Article Contents
Article Contents

# The maximal number of interior peak solutions concentrating on hyperplanes for a singularly perturbed Neumann problem

• We consider the following singularly perturbed elliptic problem

$\varepsilon^2 \Delta u-u+f(u)=0, u>0$ in $B_1$,

$\frac{\partial u}{\partial \nu}=0$ on $\partial B_1,$

where $\Delta = \sum_{i=1}^N \frac{\partial^2}{\partial x_i^2}$ is the Laplace operator, $B_1$ is the unit ball centered at the origin in $R^N$ $(N\ge 3)$, $\nu$ denotes the unit outer normal to $\partial B_1$, $\varepsilon > 0$ is a constant, and $f$ is a superlinear, subcritical nonlinearity . We will show that when $\e$ is sufficiently small there exists a solution with K interior peaks located on a hyperplane, where $1\le K \varepsilon\frac{C}{(\varepsilon)^{N-1}}$ with $C$ a positive constant depending on $N$ and $f$ only. As a consequence, we obtain that there exists at least $[\frac{C}{(\varepsilon)^{N-1}}]$ number of solutions for $\varepsilon$ sufficiently small.

Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J40.

 Citation:

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