article

Title: Multi-bump solutions for a semilinear Schroedinger equation

Authors: Lishan Lin, Zhaoli Liu and Shaowei Chen

Issue: Volume 58 (2009), Issue 4, 1659-1690

Abstract:

$We study the existence of multi-bump solutions for the semilinear Schr\"odinger equation \[-\Delta u + (1 + \epsilon a(x))u = |u|^{p-2}u,\quad u\in H^1(\mathbb{R}^N),\] where $N\geq1$, $2<p<2N/(N - 2)$ if $N\geq3$, $p>2$ if $N = 1$ or $N = 2$, and $\epsilon>0$ is a parameter. The function $a$ is assumed to satisfy the following conditions: $a\in C(\mathbb{R}^N)$, $a(x)>0$ in $\mathbb{R}^N$, $a(x) = o(1)$ and $\ln(a(x)) = o(|x|)$ as $|x|\to\infty$. For any positive integer $n$, we prove that there exists $\epsilon(n)>0$ such that, for $0<\epsilon<\epsilon(n)$, the equation has an $n$-bump positive solution. Therefore, the equation has more and more multi-bump positive solutions as $\epsilon\to0$.$