutionsAbstract
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 $\partial \Omega$.Downloads
Published
2004-06-01
How to Cite
1.
POMPONIO, Alessio. Singularly perturbed Neumann problems with potentials. Topological Methods in Nonlinear Analysis. Online. 1 June 2004. Vol. 23, no. 2, pp. 301 - 322. [Accessed 18 April 2024].
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