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Infinitely many positive solutions for the nonlinear Schrödinger equations in \({\mathbb{R}^N}\)

  • Juncheng WeiEmail author
  • Shusen Yan
Article

Abstract

We consider the following nonlinear problem in \({\mathbb {R}^N}\)
$$- \Delta u +V(|y|)u = u^{p},\quad u > 0 \quad {\rm in}\, \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \quad \quad \quad (0.1)$$
where V(r) is a positive function, \({1< p < {\frac{N+2}{N-2}}}\). We show that if V(r) has the following expansion:
$$V(r) = V_0+\frac a {r^m} +O \left(\frac1{r^{m+\theta}}\right),\quad {\rm as} \, r\to +\infty,$$
where a > 0, m > 1, θ > 0, and V 0 > 0 are some constants, then (0.1) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

Mathematics Subject Classification (2000)

35B40 35B45 35J40 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsChinese University of Hong KongShatinHong Kong
  2. 2.School of Mathematics, Statistics and Computer ScienceThe University of New EnglandArmidaleAustralia

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